A STUDY 0:â€œ: me Pmaucnom mm WW
0? RESONANT STATES IN PERIPHERAL HIGH
ENERGY commons

Thesis fer the Degree of Ph. D.
MECHEGMÃ© STATE umwzasm "
GLENN T. wmmow
1969

u .4...â€˜....

I Huts

LIBRARY

Michigan State
University

 

This is to certify that the

thesis entitled

A STUDY OF THE PRODUCTION AND DECAY OF
RESONANT STATES IN PERIPHERAL HIGH ENERGY
COLLISIONS

presented by

Glenn T. Williamson

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Physics

 

 

Â«] Major professor

Date May 1141 1969

 

0-169

ABSTRACT.
A STUDY OF THE PRODUCTION AND DECAY OF
RESONANT STATES IN PERIPHERAL-HIGH ENERGY COLLISIONS
By

Glenn T. Williamson

The current theoretical situation in phenomenological
analysis of high energy scattering is reviewed. Various
attempts are made to improve upon several models. In
particular, the effect of form factors and Regge treatment
of propagators on Born amplitudes is examined. Higher
order exchange contributions are considered as possible
solutions to the problem of energy dependence of higher
spin exchange amplitudes, It is found that a combination
of the absorptive peripheral model and the Regge pole
model is the most successful approach,-in that energy
dependence difficulties are eliminated as is the need for

conspiring trajectories.

A STUDY OF THE PRODUCTION AND DECAY OF

RESONANT STATES IN PERIPHERAL HIGH ENERGY COLLISIONS

BY

Glenn T. Williamson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1969

ACKNOWLEDGEMENTS

I wish to thank Professor Hugh McManus for his
guidance assistance and support throughout my graduate
studies.

I wish to thank Professor Walter Benenson for his
critical reading of the manuscript.

My thanks are due to Mrs. Julie Perkins for her

assistance in the typing and preparation of this thesis.

ii

TABLE OF CONTENTS

INTRODUCTION 0 O O O O O O O O O O O O O O O O O O O O C O

The Exchanged Particle

 

Decay of the Resonant State

 

THE GLAUBER FORMALISM FOR HIGH ENERGY SCATTERING

AND THE BORN APPROXIMATION . . . . . . . . . . . . . . .
ABSORPTIVE CORRECTIONS . . . . . . . . . . . . . .â€˜. . ..
FORM FACTORS . . . . . . . . . . . . . . . . . . . . . . .
A REGGE MODEL . . . . . . . . . . . . . . . . . . . . . .
ONE-MESON-EXCHANGE CALCULATIONS AND COMPARISON

WITH EXPERIMENTS . . . . . . . . . . . . . . . . . . . .

The Born Term Model (BTM)

 

Absorptive Corrections

 

ANOTHER APPROACH TO ABSORPTIVE CORRECTIONS. . . . . . . .
OTHER MECHANISMS O O O O O O O O O C O O O O O O O O O C

Vertex Correction Mechanisms

 

Two-Pion Exchange

 

ABSORPTIVE REGGE MODEL . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX
I. NOTATION AND KINEMATICS . . . . . . . . . . . . . .
II. CALCULATION DETAILS: OME AND MODIFICATIONS

FOR TT+n+wp I O O O O O I I O 0 O O O O I O O O 0

iii

15

18

23

25

30

44

52

64

76

78

86

III.

TABLE OF CONTENTS (continued)

BVB VERTEX CORRECTIONS . . . . . . . . .

IV. TWO MESON EXCHANGE IN PB+VB . . . . . .

V.

4TH ORDER PARAMETER FUNCTIONS FOR PB+VB.

iv

90
94

98

TABLE I.

LIST OF TABLES

Predictions of the BTM for Density Matrix

Elements I I I I I I I O I I I I I I I I I I I

14

LIST OF FIGURES

FIGURE 1. The reaction ab+n particles in the region where
the final state is dominated by production of two
resonances is approximated by a sum over all allowed

one particle exchanges . . . . . . . . . . . . . . . .

FIGURE 2. The two body scattering ab+cd in the center of

mass and the orientation of the x and z axes . . .. .

FIGURE 3. The two body scattering ab+cd in the rest frame

Of partiCle do I I I I I I I I I I I I I I I I I I I I

FIGURE 4. The decay products g and h of the resonance

d in its rest frame. . . . . . . . . . . . . . . . . .

