max-mm
‘ I ~ MG-PTGWCEANGE—

 

Dissertatmn for the Degree of Ph. D.
‘ MICHIGAN STATE UNIVERSITY . 1 . ' a .
GEORGE EDMOND BDHANNON

 

 

 

This is to certify that the
thesis entitled

NUCLEON -NUCLEON
TWO— P I ON- EXCHANGE

presented by

George Edmond Bohannon

has been accepted towards fulfillment
of the requirements for

Ph.D. Physics

degree in

_ 7
Major proerr

Date 5 0 74—

0-7639

  
   

    

meme .
it, "kW-“mgr

 

 

 

 

 

 

 

 

 

 

NU

ABSTRACT

NUCLEON-NUCLEON TWO—PION-EXCHANGE

BY

George Edmond Bohannon

The J=0,l,2 Nfi+fiw amplitudes were evaluated in the
pseudophysical region ti4u2 and used in a calculation of

nucleon-nucleon two-pion—exchange. The «N phase shifts,

which entered the calculation as the phases of the t:4p2

Nfi+nn amplitudes, are discussed in detail. Results are

given for several forms of the inputed I=J=0 nu phase shift

60
00

The NN amplitudes A(+)(s=u,t=0) and
a

EE'A(+)(s=u't)1t=0 were determined from fixed-t dispersion

relations to be 25.9 i .5 u—1

and 1.16 i .05 u_3 respec-
tively. These values were used as subtraction constants in
a fixed v=(s—u)/(4m)=0 dispersion relation with the phase
shift 6: as input to continue to positive t. The results
for A(+)(t=4u2) were somewhat dependent upon the form of 63

l and 34 u-l.

used; values obtained were between 32 u—
+
The values of A(+)(t=0), A( )(t=4u2) and a deter—
mination of the J=O mfiflm amplitude f$(t=0) by Furuichi

and Watanabel were used for subtraction constants along

 

with the phase
to determine in
enhancement in
Saclay Ke4 expr

For thr
partial wave d:
Nielsen‘s fixe.
requiring that
nucleon form f
with a total 0
form factors G
With experimen
required in th
d0minance of t
With exPerimen
imaginary part
with resonance

constants as c-

/9\ George Edmond Bohannon

with the phase shift 6: in partial wave dispersion relations
to determine lf2(t)l. It was found that introducing an
enhancement in the low energy 6:, as seen in the Geneva-
Saclay Ke4 experiment,2 resulted in a smaller If:(t26u2)l.
For the J=l case the subtraction constants in a
partial wave dispersion relation were determined from

Nielsen's fixed—t dispersion relation results3

and by
requiring that the two pion intermediate state dominate the
nucleon form factors. The resulting J=l Nfi+nn amplitudes
with a total of two adjusted subtraction constants produced
v v
form factors GE and GM

with experimental results. The pion form factor, which was

which are in excellent agreement

required in the calculation, was evaluated assuming
dominance of the two pion intermediate state and compared
with experimental results in the timelike region. The
imaginary parts of the J=1 Nfi+nn amplitudes were fitted
with resonance expressions to determine the pNN coupling
constants as defined at the rho meson pole. The values

found are ggNN/4n = 0.44:0.05 and f = 5.8:0.2.

ONN/gDNN
The J=0,l Nfi+nn amplitudes were also evaluated via
partial wave dispersion relations subtracted twice at t=0,
where the subtraction constants were taken from Nielsen's
work.3
The J=2 Nfi+nn amplitudes were evaluated with once

Subtracted partial wave dispersion relations. Results are

shown with and without the right—hand (nn) cut.

 

The fin
exchange, one-I
tudes. The NN
MAW-X phase sh
exchange fits5
amplitude. It
enhancement in
nent of most L

The NN
with the Hamad

Varior

CWining const

meson-dominanc

l

31. 465 (its?)
(1971). 2A' 23

3H. N:
Phys. Re:b.d'l§
(1972). SR‘ A
(1962). 6T~ H

George Edmond Bohannon

The final NN amplitude was the sum of the one-pion—
exchange, one-omega-exchange and two-pion-exchange ampli-
tudes. The NN phase shifts were compared with those of the
MAW-X phase shift analyses,4 the Bryan-Gersten one-boson
exchange fits5 and the pseudoscalar coupling nucleon box
amplitude. It was found that introducing a low energy
enhancement into the nn phase shift 6: improved the agree—
ment of most L12 phase shifts with the phase shift analyses.

The NN potential was calculated and shown along
with the Hamada-Johnston6 and nucleon box potentials.

Various methods are discussed for finding the wNN
coupling constants based on the quark model and vector-

meson—dominance.

1S. Furuichi and K. Watanabe, Prog. Theoret. Phys.
31, 465 (1967).

2

A. Zylbersztejn et al., Phys. Letters 388, 457
(1971).

3H. Nielsen, Nucl. Phys. B33, 152 (1971).

4M. H. MacGregor, R. A. Arndt and R. M. Wright,
Phys. Rev. 182, 1714 (1969).

5

R. A. Bryan and A. Gersten, Phys. Rev. 22, 341
(1972).

6

( T. Hamada and I. D. Johnston, Nucl. Phys. 34, 382
1962). ‘-

 

 

111 Pa]

NUCLEON-NUCLEON TWO-PION-EXCHANGE

BY

George Edmond Bohannon

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics

1974

Physi
problem the wc
of work which

0f Physics. 1

ACKNOWLEDGEMENTS

Physics research involves focusing onto a specific
problem the work of a large number of people. The diversity
of work which can be so focused is a measure of the unity
of physics. I have been influenced to some extent by each
of the authors in the List of References. The contributions
of some people should be specifically noted here.

I owe my sincerest gratitude to Professor Peter
Signell, who suggested this problem and provided continued
encouragement. His suggestions and remarks and his own
work in this area have been valuable assets in the comple-
tion of this thesis.

I am grateful to Dr. Jon Pumplin for helpful dis—
cussions about particle physics. I am also grateful to Dr.
Edward Yen for sharing with me some of his knowledge of
Particle physics. Their remarks regarding the electromag—
netic form factors, vector-meson-dominance, quarks and the
Quark model have been particularly useful.

Discussions with Dr. Wayne Repko have been helpful

in understanding the intricacies of field theory and elec-

trOrnagnetic interactions.

ii

 

I a1:
Leon Heller re
the two-pion-e:
cerning the re

Discu
aspects of the

I am
Stevens, Dr. G
the high energ
of the NEW; 5
IGSpectively .

The n

because of the

BY qr

Phase shift ar

I also appreciate a helpful discussion with Dr.

Leon Heller regarding the calculation of a potential from
the two-pion-exchange amplitude as well as his remarks con-
cerning the reasons for generating such a potential.

Discussions with Dr. D.O. Riska concerning several
aspects of the two-pion-exchange problem have been helpful.

I am grateful for communications with Dr. Paul
Stevens, Dr. Geoffrey Epstein, and Dr. Graham Shaw regarding
the high energy behavior of nN amplitudes, the evaluation
of the Nfienn s—wave amplitude, and the I=J=0 nu phase shift,
respectively.

The numerical calculations were much easier
because of the expert advice of Dr. Jonas Holdeman.

By quickly responding to my request for the MAW-X
phase shift analysis error matrix M.H. MacGregor and R.M.
wright made it possible to include in this thesis unpublished
combinations of their published phase shifts with the correct

uncertainties.

The accurate presentation of the calculational

results was made possible by Rick Meyers, whose computer

°°del with some modifications, was used to draw most of the

graphs for this thesis.

Finally and very importantly, I am grateful to the

National Science Foundation and its supporters, the American

Public, for making this research possible.

iii

LIST OF TABLES

LIST OF FIGURI

INTRODUCTION .

Chapter

I- DISPERI

TABLE OF CONTENTS

Page
I . ,
L ST OF TABLES . ‘ . ‘ ' O I o I a c o 0 V1
LIST OF FIGURES. . . . , . . . . ~ . . . vii
INTRODUCTION. . . . , , _ , . . . . . . 1
Chapter
I. DISPERSION RELATIONS. . . . . . . . . 5
A. Single Variable Dispersion Relations
from Analyticity . . . . . . . 5
B. Two Dimensional Representation . . . 11
II. TWO-PION-EXCHANGE FORMALISM . . . . . . 22
A. The Dispersion Thegretic Approach . . 22
B. Inclusion of the NN+nn Amplitudes . . 32
C. Two Pion Exchange Potential . . . . 36
III. EVALUATION OF THE NN+UN AMPLITUDES . . . . 44
A. Partial-Wave Dispersion Relations for
NN+nn . . . . . . . . 44
B. Pion- -Pion Phase Shift Input . . . . 47
C. Evaluation of A(+) and 8A(+) at S-ur
t: O. Q Q Q . rt 0 O O O O 57
D. The NN+mn s-Wave Amplitude: A(+)
Fixed- —v Dispersion Relation. . . . 66
E. The NN+nn s-Wave Amplitude: Omnes
Dispersion_ Relations . . . . 74
F. The p- Wave NN+nn Amplitudes and the
Nucleon and Pion Electromagnetic
Form Factors. . - 78
G. The NN+nn d- -Wave Helicity Amplitudes . 94
IV. THE NUCLEON-NUCLEON AMPLITUDE. . . . . . 98
A. Phase Shifts . . . . . . - - - 132
B. Potentials . . . - . - ' ‘ ‘ ' 126

C. Discussion . . . . . .

iv

Chapter

LIST OF REFER]

APPENDICES

Appendix
A. THE HE]
B. POLE C(
C. PARTIA]
D. POTENTi
13. DIRAC I
F. Rho-ME:

G. OMEGA-I

LIST OF REFERENCES . . . . . . . . . . . . 130

APPENDICES

Appendix
A. THE HELICITY EXPANSION . . . . . . . . 138
B. POLE CONTRIBUTIONS TO THE NN+UH AMPLITUDES. . 141
C. PARTIAL WAVE PROJECTIONS. . . . . . . . 147
D. POTENTIAL EQUATIONS . . . . . . . . . 154
E. DIRAC ALGEBRA . . . . . . . . . . . 158
F. RHO-MESON COUPLING CONSTANTS . . . . . . 160
G. OMEGA—MESON COUPLING CONSTANTS. . . . . . 164

V

LIST OF TABLES

Table

l. A compilation of s-wave nN scattering length
determinations . .

2. A compilation of p-wave UN scattering length
determinations .

3. Results for A(+)(0,t) and its derivative at
t=0 O O O O

+
4. Results of various authors for A( )(0,0)

5. A compilation of UNN coupling constant deter—
minations. . . . .

6. Contributions to the s-wave dispersion relation
with MS phase shift . . . . . . . . .

7. The p-wave NN+UU amplitudes and derivatives at

t=0 . O n o I O I o o o a 0 O 0 o

8. Values used as subtraction constants for the
evaluation of the d-wave helic1ty amplitudes .

9. Predictions for the omega coupling constants. .

vi

Page

61

62

64

64

65

77

82

95

171

ll.
12.
l3.
14.
15.

LIST OF FIGURES

The momentum labels and scattering channels for
two-bOdY scattering I I I I I o I I I

The double spectral region for equal-mass
scattering . . . . . . . . . . . .

Low order nN scattering diagrams . . . . .

The nN physical regions and double spectral
regions 0 I O O O O O I I C I O O

The Mandelstam diagram for pion-nucleon
scattering . . . . . . . . . . . .

Mandelstam diagram for nucleon-nucleon
scattering . . . . . . . . . . . .

The t—channel diagrams for NN+nm+NN . . . .

Singularities of the NN+nn helicity amplitudes
fJ(t) O O . O O I O O C O O I l C
+

Single pion production and Ke4 decay . . . .

The I=J=0 nu phase shift derived from
experimental information . . . . . . .

The I=J=l nn phase shift . . . . . . . .
+
The UN amplitude A( ) along s=u. . . . . .
— . 0
Real part of the NN+nn amplitude f+ . . . .
— . f0
Imaginary part of the NN+nn amplitude +. . .

The I=J=O mm phase shifts used in the
calculations . . . . . . .

vii

Page

12

l4

19

20

21

26

27

45

48

52
54
69
7O

71

73

Figure

16!
17.

18.

19.

Figure
16.
17.
18.

19.

20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.

The J=0 spectral function .

The pion form factor.

The J=1 NN+nn amplitudes from twice subtracted
dispersion relations

The two-pion contribution to the imaginary parts

of the nucleon form factors.

Fit
Fit
The
The
The
The
The
The
The
The
The
The
The
The
The
The
The
The
The

The

to the nucleon electric form factor

to the nucleon magnetic form factor

NN+nn amplitude

NN+nn amplitude

NN+nn amplitude

lD2 phase
1G4 phase
3
Pl phase
3

2 phase

rUI

w
'm

C phase

T phase

w
"1|

w
'1le

LS
F2 phase
F

phase
F phase
Pl phase
F phase
phase

D phase

singlet even central potential

shift
shift
shift
shift
shift

shift

shift
shift
shift
shift
shift
shift

shift

qa/zlrll O C I O

3/2
q IPZI
qs/Zlfil

phase shift.

viii

Page
79

80
83

85
88
89
92
93
97
99
100
101
102
103
104
105
106
107
108
109
110
111
112

115

 

 

 

T

Figure
40.
41.
42.
43.
44.
45.
46.
47.
48.

49.

50.

The triplet odd central potential .

The T=l tensor potential

The T=l spin—orbit pOtential.

0

The long-range T=l spin-orbit potential .

The singlet odd central potential .

The triplet even central potential.

The T=0 tensor potential .

The T=0 spin-orbit potential.

The long range T=0 spin-orbit potential

Imaginary parts of the NN+nn amplitudes Ti

resonance fits. . .

Meson-nucleon quark diagrams.

ix

C

Page
116
117
118
119
120
121
122
123

124

163

165

     

- flint-.115 3.5:“!

u.¥|

55.50 sea-1.73:"

L.-' ' il|=‘

 

The d
nuclear force
physics reseaa
on'expressions

better knowled
standing of m
There

properties of

 

Although it 1.
determine the
a few parametr
introduce somr
Phenomenologil
to isolate thl
interest from
The 1:
interaction 0
9Xperimental
areas of part

enhance the k

 

INTRODUCTION

The desire to understand various aspects of the
nuclear force is the motivation for a vast amount of
physics research. Most nuclear physics calculations depend
on expressions for the nucleon-nucleon interaction, and a
better knowledge of this interaction may aid in the under-
standing of nuclei.

There have been many attempts to understand
properties of the nuclear force from a fundamental approach.
Although it is hoped that such an approach will eventually
determine the nucleon-nucleon interaction in terms of only
a few parameters, one is at present usually compelled to
introduce some phenomenological content into a calculation.
Phenomenological input to such calculations has been used
to isolate the aspects of a problem which are of immediate
interest from those not yet amenable to calculation.

The two-pion~exchange part of the nucleon-nucleon
interaction offers an excellent opportunity to combine
experimental information and theoretical ideas from several
areas of particle physics. This unification will hopefully

enhance the knowledge of each area.

 

 

. oulation of

      
   
  
 
 

lac-r ".sSewer1
order (in the
dontrihution
lemon1 showed
Binstock3 has
exchange pole
boson-exchange
culations is

except in the
ignored pion-
iind that the

two-pion-exch

 

The d
iations of Am
amplitudes to
authors used
by which they
s-wave energy
scattering th
rescattering
Thrive an int
approach has
discussed by

Using
exchange Brow

in two resper

 

Several authorsl' 2 have calculated the fourth—
order (in the nNN pseudoscalar interaction) perturbative
contribution to nucleon-nucleon scattering. A recent cal-I
culation of the fourth-order potential by Partovi and
Lomon1 showed that this approach can be fairly successful.
Binstock3 has added the 4th—order amplitude to meson—
exchange poles to obtain an improved version of the one-
boson-exchange model. The obvious weakness of these cal-
culations is that they ignored pion-pion interactions
except in the sharp resonance approximation and usually
ignored pion-nucleon rescattering. Nevertheless, we will
find that the 4th-order contribution is the largest part of
two—pion-exchange.

The dispersion-theoretic two-pion-exchange calcu-
lations of Amati, Leader, and Vitale4 allowed the NN+nn
amplitudes to be included in a more realistic form. These
authors used the Bowcock, Cottingham, Lurié5 nN amplitudes,
by which they included the fourth-order contribution, an
s-wave energy—independent mm interaction, and nN re—
scattering through (1) the A33 and (2) an effective
rescattering constant. The calculations also included the
p-wave nu interaction via a rho-exchange pole. The ALV

approach has also been used by Durso6 and Kapad'ia,7 and

discussed by Signell.8' 9
Using the ALV dispersion treatment of two—pion—

exchange Brown and Durso10 have improved the calculation

in two respects. First, they included a finite width

 

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In an
experimental
MPH continu
relations fro
region of the
amplitudes we
more reliable
mental determ
shift have sh

questionable .

 

has been used
exchange.l3’
Evalu
using the 1m ‘
iorned by Nie
Vinh Mau et a
use of these
of NPP appare
phase shift a
tides were ca
i=J=0 mm phas
phase shift i

In energy whe

exchange.

 

rho-exchange and, second, they introduced an axial-vector
nNN coupling correction, consistent with PCAC,ll into the
Nfi+nn amplitude.