FIGURE 5. The absorptive model diagram for the scattering

ab+Cdo o o o o o I o I o o o o o o o I o o o o o o o 0

FIGURE 6. Qualitative sketches of the tâ€”angular
distributions for (a) Hp+pp at 4.0 GeV/c and

(b) ï¬‚n+wp at 3.25 GeV/c. . . . . . . . . . . . . . .

FIGURE 7. Sketch of s-dependence of the total cross
section in a quasi two body high energy scattering

involving quantum number exchange. . . . . . . . . . .

FIGURE 8. Comparison sketches of the BTM and experiment

(EXP) for (a) np+pp and (b) ï¬‚n+wp. . . . . . . . . . .

vi

18

31

31

32

LIST OF FIGURES (continued)

FIGURE 9. The Twp-vertex in the pâ€”meson rest frame. . . . 32

FIGURE 10. Sketches showing the experimental decay
distributions for the decay p+nn in the reaction

np+pp+nnp assuming 0- exchange (a) W(cosa) (b) W(8). . 34

FIGURE ll. Sketches showing the experimental decay
distributions for the decay p+nn in the reaction

ï¬p+pp+nwp assuming 1- exchange (a) W(cosa) (b) W(B). . 34

FIGURE 12. Sketches of the experimental decay
distributions of w in the reaction nn+wp

(a) W(cosa) (b) W(B). . . . . . . . . . . . . . . . . 36

FIGURE 13. Comparison sketches of (a) BTMF and (b) BTMR
Wlth experiment for the reaction np+pp. The dashed
curves are the experimental results. 3%(t=0)is

matched to the theoretical calculation at t=0. . o . 37

FIGURE 14. Comparison sketches of (a) BTMF and (b) BTMR
with experiment for the reaction nn+wp. The dashed
curves are the experimental results. 3%(t=0) is
matched to the theoretical calculation at t=0. Theory

and experiment are on the same scale. . . . . . . . . 38

FIGURE 15. Comparison of theory and experiment for

the reaction pp+nA3/2. . . . . . . . . . . . . . . . 50
FIGURE 16. Comparison of pp elastic scattering with
theory giving best fit to panA 51

3/'ZI I I I I I I I I

Vii

LIST OF FIGURES (continued)

FIGURE 17. Diagrams showing (a) the BTM and (b)+(c)
some vertex corrections to the NpN vertex for

ï¬n+wpI I I I O I â€˜l I O I I I I I I I I I I I I I I I

FIGURE 18. The real part of the structure function
K[yz] over the physical range of t for the reaction

TTIT'NDp. o o o o o o o o o o o o o o o o e o o o o o 0

FIGURE 20. Simple two pion exchange diagrams for the
reaction nn+wp. . . . . . . . . . . . . . . . . . . .

(I)
l

of t for the reaction nn+wp at 3.25 GeV/c. . . . . .

FIGURE 21. Re 1:1) and Im I over the physical range

FIGURE 22. Re(Il-Il) and Im(Il-Il) over the physical
range of t for the reaction nn+wp at 3.25 GeV/c. . .

FIGURE 23. Â§% vs. t for the reaction Hn+wp assuming

2n-exchange as the dominant mechanism. . . . . . . .
FIGURE 24. Sketches of (a) n pole amplitude and (b)
experimental t angular distribution for np+pn

at 8 GeV/Co I I I I I e I G I I I I I O I I I I I I I

FIGURE 25. Sketches of p and A contributions to (a)

2

n=0 and (b) n+0 amplitudes in the reaction np+pn. . .

do

FIGURE 26. H? VS' t for np+pn at 8 GeV/c. (Data are from
G. Manning et al A.E.R.E. Harwell, England. . . . .
FIGURE 27. %% vs. t for pp+nR at 7 GeV/c. (Data are from

P. Astbury et al. Phys. Lett. 16(1965)328). . . . . .

viii

52

54

57

60

61

62

68

69

72

75

LIST OF FIGURES (continued)

FIGURE 28. Feynman diagram of OPE contribution

to ab+cd scattering. . . o . . . . . . . . . . . . . 78

FIGURE 29. The s-channel center-of-mass system for

ab+cd scattering. . . . . . . . . . . . . . . . . . 79

FIGURE 30. Coordinate system for OME calculation

Of ï¬‚+n+wpe I I I I I I I I I I I I I I I I I I I I I 86

FIGURE 31. Diagrams giving vertex corrections to

NON pOint verteXI I I I I I I I I I I I I C I I I I I 90
FIGURE 32. Two pion exchange graphs for N+n+wp. . . . . 94

FIGURE 33. Configurations for two-meson exchange

graphs for n+n+wpI I O I I Yâ€˜3 C I I I I I I I I I I I 99

FIGURE 34. Configuration of NeN-vertex correction

graphs for nn+wp. . . . . . . . . . . . . . . . . . . 102

ix

INTRODUCTION

In a large class of inelastic elementary particle
reactions the experimental data reveal the following promo-
nent general features:

1. One can usually group.the final particles into
two quasi-states, the centerâ€”ofâ€”mass of each one nearly
maintaining the direction of one of the incident particles,
RRCM is strongly peaked either in the forward or backward
direction or both.

The reaction ab+1+2+3+...appears-as if there were

only two particles-in the final state.~ Examples of such

reactions are

Np+nnN
+HUHN
KN+KHN

+KHNN

We refer to these-â€particles" as c and d and consider
quasi two-body interactions ab+cd-in which c or d may be
multiparticle states.- They are,.in fact, the decay products
of resonant states and are no different from the resonances
themselves kinematically.

The higher the inCIdent particle energy and the
simpler the compOSItion of c and d, the more evident the

forward or backward peaking becomes.

2. -In almost every case,-at-certain energies the
final states do not appear to be directly produced.- The
reactions proceed via the production of strongly inter-
acting-(and very short-lived)~resonant~states which under-
go strong decay-into the final state.- For example

HN+ON+HHN
KN+K*N**KHNN

3. The total cross-section O (s+w) behaves like

tot
s-n for small positive n.- While.the density of experimental
data available over a wide range of reactions and energies
is light, it is possible to note a rapid decrease (in
most cases) in the total resonance production cross-section
with energy. As we shall see later it is this fact that
proves to be the most difficult to handle theoretically.

These are peripheral collisions, and for lack of a
better definition we shall call theoretical models for
these processes peripheral models. The idea is that a and
b do not collide head on; they undergo a glancing collision
in which the two particles barely touch each other. We
do not expect the trajectories of either of the final
particles to deviate from the directions of the incident
particles.

One expects these peripheral processes to be
dominated by long range forces. Because of the energies
involved the treatment must be relativistic and there is
no reason to suppose that we can ignore the intrinsic

Spin of the particles. From the point of View of field

theory these reactions proceed via the exchange of the
virtual quanta that will induce the longest range force,
the quanta of lightest mass. The problem has been

reduced to consideration of a sum of one particle exchange

graphs as indicated in Figure 1.

Q m

Figure l. The reaction ab+n particles in the
region where the final state is dominated by production
of two resonances is approximated by a sum over all allowed
one-particle exchanges.

In its simplest form, this "Born Term Model" leads

to a matrix element of the form
<cd|Blab> = Baec(tâ€™mc);:Lmâ€”Z Bbed(t,md)
e

for the production of the intermediate resonant states
where t=-(b-d)2 is the invariant 4-momentum transfer,
p/(t-mi) is the prOpagator of the exchanged particle and
Bi (t,M ) is the vertex function. At t=M2, B (M2, M2)

en n e zen e n
is the matrix element for the physical process 2+e+n. If
one regards n as a particle, B is equal to the (Zen)-

coupling constant multiplied by a known kinematical factor.

These vertex functions.are obtained in the usual
way from interaction Lagrangians or by writing down the
most general function consistent with Lorentz invariance
and then using hermiticity, CPT.and other strong interaction
invariances to limit the form.further.
Since small physical values of t correspond to
small center-of-mass scattering angles it is clear that
if the vertex functions are not pathological in their
behavior this Born approximation ought to beta good start
to explaining peripheral collisions. And because of the

"factored" structure of <chB|ab> we have a basic test

229.
d9

between the planes defined by ac and bd.

of the model: should be independent of the angle

The Exchanged Particle

Consider any specific reaction and the usual
invariance principles determine which exchanges are
allowed.

Example: KN+KN*

1. Conservation of baryon number tells us that e
cannot be a baryon.

2. None of K,N,N* are eigenstates of the Gâ€”parity
Operator so we obtain no Specific information from G-
parity conservation: Ge is arbitrary.

3. Since strangeness is conserved at both

vertices the exchanged particle must not be strange: Se=0'

4. Isotopic Spin conservation at the NeN* vertex
requires l/2+I=3/2, which gives Ie=1,2. Consistency with
the KeK vertex requires that the exchanged quanta be a
member of an I-Spin triplet: .Ie=l.

5. Consider angular-momentum and parity conserva-
tion at the KeK vertex in the outgoing K-meson rest frame.
The K has Jp=0- so the addition theorem requires

(_)Â£+l

_ 1 _ _
0 +2( ) +JÂ§e=0 . This reduces immediately to Â£ +Jge=0 .