In an attempt to determine the NN+nn amplitudes from
experimental input, Nielsen, Petersen, and Pietarien12
(NPP) continued nN scattering amplitudes via dispersion
relations from the nN physical region to the physical
region of the two pions. As NPP pointed out, their p-wave
amplitudes were in strong disagreement with the earlier,
more reliable, results of Hohler et al. Recently, experi—
mental determinations of the I=J=0 nn scattering phase
shift have shown that the s—wave NN+nn NPP result is also
questionable. Nevertheless, the NPP s-wave NN+nn amplitude
has been used in several recent calculations of two-pion-
exchange.l3' 14

Evaluations of the NN+nn s- and p—wave amplitudes
using the nn phase shifts as input have recently been per-
formed by Nielsen and Oades15 and by Epstein and McKellar.l6
Vinh Mau et a1.14 and Epstein and McKellarl6 have shown that
use of these newer Nfi+nn s-wave amplitudes rather than that
of NPP apparently results in somewhat poorer agreement with
phase shift analysis. Both of these newer s—wave ampli-
tudes were calculated using very similar forms for the
I=J=0 nn phase shift 6: as input. As we shall see, this
phase shift is not free of uncertainty, particularly at low

nn energy where it is of especial importance in two-pion—

exchange.

 

E’Ein’pht, and us

begin with an

derived from

statements re
for fixed-t 11
do not have

from Mandels
describe the

into the two
describes th
discussion 0
gives the res
and potential

Equat
units where ‘1'
u=1 and the r
required in d
the evaluatit

exchange we 1

quoted masse:

 

In this thesis we describe how the s-, p—, and d-
wave Nfi+nn amplitudes have been evaluated using subtracted
Omnes dispersion relations with the nn phase shifts as
input, and used in the two—pion-exchange calculation. We
begin with an elementary discussion of dispersion relations
derived from analyticity assumptions followed by a few
statements regarding the Mandelstam representation. Except
for fixed-t nN dispersion relations, the relations we used
do not have known field-theoretic proofs but all follow
from Mandelstam analyticity. In the second chapter we
describe the procedure for inputing the NN+nn amplitudes
into the two-pion—exchange calculation. The third chapter
describes the evaluation of the NN+nn amplitudes with a
discussion of the nu phase shift input. The fourth chapter
gives the results for the two-pion-exchange phase shifts
and potential. The appendices contain related matter.

Equations in this thesis are always written in
units where h=c=l. In most cases we also set the pion mass
u=1 and the nucleon mass m=6.7222. Where the masses are
required in MeV we used u=l39.576 MeV, m=938.259 MeV. For
the evaluation of the NN+nn amplitudes and the NN two-pion-
exchange we have assumed isospin invariance with the above

quoted masses and giNN/4n=l4.6.

 

hppl

9(2) analyti .

complex plan

If fiz) is a

no worse than

.1
h=l
Md: provided

contour can t

 

 

I. DISPERSION RELATIONS

A. Sin le Variable Dis ersion Relations
from AnaIyt1c1ty

Applying Cauchy's integral theorem to a function
giz) analytic within a closed contour C in the upper half

complex plane gives

._ 4 (2’
3(2) .. 37;; f .131. 42'. (1.1)
C 2 ~

If f(z) is a function also analytic within C and diverging

no worse than I2IN_n for positive N and n as |2I+co then

3(2)
nut—“(2-2.1, “’ O as izl-e w, m)
k=|

 

and, provided 2 are outside the original contour, the

k
contour can be expanded to give

 

’—

 

-F(Z) — ’ a. {(3') 32’

fr (2‘20 2n:i fl. (z'—z)frl(z'.zh+i<5)
3‘ S

._. 4.: Hz» +47%” fandz'
2 r 2m _ (2'4thng

*r ‘ kn

 

(1.3)

 

 

The real parh

relation wit]

fl:)=

 

where the first term in the last line is a sum over poles
on the real axis.
Now letting z approach the real axis by writing

z+z+ie and letting e+O+ we obtain

 

C )
Tl‘ — r — 2,-2 2 *"

 

. ~—'—.Pi M) 4"" '
27m -. (sz)TY(Z'-Zh) (1.4)
k

 

her
+ TrCz—zt) ”fl. (1.5)
T “(212)133’40

The real part of this equation gives the usual dispersion

 

1T (z-zh
flab Z ”r k f(zr)
" Z-Z
21r( :- in)
+ {Hz-2,.) ”w . (1.6)
#7 (sz) 'n' (It-2h)

 

For N:

nisthe dispe

it)

Letti

0fthe twice

integrand de‘

t0 Pronerly

For N=2 with both subtraction points on the real

axis the dispersion relation takes the form

 

Hz): (2 2:) its) + (z- 2) )itzz

 

 

(Z.‘Zz) (22 -2')
00 I . (
+ (Z'Z.)(Z-21) Im if“? )ciz .(I.7)
“3 _.°(2’-2)<2’—2.hL2’-22)
Letting 22 approach 21 gives a commonly used form

of the twice subtracted relation:
.1 z
4312) = ta.) + (2-2.) a2,4”(2 )tz’i,
(LBJZJ’OO Im Ctz’) 42’
-fi (24—2)(2’-2.)z '

In (1.6, 7, 8) it is understood that each factor in the

(LB)

 

 

integrand denominators has a small negative imaginary part
to Properly reproduce the imaginary part of f(z). The

following points are emphasized:

1. It was necessary that both the real and imaginary
parts Of f(z) behave no worse than lle—n as lz|+oo in the
upper half plane.

2. Analyticity was assumed only in the upper half

plane.

he
3- It was not actually necessary to let 2 approach t

. - used
real axis. Thus, the disper51on relation (1.6) can be

 

to continue f

sider the uns

he) :

The real part

Re “2):

to continue f(z) into the complex plane. For example, con—

sider the unsubtracted version of (I.3).

4”!) = —’— [wifl Jz’

The real part of f(z) is
°° Imflz' 2’42 2 +12 ftz' I 2
Pe.¥(z): 2%? —_____l%___:_%__:}__l_li dza
-. (z—z)(z—z ) (1.10)

Because the integration can be considered as around the

upper half plane, the last term in the numerator gives

_I_IJ‘°° Re f(z’ Inez

I
’ .. 12 H (1.11)
21': “00 (21-2)(2L2*) J? - 2 e z) .

Combining with (1.10) we find

 

_. _‘_. a. IM‘CG’) IMZ I ( .12)
720mg) _ W I, (2’-2)(2’—2’) dz ' I

AS in (1.11) one can show that

 

In f(z):

Adding (1.12)

version of (I

)C(z‘

In pr
discontinuitj
Provided the
IaPidly one I
SingUlaritie:

diSPEISion i,

Adding (1.12) and i times (I.le) gives the unsubtracted

version of (1.6):

 

60 ‘ I
£(Z) = a-L rm (2) 42’ (1.14)
TB Z/__2 .

—Os>

In practical calculations ones knowledge about the
discontinuities, 2Imf(z), is limited. Nevertheless,
provided the dispersion integral converges sufficiently
rapidly one may find it useful to include only the "nearby"
Singularities. Consider the large 2' contribution to the

dispersion integral in (I.6),

(I.lS)

 

 

N
I!" (2*2h) foo Im £(zl) 42’

773 L (2L2) fr (222.)
hzl

If L is much greater than 2 and all the subtraction points

2k then we may expand (I.l5) as follows:

10

 

Tl'lz-Zh) J” Im-Fdz‘) dz'
1. (2’)

1: M?! 1-1 -2
( 2,)Tr(( 3'3)

 

 

 

_ )sz-zb) °° 1391166.“?!
‘ +

24-232 /
TC L (Birds-I h +...}JZ . ”'16)

2!
Applying (I.2) for zzp', where L': L, we have

In»; fitz) s {3° 2""). > / (1.17)

where fO is a constant.

For L<z<L' we define an upper bound on Imf(z) as

Imac(z).<(3, ,L<Z<L’ (I.lB)

where fl is a constant.

An upper limit to (I.lS) is, therefore,

 

N
232(2‘25) £0
TC ”(Ln

 

+ I? [d— (%)N]+O(E¥h)} (1.19)

Thus, the dominant contribution neglected by terminating the
dispersion integral at z'=L is, for L>>z,zk, an Nth order
POlynomial.

The two—pion—exchange calculation to be described

later is based on an unsubtracted dispersion relation where

 

the variable
contribution
expected to b

primarily to

In!

The t
Mandelstaml7
than the one
discussed.

We me

Then the vari

11

the variable of integration is the momentum transfer. The
contribution from the neglected distant singularities is
expected to be approximately constant and to contribute

primarily to the low NN partial waves.

B. Two Dimensional Representation

The two dimensional representation introduced by
Mandelstaml7 in 1958, provides considerably more information

than the one dimensional dispersion relations which we have

discussed.

We may consider the 2+2 process shown in Figure 1.

Then the variables

S = ‘(P1 - nl)2 I
t = —(pl - p2)2 , (I.20)

are the barycentric energies squared in the s-, t-, u-

channels, respectively.

Mandlestam postulated that the only singularities
in 5r t, and u of a properly chosen amplitude are those
required by unitarity. The two—dimensional dispersion
are

relation has the following form if no subtraCtionS

required

 

 

hchannel e

Fisure ;

channelg

12

  
 

t-channel +

 

+

s-channel

Figure l. The momentum labels and scattering

channels for two-body scattering-

 

Shall negati.

real and the;

l3

 

 

 

 

‘7'; jdt' (Ju' fl— (1.21)
(.t‘-i’)(u’-u) '

It is understood that each factor in the denominators has a
small negative imaginary part. The spectral functions are
real and they are nonvanishing only when their arguments
are at least as large as the mass squared of a physically
realizable system.

As an example consider the scattering of two
particles of equal mass. The double spectral function
pst(st) is nonvanishing only in a restricted region where
S and t are both greater than (2m)2.

The boundary of the double spectral region is
obtained by considering the Feynman diagram involving the
lowest mass intermediate states in the s and t channels.

The boundary (Figure 2) is found to be st-4m2(s+t)+4m4=0.

 

4n)2 _

Figu

equa

14

 

Figure 2. The double spectral region for

equal—mass scattering.

 

The double 5;
singularity a
identicle bos
and the three

The d
valid for an
ities, which
intermediate
because it ca
0f analyticit
however, the
theory for a:
Mandelstam, 1"
rePresentam
Sional repres
aPproximatioI

We me
0(1) the dom
m particulaJ
Oru. The s-
ihdginary pa)

denominator s

l

%*P.

i

absorptive pa

15

The double spectral function pst has an inverse square root

singularity along this boundary. If the two particles are

identicle bosons the amplitude is symmetric in s, t, and u
and the three double spectral functions are identical.

The Mandelstam representation is expected to be.

valid for an amplitude which has only dynamical singular-
ities, which are due to the presence of real, on-mass-shell

intermediate states. The representation is very attractive

because it can incorporate the important physical principles

of analyticity, crossing symmetry and Lorentz covariance.

However, the representation has not been proven from field

theory for any process. It has been shown, originally by

Mandelstam,17 that the 4th-order diagrams do satisfy the

representation. Furthermore, all uses of the two dimen-

sional representation have been in a "nearest singularities"

approximation.

We may now reduce equation (I.21) (we shall consider

only the double—spectral distributions) to one—dimension.
In particular we may write down relations for fixed 5, t,

or u. The s-channel absorptive part is defined by the
imaginary part of A(s,t,u) in the s—channel where only the

denominator s'-s may vanish. Then with the prescription

1 1 . , .
§72§:IE + P ET:§ + 1m6(s -s) we write the s—channel

absorptive part as

 

         

mister-m? Isa-13:99- alduoh and: w 'H . - ..

 

. I 5913mm on

sat a): lysine-(axe .3;- aorzs+crsee'r‘gs't mialobnsn ad'i‘

"" '~-‘i3’-’1—"-‘-“- “Fifi-“'7” .15 Y'l‘m are:

: ‘ ""'R"'"'-W-- ms :0”: bum . . 7"".-
r 1 I W _ t . Ada”).
- ' r .._,_-. .;_I- .~_. 3.“. ‘:;Le_ 31

' murmur

The

Me
At

A(

16

A S,(Stu)= Tfsdélw 95*(S£)+ “Jul psuCSU')‘(I.22a)

U-t u‘—u

Similarly the t- and u—channel absorptive parts are

At(s,f)u):7c4—fs 45’ fstklh) ,L Ju’iL‘uul)

 

'5’ S at he'— u ) (I.22b)
: __y (4(5 IL!) I '
A (539(1),;L (is _:;__s__ + ~filt ?:“___’____::“) ‘ (I.22c)

The fixed 5, t, and u dispersion relations then

have the following forms:

A(stu) : .714? t (“:1 A); (24:3 “'75,?”
5

 

 

 

t’—'t
_L I A (S,£"($,U' I
+ T: gust“; J—m—lfl') . (I.23a)
A(s,t,u)-:71:- 45‘ As(S’,t,u"(s,’+)
5t g L
1.1;. du’ A (S"(t,’u), 4: u')
at u1_(1.23b)
ALSJ: u): J—[ ds' As(5l,t"(sl,u),u)
' TC Su 57.
’ "t’ t u)
'f' __ at! At‘s ( ,U), J (I.23c)

 

 

we have intr<
In va
distinguish i

the first ca:

_~

5’- 5
leads quicklj
The
be surprisin
ables. The
irltegration
he merely re

‘dt" and tha

l7

i—s-t', etc.

we have introduced the notation u"(s,t')E£m
In varifying these relations one must be careful to
distinguish between primed and unprimed arguments. Then in

the first case, for example, the identity

 

 

I 4 I -I
I +—-—'—- : _______..—o-—
s’-s [H M (stew-u)

leads quickly back to equation (I-Zl).

The form of the one-dimensional representations may
be surprising in that each involves two integration vari—
ables. The equations may, however, easily be cast in one
integration variable. For the fixed-S dispersion relation
we merely realize that t"(s,u') = Zmi-s-u' implies du' =

~dt" and that u'-u = —(t"-t). Then we can write

 

A(si’u) —_- LL: CHJ Ae(5)t"u”(s,tl))

 

J. -... . A st' " s,e')
+ (t A. .(,,u( ) (1.24)

777 (Slus) tl-t '

The lower limit of the second integral is now a function of
8- These two terms correspond to the right- and left—hand
cuts of A as a function of t at fixed 5.

As already mentioned the forms (1.23) may in some
cases be simplified by crossing symmetry. An example we
have seen is nN scattering where one may write As(s,t,u) =

:Au(u,t,s), the sign depending on the isospin state being

 

considered.

written

A(s,t,u) '~'

The

spectral reg
scalar natun
°H1Y an odd
similarly fc
Figure 3, g
S=(m+2u)2, 1
at s=(m+u)21

in Fi(lure 4

18

considered. The fixed-t dispersion relation can then be

written

A(s,t,u) = #j ds’ Asisft,u'(s,‘e))[——s.1s 3'- ‘5’!“-

(MP

The diagrams determining the mN scattering double
spectral region boundaries are complicated by the pseudo—
scalar nature of the pion, which disallows vertices having
only an odd number of pions. The boundary for pst and
similarly for put is determined by the two diagrams in
Figure 3. The diagram in Figure 3a has branch cuts at
s=(m+2p)2, t=4u2 while the diagram in Figure 3b has cuts
at s=(m+u)2, t=16u2. The double spectral regions are shown

in Figure 4 and dynamical singularities are shown in Figure

5.

 

 

 

l9

 

(b)

(a)

Figure 3.

Low order mN scattering diagrams.

 

 

ut

u‘Chfnnn.

nN-an

fi

20

t-channel
Nfienn
put ‘ pSt

u-channel s-channel

    
 
 

 

fiN+mN mN+nN

SL1

Figure 4. The nN physical regions and double

Spectral regions.

 

 

Figure 5.

Scattering.

21

N TTN nN N

The Mandelstam diagram for pion-nucleon

 

 

 

Th
pion-excha
For comple
nology the
including

Th

Figure 1.

”(15.011 are

anlie e , (

II. TWO-PION-EXCHANGE FORMALISM

A. The Dispersion Theoretic Approach

The basic dispersion theoretic treatment of two-
pion-exchange is described in the literature.4'7’8’9’l3’l4
For completeness and to establish the notation and termi—

nology the formalism is repeated here. The procedure for

 

including the NN+mn d-wave amplitudes is also introduced.
The nucleon four-momenta are labelled as in

Figure 1. The Mandelstam variables are defined by

2

W = '(Pl+nl) I

— 2 (11 1)
t — -(nl-n2) , -
E = ‘(Pl‘n2)2 I

which are written in terms of the center of mass scattering

angle 9, energy per particle E, and momentum p as

w = 4E2 = 4(p2+m2) :

t = -2p2(l-cose) , (11.2)

—2p2(1+cose) .

rH
ll

22

 

is usual, t

the sum of

he define

and the se

whiCh haVl

23

As usual, the sum of the Mandelstam variables is equal to

the sum of the squares of the external masses.

2

w+t+E = 4m (11.3)
We define the four-vectors
N = (nl+n2)/2 ,
P = (pl+p2)/2 , (11.4)
A = (pZ—pl) = (nl-nz) .
and the set of orthogonal four vectors
M = N+P ,
K = N-P ,
(11.5)
A ,
Ll = Auvau v D '
which have moduli given by
N2 = P2 = —1/4(w+E) .
M2 = -w ,
K2 ___ -E (II.6)
A2 = —t .
L2 = —wtt .