Je

Clearly we must have Â£=Je and pe=(-)Â£=(-) which means

that Jge=0+ or l-,2+,... Examination of the NeN* vertex
provides no information which further restricts Jge.

In summary, for KN+KN* the exchanged quanta must be
a non-strange member of an isospin triplet of mesons with
Jp=J(-)J. We choose from among the known low lying meson
states and conclude that 0 exchange is likely to dominate.

One proceeds in just this way, applying the conserâ€”
vation laws at each vertex in.turn for each reaction to be

considered. This procedure rarely limits the exchanged

quanta to a Single state.
Decay of the Resonant State

Recall that particles c, d or both in the reaction
ab+cd are in most cases pseudo-particles, resonant inter-
actions of several elementary particles. So far we have
discussed in general terms the amplitude <cd|M|ab> for the
production of the resonant state, particle d. (For the
moment c will be just another elementary particle, eg.

HN+NN*). We will consider decay into two particles: d+fg.

If we had particle d, with Spin Sd at rest in the labor-
atory, its decay would be governed solely by the transition
matrix for the process d+fg:

<Â§Â§>

~Â£ <fg|M|d>|2.
decay fgdl

Given an ensemble of states d at rest with Spin

polulation A we have no reason to expect that the popula-

d
tions of each projection state are different. It is clear
that this is not the case in our Situation. We do not
start with particle d; we actually observe the process
ab+cfg which proceeds via the intermediate state d:
ab+cd+cfg. In contrast to the above, we do not expect the
resonance d to be produced with an isotropic spin distri-
bution. It must be assumed that the production matrix
elements are spin dependent until we know otherwise. One
can still obtain the decay angular distribution from the
formula given above, but the initial state for the decay

matrix element must be weighted by the probability for

production of the Spin state

<fgc|MIab>=X <fg|M Iab>
S

- deca |d><Cd|M
pin Y P

rod.
and therefore

(10' ~ I
(dâ€”SI)d->fg Z<fg|Md|d>*<fg|Mdld >gga

b<cd|M |ab>*<cd'|M |ab>
p p

Considering only the angular part H of the decay
matrix element and normalizing, we arrive at a formula
for the expected angular distribution of the decay proâ€”
ducts f, g of the resonance d produced in ab+cd. For
convenience everything is done in the d rest frame.
W(a.8) = N ZH(fgd)*H(fgd')odd.

where pdd' = N Z<cd|Mp|ab>*<cd'|Mp|ab>

2

1 determines N2

Various symmetries can be used to obtain relations

Trp

among the production density matrix elements pmm" To

find these it is necessary to examine W(a,B) in more detail.

Coordinate system

 

Since it is convenient to do calculations in the
center-of-mass frame, we start there.. In all of our calâ€”
culations we have defined the production plane to be the
X2 plane with the momentum vectors aligned as Shown in

Figure 2.

[â€”I |>>
Dr)

d
TB
:74â€œ

Figure 2. The two-body scattering ab+cd in the
center-ofâ€”mass and the orientation of the x and z axes.

Because it is easiest to look at the decay of d
from its rest system, we perform a Lorentz transformation

into this system as indicated in Figure 3.

Figure 3. The two body scattering ab+cd in the
rest frame of particle d.

The 1 direction, the normal to the production plane, is
axc

defined by j_ = TERI"
The decay angular distribution

Now add the decay products f and g to the diagram,
and delete a and c to get them out of the way as indicated

in Figure 4.

LI
I
a

r
I
/
x

I

I

l

l

l

l

l r:
IX)

Figure 4. The decay products g and h of the resonance
d in its rest frame.

Two angles are needed to locate the decay products. Momen-
tum conservation requires f and g to move in opposite
directions. We use a polar angle a, which measures deviaâ€”
tion of the line of motion from the]; direction, and an
azimuthal angle 8 measuring deviation from the production
plane. These definitions correspond to the usual definihg
angles for the spherical harmonics YÂ£m(a,3). AS long as we
choose the x axis to be the spin quantization axis for the
resonant state d of Spin jd' the angular part of the decay

matrix element will be |2m> coupled to jf+jg to form jd.

An Example:

This is a l-+0â€”+0_ transition. In the p rest frame
the intrinsic spin of the p meson becomes the relative
orbital angular momentum of the two pions. The angular
part of the decay matrix element is proportional to YÂ£m(a,B).

We write

W(a:8)â€1Â§mqum(arB)Yimu(01:8)Dmm.
where we normalize fanpW(a,B)=l.

Symmetries of pmm'

It is useful to take advantage of available symmet-
ries to simplify the density matrix as much as possible.
We require that p be hermitian, p+=p. Application of
parity conservation to ab+cd leads to additional relations
among the production amplitudes. Substitution of these
relations into the formula fortnimmediately Shows that

=(-)â€™â€œ"â€œ'

D 0

mm' -m-m'
These two relations plus the trace condition enable

us to write

a TD
iâ€™1/2(1"â€™oo) 010 p1,-1
J=1 _ * _ *
pmml _ p 10 poo p 10
01,-]. -p10 l/2(1-p00)-J

 

 

10

Using this matrix we obtain an expression for Wp+2"(a,8):

i 0+O_3 _ _ 2 _ . 2 _
W(Q,B)-EEI3/2(l p00)+1/2(3p00 l)cos a plâ€™_131n acosZB

V2Rep1051n2acoss]

One also obtains an expression for the polarization of

particle d by using P=Ter Px and P2 are found to vanish,

d'
so the polarization vector of d is perpendicular to the
production plane.

The decay of a spin jd into spins jf and jg can be
analyzed in a similar way but it is necessary to use the
angular momentum addition theorem to write the prOper
angular functions. Once one has an expression for W in
terms of density matrix elements, he again examines the
effect of invariances at the vertices on the density matrix
elements to obtain model dependent predictions for the
angular distributions, assuming a jge for the exchanged
quantum. To Show the method and complete the p-production
example, we look at the p+wn decay.

Assume the dominant contribution to nN+pN is the
pion exchange force and ask what this means for the decay
distribution of the produced mesons. Consider the
vertex in the p-meson rest frame.

A unit of orbital angular momentum is picked up in
p-formation as intrisic spin and jz for the 2H system is
zero. Note that parity is conserved. Conservation of Jz
implies Jz=0 and the jz=l Spin projections of the p are not

produced at all.â€˜ The only non-vanishing density matrix

ll

element is p00 which, because of the normalization condi-
tion, must be unity. Substitution into W(a,B) gives

deW(o,B) = 3/2 cosza

W(a,B) = 3/4ncosza W(a)

W(B) l/Zv

We look for this behavior in the experimental data as
evidence for n-exchange. This result is quite general;
any reaction of the form PB+VB proceeding via pseudo-
scalar exchange will lead to the above angular distri-
butions for the decay of the produced V-meson.

Similar analyses may be performed for the other
allowed spin-parity assignments of the exchanged quanta
and for arbitrary Spin of the produced resonance. Table A
summarizes the results of analyses for other relevant
cases.

When doing calculations helicity eigenstates are used
rather than spinors quantized relative to the z-axis.]'These
are states in which each spin is quantized relative to
its own direction of motion. Such states have very simple
Lorentz transformation-prOpertieS-making-them very easy
to use. We calculate amplitudes and cross-sections in
the production centerâ€”ofâ€”mass system and decay angular
distributions in the decay-center-ofâ€”mass system. â€˜Before
we can use the-density matrix it is necessary to perform
â€˜a rotation so that spin-quantization is relative to the
zâ€”axis, a more transparent-situation for comparison with

experiment.

12

In the production CM, the angle of rotation for

c,d is clearly 6cm' In the C(d) rest.system we have

"I =g_- ' ' I. = .
Sinwc ac Sinecm (sinwd %a81n6cm)
. _ 2 2 2
where, With A(x,y,z) - x +y +z -2xy-2xz-2yz
q=-l-%l/2(s,m:,mÂ§) An incident channel CM-momentum
2fs'.â€˜

production process

1 A'l/2(t m:,m2 )

ac=2mc

momentum of a in c rest system

_1 1/2 2 2 . g p. _
bd-Eï¬â€”A (t,mb,md) momentum of b in d rest system

d
This transformation formula for the angle is very easy to
derive. The component of a 4-vector perpendicular to the
direction of a Lorentz transformation is unaffected by
that transformation.

5(J)

Let %EL be the helicity state density matrix for

production of the resonant state c of Spin jc. The density
matrix for quantization along the Z-axis in the c rest

system is then given by

(J) J
C d? (-w)
A lAl c

which is a function of s and t for the production process.

(J)
c J
p .=dC (11))01
AcA c ' AcAl c All 1

It is this pmm' that goes into the angular distribution

formulas.