 

 

Tl

terms of 1

The Dirac

T]

exPressed

Now the 1

Satisfy t

24

The invariant amplitude Mfi for NN+NN is defined in

terms of the S—matrix by

 

4 I
S;— = C) ' +1: c)? I M /Z
1 (. 2+"z‘ rm) 4 M. . (II 7)
F P (21:) EIIEneEhE": it
The amplitude M is defined by
777,; : (I(/>,_)(:(n;) M tel/D.) u(n,) , (11.8)

Mfi is a matrix in the isospin space of the two nucleons.

The Dirac algebra is defined in Appendix E.

The spin and isospin structure of M is conveniently

expressed in terms of the spin operators

p2 = C(Y“)- T" + Y"’-N)r
133 : _ Y").N ‘1‘“).P , (11.9)
P4 : Y‘a. Y‘z) I
- 0
_P5 1 Ys‘ 11’2?"
and the Pauli isospin matrices.

Then

5'

772 = E} (_st (w H?) L2,); (wit) T‘Q’r’m] Pk .(11'10)

_ +
Now the longer range contributions to p; are assumed to

Satisfy the dispersion relation

 

 

 

(Dihvti: '
where the

P; (OPE)

Tl
branch cut
Figure 6.
spending 1
the other
We will 1;
amillitude

T)

(his in (
Process Ni

W" Pions

and illus

25

r (II.ll)

where the one-pion—exchange contribution is

+ - 2
Ph‘OPE)=O ) PthE) 2' fl?” CS‘LS' . (11.12)

The integration in (11.11) begins at the two pion
branch cut, clearly seen in the Mandelstam diagram in
Figure 6. One sees that there are in fact two cuts corre—
5ponding to the two pion state, one beginning at t=4 and
the other at t=4. For now we include only the cut in t but
we will later retrieve the other by antisymmetrizing the
amplitude.

The two pion cut is isolated from the more distant
cuts in (11.11) by applying unitarity for the t-channel
process NN+NN and restricting the intermediate states to

two pions. The kinematics in the NN channel are defined by

P = P2 r

p'= '91 .

n = n1 . (11.13)
n' = —n2 I

Q = (q-q')/2 (11.14)

and illustrated in Figure 7'

 

 

Figure 6.

Scattering.

    

26

\ \ t—channel
NNeNN

NN+NN

Mandelstam diagram for nucleon—nucleon

MeV

w—channel

 

 

F iiiii

27

 

p. P p' p
\ /
l l K a
f q q"? ”XR
l q/ \q
n n' n nl
(a) (b)

Figure 7. The t-channel diagrams for NN+mm+NN.

 

T1
momenta a)

notation.

Tl

(conventi

is

((115

28

The nucleons have momenta n,p while the antinucleon

momenta are n',p'. Note the minor deviation from the ALV

notation.

The unitarity relation for M readsl9
. T
“[W-fiz—mfl] :
4 4 . m~ 'l'
1 -
? E (2711—) 6 (RF Pn)(l _i‘) W43“ mnl' (11.15)

Ln) (n) V

where n. = fi— for intermediate state bosons with energy
3 Q3

ha and (3) 9L means to include only the intermediate state
particles in the product. Of course V is the quantization
volume, and c is a statistical factor to avoid multiple

counting in the sum. Designating the NN+nn amplitude by T

(Convention), the two-pion intermediate state contribution

is

 

X 54(H+hl—2~2’) I (II-l6)

Where 0,8 are the pion isospin indices.

The usefulness of (11.16) depends on a continuation
0f the NN+NN unitarity relation from the physical region to
far below physical threshold.9 The possibility of such a

continuation follows from the assumption of Mandelstam

 

 

analyticii
indicates
to be rel:
points he
of the don

Tl

tude is g:

(mm

K

and T

The ampli

variables

home nta a

29

analyticity. Furthermore, the Mandelstam representation
indicates how the quantities <manlNN> and <NN|T+|mn> are
to be related after continuing across the various branch
points below the NN threshold, provided one remains outside

of the double spectral regions.

The spin and isospin structure of the NN+nn ampli-

tude is given by CGLNZO:

was. 11M) = 5.FTM+%[TF,1JTH um

and T‘t) -: -—A(t) + [‘0’.QB(1—) . (11.18)

The amplitudes A(:) and B(:) are functions of the Mandelstam

variables for nN scattering defined as

5n = -(n-q)2 .
un = -(n-q')2 I
s = -(p-q')2 , (11.19)
p 2
up = -(p-q) .
— — t — —(n+n')2 = -(p+p')2 -
tp — tn — -

In the NN c.m. frame the magnitudes of the nucleon and pion

momenta are given by

p2 t/4-m I

(11.20)
2 t/4-l .

e-Q
l|

 

 

The c.m.

CC

so that

 

COS en — Sn‘uk
4P1
(11.21)
M 9 = 3:31 ,
7 41").
so that
Sn : " (132+11‘2FLQSD” )1
U.‘ = ‘(F +7:- +2rz c059,“)
$7; = - (f +17“ ‘21’7, 959?), (11.22)
Ur 2 -(F1+12+1FL COSBF) I

t : 4(P1+m2)=4(22+/)-

Before introducing these variables into (11.16) we

Will make the inclusion of both diagrams in Figure 7 more

apparent by writing

ICT _ _
f). hm): gfit’.w1"‘):(~n (,8)

From (11.10, 11) we find
NWIMT—WUND =
is 3 _ [2 ti: (11.24)
:1): (of: (we) -(1) it“ )(

+ 2{ shim?) +(—))h(>(;(e+)(T"31(2)]Pe .

 

Collectin

where ()
Expressin
gives the

amPlitude

31

Collecting (II.16, 17, 18, 24) we can write

5 (t
Z; 1RUM?)I<~0k9fl€€fln =

 

; I “‘11. (1 + , . at
’28 7‘2 :5 {Id—(1‘ [‘A ) (Sr-t) +‘X(r)'QB(t)(Srt’)]

x [‘Ammn 11’) l” [ {(3)4} 8mm 9)] , (11.25)

where QQ is the fin solid angle in the NE c.m. frame.

Expressing the right side of (II.25) in terms of the P
(i)

k

gives the spectral functions pk in terms of the Nfi+nn

amplitudes.

(o.+2mc_+m2e.) be:
-({>+m43 fizz
e h=3 (II.26)
~43 Iz=4
o Iz=§

 

32
(1:411 in If” (3,1) Ame, w Mar
1:: 4711£N[Rel-13m(Srgt')A(t)(M')][&—M+Q $1431 .

c. : 47ZZ£N [1248"5J)WI)A‘“(5,U)][ -%]Jno ,

 

 

 

1"-= WfifBu’avmms. 12‘)[o-M‘”’(Q") ] 412,,

4 I.
N: 547,-; "'5‘?! . (11.27)

B. Inclusion of the Nfi?nn Am litudes

 

The formalism of the preceeding section allows one
to calculate the two-pion-exchange NN amplitude given the
Nfi+nn amplitudes A(i), B(:). The amplitude B(i) is

Separated into Born and non-Born parts (the Born terms are

defined by pseudoscalar coupling theory):

 

 

B“
”8%

Recall th

Inserting
the nucle
'I

are easi]

 

 

‘2 (t "’
B ) ‘= 35) +13 (i), . (II.28)
t 1 — l

Recall that there is no Born contribution to A(i):

(+) _
AB- — 0 . (11.30)

Inserting (II.29, 30) for A(i), B(:) into (II.26) produces
the nucleon—box two—pion-exchange spectral functions.
The results for the nucleon-box spectral functions

are easily written in terms of the following functions.

 

 

k '-' 3‘2 , (11.31)
F?
‘ t“? 2 7- -_ . ( .32a)
7. : W12 [412.]; _ (1-2)? , (11.3210)
I,(L-): if. 4H1? .
x bj[~——+_JT] for 300
= 4:5 {or 31:0 ,
: [1:715 ant” ___.t: £9,- 94<0 , (11.33)
Xlt) 1‘ 35-1“— (l... h mtan‘lfi') , (II.34a)

Mai): Xlt)+ 47CU—ZA antaufi) +- (£22)Z Io(t), (11.3413)

 

 

547:3 T - (11.35)
‘1) 2N4 3 41
Pa 9 (we): 1‘ “Tzi[%*—t me)
2
+ 3:713}: Xtt) +czz(w-t)1:.(e)]r (II 36a)
<3) _ m4 4 .2 z
12¢ 9?. (Wi‘) - 1‘ fi- 2+ Mu”
+ W;2k X(t) -4F212 I0z£)], (II 3613)
Re figs-(wt) - # Re. Q,(r)(wf) . (II.36C)
(1:
Re (24in): mfg—men 35 X(+)
+ 12 IO (f)] , (II.36d)
(eéthwt) =0 . (II.36e)

The more general expressions for other box diagrams were

' 4 ' ' f ALV
given by ALV . When u51ng the general expre551ons 0
One must remember that ALV used the t-channel nucleon
momentum rather than the conventionally chosen antinucleon
momentum as the quantization axis.

' ' (+) (+) intro-
The non—Born contributions A — , B — are

dUCed via the helicity expansion (A.3). Because the t—

Channel partial waves in (II.27) do not interfere, we may

write

 

where pkg

I
with the
from (A.E

Born app:

We write

35

(t) , (t) ct)
?h (Wt ): 'fl‘SB (wt)+ QMSPJUNH
(1:)
- fi’hfispUW‘L'), (”‘37)
where pfiég is derived from (II.29, 30), pk;spd from (A.3)

with the first three partial waves included, and pk-Bspd

from (A.3) with the first three partial waves included in

Born approximation.

The partial wave contributions are summerized here.

We write the expansion (A.3) more conveniently as

 

 

A”) = at), +09, +a22(5;“)2+(124), (II.38a)
A") : oz, ( 2.75) + (3‘23), (II.38b)
‘8“) = F2($;u) + (:24), (II.38c)
—B(-) : IS' + (3.33), (II.38d)

where the first subscript on each term denotes the angular

momentum J. From (II.27) we find the J=0,l,2 contributions
) 2 at
we,“ (wt’) : NTC[](X,‘2 +161“! + émfRd/Bzdu)
.*
11124012. dzz) - ‘fflgmzmdpz 0‘22.)

4 m 4 z , (II.39a)
+ 40 % [dZZIZCW- 3H1) + T5: 0le j

 

the Oak

Procedm

f15M

unitari‘

 

4+) l ._ JM. 4 *
(+)
= 4 4 2

93 (wf’) Nng //32/ , (11.39c)
(+) l __ W

(>4 (w-l: - NTC E (1:4 [Fa/2 , (II.39d)
(-) -

’?1 (Wt’) - ”N75 ‘24 41,2 1°14): I (II'39e)
L-)

Q), (wt’) = -NTC ‘é‘ 121% («f/5,), (II-39f)
C-) t,

Q3 (W )‘= 0 . (11.39g)
(~)

1+ cwt‘) == ~~Tc % “1.2 We (IL...)
(1:) I

Evaluation of the J=0,l,2 Nfi+nn amplitudes completes
the calculation of the two-pion-exchange amplitude. The

procedure for projecting out partial-wave amplitudes is

discussed in Appendix C.

C. Two-Pion Exchange Potential

Recall from Equation (11.7) the definition of

M which has the NN elastic

fi E W (grep/2, (gray/2)

21

unitarity relation in the c.m. frame:

 

1th

satisfie)
Th1)
Using k .
Thaw

Thus, th

37

mun-m1.) (;—;——;’::. (4’7, 7;) 71:,1W l2)c§(2El-2E)

(II. 40)

where E = (k2+m2)l/2 and E —

q _ (q2+m2)l/2.

The scattering amplitude T(k',k) occurring in the

Lippmann-Schwinger equation

 

T(k',k)=\/(Hh)-—

 

(1,4433%? Egg; 2... 1(2).)

(II.41)

satisfies the unitarity relation

Tune) flue): )Tgfia’i T(ef1)T(z.H5(-i.:~§).

(11.42)
Using k2=k'2 = Ez—m2 (II.42) becomes21

T (h’h)— T( Hi):

 

Eff) ZT‘h‘L) T(piz)c)(2ez— 25)

(II. 43)

(211)7‘E

Thus, the relation between M and T is

(z .
”(71,12)- —-(—3%') 7W?” “-4“

The one-pion-exchange potential is defined by

 

Where Mh is the one—pion—exchange'amplitude

 

 

 

The coor.

V‘P’)

vah

1H the

38

M10 = Tf-Tn Y; ‘15” fig . (11.46)

and t = —(£'—£)2,

Explicitly, the potential is

 

/ _ N72 5 T 2
VE‘A)A)‘ (Eh’Eh) 11” Zn 7;“); m
x $‘(g‘lflj 93(Eté') 3": ’Xn . (HA7)

W") = (2:1)“f ed” ii“ War/2H 1.11.1

(II.48)

 

If V(Eifi) is a function of only the momentum transfer

E"£ then V(E',£) is local and V(E‘,£) = V(r) 63(£'-£),

 

V(r): ’ (5‘93" V(A)ABA (11.49)

(27f)3

where A = k'-k.

The one—pion—exchange potential (II.47) is clearly
not local. Partovi and Lomon1 have shown how to calculate
one-meson—exchange potentials by expanding the square—root

factor in (II.47) in powers of P = l/2(£'+£) while avoiding

an eXpansion in A. Their treatment is useful for ma551ve

eXch m 3 m but unnecessary for w-
anges ( meson nucleon)

exchange. The one-pion-exchange potential is obtained by

 

 

and the

V]; (P)

The term

"here M,

iterate

39

nonrelativistically reducing the square-root factor in

(II.47) to unity.

The momentum-space potential is then
T r 37‘
V-mM») - '1': X, '1??? 4m2(A7'y‘2)
x (frag gm; ’KT’XV, (11.50)

and the coordinate space potential is

 

 

The tensor operator is

A A
$12.: Bgf'r~"‘r‘qf"g" .

The two-pion-exchange potential is defined by

 

.1 y _. , , -.
V2,: (1:1): ”(Era )’- urUg) u,(—é ”1121? ur(é)u.,( L)

T (hilt) (11.52)
_' 1r: 1

- ' he
where M2 is the two—pion—exchange amplltude and Tnz 15 t

1r

iterated one-pion—exchange,

 

 

(11.44) ;

Have fun.

we find

40

T12 (W3): (2 21:)3 g. cm. 21W (LVTBU‘IIZ) Eff-271:6 V1112)”-
Srn

 

 

(II.53)

We may reduce the square-root factor in (II.52) to
unity. However, it is necessary to retain the square-root
factor in V", using (II.47) rather than (II.50) in (II.53).
The reason is clear: the intermediate state integral in
(II.53) involves a large momentum q so that a nonrelativis—
tic reduction of VTr is inappropriate.

It has been pointed out by Leon Heller22 that

(11.44) is (except for the sign) merely a change in the

wave function normalization. Defining

Vw(1IM=' .(E”" E: 1%) (35.1.), (11.54)

2

Tn'zéik) = "( 5:5). ’1 711(1),”, (11.55)

 

. (21).-» .

Equation (II.56) also follows directly from the

. 23 24 -
Blankenbecler—Sugar-Logunov—Tavkhelidze ' equation,

13

Which was used by Chemtob, Durso, and Riska to obtain

 

41

the iterated one-pion-exchange contribution to V2"
(Appendix D). It is most convenient to discuss the results
for T1T2 along with the complete potential V2".

Before transforming the potential to coordinate

space we shall rewrite the amplitude (II.10) in terms of

the following set of spin operators:

 

2.
_ 2
T ‘ Q ~r‘9’»-3vreg.4
55 ‘2 ‘51 srP‘Srk
f1... = “F.(gag)g;.(£.;~)) . (11.57)

These operators are of convenience because, except

for 9 they Fourier transform directly into the following

802’

set of operators:

W

I
)

r)

42

We therefore write

E: Qi‘t)(Wb"¢-)Ph = “Z, a/(t)(wth)fia

The transformation from pi to :1 is given in Appendix D.

The coordinate space potential is now

 

=/( t1; =(wtlz ~
'1). w m.

X
II
where B = l/2(E'+E). The :a form the iterated one-pion-
exchange contribution (Appendix D). The presence of t =

4m2-w—A2 and w=A2+4P2+4m2 in the arguments of the spectral

functions complicates the Fourier transforms in (II.60).

A similar problem with the t dependence is encountered in
Appendix C where partial-wave amplitudes are projected out.
As in Appendix C, and with the same justification, we set
EEO. As for the w dependence, the sum :2+:; is only mildly
dependent on w when lk'|=lkl, although :1 and 7a are indi-
vidually strongly w dependent.l3 We therefore set w=4m2.
Furthermore, the dependence on P-A is expected to be weak
at long range1 and it is neglected. The only nonlocality
then comes from the nu.

Expanding the coordinate space potential in the

operators (II.58) and simplifying the notation gives

 

 

.i'i‘

 

where x-

43

Z; \<((r)'J)'u _H(2173 5: 5‘J3A) 8-542:

 

°° =’ I = I ~
x {Ff (“I ‘9“ (t) 49’1““) D ’ (11.61)
4 t’f-AZ °‘

' where Q& have been integrated over g. The Fourier transform

can be easily performed to give

 

(1‘)
\/ (r): 7;). .rgf’ [(S’We') ”72% Jada) e7 ,

where x = Vt' r and

RC = i,
I
1250 : "E"(’+ Ix)!