13

A look at the experimental data provides clues to
the production mechanism. We invent a model, calculate
production angular distributions, total crossâ€”sections,
production density matrix elements, and compare with the
experimental-data. Proceeding in this manner, it has been
possible to explain a-large-number of high energy-reactions,
though in most cases not-without modifications, to the

basic Born Terms.

14

TABLE I

Predictions of the BTM for Density Matrix Elements

1. Vector Meson Production (Jp=l-) [decay:1'+O-+0-]
significant density matrix elements: â€900:01 -l'plO
I

(p10 is complex).

a) Pseudoscalar exchange (Jp=0-)

prediction: p00=l, others zero.

_J
b) Natrual parity exchange (JP=J< ) )

'arbitrary,.others zero.
(_)J+l)

redictio :
P n - p1,-1
c) Unnatural parity exchange.(Jp=J

noeprediction.

2. N*3 production (Jp=3/2+) [decayz 3/2++1/2++o']

Significant density matrix elements: 033â€™03â€”1'031
com 1 x
(03,-1'031 P e )
a) Pseudoscalar exchange (Jp=0_)
Prediction: all above elements are zero.

b) Vector meson exchange (Jp=l-)

prediction: Rep3l=0, p33=3/8, Rep3 _l=3/8.
I

THE GLAUBER FORMALISM FOR HIGH ENERGY

SCATTERING AND THE BORN APPROXIMATION

We start with a potential V(Â£) of finite range
dnd use the Schroedinger equation to solve for the wave
function, assuming high energy for the incident particle.
By high energy we mean the incident particle energy e
is much greater than the absolute magnitude of the potential
V0, and that the wave length is much shorter than the

range r of the potential:

0

V0<<e kr0<<l.

Since, relative to e the potential is weak, we do not
expect the wave function to deviate significantly from

a plane wave. We write

M is assumed to vary slowly with Â£'
Substituting in the integral form of the Schroedinger

equation
eiklrâ€” r'
WU?) =8 iâ€”â€˜AITiâ€”{r'f- V(Â£ )KHIâ€˜ )dr r
we easily obtain

iklr- r'lâ€”ik (râ€” r ')V V(r

my: l-Ae ._'>M<_r_'>dÂ£'
|Â£-1"|
â‚¬10â€œ? â€li'lâ€”Zi) ,
- l-Al-r: V<Â£-_r_l>M<Â£-_r_ lldil

15

l6

2
l ldrld(cose)dp.

Assuming V

where d5 =r

OM changes appreciably over distance d

but varies slowly over a wave length, we obtain by partial

integration
2 eikrl(l-cos6) cose=l 1
M(Â£)=1+Afdrlrld6[IEEâ€”â€”- v<57Â£l)M(Â£gÂ£1)1 +6<an
l cose=â€”l

Atu=-l(rl antiparellel to k) the exponential
oscillates rapidly and this term is also of order l/kd.

The leading contribution is
Ik

lI
A1:

r
M(Â£)=1â€”iÃ©fder(Â£-Â£l)M(Â£â€˜Â£2)Igz

Choosing k to be in z direction we obtain
M(g,z)=1-iA/kfdz'V(Â§+gz)M(g+gz')

-â€˜ Z I A u
52% =â€”%de'+M(-k3,z)=e lA/klwdz V(I_)_+_]EZ )

where we have used as boundary condition M(b,z=-w)=l,
which comes from the requirement that we start with an
incident plane wave.

The final form for the wave function is

x(b z)=eih'(Pfizlâ€˜iA/kfde'V(2+Â£z).

which is valid in the region of the potential.

The limits of applicability of this approximation
have been given by R. J. Glauber.2 It is sufficient to
know that this formalism will provide a good description

of the scattering in the forward and backward cones. We

17

use the wave function x(b,z) in the formula for the
scattering amplitude in place of a plane wave. The idea
is that if the potential is-well-behaved and the incident
energy is high enough, the incident wave will be only
slightly distorted in the region of the potential. We
have calculated the first order correction to the wave
function due to this distortion. This is called the
Distorted Wave Born Approximation and was used by

J. D. Jackson and collaboratorsB'4 as the starting point

in developing the absorptive model.

ABSORPTIVE CORRECTIONS

We are studying inelastic processes of the form
ab+cd, which we assume to be due to the interaction V in
lowest order. Now we allow a and b to interact elastically
before they interact via V to produce c and d, which also

interact elastically, as shown in Figure 5.

 

Q, C.

 

b A

 

Figure 5. The absorptive model diagram for the
scattering ab+cd.