 

 

amplituc‘

branch (
intermec
corre6p(
left cu)
(IN sca'
bY the 1

t=0 the

mediate

require

III. EVALUATION OF THE Nfi+flfi AMPLITUDES

A. Partial—Wave Dis ersion
ReIations for NN+nn
The analyticity structure of the Nfi+wn helicity

amplitudes25

 

fi(t) is shown in Figure 8. The right—hand
branch cut begins at t=4 and corresponds to the two-pion
intermediate state. Further branch points lie on this cut
corresponding to 4, 8, ... pion intermediate states. The
left cut is produced by the s- and u-channel reactions
(“N scattering). The branch point at t=4-l/m25a is caused
by the partial—wave projection of the nucleon pole. At
t=0 the nN states produce an additional branch point.

Since for 4<t<16 the only t-channel physical inter-

mediate state is two pions, the unitarity condition

requires that

Im-ff(t) 2‘ fig-"(H e Isl/n28: , (111.1)

Where 6; is the appropriate WW scattering phase shift. In
other words, the phase of f: is the flfl phase shift, modulo
W. Although this relation Is strictly valid only below
t=16 it is expected to be useful as long as the pion

scattering for angular momentum J is essentially elastic.

44

 

 

 

Figur

ampli

45

 

Figure 8. Singularities of the Nfi+nn helicity

amplitudes fi(t).

 

is an a:
extendi:
that the
t<4-1/m:
inelast:
followi)

Subtrac‘

The J=0

may Wri

46

26

The Omnes function, defined as

 

.. :1 °° 81 t’

D3“) _ “PL" L twinge—5):) AF] (HM)
is an analytic function except for a cut beginning at t=4
extending to t=w. The phase of DJ(t) along this cut is such
that the product DJ(t) fi(t) has only a left cut at
t<4—l/m2 and a right cut-where J-wave fifi scattering becomes
inelastic. With this knowledge of the analyticity the

following dispersion relation can be written, assuming no

subtractions are required:

a 'D;(t') Iw$f(t’)
H—t

 

Jé' _ (111.3)

"03111) Fife) = 3%]

$

The J=0 amplitude has a kinematical zero25 at t=4m2 so we

may write

 

 

’Dottii’fle) _ _Lf“ D.Lt’)rm££(e') Jt’
1'14"” 7E -.. (£14m2)(+.’—t)

The helicity amplitudes f: and fi were evaluated by
15

Nielsen and Oades in 1972. Their procedure was based on

(III.3) with the discontinuity on the left cut determined

l
(+), B(+) fixed-t dispersion relations.

by Nielsen27 from A
By introducing the Omnes function (III.2) the right cut was

effectively moved out to t z 50. Nielsen and Oades could

 

 

then ex
unsubtr
from t<
with se‘
and Oad
but onl

shown h

47

then extrapolate a discrepancy (Dofg/(t-4m2) minus the

unsubtracted Omnes relation (III.4)) by conformal mapping,
from t<4 to 4:t<50. The s-wave amplitude f2 was evaluated
with several different s—wave nn phase shifts 6:. Nielsen
and Oades preferred a 6: with a negative scattering length

but only their solution using the Morgan-Shaw28 60 will be

0

shown here.

B. Pion—Pion Phase Shift Input

As pointed out in the discussion of the Omnes
relation, generalized unitarity equates the phases of the
Nfi+nn and nn+nn amplitudes in the same angular momentum
states as long as the NH waves are elastic. For this
equality to be useful, an accurate knowledge of the phase
shifts for WW scattering is required.

There have been a very large number of analyses29
of the reaction nN+nnN to determine the WW phase shifts.
Much of the early work relied on measuring the s—p inter—
ference in the nu system.30 If only 5, p, and d-waves are
Significant, then the angular distribution in the dipion
center of mass can be written in terms of the nu scattering
angle as approximately W(w,t,cose) = ao(w,t)+al(w,t)cose+
a2(wrt)cosze. In this expression w is the dipion energy
and t is the negative mementum transfer squared to the

dipion. Figure 9a depicts the relation between nN+nnN and

nn+nn.

 

 

 

48

N(p-) ”(kl) "(k

 

N(p) 11(q) \

 

Figure 9. Single pion production and Ke4 decay.

(a)

(b)

 

The int
an asym
(forwar
and B a
reSpect

be writ

The asy

w = —(kl+k2)2

an asymmetry between the number of events with cose>0

(forward, F) and those with cose<0 (backward, B). Since F
and B are proportional to fiW d(6059) and [EW d(COSG)I
respectively, the asymmetry defined by g = (F-B)/(F+B) can

be written as

Al .

g:
273 A2+2Ao

The asymmetry can be smoothly extrapolated to t = “2 where
the pion scattering is on-mass-shell and 9 can be written
in terms of the s, p, and d-wave phase shifts. The
Procedure requires a knowledge of the p-wave phase shift to
extract the less well known s—wave phase shift. In
addition, there is a two-fold ambiguity 33+n/2 - (cg-5%).
Similarly, one may concentrate on the minimum in

the NW angular distribution and determine the s—wave
amplitude which will interfere with the input p-wave

. . 31
amplitude to produce such a minimum.

An alternate procedure involves assuming that the
process in Figure 9a dominates the physical single pion

Production at small It] (nearest the pion P01e)~ This

 

 

treatme

evaluat

the sug
polate
(t=w2)
extrapo
pansion
napping

increas

have be
0
do abow

resolve

50

treatment allows, with a sufficient number of events, the

evaluation of all of the relevant phase shifts.

The latter procedure has been improved by following

the suggestion of Goebel32 and of Chew and Low33 to extra—

polate the fin differential cross-section to the pion pole

(t=u2) where the phase shift analysis is performed. The

extrapolation has been performed34-36 by polynomial ex—

pansion in both t and in a variable obtained by conformally

mapping the t-plane such that the domain of convergence is

increased.

5: above the rho mass. This ambiguity has recently been
resolved in large part by the Berkeley analysis of Proto-
popescu et al.36 The authors included, approximately, the

effect of the KR threshold in a dipion energy dependent

analysis of the reactions

The Berkeley group also performed an energy independent
analysis and found a similar but slightly larger 6: with
larger uncertainty.

The CERN-Munich37 energy independent analysis
resolved the ambiguity (their two solutions also differed

below the rho) by comparing the PrediCted "ONO mass

 

 

spectru)
the exp)
below 8

Berkelej

also su

CERN-Mu

Particr
energy
Situatj
higher
down a;

due to

51

spectrum, which involves only even partial waves, with

the experimental spectrum. However, the preferred solution

below 800 MeV is consistently about 11° higher than the

Berkeley energy dependent solution (Figure 10).

Other experiments38 to determine the n°n° spectrum
also support the preferred solutions of the Berkeley and
CERN-Munich analyses.

The vast experimental effort on the nu problem has
thus given a good qualitative picture of the on phase

shifts between 450 MeV and 1 GeV. Quantitatively, the s-

 

wave 6: is uncertain by about 15° between 450 MeV and 770
MeV although individual analyses quote error limits of
about 14° and less.

A look at the Omnes function (III-2), regarding in
particular its once-subtracted form, reveals that the low
energy Wfl phase shift is very important. The experimental
situation below 500 MeV is at least as uncertain as at
higher energy because of the emphasis on resolving the up-
down ambiguity and the large total number of events required

due to the presence of the A and p.

33

. 30
The early analys1s of Jones et al.

by one-pion-
31 ' d' ated a

exchange dominance and of Walker et a1. , in 10

Smooth continuation of 6: from higher energy to threshold.

39,40

The phase shift 68 has been studied at low on

energy Via Ke4 decay (Figure 9b)

++-
K++eVTTTT‘

 

ii
i

 

en

DEGREES

52

. 8. 12. 16. 20. 2%. 28. 32. 36. HO.

2
i304.)
Figure 10. The I=J=0 no phase shift derived from

experimental information.

 

H4.

 

 

The Gene
phase 31
of a0 =
as 0.2,

Pennsyl)

two of ‘

G resu

beCause
Shift a

Shown a

53

The Geneva-Saclay collaboration39 found a rather large

phase shift at.low energy, and preferred a scattering length

of a0 = 0.619.25. They will not rule out a value as small

as 0.2, however. A similar experiment was performed by the

Pennsylvania group40 who obtained very similar results at

two of the three no energy bands.

Morgan and Shaw28 had earlier combined experimental

6: results and the ratio of I=0 to I=2 scattering lengths
. . o .
ao/a2 with on dispersion relations to determine 60. Their

results for 63 at low energy are considerably below the

39

Geneva-Scalay Ke4 decay results and somewhat lower than

36
the n-production experiment results from Berkeley, and
CERN-Munich37 (Figure 10). The latter two analyses, how-

ever, were for energies above about 500 MeV. The scattering

length found by Morgan and Shaw is a0 = 0.16 u-l. This
value of a0 is strongly supported by the results of
Pennington and Protopopescu4l who combined analyticity with
newer experimental results.

The p-wave phase shift 6% is much better known
beCause of the strong p-resonance. In Figure 11 this phase

shift as found by the Berkeley36 experimental analysis is

Shown along with two parameterizations of the form

 

V)»: at 8' : (4_av)(w+bv) . (111.4)
v c ’ c v

28
This is the form used by Morgan and Shaw Where the

Parameters were found to be

 

  

(DEGREES)

Inc.

120.

1m.

80.

60.

MO.

20.

u. a. 12.

Figure 11.

54

I": 120 mv
P2140 HIV

16. 20. 2'4. 28. 32. 36.
’09!)

The I=J=1 nn phase shift.

 

‘40.

'49.

 

l18.

 

 

 

These 1

1680113)

T 2-
(p m‘

55

a = 0.1536 ,

b = 0.028 , (III.5)

— 0.035 .

O
I

These parameters are related to the scattering length a1,

resonant mass mp (where 61 = 90°) and resonant width

:_ 3 1
(Pp- mp 53 cotdl) by

(1,: c.
mg: 2 a%’. (111.6)

 

2c;
F? = W(a+l)(a+b) .

Thus, the Morgan-Shaw 6% has a resonant mass of 765 MeV
(m1T = 139.576 MeV) and width of 119 MeV.

Another form for 0% given by Olsson42 and used by
Nielsen and Oades15 is completely equivalent to (111.4) and
has a width of 120 MeV. and scattering length 0.035.

The second curve in Figure 11 is (111.4) with a and

C as in Morgan and Shaw but with b changed to 0.00076, or

T

140 MeV. This second curve is in considerably better
agreement with the experimental phase shift. However, of
primary importance to us is the Omnes function which

results from this phase shift. This subject is reserved

for the section on the p—wave Nfi+nn amplitudes.

 

 

However
uninpor

is that

which

Munich
The CE
inelas
41 dec
N41)”
EXpres

degree

56

The d—wave phase shift is rather poorly known.
However, we shall see that its precise value is rather

unimportant because of its small size. The form used here

is that of Morgan and Shaw28

9 (:01: (S: : (4" 0'0524V)(‘1-D.ZD4V+0.0015'v1)

0+) 0.0015‘V1

 

I

(111.7)

which exhibits a resonance at the f° mass, t=80. The CERN—
Munich37 nn analysis found the d—wave resonance at t=83.
The CERN-Munich analysis, however, found considerable
inelasticity, n=0.6, at resonance. The situation regarding
4n decay of the f° is still hazy, but the branching ratio

F(4n)/F(2n) is generally agreed to be very small.43

The
expression (111.7) indicates that 03 reaches about eight
degrees at 1 GeV.

A meaningful application of the unitarity relation
(III.l) requires that we know where the fin amplitudes
become appreciably inelastic. All analyses have been
Consistent with elastic s— and p—wave amplitudes below the

34

0 mass. Baton et a1. analyzed the data with and without

0
the elasticity constraint on the s—wave 60 and found that

the now—preferred down solution remains elastic up to 800
MeV where inelasticity gradually sets in. More recent

36,37

analyses have found the s—wave amplitude essentially

elastic to 1 GeV where it becomes very strongly inelastic.

The p-wave 0% also appears to be essentially elastic to

1 GeV.

 

 

6w bra

that t

(1:50)

consta
Omnes

isospi

relati

field

(+)

57

These results indicate that the 4n and possibly the
Go branch cuts in the Nfi+nn helicity amplitudes are weak so
that the unitarity relation may be useful to about 1 GeV

(t;50).

(+)

c. Evaluation of A and 3A(+)gat
at s=u, t=

In preparation for determining the subtraction
constants to be used in a J=0 twice subtracted form of the
Omnes dispersion relation (111.3) we first evaluated the

(+)

isospin-even nN amplitude A and its derivative at s=u,
t=0. The evaluation was performed via nN fixed-t dispersion
relations, which have been rigorously proven in quantum
field theory.44
The dispersion relations were written in the vari-
able v = Eifi and subtracted once at Vc = 1+I% where the
subtraction terms could be written in terms of the nN
scattering lengths.
(+)

The amplitude A satisfies the relation

)
Aucvt)- - A“)

 

 

It is necessary to define

 

) v (+)
AIC+ (vii) -_- A(+)(v’f) + [+t/(4nfi)B (V)t)’ (III.9)

 

 

which

(III.8

tudes

The am

l+~ a

includ

58

which also satisfies a dispersion relation of the form
(III.8) .
The scattering lengths for the isospin even ampli-

tudes are written in the following notation:

a(+) s-wave
0 I
a(+) — 'th J=1+1/2
a1+ p wave W1 _ I
a(+) d-wave with J=2+l/2
a2+ _ .
- (+) ' (+) . _
The amplitudes A and A can be written for v—vc=
l+z% as follows. For completeness the expression B<+> is

included.

 

(+)
A (Vc't) = 2m+l(a° (+)+ lamt)

- m (aft) — 0.73.) + 2 301;- _ (12+))t )] (111.10a)

c+ (+) (+) (+)
+ m( a..-) - a.” + —(a 23.0.. )t)] (111.1016)

(Vc’t) - 4T5 7;; (1° +(8m1+2a’1-

4' onl" )t + 0119)] (111.10c)

 

 

tract:

59

The presence of the d-wave scattering lengths makes

A”) (vc,t) useful only for t=0.

we note that at v=0 crossing symmetry implies that

 

3 I“; f) 3 AIM) ( ll)
— V, = — (H- 111.
>t cheé“ at ‘ ) $Sti S.

 

The following expressions are the necessary sub—

traction terms.

 

 

(+) +I
A “10) = 815m 34:2 a:+)+af:)_af:’] (111.12a)
(+) (+)
: 31Em [0008 a;+)+0.+ “ 0%-]
(+) (+) (+)
.ibt‘Ald’zvpt) : 41; ”Trad [40‘01 + i- 0... + a,,.](111,12b)
t=o m

(+) (+) (+)
:- 47E %fl[o.oozs a, + 150,- +61. J

As we shall see, a

é+) is quite small compared to

a{:) and the subtraction terms are only weakly dependent on

its precise value.

In terms of the scattering lengths a2T and a2T,2J

f0r S- and p-waves of definite isospin and angular momentum,

the isospin—even combinations are

0.2” = than-2013):

(if? = 43- (an +2613)

 

 

 

poorly
We use

Carter

althot
larges
inTn
have j

binati

and

60

(+)
0"” : 301134-2433) , (111.13)

The s—wave scattering length aé+) is particularly

poorly known. A partial compilation is given in Table 1.

We used the value found recently by Bugg,

Carter45 from forward nN dispersion relations.

Carter, and

The isospin even p-wave scattering lengths,
although much better known than the s—wave, produced the
largest uncertainty in our results. A compilation is given

in Table 2. It is seen that the relevant combinations

(a +2a

13 33) ~ (all+2a3l) and (a13+2a33) + 1/2 (all+2a

31)
have just less than a 3% variation. In the latter com-
bination, however, the results of Hamilton and Woolcock46
and of Hohler et al.47 are about identical. The 0.7% error
limit given by Bugg et al.45 for the p-wave combination

- ' 11. Their method
(a13+2a33) (all+2a3l) is remarkably sma

for determining this combination involves evaluating the
. 2

B(+) dispersion relation at threshold uSing the f found at

finite energies. However, the quoted p-wave scattering-

1ength uncertainty is smaller than that produced by their

uncertainty in f2 alone. We have used the values

((1)3 +2 033)—(a,,+2a3,) = 0.5"]: towlo (111.14a)

and

‘ 1‘ . 0 . (111.14b)
(a.3+ 20.33)+fi(a“ +2a3,)-o.310 - DD/

 

 

 

Table

0.055'

'.002

-.005

'.014
+003
2023
'.073
~.021

..070

‘.08(]

diffs
.IOOE

61

Table l.--A compilation of s-wave nN scattering length
determinations.

 

 

 

a1+2a3 First Author (Ref.) Method
0.0557 1 .0155a Lovelace (56) backward d.r.
-.002 i .008 Hamilton (57) low energy 0: p
data
—.005 + .009b Hamilton (46) forward d.r. and
— sum rule + Pan.
ratio
-.014 i .005 Bugg (45) forward d.r.
+.0031 i .008 Samaranake (58) forward d.r.
-.023 i .015 Hald (59) forward d.r.
-.o73c Engels (55) forward d.r.
-.021 Hohler (47) partial wave d.r.

________________________——————

aValues quoted are al 0.1957 1 .0111 and a3 =

-.0700 i .0054.

bValues quoted are a1 = 0.171 1 .005 and a3 =
“-088 i .004.

cSeveral valuesquoted in reference corresponding to
different applications of forward d.r. Values range from
-0006 to -.0730

62

Table 2.--A compilation of p-wave nN scattering length
determinations.

 

First Author
(Ref.) Method

+ 2a + 2a

all 31 al3 33

 

-.l77 : .012a 0.401 1 .011a Hamilton (46) forward d.r.
+ sum rule +
Pan. ratio

-.168 1 .02a 0.396 1 .02a H6h1er (47) partial wave
d.r.
_ + .009 + .014
.173 _ .014 0.390 _ .016 Engels (55) forward d.r.

The result of Bugg (45)

= -.570 i .004 is consistent

+2a - +2a

all 31 (a13

with the values above.
__________________________________________________________

33)

aValues quoted by authors are for individual
Errors given here assume no correlation in quoted

azr 2 °
errorg and therefore may be inaccurate.

 

which

by the
shift
These
severe
its t-
were 1
set.

exten(
energ}
Barge;
diSpe;
input

sets,

63

which are consistent with all values in Table 2.

The imaginary parts of the UN amplitudes required
by the dispersion relations were calculated from phase
shift solutions as supplied by the Particle Data Group.48
These included Glasgow A and B,49 Saclay,50 Roper,51 and

52

several CERN solutions. The results for both A(+) and

set. Except for the Roper solution, these phase shifts
extend to about 2 GeV center-of-mass energy. Above this
energy we have used the two-Pomeron pole parameterization of
Barger and Phillips.53 However, the once subtracted
dispersion integrals were insensitive to this high energy
input. Table 3 gives the results for several phase shift

sets. The following values summerize this work.

A(+)(s=u,t=o)= 25.9 20.5" (111.1581)
( _ + (111.15b)
3%; +)(s=u,t)(t=o— (J10 - 0.05

The value of A(+) is compared to other determinations in
Table 4. The slope is consistent with that of H6hler
et al.54 (1.15 i 0.1), Engels,55 and Nielsen.27

Since in many cases the nNN coupling constant was

determined along with the scattering lengths, we have

2

included in Table 5 a compilation of the f results-

 

 

Table

CERN
CERN
Roper

Roper

64

Table 3.—-Results for A(+)(0,t) and its derivative at t=0.

 

 

Phase Shifts A(+)(s=u,t=0) 5%A(+)(s=u,t)|t=0
CERN EXP (Kirsopp or Wagner) 25.9 1.16
CERN (Almehed, Lovelace) 25.8 1.16
Roper + Glasgow A 25.9
Roper + Saclay min. path 25.8

 

+
Table 4.--Resu1ts of various authors for A( )(0,0).
M

First Author (Ref.) A(+)(s=u,t=0)
____________________________________________________.__________
Samaranayakea (60) 24.6 i 0.4

Hahler (54) 26.1 i 0.3

Engels (55) 26.0

Nielsen (27) 26.1

Adler (11) 26.15 i 0.2

This work 25.9 i 0.5

____________________________————————

aSee Ref. 58 for comments on the input to this
calculation.

 

Table

0.078

0.076

0.078

0.077

0.076

0.081

0.079

0.081

0.083

0.074

hoec

65

Table 5.—-A compilation of nNN coupling constant deter-
minations.

 

 

f First Author (Ref.) Method
0.0787 i .0034a MacGregor (61) N.N. Scatt., no dist.
for Ni, n°
0.076 t .005 Bilen'kaya (62) Same
0.078b Cutkosky (63) analytic approx theory
350 MeV
o.o77° Cutkosky (63) 400 MeV n—p data w/
Form Factor
0.0763 : .002 Samaranake (58) forward d.r. for lab.
amp.
0.0816 : .0029 Hald (59) forward d.r. for lab.
amp-
+
0.0790 1 .0010 Bugg (45) d.r. for B(—)
0.081 + .002 Hamilton (46) forward d.r. + sum
— rule 1 Pan. ratio
+
0.083 + .005 H'o'hler (47) d.r. for B( ), CERN
_ forw. amps
0.074 + .01 Hohler (47) d.r. for F"), CERN
_ forw. amps, CEX forward
cross sections
0.0803 + .0010d Lichard (64) analytic continuation

of F(—)(w)/w

___________________________——

a
Value uses 0° mass.

Value by

bQuoted as g2/4n

CQuoted as 92/40

dSee Ref. 58 regarding the uncer

7%.

this result.

Charged pion mass raises

14.1 (+1.4,-0.8).

13.9 (+1.6,-1.7).

tainty quoted in

 

 

An ex

Pilku

,0)

tribu

where

Parts

66

An extensive compilation of nN parameters was given by H.

Pilkuhn et al., Nucl. Phys. B65, 460 (1973).

D. The NN+nn s-Wave Am litude:
A(T) Fixed—v DisperSion Relation
The NN+nn s—wave amplitude contributes only to
A(+). In terms of the helicity amplitude f: this con—

tribution is

(+) o
A =O(U,t)= ‘73:}; 'F+(i‘) , (111.16)

where p3 = mZ-t/4.

The values (111.15) of AH) (0,0) and ’53? AH) (0,t)l
t=0, previously discussed, have been used in a fixed-v
dispersion relation for A(+) to determine the low-t f:(t).
As was the case for the partial-wave Omnes relation (111.3),
we introduced the s—wave Omnes function Do(t) and dispersed
for Do(t) A(+)(0,t). This function maintains the right cut
at t>4 which can, however, be evaluated with the d—wave
amplitudes f3 yet to be described. We assumed that all
3:4 partial wave amplitudes are adequately represented by
the nucleon pole (Born) amplitude. The smallness of the
JZZ-wave nn phase shifts allowed us to accurately approxi—
mate the J12 partial wave amplitudes f: by their real

parts. The twice subtracted dispersion relation reads

 

where

where

Then

Dot (t = (00) _7 (+)
()A 0,) A +tat [mum (0,101

[:0

ll ~Ld, 0(4' H) 9
+IL+€11 WlDlHPeI )AJ,Z(°,t)]Smd°(£')°/tl
K ,
‘ (t) (t’-t)

(111.17)
where

The contribution from the left cut beginning at t

4mu;-27 is represented by IL. This contribution is small

for 0:t_<_4 because of the distance of the left cut and the

two subtractions in (111.17) . Nevertheless, IL is necessary

for an accurate evaluation of f3. Because of the weak
dependence on the left cut discontinuity and the expectation
that only its leading edge (t;—4mu) will be important, we
introduced a simple parameterization and extrapolated from

negative to positive t. We represented the left cut dis-

continuity by

736(1) Abs Afiiot): or -t-l> <IIL18)

I

where b = 4mu and ti-b.

Then fifth IJ-—-—1. t’-(> (“I
It : a 3“ .. (were—t)

t
“(REF—57f) . (111.19)

 

 

To 06
fixed

With

Tigun

(+)
Angz

the i

evah
in a
disp(
sens
of f

mina

68

To determine a equation (111.17) was compared with the

fixed-t dispersion results of Nielsen at t=—5 and -10.

With the Morgan—Shaw28 63 we found a;29u-2

Results for A(+)(0,t) and f:(t) are shown in

Figures 12, 13, 14.

Aé:; from A(+). We have used t=80 for the upper limit in

For t>4 f: was obtained by subtracting

the integrand (111.17).

An important product of this calculation was the

evaluation of A(+)(0,4) = peg? f$(4), which will be used

in a second calculation of f:(t). The s-wave Omnes
dispersion relation subtracted at t=0 and t=4 is very
sensitive to the value at t=4 so an accurate determination
of f:(4) is required. We list the input to the deter-
mination of f$(4):

a. s—wave nn phase shift 6:. The Morgan—Shaw pre-
ferred solution is used to t:50 where 68 is turned
downward and extended to w as l/t.

b. Subtraction constants A(+)(0,0) = 25.9 and
5% A(+)(0,t)lt=0 = 1.16.

C. Left cut discontinuity approximated as described
above.

- (+)
d. The higher partial wave amplitudes AJ>2 from

partial wave Omnes relation (J=2) and the nucleon

~ ~2
Pole (J14). The value fi(24) was used for f+(t:24)'

 

.-_r

 

l10.

30.

10.

Figure 12.

69

-60 -ua —2 0. 20

t 0.2)

The nN amplitude A(+)

along s=u.

 

70

All curves calculated
with Morgan-Shaw 6:.

150.

«100.
(~—-—~) Eq. (111.21),

0 _ 2
Dof+/(t 4m ).

50‘ (--—-) Eq. (111.17),

With 1L=O.

o
( ----- ) D.r. for Dof+
subt. twice at t=0.

 

0.
q. 8. 120 160 200

t 0B)

Figure 13. Real part of the NN+nn amplitude f2.

 

 

Figure

 

14.

71

Notation is as

in Figure 13.

8. 12. )6. 20.

t (y)

— . 0
Imaginary part of the NN+nn amplitude f+.

 

The (
a'bov)
we h

t=15

72

e. The analyticity structure of A(+)(O,t) and
f2’2(t). Mandelstam analyticity is sufficient.

With this input we found

f2(4) = 114. + 2. (III.20a)

The error limit is intended to include uncertainty in the
above inputs (b) through (d). As an alternate phase shift
we have used the Berkeley36 energy dependent results above
t=15 appended to the low energy phase shift predicted by
the Morgan—Shaw65 model. This phase shift is shown in
Figure 15. The result with otherwise the same input was
f2(4) = 115. On the other hand, extrapolating the cal-
culated f3 linearly to t=80 reduced f$(4) to 113. Com—
pletely neglecting the left out in (III.1) gave fi(4) = 116.
Using the Morgan-Shaw 6: extended beyond t=50 as l/t3 gave
f2<4> = 113.

In Figures 10, 15 we have shown a 62 constructed to
agree with the Geneva-Saclay39 Ke4 decay results and join
smoothly with the Berkeley36 energy dependent phase shift
at t=20. With otherwise the same input as for (111.20) this

phase shift, designated GS+LBL, gave a=29 and
ff(4) = 118. i 2. (111.20b)

The values (111.20) of f:(4) can be compared to

determinations by fixed-t dispersion relations. Engels55

47

and Hohler et a1. each obtained A(+)(0,4) = 31.5, f$(4) =

27

111. Nielsen's tables can be extrapolated by hand to

WNUNNUWQ

DEGREES

 

73

<—-) Ms
(- — -) M 5 Model

'4. 8. 12. 16. 20. 2'4. 28. 32. 36. '40. Lil}.

1:942)

Figure 15. The I=J=0 n0 phase shifts used in the

calculations.

 

imply
are 0
of (IN
Niels
and 2
ampli
<_)p_t_ip
extra
one (-
erro:

the (

GS+Lj

74

imply A(+)(O,4) = 31.6, f$(4) = 111. However, such values
are of questionable accuracy since they involve continuation
of nN amplitudes by Legendre polynomials to the branch cut.
Nielsen appropriately terminated his tables at t=3. Chao
and Zia66 have studied the problem of continuing nN
amplitudes to the on cut. They found that using an
optimized expansion they could place an upper bound on the
extrapolation error. The upper bound rises very sharply as
one approaches the cut. With a Legendre expansion the
error bound may be smoother and probably larger than for
the optimized expansion.

That we found the same a for both the MS and
GS+LBL phase shifts means that the results for A(+)(0,t<0)
are very weakly dependent on the s—wave 00 interaction.
However, we note the nonlinearity of A(+)(0,t) for 0<t<4.
The 4% change in f$(4) resulting from the change in 6: is

significant for the NN two-pion-exchange problem.

E. The NN+nn s—Wave Amplitude:
Omnes DisperSion Re atlons
The partial wave Omnes relation provides a second,

more conventional, way to obtain f$(t). The left cut, as

2-

shown in Figure 8, begins at t=aE4—l/m ~3.97, very near the

Pseudophysical threshold, but the discontinuity can be
calculated for t>-26 from nN phase shifts. Because of a

kinematical zero25 at t=4m2 the dispersion relation may be

written for Dofi or for DofS/(t-4m2). The latter is

 

 

exp¢

rel¢

75

expected to converge more rapidly on the left cut. The

relevant disperion relation is

DalfHfit) . Lt~4)£+°(o) + tau) m4)
t—4m7' MamZ 4(4-m‘)

 

 

fled) a m,(t')ImC+('t) cH-I
t’(£’-’-4)(i 4m) )(éLf)

 

(111.21)

 

Twice subtracted dispersion relations such as
(III.21) are quite sensitive to error in the subtraction
constants because of the multipliers proportional to t.

The large size of f$(4) produces a particular problem where

a 1% uncertainty in f2(4) produces about 10% uncertainty in

|f$(20)l2. The value of f:(0)=-2.8 was found by Furuichi

and Watanabe67 using the partial wave projection of a once-

subtracted fixed t dispersion relation for A(+). The sub-
(+)

traction constant in that calculation, A (0,0), taken from

Reference 60, can be replaced by the value (II.15a) to give

12(0) = —2.4 (111.22)

which was used in all of our evaluations of (III.21).
The imaginary part of f:(t<a) was obtained from

27 tables. We have cut off the integral in

Nielsen's
(III.21) at '=—35, extrapolating Im f:(t) by hand from
t=-25 to t=-35. This extrapolation is a reasonable com—

promise between cut off error and uncertainty in

~o
Im f+(t<-25).

 

 

the pha

the dis

are sho
Figures
with th
labelle
118. w

t>2o.

76

The results from (III.21) are somewhat sensitive to
the phase of f:(t>50) which is unknown. We have evaluated

the disPersion relation with t>50 phases of the form
6(t) = (SO/t)n 6(50).

The results for the s-wave spectral function

 

(+) _ 1;_-_4 4::le
9° " 8”: t 4m2-t

are shown in Figure 16. We have also shown results in
Figures 13, 14. The curves labelled 81 have been calculated
with the Morgan-Shaw phase shift 6: and f2(4)=ll4. Those
labelled 82 were calculated with the GS:lBL 6: and f$(4)=
118. We see that the t>50 phase affects If$(t)l mostly at
t>20. Although the GS:;BL phase shift gave a larger

f$(4), it gives a smaller If:(t>5.7)l.

There is a considerable amount of cancellation
among the three terms of (III.21). Table 6 shows these
three terms at various values of t. We see the importance
Of f:(4) and the relative unimportance of Ref:(0).

In Figures l3, 14 we have also shown the results of
a calculation using a dispersion relation written for
Do(t)f$(t) subtracted twice at t=0. The input to this

calculation included the M—S 6:, Nielsen's27

Imf:(t<a)
extrapolated to t=-45, Refi(0)=-2.4 as discussed previously,
and the derivative of Ref:(t) at t=0. For the derivative

we used

 

 

 

which wa
the t>50

smoothly

Table 6.

10
20

30

77

a
fiRe f3(t)[t=o = 3.2 ,

which was obtained from the tables of Reference 27. For

the t>50 f: phase we used n=l but allowed the phase to turn

smoothly over at t=50.

Table 6.--Contributions to the s-wave dispersion relation
with MS phase shift.

 

 

t=0 t=4 Disperion
t Subtraction Subtraction Integral Total
5 —102.5 —l9487. 13110. -6480.
10 -580.7 —36788. 29103. ~8266.
20 -l372. -65212. 55427. —11157.
30 -l961. -86027. 75355. —12633.

W

The derivative value quoted above is much larger
than that used by Epstein and McKellarl6 who found the
value 2.0 by forcing the disperion relation to give
f2(4)=lll. In addition, Epstein and McKellar used Ref:(0)=
“2.8, the value found by Furuichi and Watanabe67 using
A(“(0,0) from Reference 60.

With our calculation, derivative=3.2, we found

f2(4)=ll7, somewhat larger than we found from Equation

(111.17) with the MS 5:.

 

78

On

 

0v

Om

lo
/ ./.;:c//
to . ll ’
/ ltd. II
I: a III/o I,
o OIZIui/o Mu: _m III/o I,
I l/ I’I/l
0 II /
II III
I
I
I
I
I
II

.cofluocnw Hmuuommm cum on&

:3 I
ON 9

.ma musmnm

siderati
They are

the nuci

the inw

79

F. The p-Wave Nfi+nn Amplitudes and the
Nuc eon an Pion E ectromagnetic
Form—Factors

The p-wave Nfi+wn amplitudes require special con—
sideration because of the presence of the strong p resonance.
They are of special interest because of their relation to
the nucleon electromagnetic form factors.

The p-wave Omnes function Dl(t) is approximately

. 68
the inverse of the pion electromagnetic form factor F“.

 

... i °° 8“,) I] (111.23)
Fm“) - exF[rL t’(£’-t) (it

The peak of FTr near the p mass is sensitive to the phase
shift 61. In Figure 17 we show IFNI evaluated with two
different rho widths in the expression (III.4). Also shown
are the results of the colliding e+e_ beam experiments at

69 and Novosibirsk.7o The value F = 120 MeV was used

Orsay
by Nielsen and Oades15 and is very close to that found by
Morgan and Shaw.28 We used the value F = 140 MeV. We have
already noted that a rho width of 140 MeV gives a good
approximation to the experimental 5% for t:50. The t>25
form factor, however, is sensitive to the phase shift above
t=50. We also note that (III.23) is approximate in that it
neglects all but the 2H intermediate state in the flfl+Y
unitarity relation.

The amplitudes f} were assumed to satisfy the

fOllowing twice subtracted Omnes dispersion relation.

 

 

8.

ll.

80

 

 

 

Figure 17. The pion form factor.