The incident and outgoing channel elastic interactions Vi
and Vf are treated exactly. The result is a modification
to the Born term for the inelastic process. At the inter-
action V, a and b are no longer plane waves; they are
gradually distorted by the elastic interaction as they
approach the inelastic interaction region. We require

that Vi and Vf and the incident energy satisfy the restric-
tions implied in the Glauber wave function. The incident
and final wave functions xi and xf are

)<"â€˜f(13r2)=eâ€”lgL .Ee-u/q [zdz VfUPfEE)
. , z ' A '
(11>.2)=e19~'Â£e'â€/q dz Vi(]2+iz_ )

iâ€”â€” â€”00

18

19

We use these wave functions to calculate first order
contribution to the inelastic scattering:

iO-b

<cd|M|ab>Zfd2bdze â€” -<CdIV|ab>e

â€”iA/qudz'Vi-iA/q'fmdz'Vf
~00 z

ZZHfjbde (6b)ei/2xi(b)B(b)ei/2Xf(b)
where B(b)=fmsz(b+Â£z) and gagâ€”3'

â€”a>

The range of V is assumed much smaller than Vi or Vf and
the phase shift of wave traveling through V at impact

parameter b is given by

xn<b)= â€”Â§h{: dz'V<BTEz) (n=i.f)

Compare this expression for the amplitude with the partial

wave expansion.
<|M|> = Z(Â£+l/2)MÂ£ P2(cose)=fxdxM(x)JO(wx) (w=25in8/2)
, 2

Note that the expansion in impact parameter is
approximately equivalent to the partial wave expansion.

To obtain a relativistic version of this formula
and include spin, we assume that the partial Born amplitude
<ACAd|MjIAaAb>, where Ai is the helicity of the 1th particle,

.J..'
is replaced by el/ZXIMJel/ng.

We start with the relativistic partial wave expansion
A=Aa-Ab

_ , j j
<AcAd|M|AaAb>â€” Â§<3+1/2><ACAdIM 'Aaxb>d Au(e)'u=x _(
C

. d
invert it to obtain Mj

<A A

C dIMJIAaAb>=fd(cose)<ACAd|M|AaA >dj (e).

b Au

20

Including the distorted wave correction DWj and resumming
theâ€˜serieS'we'obtain

| . â€˜ =2. ' V. '
<AcAd|M IAaAb> j(j+l/2){Dijd(cose)<ACAd|M|AaA

j j .
b>dAu(e)}dAu(e)'

In actuai practice this procedure is avoided when-
ever possible by making use of a few mathematical tricks.
For Singleâ€”quantum exchanges, retaining only the two lowest
terms in n=|u-AI, the amplitudes can be put in the form

a'

_" l .
Mnfe 2â€˜(xanw +x w ).

 

It is possible to write

l='l' 1
t-m: qq' â‚¬2+w2

 

 

where 82 is composed of angle independent kinematical
quantities. Then using
n
w

â‚¬2+w2

 

â€” e foxden(wx)kn(ex)

we are able to write each amplitude as a partial wave
sum and all that need be done is to insert the distorted
wave factor DW(X) under the integral.

It should be mentioned that for all of these high
energy reactions, large numbers of partial waves contribute.

Since we are limited to small 6C we can make use of the

M

approximation

 

j ~ 2 _

dAu(8)~JnÂ«J+l/2)w) n In A

Substituting in the Jacob-Wick Expansion and approximating
the sum by an integral we obtain

<ACAd|MIAaAb>n = f xdx<AcAd|M(x)IAaAb>an(wx).

Jo

21

Comparing this formula with the one obtained by using
the approximate amplitudes M: and using the integral formula

above,we write

wn

a _ 6 n-
Mn I qu[(xanâ‚¬ 232an):2:;7 + w x an]
9

= Â§Â§T[(x -62Â§an)foxden(wx)Kn(ex)+wnÂ§an].
Adding DW(X) under the integral in M is essentially
the correct procedure for adding the distorted wave correcâ€”
tions.
It is possible to write the absorptive amplitude
for a process in a way that explicitly separates the
correction term from the Born exchange contribution. We

begin with the Sopkovich formula for absorptive correc-

tions:

A _ Â° .
<ACAd|Tj|AaAb> _ e j<ACAd|T.