 

From Fi(
the 9 m(
180. I:
subtrac
Present
fact ha

the p-w

81

l 4 '
“Mid“ = fem +t[%uv.w)¥.'w)].,

 

+ mi?“ 73,051) I“ #179)
-.. (Ufa-Lt)

Jé’ . (111.24)

From Figure 17 we see that D(30) = 1/1F"(30)|:6 so that at
the 9 mass the error in ngilt=0 is amplified by tDl(30)z
180. In fact we found that the J=1 dispersion relations
subtracted twice at t=0 are not useful above t=10 with
presently known values of the subtraction constants. This
fact has also been noted by Epstein.l6 When considering

the p—wave amplitudes it is convenient to define25’5

/
n“) = fi‘ (4.1%) —$7" 41%“) , (III.25a)

 

em = 2,1. (the) + 7%— mw) .
These amplitudes are also expected to satisfy dispersion
relations of the form (III.24).
From (III.20) we have calculated F2 and f} for
t<10 using as subtraction constants the values in Table 7.
The non-Born entries were found by interpolation of

27

Nielsen's tables and the Born terms were calculated from

the expressions in Appendix B. We have used a lower limit

of t=—45 in (111.24) and let Imf}(t) Imfh-ZS) for -45<t<-25

(Similarly for ImF2(t)). This lack of knowledge about

1

Imf and ImF

_ 2 below t=-25 is a second reason that (III.24)

 

 

Table 7

tudes j

ralati(

D (

82

Table 7.--The p-wave Nfi+nn amplitudes and derivatives at

 

=0.
Born Non-Born Complete
Re r2(0) 0.02394 -.0185 0.0054
Re f}(0) -.2280 0.328 0.100
Re T: r2(0) -.00566 -.0004 —.0061
Re r:- 12(0) 0.0554 0.001 0.056

 

fails for tilo. The results for t<10 are shown in
Figure 18.

A more satisfactory approach to the p-wave ampli—
tudes is to subtract at t=0 and mpz. The dispersion

relation for Pi takes the form
D can (1) - ___—on) +—-z>,(1 ) flu.)

f(t- t.) Deal) 1:... PM?) Jfi’
-!:'(t’— t )(tlet)

(III.26)

 

+

 

This form of the dispersion relation is difficult to use
because the Born contribution to Imrz iS rapidly varying

near t=a. This rapid variation can be suppressed by sub—

tracting a similar relation for rBi’ the Born amplitude,

_ t ,
r13; : at: FBI“) +7; F3150)

tlt—t,) 4» I'm Fazéf’) clt’. (111.27)
7- -oo £’(£I‘£9)(£Lf)

 

 

 

 

83

   

04315
Twice subt. at t=0.
Subt. at t=0, 30
to fit nucleon
0.010 form factors.
0.005
O.
8.
6. 1
fl.
2.
0.
Q. 8 8. 10.

t 942)

Figure 18. The J=1 Nfiann amplitudes from twice

subtracted dispersion relations.

 

The res

In pra<

consta:

becauS(

Small

84

The resulting dispersion relation is

3,11) me) = F341) + %[D,(t-,)Q(t°)~F‘Bl-_(to)]

 

 

1 1,-t ~
12., ,7“)
+ tit-t.) “ Dd’c’hml‘w') +ED,I£')—(]1:.Fe;‘*' 41’
v—,—-\_ .
77‘ -.. t {t’-t.)(t’—t)

(III.28)

In practice we used to=30. To determine the subtraction

constants at tO we turned to the nucleon form factors.

The nucleon electromagnetic form factors are useful

because the photon isolates the J=1 Nfi+nn amplitudes.

Previous work5’7l'72 has supported the postulate that for

Small It] the form factors are determined by their nor-
malization at t=0 and the nearest part of their branch cut
at positive t. The discontinuity along this cut is given
by unitarity as depicted in Figure 19 where only the two-
pion intermediate state is shown. The I=l (isovector) form
factor can have contributions only from an even number of
Pions in the intermediate state. We assumed that only the

two pion state is important.

The form factors are defined in terms of the

. 73
nucleon electromagnetic current 311 by

 

85

<:::fé:;:>
I \

I \
1T, ‘1]-
/

\
I \
‘ N fi \

Figure 19. The two—pion contribution to the

imaginary parts of the nucleon form factors.

 

 

(fl

Q

The i501

given b)

where 1

Tagneti

where .

86

I . 7. ' " l I.
(F IJ)‘ I P> = elf—1%); (AMP )[F/i )§11+]:2(%2)°;w(F-P')JU(P)
(III.29)

The isovector (I=1) and isoscalar (I=0) combinations are

given by

_ 5 v

S

[:2 ‘2 F:2

(111.30)

'+ ATE ’2év I

where 13 is +1 (—1) for proton (neutron).

More commonly encountered are the electric and

magnetic form factors GE and GM defined by

 

6; = F, -' 2m T F . (111.3111)
GM -_-_ I." 4. 2m [:2 , (111.3115)
p, -: G: + TQM , (111.31c)
H- 'T
-55
F = 6’“ 111.3111
2 l +,7. , ( )

where 15 1%:-

The form factors are normalized to the values

V
l _ . 2
G2,“): 2. ) GM“) ’ ‘zL+3 ' (III 3 a)

V
F'Vco) 2% ) 2mF2(0)23 , (III.32b)

 

 

where g

magneti

dispers

The T.
1
couplir

vector

87

where g = 1/2 (A —An) = 1.8530 and AP, A

p n are the anamolous

magnetic ratios.

The Nfi+ww amplitudes Ti are related to Fi by the

dispersion relation25

 

go , 3 I f» r

v .1 (1-4) / 1: (y nave/t

Fllth/‘WEJ [mfg " ,) , .
4 t (t ~f) (111.33)

The Pi are just the Nfi+wn amplitudes for vector and tensor

coupling. They are, therefore, related to the rho meson

vector and tensor coupling constants, as shown in Appendix
C.

Without detailed fitting, because of the large
amount of computation involved, we found that the form

factor data were reproduced with the following values of

Yi E [Dl(30)Fi(3O)-TiB(30)]=

yl = 0.136 ,

(111.34)
y2 = 0.0116 .

Since m§;30 and ReF(m§)=0 we found, with Dl(30)=0.168, that

Im 11(30) = —0.121 ,
(111.35)
Im 12(30) = -0-0544 -
The resulting form factors are compared with experimental
values in Figures 20, 21. The relevant experiments are
cited in Reference 71. The calculated curves are in 900d

agreement with the data for t>-40.

 

 

 

...g‘

88

0

.v

 

.Houomw EMOm Oflnuooao cooaoss ecu on uHm

c3 T

On ON

.om enemas

O_

0.0

 

m.N

 

 

89

0V

 

I!

.Mouomm EHOM oaumcmma comaosc map on new

$3 T
0m 0m 0_

.HN mnsmnm

0.0

0.0

0._

0._

ON

ON

 

with t]

Schwar
the p,

tude a

90

The results for Imfi are shown in Figure 22 along

with the Nielsen—Oades15 results.

The nucleon form factors were calculated by

74 by a procedure which rather elegantly included

the p, N, and A.

Schwarz

Schwarz wrote the NH scattering ampli-

tude as

(cf/£11)" £rn.(f) = 2:531;ng (111.36)

1,111) = 433/1]. +b(+)-)z(m;)-

 

+ ("113.-t) [HmSj-Hl/xf] + ’Vmfll‘flz' (111.37)

Lit)= [zit/(Ifh)] [313(CL'Fé'L'l/z) , (111.38)

where q=(t/4-1)l/2, and yp determines the rho width. The
Parameter A is an arbitrary entire function; Schwarz

found an acceptable pion form factor with h=0.

Since b (t) has no left cut, the pion form factor
00

is just
{- , (111.39)
F7...- (H = {37‘7” VERA—(o)

The Nfi+wn amplitudes

Edi}: -2/—', (t) , (111.40a)

 

where

where

has nc

b-(t)

Where
b‘.

1 15
rho 0(

went -

and A

n“Cle

91

{32(19): -('23'-"-) I; (19) , (III.40b)

where g=1.853, satisfy the dispersion relation

°° (1'3 .(t' A *1. y
12,-(+) 25.71) mi] ) é‘ ) F l )Ji‘,
L 4 (+1)" (f’~t)

(III.41)

where biL(t) is the left cut contribution. Since bfln(t)

has no left out Schwarz could solve this equation for

bi(t) to obtain

Jo; (e) 2 (53111) +— {1mm [11(1)
433]“ M .11 J (111.42)

4 (1)72- (1(1)

where u(t) is an entire function. Of course, the phase of

bi is equal to the phase of b . Assuming universality of

fin

rho couplings, Schwarz required u(0)=l, and the require-

ment that bi(t)+0 as t+w ensured that u(t) is a constant.
The left out contribution b§ was approximated by N

and A exchange. Schwarz then proceeded to calculate the

nucleon form-factors.

Unfortunately, the Nfi+wn amplitudes bi were not
Shown by Schwarz. We have, therefore, calculated the Ti
from (III.40, 42) using 1:0, Yp=l.53 (Pp=120 MeV), PA:
120 MeV (A parameters from (B.6)), and a cutoff at t'=150.

The results for [F11 and IF shown in Figures 22, 23 are

2),

smaller than we have obtained from (III.28). This fact 15

 

92

   

3/zlr1l-

Figure 22. The NN-mr amplitude q

 

Figure 23.

93

The NN+ww amplitude q

 

3/2|P2l'

 

 

consisten
nucleon f
relation
mediate 5
However,
(left cu‘
proceduri

logarith

94

consistent with Schwarz's observation that the resulting
nucleon form factors calculated from a subtracted dispersion

relation are too large. Possibly the higher mass inter-

mediate states (KE, nw) are the culprits, as Schwarz felt.

However, using the N,A—poles to represent the 1N scattering
(left out of Pi) and pushing it beyond t=100 is a dubious
procedure. The A—pole contribution to F1 is, in fact,

logarithmically divergent, as Schwarz pointed out.

G. The NN+NW d—Wave Helicity Amplitudes

The once subtracted Omnes dispersion relation for

2 .
f+ 15

D2 (:2 Z t-t, Q'D (t)Im-€t2(t)<it'
t -t : “'t 4:t tb+ -——-° 2

(111.43)

As was the case for Fi’ we have subtracted from this
relation a similar one for the Born amplitudes, in this
case to suppress the effect of rapid variation in

Imf (t') near t'=a. Having no right cut, the Born ampli—

tudes satisfy

2 , I
1... 15,3 (é) elf

. 11.43
Multiplying (III.44) by D2(t) and subtracting from (I )

gives

 

 

Niels
ca10u

POint

95

(32(1) 103%) 2 02(16):; (t) +[Dz(t.)-D.<e)]£f.,(t.)

0' I ~ I
+ £451 'Dzu. )rmcfu ) +[Dzlt’)-pz(t)]1-M 5:11)]
7: \\~ t"
-:o (thoMt-Lt)

(III.45)

The function D2(t) is a very slowly varying function of t

for values of t of interest here. Therefore, the factor

[D2(t')—D2(t)] considerably reduces the contribution of the

Born amplitude to the dispersion integral.

 

The non-Born amplitudes Re fi(to) interpolated from

Nielsen's27 tables are given together with the explicitly

calculated Re fEB(tO) in Table 8 for two subtraction

points.

Table 8.-—Va1ues used as subtraction constants for the
evaluation of the d—wave helicity amplitudes.

 

1 0.0156 —.01079 -.00719 —.3517

3 0.0170 -.09903 -.00600 -l.055

Calculations with (III.45) are increasingly

dependent on the dispersion integral cutoff as t increases.

The values of Im fi(t) can be extrapolated from Nielsen's

tables to t=-30 with some confidence. In an attempt to

 

furths

extra]

96

further decrease the cutoff error we have used a hand
extrapolation to t=-45.

The results for qglfil, where q2=t/4-1, are shown
in Figure 24. The f3 amplitude is quite close to the Born
term. The right cuts of the helicity amplitudes are
introduced through D2(t). Since the input phase shift 63
is rather uncertain we have investigated the dependence on
03 by setting D2(t)=l, thus eliminating the right cut. The

effect of the right cut is greatest on f: but not large in

any case. The fi(t) results with either to=1 or 3 are very

nearly the same.

 

 

 

97

 

(—-) camp'ete
(""") with 63:0

gala!

 

w. a. 12 16. 20. 2t).

1 0.2)

Figure 24. The NET—+1111 amplitudes q5/2 IfEI .

 

one-p
pole
stant
show
31 we
label
we he
inelt
Some)

the 5

IV. THE NUCLEON-NUCLEON AMPLITUDE

AND POTENTIAL

A. Phase Shifts
To the two—pion-exchange amplitude we have added a
one-pion—exchange pole with gz/4w=l4.4 and an omega-exchange
pole with gi/4n=4.7, fw/gw=-0.12. The omega coupling con—
stants are discussed in Appendix G. In Figures 25-38 we

show results for two ww s-wave inputs. The curves labelled

51 were calculated with the Morgan-Shaw28 6: as input; those
labelled 82 used the GS+LBL39’36 6: as input. In each case

we have used n=3. The spectral functions have been

. o

Included to t=45. The results from the GS+LBL 60 are
somewhat less attractive than 51, as one would expect from

the smaller [fo

+I this phase shift produces.

The theoretical phase shifts can be compared to the
energy independent phase shift analysis results of Arndt,
MacGregor and Wright75 (MAW—X), shown as data points in the
Figures. We have also shown the Hamada-Johnston76 phe-

nomenological potential values as small squares in the

Figures. The pion + fourth-order contribution is desig—

nated N and the pion-exchange by w.

98

 

u:

bk a IT NWT—in. ND .

 

99

 

.
u.

harm most «a.

MO. 60. 80. 100.

20

E (HEV)

Figure 25. The 1D2 phase shift.

...“..HIW what... .10.

 

 

PHHSE SHIFT

'
54

Figure 26.

The

1

G

100

100. 150.

E [HEVI

4 phase shift.

200.

 

250.

 

Flu—HI“ “mun-In-

 

3P, PHRSE SHIFT

Figure 27.

The

101

20. 30.

3

E [HEV)

Pl phase shift.

“0.

 

 

bun—am mun-In. on n

 

3%; PHFlSE SHIFT

Figure 28.

The

313

2

102

 

E (HEV)

phase shift.

 

Plath Mum-“at. c.m.. 0

 

 

3% muse SHIFT

Figure 29.

50.

The

F

103

1000 150'

E fMEV)

C phase shift.

 

200.

 

 

 

 

)-,
u.
H.
h
11
ET
E

3;?

104

0.0

term mmqra Mn

no
I“
.

 

-l.5

200.

150.

100.

50.

E (HEV)

T phase shift.

3a

The

Figure 30.

#0310 00.1.1. 3&0

 

 

3?“ Pl-IFISE SHIFT

Figure 31.

The

3-

F

105

 

E (MEV)

LS phase shift.

 

 

 

E
a
w
i

3 F2

3?: PHRSE SHIFT

0.

Figure 32.

so.

The

3-

106

100. 150.
E (HEY)

F2 phase shift.

 

200.

 

FLHIW mania. . «....»

  

 

3F3 PHRSE SHIFT

Figure 33.

50.

The

3

107

100. 150.

E IMEV)

F3 phase shift.

 

 

 

 

 

3?, FHHSE SHIFT

3.0

2.5

N
D
O

m

D

0.5

Figure 34.

The

3E

108

100. 150.

E [HEVl

4 phase shift.

 

200.

 

 

 

 

 

 

'F, PHFISE SHIFT

0. 10.

Figure 35.

The

109

20. 30.

E (HEV)

1P1 phase shift.

“0.

 

50.

 

 

 

 

110

-l.

 

Q 3

hmmrm wmcxi

 

200.

150.

100.

50

E (HEV)

Figure 36. The 1F3 phase shift.

 

 

111

 

30.

25.

U
2

 

5 0.

0|-

Eazm use... no...

5.

MO. 60. 80. 100. 120. THU.

20

E (MEV)

Figure 37. The 3D2 phase shift.

363 FHHSE SHIFT

Figure 38.

50.

The

3.-

112

100. 150‘

E (HEV)

D3 phase shift.

 

200.-

 

 

 

113

We have also shown the results of a recent boson-
exchange-model fit, labelled BG. These results, the fit C
of Bryan and Gersten,77 were obtained by fitting with the
MAW-X phase shift analysis error matrix, including the
s-wave phase shifts.

The singlet even state is displayed by the 1D2 and
1G4 phase shifts. The theoretical result 81 (MS 6:) is
somewhat too attractive above 25 MeV in the 1D2 case. As
mentioned, reducing the [£2] input as in 82 (GS+LBL 6:)
decreases the attraction. We have removed the Coulomb
effect in the phase shift analysis lD2 values via the
Hamada-Johnston76 potential (see Ref. 78).

The excess attraction is also present in the triplet
odd state, as evidenced by the 3FJ phase shifts. The failure
of the theory to produce the L=l phase shifts should possibly
not be taken seriously since the dispersion relation (II.11)
may require one subtraction. As noted in Chapter I, if one

Subtraction is required then the neglected high-t' dispersion

integral would be a first order polynomial in t. We have

3

shown the 25 MeV PJ phase shifts, Coulomb effect removed,

which were found in a recent analysis.78 The 3FJ phase

shifts are also shown in linear combinations which approxi-
mately separate the central, tensor, and spin—orbit compo-

nents of the amplitude. The combinations are defined for

arbitrary L as

3Lc = 35:3)L72L-1)5L.,+(2L+05L + (2L+3) JL-H] ,

3LT: -(zL-i)(2L*32 Bugg“ -(2L+I)5,_ + ‘- 614-1],

(2 L.(2L.H)(L+i)

 

 

114

--1
3L“ : 2L(2L+I)LL+I) (1L"XL+I)5L., *(2L+/)5L’L(2L+3) 511,]

 

The theoretical L=3 central component is much too attractive,
but the tensor component, which is not directly dependent
on the s-wave an interaction, is very well produced. The
spin-orbit component is insufficiently attractive, but
superior to that produced by the Hamada—Johnston potential.

The singlet odd phase shifts are apparently too
repulsive, although the 1F3 case is somewhat unclear. For
the singlet odd state we found that using the GS+LBL rather
than Morgan—Shaw 6: input does not improve the agreement
with nucleon—nucleon phase shift analysis.

3 3—

The triplet even state is shown by the D2 and D3

phase shifts. The theoretical 302 phase shift is too

attractive, as is the BG boson-exchange fit. The D3 phase

shift is very sensitive to the high-t spectral function

input. For example, changing the dispersion integral cut—

off (Equation (11.11)) from 45112 to 35112 results in a 23%
3..

change in D3 at 210 MeV but a less than 3% change in

other L>l phase shifts.

B. Potentials

 

The potentials, calculated with the same input as

for the phase shifts, are shown in Figures 39-48. The

76

Hamada-Johnston potential is also shown, and one may

consider the phase shifts produced by the HJ potential in

 

 

 

SINGLET EVEN CENTRRL

Figure

39.

115

 

ll 2.

R (FM)

The singlet even central potential.

 

 

TRIPLET ODD CENTRRL

Figure 40.

116

 

R (FM)

The triplet odd central potential.

 

 

T=1 TENSUR

Figure 41.

117

2. 3.

R (FM)

The T=l tensor potential.

 

H.

 

 

.H NET-lid.“ 1 .- (in...

  

T=1 SPIN-ORBIT

Figure 42.

The T

118

 

R (FH)

l spin—orbit potential.

 

 

T=1 SPIN-ORBIT

Figure 43.

potential.

119

 

R (FM)

The long range T=1 spin-orbit

 

SINGLET ODD CENTRRL

Figure 44.

120

 

R (FM)

The singlet odd central potential.

 

 

TRIPLET EVEN CENTRRL

Figure 45.

121

 

l. 2. 3. Q.

R (FM)

The triplet even central potential.

 

 

 

0 TENSOR

T:

-M0.

-60.

122

 

Figure 46.

l. 2. 3.

R (FM)

The T=0 tensor potential.

 

 

0 SPIN—ORBIT

T:

Figure 47.

123

 

R (FM)

The T=0 spin-orbit potential.

 

 

 

0 SPIN-ORBIT

T:

Figure 48.

potential.

124

 

R (FM)

The long range T=0 spin—orbit

 

 

 

125

judging its correctness. We do not imply that the two-pion-
exchange + w + w potential should necessarily reproduce the
HJ one.

In Figure 39 we see that the theoretical singlet
even potential is somewhat more attractive than the HJ
potential, consistent with our conclusions from the 1D2 and
164 phase shifts. The 4th-order + 1 potential, labelled N,
is remarkably close to the HJ potential, a fact noted by
Partovi and Lomon.l It is interesting that increasing the
low—t we s—wave enhancement brings the theoretical potential
closer to HJ and N.

The tendency to be overly attractive is also seen
in the triplet odd central and the triplet even central
potentials, although the agreement with the HJ triplet even
is good at long range. The discrepancy with the HJ triplet
odd central potential is remarkably large.

The tensor potentials are in good agreement with HJ
in to one Fermi. The tensor force is not directly dependent
on the s-wave no interaction.

The 3FLS phase shift indicated that at low energy
the HJ spin orbit force is less attractive than the 4th
order + n or the theoretical values, but that at higher

energies (above 200 MeV) the HJ 3F increases rapidly.

LS
The HJ T=1 spin—orbit potential is actually more attractive
than the 4th—order + w or theoretical potentials even at

two Fermis. The HJ potential becomes repulsive at ~2.2

 

 

red‘

126

Fermis. The L=3 phase shifts at energies less than 200 MeV
are very sensitive to the potential at r>2fm.

The HJ T=0 spin-orbit potential is in strong dis-
agreement with the calculated potentials. None of the
curves changes sign between one and four Fermis.

The singlet odd state is interesting in that the HJ
non-n-exchange potential is very long range. The theoreti-
cal potentials do not show the very long range behavior.
However, the calculations 81, 82 do differ considerably

from the 4-order + 1 potential.

C. Discussion

 

The largest uncertainty in TPE is the magnitude of
the s—wave NN+ww amplitude f:. If the uncertainties in the
NH phase shift 0:, especially at t<20 (Mnn<620 MeV), were
reduced, the magnitude lffil could be evaluated with corre-

spondingly greater certainty. We have seen that the

O

enhancement of the near—threshold 00,

as reported by the
Geneva-Saclay39 group, gives much better agreement with
most of the NN phase shifts than the no-enhancement Morgan-
28

50

Shaw 0‘

Although perhaps it is an insignificant fact,
we note that the GS+LBL phase shift is very similar to the
Berkeley36 case 3 solution at t<10 (interpolation between
t=4 and the t>15 fit). However, the improvement produced
by the GS+LBL phase shift is not in all states. In par—

ticular, the singlet odd central and long-range T=1 spin—

orbit forces suffer Somewhat from this enhanced low—t 0:.

 

 

 

exac

ener
aTT

sea)

effl

acc'

127

The dependence of |f$| on the t>50 f2 phase is not
very serious for most NN higher partial waves. Perhaps
combining the Omnes dispersion relation approach with

15 could decrease

conformal mapping, a la Nielsen and Oades,
this uncertainty once the correct t<50u2 phase is known
accurately. Lack of an accurate knowledge of Ifil is now
the only thing preventing a reliable calculation of the on—
shell TPE amplitude.

One— and two—pion-exchange cannot be expected to
exactly produce the NN amplitude, particularly at higher
energies or lower partial waves. It may be possible to use
a TPE calculation in a phenomenological analysis of NN
scattering. Provided one is confident of the TPE calcu-
lation, he could perhaps study the higher mass exchange
effects, determine the mNN, nNN coupling constants more
accurately than at present, etc. We note here that Barker
et al.79 have shown that nn exchange may be important.

Before we can claim to have a real understanding of
the intermediate range NN force it may be necessary to
calculate the low-t three-pion—exchange in at least some
approximate way. Because of the broad a and p distri—
butions, and the small pion mass, we must be very cautious
before neglecting three-pion—exchange, especially at, say,
one Fermi. A three-pion—exchange calculation should
include the J=0,1 NW interactions.

The exchange of the NH p-wave system is probably

accurately included. We have obtained very good agreement

 

 

 

 

128

with the experimental nucleon form factors using our

V
E

only two parameters, Im Ti(m§), gives some support to our

NN+wn p-wave amplitudes. The fit to both G and G; using
belief that the two pion state gives by far the dominant
contribution to the small It] nucleon form factors and is
consistent with p—wave elasticity to well beyond the 0. We
have found that the nucleon form factors are more sensitive
to the Pi(t;30) than are the higher partial wave NN ampli-
tudes below 200 MeV. Thus, the form factors provide a
stringent test of the validity of the p-wave input.

Our evaluation of Fi(t) depended on the pion form
factor Fn(t). The experimental situation for timelike t
(t>0) is not yet very satisfying. The form factor which we
have used is in good agreement with the Novosibirsk70
experimental results. When better experimental Fw results
are available they may be used in the study of the nucleon
form factors. We do not expect better FTr results to signi—
ficantly affect the TPE calculation.

We have accurately determined the d-wave NN+ww
amplitudes at 4<t<25, the most important region for TPE.
We do not, therefore, have to rely upon theoretical models
for the d-wave exchange. The nucleon pole contribution is
expected to adequately represent all higher-J no exchange.
However, theoretical models can be tested against our
amplitudes. We have, for example, shown how one can
approximate the d—wave exchange by nucleon and A33 poles.

The d—wave wn interaction was shown to have a rather small

 

 

 

 

 

 

 

129

effect on the Nfi+wn amplitudes. We can conclude that
Vinh Mau et a1.14 were justified in neglecting the J32 t-
channel branch cut.

Finally, we can conclude that the Mandelstam
representation has once again shown its great value to

particle physics calculations.

 

 

 

 

 

LIST OF REFERENCES

 

 

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“ /
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In our Equation (II.42) we interpret k2 as Ez-m
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134

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___

 

     
 

. . - . .- . -.'1-.
1' l_ . In .fiJ‘f-J‘I. 0' [

 

'1):
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CERN .

 

 

 

 

 

APPENDICES

 

 

 

APPENDIX A

THE HELICITY EXPANSION

 

 

APPENDIX A

THE HELICITY EXPANSION

The momenta for the NN+wn process are labelled as

in Figure 3. The corresponding Mandelstam variables are

S = "(P _ (1)2 r

, 2
t=-(p+P) I (A.l)
u: _(p_q')2 I

and the nucleon momentum, pion momentum, and scattering

angle, all in the c.m. frame are

p2 = t/4 - m2 ,
2 — - (A.2)
q " t/4 l I

cosG = (s - u)/(4pq) .

2
In the pseudophysical region (4: t<4m ) the nucleon c.m.
momentum is positive imaginary while below the pseudo-

physical threshold (t<4) both nucleon and pion c.m. momenta

138

 

 

 

139

are positive imaginary. It is customary to define the

variables p_ and q_ as p = ip_ and q

iq_.
Taking the quantization axis along the antinucleon

momentum p', Frazer and Fulco25 gave the helicity expansion

for A(:) and B(+):

A(t)

 

- -Z'. (3‘+' ;) $(f1)1{ PJTcOwHIfH)

 

. meow P; (c059) £31(A.3a)

43111-1)

 

i 9 (1+’~ 1.4 .
BI

Generalized Bose—Einstein statistics requires that

the t-channel isospin is 0 (1) when J is even (odd). We
recall that the superscript on A and B is + (—) for t-

. J . .
channel isospin 0 (l). The subscript on f+ indicates that

the nucleon and antinucleon helicities are the same (+) or

 

Opposite (-). The partial wave projections of A and B are
T 4 ' - 2 A m
: ‘——'"’ ‘ + :4 + ‘
‘FFRJ 3'3 (FL): I (KIT-(MP1,) I”- 0 5.1+!

+- 3- 83., 3 ]' (A.4a)

£1“) = .1; JJ'LJ'E(:_&:T { 3.7-! ~— BJ-qu (A.4b)

 

 

 

140

where

 

 

 

 

 

APPENDIX B

POLE CONTRIBUTIONS TO THE

Nfi+nn AMPLITUDES

 

 

APPENDIX B

POLE CONTRIBUTIONS TO THE

Nfi+nn AMPLITUDES

The contributions of s- and u-channel poles to f:
are found by putting the poles into the appropriate term of
the s— and u—channel partial expansions of A and B, and

applying (A.4) .

The following forms for J=0,l,2 are most convenient

“$00“ = —'—[I‘— 6’4")»: u f + m 6‘;“U» arctan 1.1, ‘ ’)] (13.1)

+ 4.: Z
- - ’— P) 2 ‘1‘ I .2
p;(t)=é[—7:rgg)(,-maflt)+F-E B (Ar-Arman“); (B a)
rp'bf): .JL'L‘fl G15) (""1) “"t"‘T,L ‘ "'r , (B.2b)
‘ 4“ P-Z ' ,
fifth): 8761—3— GI’JZ(3J1EH)M‘CI« ’5: " 311,-] I (B.3a)
Fl
...—A ' I
4120:) 35E); G5”??? +J1,2~(hr3+J1,.)4rctm‘;‘ , (B.3b)

141

 

 

 

142

Where J1,: t-ZJM‘H-mfi)
“N

4M

and m: is the pole position in s and u.

The nucleon pole (Born term) residues are

(35:0,
a: = g2 = 4fl(l4.6) .

For a "zero-width" A(1234) they are

     

M Ian: ml". 3( mm} _f_:_ m —M
6 : hum-..“— I-A~T. (If 2) l I I
A 3 (MA+M)‘-f 2‘16 (rm-n!) -

(+)- lemma, 3 "H _ ’{
G5 - “"3 [(M‘rmh ( 22,4) mfm‘J ,

where

CL: = IMf-wrllz]ln7:—<M-f)‘]/ MM?)

With mA

T 120 MeV we find

A

(B.4)

(B.5a)

(B.5b)

(B.SC)

1234 MeV, m = 139.576 MeV, m = 938.259 MeV, and
n

 

 

 

143

4n(43.7 + 1.96 t) ,

19+

(B.6)
4fl(-15.0 + 0.126 t) .

[11+

Often one will see the residues for the A deter-
mined not by the physical width but by other criteria, such
as the NN scattering lengths or the integral of the partial
wave amplitude over the region of the A. The latter method

was used by Epsteinl6 to obtain

 

G;+) = 4n(36.0 + 1.6 t) ,
(B.7)
Gé+) = 4n(-12.2 + 0.1 t) .

Note that with the masses used in (A.10) one should
maintain the following relations regardless of how the

residues are determined:

Gé+) = 4n(l + 0.0449 t)x ,

(B.8)
Gé+) = 4n(-0.343 + 0.00288 t)x .
with x to be determined.
. 25
Also useful are the p-wave amplitudes
[w - _l3. (1; ’t _,._§._. J:'ct))r (B.9a)
’(é) " I); +() 4J7," .
' ' , -—31- ' ' t ) (B 9b)
: “M -— ’ _‘l I ~
Flu) 2,24 .Cfurw— 7 J

where, again, p2 = t/4 — m .

 

 

 

144

For reference, the following relations are also given-

 

 

 

 

 

 

2
¥+o(4) = FL~6A . (13.10)
mat-tr)
I V Z L" (‘)
+1” : ji- GA +' m 63 (B 11
3ft(4‘trJ2 amc4~tr1 ' ' a’
4 m I‘)
‘F. (4) ' — 2 ~95 , (3.1110)
3.114 ~tr)
Z, _ z ‘*’ 1*)
{-4 L4) '- 7 “__Li'fl-“t + 2.51.9.5...— , (3.125))
37r(4 -t,-)" [LEM-tr)
. ~w L+)
1- . _ [(9 6
J:. H) ’ -»-—~—- L 2 , (13.1210)
4i€{4-tr)
where tr = 2(m2 + l - mi) and p3 = p3(4) = m2 — 1.
For t<4 — £5 the Born terms in f: become complex.
m _

Here the J=0,l,2 real parts are written in terms of

logarithms.

 

 

 

 

 

145
0 3 3113: "‘
Ream 41: [’ + Q 403 fig , (3.13)
' = £32.. -
Re ‘F'J- (t) 419%“)[0 + QzLOj 4%; I (B.l4a)
' : __Sf__. _filzz {-0
Re L- (t) 4itf2—y/D—Z [’ Q + [03 W I (B.l4b)
— _3:._)_ _L 2
Re 15,12 (t) 811 ‘f-Z~ [z (0" 30 3W9. m9 —3Q ],(B.15a)
'7- - 1. Z "2. Z 1 3. :9
126 1C. “9" W2[ 3 +0 +Z(0 QHDSM),
(3.153)
9 ,2.
In: Pitt) ‘3 11133" Q , (3.16)
I - -mz' 2'
Im 1C*(t)= _‘L Q , (3.17:1)
3H-
Im J:_ Lt) — F32 (02,4) (B.l7b)
”'F-Z
I'm fifct) = 113-— (3&3 -Q), (B..l8a)
H. (Afl-
, (.2
IM Lth) = ESL—z (0 -Q3) , (B.18b)
”(F-5‘)
L 7. 2.
-: [ILL ’- (3:1)): . a
I»: l",(t) 3F31‘[Q + . 5m“ , (3 19)
Im C(t) : 9135-— (f—BQZ') , (3.193)

 

 

 

146

2—t
4p_q_

These equations may be obtained from (A.5,6,7) by the

where Q =

 

substitutions

h—a (1Q,

(B.20)

retina—fiw/2+Lébfifl

pole approximation to be adequate as given in (A.5). The

+ . .
A( ) s-channel dispersion relation is rather slowly con—

vergent and one can expect f: to be sensitive to the large
8 behavior of the absorptive parts in an unsubtracted
relation. One can, however, project f: from a once sub—

+ .
tracted relation for A(+) (unsubtracted for B( )) to obtain

,—

(H

1C3“): fiinA (gunfit) —Z/>_ZE:,[J‘/I,Mtfl}hr

t-Z

W. 1 (4- (“r*”)0'~t“/lhr)J
2(M,.‘—m‘)LM,."—m—z+t)

 

_ m G;+) (’_ hr arctxm. {/Ar )f , (B.2l)

t—2
4p_q

Brown and DursolO have obtained a particularly

 

. . 2
where the subtraction is at s = m and h

simple expression for f: by keeping only the Born contri-
. + .

butions to B(+) and approximating A( ) by its current

algebra value g2/m at the Adlerll point. They obtained

,1
If“) : ..EIZCLUq ar.ta~}h- 33,31) , (3'22)

 

 

 

 

APPENDIX C

PARTIAL WAVE PROJECTIONS

 

 

APPENDIX C

PARTIAL WAVE PROJECTIONS

To compare the theoretical NN amplitude with
experiment we project out the partial wave amplitudes.
The bar phase shifts and mixing parameters are

defined in terms of the oc—matrix,80 where id = S.

singlet: Ld’l : euép (C.la)

. 2&5 1 C.lb
triplet uncoupled: Lat/'1 ‘3 e 1’ ( )

triplet coupled: (C 1c)

The a—matrix elements are written in terms of the singlet-

. 81,4
triplet partial wave amplitudes Tij(£) as

(11' 21+! ' "‘

a”: 27:1[7-(12) 1.)-(1+2)(1-4)T,‘,(1) HFTMW]

147

 

 

 

148

_ 13 ’
OW“ ' (manna) I (1+2) 77‘”) +1?) UH) 77°”)

 

4.111.414) 77-,(12) -197 1am) T.,(2)
+ (1+1) 7;. (2)]

k.

at _ — ——~~— ' ,
1.1) ~ (21.))(21-,)Z”"’77‘W #71 (1+1) 77012)

+ (.£+2)(£+/)(£-/) 774(1)

+1.?(1wM1-x) 7;,(2) +1 723(1)

 

a u; : ~—kL(1f/)(ff2)]’z

M

(Lflf0122f3)

 

[7710-1271 710(2)
+ 1.134} 77-1‘2)’J71 731(2) " 733(1) (c.2)

where k is the NN center of mass frame momentum.

The partial wave amplitudes Tij(£) are given by

 

 

4
T- : 21H (Z-m)! m
L) M) 2 1.2m)! 7‘51””) P! (”59) du’"; (C'3)
-1
Where m = Ji-jJ and

P c J

773(939) = '82: Vg-rwvse){Urtwwalfdw'é}

_ _ , "1
+ H) “ U,t(W,-m9)/zf(wét)} (J—cos‘Bf (c.4)

 

 

149

where T is the NN isospin and pt = pi.

The matrices VF. and U are as follows:4
1] rt

   

A r _

iE—VLJ'LWJ)

1)

85

H o )[Mwmfj 2,0,7 M-(A-«M o
H o MA-t) o 4-). o
‘0 501-1) Jam—7) o {gym-1) o
01 o EyMA-J) 4702-1) mix/(M) 0
oo ZyU-A’) 23-3432] 0 219414))? 0

where A = £3 and m = nucleon mass.

 

 

150

    

r ‘3" 13:2 i=3 t=4 t=5
‘ 2m-47Jzz/m -3m2-4kz(4+y)

2 '2 "1 1119-4]:2 2, -4
3 2 ”'- “3m2-4Jzz -Z -1

4 J 1 -2m ~4h2/n’l m2 o y

5 J ‘ *bm- 4y/3/m 5m2+4kz+4ylzz 4 1

where k2 = W/4 — m2.

The amplitudes pt(wtt) were calculated via the
diSPersion relation (II.ll). Introducing (11.11) and (C.4)

into (C.3) and interchanging integrations yields

 

T9111) 2 (OPE) + Uri (1%))!de dx

 

__ 37 r-
. 32%.) U ‘1‘ 22: v.-.w1)
lk‘éy“) " t J ’
xi \thL‘VH—J féu/t £1x1)+(.1)r+~r(x—+ ~)~)} (C.5)

We have defined y(t') = l + t'/2k2 and x = cose = l + t/Zk .

The x-dependence of p' generates some difficulty

in evaluating the integral on x. Recall from (11.23) that

(r) t (t -
tau) ‘-‘ It Lwt’) :(4) Qt )(th) t’) _ (C.6)

 

 

 

151

The Mandelstam representation assures us that4

 

‘33"; ti) = 7113‘] :12 H“: *3). (c.7)

Now for physical NN scattering t is negative and, at low

energies, small:

—4k2<t<0 .
Thus it is reasonable to neglect t With respect to 4m , the
— 2
lower limit in (c.7). We have, therefore, set t=0 (t=4m -w
+-4k2) in (C.5). This prescription for handling the x-
4

dependence in (C.5) is not that used by ALV. ALV replaced

X by y(t'), equivalent to replacing t by t'. The prescrip-
16

tion E50 was also used by Epstein and McKellar and by

Chemtob et al.13

The x-integration in (C.5) can now be performed.

We note that the product v:j(w,x)Urt(w,x) is a second order
Polynomial in x. The integration can, therefore, be written

in terms of the associated Legendre functions of the second

kind. Define

and

 

 

 

152

For the case [#0 the P? satisfy

LIV fitwue 1): P2 (1) A).
7 )1.

— W;§'(w,y)(f-I)%'thy)

_2_._
3
— 3L #5 ( 5M0 +2) (Fm!) 5]] . (C.lO)

ij

 

O MHZ)

O O

aria-I) Jim->5“)

 

0! 12111-1)“ 217mm» mam) 21310-1) JI‘U-A‘I
. (M1)z +3A-n’)(A-l)

00 2mm 4mm“) 2m‘U-4A‘) O NH")
xLl- 2')

 

 

 

153
I011: .* _
m. 9‘} "
Li _t:3 £24 £35
Sb 0 o O

 

H (Hf 2mm») 112(4)?» 210-!) HZ
xu-A)‘ —I)JA-4)

 

J-J O o o O 0
JO 0 o o O O
04 O o o o 0

0° 20x4)" 4m(4+2,\) 2m‘(4+2> 41H) 2112-I)
114-W -3,\2.4)3+4>"J

where m = nucleon mass.

The partial wave amplitudes are finally obtained

by numerically performing the t'—integration in (C.5).

 

 

 

 

APPENDIX D

POTENTIAL EQUATIONS

 

 

APPENDIX D

POTENTIAL EQUATIONS

 

The iterated one-pion—exchange contribution to the

13

potential was calculated by Chemtab, Durso, and Riska and

earlier by Partovi and Lomon.l CDR wrote the iterated one-

 

pion—exchange contributionU1T2 of (II.55) in terms of

spectral functions

7.2(311) =

 

__ A , 5 ‘0 ' f: l l .- ' :- hltl -
J=1 f tl‘t (13.1)
where
Q‘ 2 ll
- P A
an - y, x, .
- P d (D 2)
CD; ' Y4 f’rf I -
Q4 : lP-ln ,
35 : Yb: ’5" ‘

 

 

155

+ .
The n; were given by CDR and are not repeated here.

However, we correct a printing error in CDR's (4.29):

T2 (Fay) = ‘viv. S,(pzt’)

l

 

 

(rah?) kflml [32 2 TT‘de) - f(d) I (13.3)

where p2 is the same as our k2. Also there should be a

closing bracket immediately preceeding the To in CDR's

(4.31b).

Another form for T2 is

Z

(kf‘PZJW (”3 “J, (3.4)

ane n: (hr‘h:)/(RE‘FJ< J.

 

Re T2:

+ =4 . .
The transformation from n— to the n— occurring in

the potential (II.62) is

”rt“ : fl de 72.1 . (D.5)

J

~

2 _ .
The matrix XQQ evaluated at w = 4m , t—O is

 

 

 

 

 

   

156
’XQJ’L
a(J' _
J=3 J=+ 1=5
2. o o
I
O 7;? 0
_-_.L __L.
0 12m1 I2m‘
—f -I
O 4""; Izm‘
——’—; o 0
9m
The transformation from p:r to :1 is
=1 __ E )2?“ 6.: . (D.6)
ea; “ .3 «j J

The matrix X522 evaluated at w=4m2, t=0 is

 

 

157

   

 

 

 

 

APPENDIX E

DIRAC ALGEBRA

 

 

 

APPENDIX E

DIRAC ALGEBRA

The Dirac algebra that we use is the Dirac-Pauli

representation:

,Y,(=25/w
'o 41071
(3k .),
I o
(. -,),

(E

(E.

(E.

(E.

(E.

.1)

2)

3)

.4)

6)

 

 

 

The free particle Dirac equations are
(iYoIJ +m) Lula) : O
(EX-F~m)1f(r)=0 .

3131(sz +m) : o ,

.-

where

31./3’1); 3

summarized by Sakaurai.

’U‘Lf) (if-lb ~31)

_—

+-

19

159

E?”
Zn1

The Feynman rules in this representation are

—

.933.

.Erm

)3

(E.7)

(E.8)

(E.9)

(E.10)

(E.11)

(E.12)

(3.13)

(E.14)

 

 

 

 

APPENDIX F

RHO-MESON COUPLING CONSTANTS

 

 

APPENDIX F

RHO-MESON COUPLING CONSTANTS

The p-meson coupling constants are defined by the

Lagrangian density for interacting nucleon and p—meson

fields:

. “‘ 1’
é : L3QNNW [SIN-#1“ XuY ~— M 01;" T.()V¥H)Y I (F.l)

where ¥ is the nucleon isospin operator; gpNN and prN are
the vector and tensor coupling constants respectively. An

equivalent expression is

0t : '4qu *¥‘m);'13- E, if“? - .2331 $ q'Uf’IIfL‘fiIJfiJ' (F.2)

where p(p') is the initial (final) nucleon momemtum
operator.

As noted in Chapter III.F, the p coupling constants
are related to the amplitudes Pi. We shall define the
Coupling constants in terms of the residues of Ti at the

p—meson pole.

160

 

 

 

161

We write the ImPi(t) near the pole position as

 

 

 

f7 =(R)z‘(4T‘ )7?- 99»)!

 

7.1-1) 3.2-(7,24) t—qu *‘Bo . (F-3a)
"/ 41'; y 1r: ,, .0
2MP : ( TR 4(-——---~* 2 ——5-—" , -
2 72-4 arena-4) t-TP; + 52 (F 3b)

where Bi are regular at t=Tpi' For Tpi we use a form
suggested by J. Pisut and M. Roos, Nucl. Phys. 29' 325
(1968):
13 We"; ‘
7r ”12.12
t
2 _ -
1 ‘ t/4' ’ I
4 - 'T -’
1p ' R/4 ‘

The Bi are expected to be approximately real (the nucleon
pole terms are exactly real and the contributions from
narrow s- and u—channel resonances are approximately real)
so we shall work with the imaginary parts of (F.3). From
the mass and width of the p we expect TI/TR<O.2 so that
We may replace Tpi by TR in the numerators.

The imaginary parts of (F.3) with Bi neglected
and Tpi replaced by TR in the numerators have been fitted

near the pole to the Pi calculated in Chapter III. The

resulting parameters are

TR = 29.2 i l. , (F.5a)
TI = 5.50 t 0.2 , (F.5b)
R3 = 0.0008 , (F.6a)
R: = 0.0652 , (F.6b)

 

 

Thus

 

 

The fitted ImI‘i are Shown in

162

= 2.35 i 0.10
= 13.6 i 0.06
= 0.44 i 0.05
= 5.8 i 0.2

Figure 49.

I

(F.7a)

(F.8a)

(F.7b)

(F.7c)

 

 

 

 

 

 

 

 

163

   

20. 25. 30. 35. '10.
t 94")
Figure 49. Imaginary parts of the NN+nn amplitudes

Pi and resonance fits.

 

 

APPENDIX G

OMEGA-MESON COUPLING CONSTANTS

 

 

 

 

 

APPENDIX G

OMEGA-MESON COUPLING CONSTANTS

The w—meson coupling constants are defined by a
Lagrangian density similar to (F.l) but without the isospin
factors. One can attempt to determine the coupling con—

stants in several ways.

The simple quark model82 relates the w and p

coupling constants. The quark structures of the p° and w

are
a - ~“ _ G.la)
Q - GT' ([7? nr: ,
' 7 +)1E (G lb)
1*) : M( J .
’2' F/ )1
Where p and n are the (isospin, charge) = (1/2, 2/3) and
(“l/2, -l/3) nonstrange quarks. The meson—proton coupling

is depicted in the quark diagrams, Figure 50. The proton +

/%(p+w) vertex is twice (because the proton contains two

1 .
p-quarks) diagram 50a and the proton + /7(p-w) vertex lS

diagram 50b. Each diagram has equal weight so gwpp
Thus

3g . Similarly one finds gwnn ‘3gpnn°

OPP

2 - 2 (c.2)
ngN " 9 gpNN '

164

 

 

 

165

 

—:740+w)

 

nPP

(proton)

PPn

(proton)
(b)

Figure 50. Meson-nucleon quark diagrams.

 

 

166
From (F.5) we find
g2
wNN
-z;- = 4-0. (G.3)

In the simple quark model the ¢-meson is composed

of only the (I,Q) = (O,-1/3) strange quarks:

'9-
II
>2
>4l
.

(G.lc)

Thus 0 does not couple to nucleon in this picture.

One can also use the quark model to predict82 the

electromagnetic couplings of the p,w, and 0. These con-
stants are defined by the current—field identity83 in terms

of the vector-meson electromagnetic current j(:VV) and the

(V) :

meson field ¢U

.(va) ‘ my (V)

- . (G.4)
J/u 23V J“

 

The corresponding interaction Lagrangian density is

. (YVV)

£21 J/U Afl~ (G.5)

Coupling the photon to the individual quark charges we find

2 , g Q .2
3 3 *—‘ G.6
3'» 139 2 34’ < >
. 83 . d
The vector-meson-dominance (VMD) model applie to
the nucleon form factors postulates that the photon couples

to the nucleon only through an intermediate vector-meson

state. A rather strong statement of the model is that the

 

 

167

nucleon electromagnetic current is proportional to the

nucleon's vector-meson current J“:

.004») _L. w)
J» = 23v ANN )3}, (6.7)

where Afig)(t) is the vector-meson propagator normalized
such that A(V)(0) = 1.
Ln)
Considering, for simplicity, only the I3=0 com—

ponents of (F.l) we can write

<PIJ‘ My): Z}: Li P')[9VW/‘

+ 4w»): @vW-f’b] (”M/3) . (G.8)

2m

 

Thus (G.7) relates the form factors in (III.29) to the

coupling constants in (G.8). The relation is

(V)
FIRM” : 231 M A/uvlt) (G.9a)

v Zjv
" ‘V’ ( .9b)
2m Fflt) = 2:1 #2”: AM“) G

the p (and p') if x=V(isovector). Higher mass mesons may

be included if they exist.

 

 

 

 

 

 

 

168
The normalizations
S
F' (0) ‘3 5" (G.lOa)
s .
2m F2 (0) = 3123—1 2 -D.O(o (G.lOb)
and (6.10) give

.3921! ‘w
j,» + §3—; : I (G.lla)

{10.3w {-.me

.. +- : -D./Z .

d” 3*? (G.llb)

From the simple quark model we have g¢NN=f¢NN=O' Then

 

ngN = gw , (G.12a)
and

f

wNN = -.12 . (G.le)

ngN

Then (6.2) follows from (G.5) and (G.l3).
It is not necessary to rely on the quark model

prediction g¢NN=O‘ The VMD model for the isoscalar form

factors with narrow-width w, ¢ is

 

 

 

 

S u I Benn I
F t = 3”” ————-, +
M) 2.30 I—t/mf, 23*, (‘t/M: (G.l3a)
- I ‘Fpnn l-.-
’ F t) : £33.51”. ”___. + ¢(G-13b)
AM 2‘ 25... l—t/MS 434» "t f

 

169

ngN

w
from the t=0 normalization of F: and the derivatives at

 

The coupling constant ratios , etc. can be determined

t=0. The form factors near t=0 were analyzed by L. H.

Chan et al.84 with the results

 

 

 

 

 

 

JG? -2
Tiff? -0 3: .2-95; C;e\/ I
LG; - 0 454 .
4t t=o - - °
P
id? lt=o : 7.98, ,
JGJI : -5,bl . (elm
at t=~>
From these values we find
S
AF. It : /.2 7 GeV? . (G'lsa)
at ‘°
5 __ (G.15b)
- . OD °
2”! iii-Z ltzu 0.0?

Solving (6.13) for the coupling constant ratios gives

 

3”“ : 2,33, (G.l6a)
v)

£315 : -0.0b7, (GJfib)
300

_fof. : _),33n (G.l6c)

 

 

 

. .rulu.

170

44%.." = -—0.053° (G.l6d)
3+
If, in addition to the w, higher mass mesons also
contribute to the isoscalar form factors (D.10) then such
high mass contributions are somewhat suppressed in a sub-

tracted dispersion relation, which we write as

In). 555115,) (“I

. ch
5 s f:
FL.(t)=L"1(o)+‘7z—L t”(t’of) 1 ((3.17)

In the narrow resonance approximation the w contributions

 

to Im F: are

you 2 ; ,a
111,511: 111131) 111—11.) 11>

2
2m I». Fzsw) -.- 7: mj(‘:;':’> (Sat-m”). (G.lsb)

 

 

so that
_S w- 1‘
)—,(t) a é: + 23.. ,3... «mm
42......) t (G.l9b)

 

Again using the derivaties (G.l6) yields

 

ngN = 1.56 , (c.20a)
gu)

 

 

171

'.

— -.0983 . (G.20b)

The YVV coupling constants gV can be determined85
directly from the V+e+e— rate measured in e+e— colliding
beam experiments. Combining (G.4) with QED one finds the

partial decay width

 

 

953
'4'? = 4.7 (G022)

which also gives giNN/4n if we accept (G.lZ).

Table 9 summerizes the predictions for the omega

coupling constants.

Table 9.--Predictions for the omega coupling.

 

2
- g
Source Equations :NN waN/ngN
(G.3) 4.0 -
(G.lZ), (G.22) 4.7 -.12
(G.ll), (G.22) 26.6 -.028
(G.20), (G.22) 11.4 -.063

W

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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