Writing Sj=l+2iNTj where N=norma1ization and energy

momentum conservation, we readily obtain

A _ B
<ACAd|T IAaAb>n -<ACAd|T IAaAb>

+iNF. E (2. +1)dj (e)<AC A â€IT |A1A2><r1 A IT? |Aa Ab >
jAlA2 2

+iNI z (2. +I)dju (e)<AC A IT? |A1A2 ><A1A |TÂ§ |A A >
jAlA2 j d 2 a b

_ j
NiNF 2(2j+1)dM1 (6)<A cAleg lkllleg |A3 A4 ><A 3A4|T3' IA aAb>+. ..

 

I|l{1JI \â€˜Iâ€˜

22

where the Jacob-Wick expansion has been used to obtain the
full amplitude. If we approximate the initial and final
state elastic scattering amplitudes by spin-independent
functions we recover our simpler, less general, formula

2(2.+1)[1+iTFNF+iTÂ¥NI-TFTÂ¥NFNI+...]
j J 3 J 3 3

A
<AcAd|T |AaAb>

, B j
<Acxd|Tj|AaAb>dAu(6)

g(2j+l)DWj<AcAd|T?|Aalb>dgu(6).
The general formula is useful as a starting point for
more sophisticated calculations, eg. spin-flip in the
elastic scattering which would allow one to take into
account the non-zero polarization found in many elastic
collisions, and to see more clearly the effects of absorp-
tive corrections on the exchange amplitude. In practice,

calculations using this form require large quantities of

computer time.

FORM FACTORS

One of the first ideas for improvement of the Born
Term Model was to replace the simple point vertex functions
and the propagator by functions which incorporate higher
order effects. The existence of such effects was used to
justify replacement of the Born amplitude B by BF, where
F is some-phenomenological function depending on t and
the masses and as few arbitrary.constants as possible.
If one chooses such-a function and obtains a best fit to
the data at some incident energy and then-uses-this function
to calculate at other energies, he hOpesâ€”to succeed in
comparing with experiment. -If-success-does come, one has
a-very simple method of calculating.

J. D. Jackson and H. Pilkuhnfsused an exponential
-form factor eAt-to improve the angular distribution in fits
to KN+K*N and K*N* over that obtained from simple vector
meson exchange.â€” Later we will see-that such a form
factor works well for ï¬‚N~inelastic-collisions~dominated
by vector meson exchange, but that another form-factor

F(s) is necessary to fit the energy dependence.

23

24

Form factors are also able to improve on the Born
modelfor reactions dominated by the exchange of pseudo-
scalar particles. Amaldi and Selleri,â€˜5 two of the early
investigators of peripheral collisions, were successful
in fitting the data on np+pp with.the function

F(t)- 0572 2 + â€˜0528 2 2
l+(Mï¬-t)/4.73M1T l+[(Mï¬‚-t)/32Mï¬‚]

A REGGE MODEL

Another approach based on an idea-first suggested
by Gell-mann, Frautschi, and Zacharisen,7 draws on the
Regge Model of high-energy~scattering. One starts with
the t-channel process, writing down a partial wave expan-
sion for the amplitude.- Assuming a~resonance in a partial
wave one performs an analytic continuation via the
Sommerfield-Watson transform to the 5 channel. This
amplitude, the contribution-to the s channel amplitude due
to the virtual exchange of the tâ€”channel resonance, is
used to calculate cross-sections.~ One obtains a one-meson

exchange model in a different way; the exchanged particle

has a spin a(t). One can show-that the characteristic

 

prOpagator form 1 2 is contained in the amplitude.
t-M * '
e
Many authors have done calculations based on this

model.8 The-idea is to replace the faCtor â€”Eâ€”Â§-in the one-
. â€œ t-M
e

meson exchange amplitude, the one obtained field theoreti-

cally and not the one crossed-from t channel by a phenom-

enological function of Regge form:

 

 

-iï¬‚a(t) , , d(t) , d(t)
F(t)' K(2dÂ§t)fl)(l+TeÂ»A )[sâ€”u I .[qtqt
t-M: 2 sideTr 4qtqÃ© Ma/MbMd

25

r Tâ€œ w .ï¬‚lLï¬ï¬‚

26

The trajectory d(t) is determined experimentally. This is
done for each exchanged particle. The extra factors have
been introduced to make the formula dimensionally simple.
The basis for this replacement is that had we started with
the t-channel amplitude Tt(s,t)=kÂ§(2Â£+l)TÂ£(st)PÂ£(coset),
assumed dominance of a resonance of spin R, performed the
Sommerfield-Watson transform and continued tO the 5 channel

we would not get a factor â€”%â€”-. Instead we would have
M -t

obtained-something like

 

t)
-6(t)- 5Â°â€œ A _ _ 2
Slnï¬‚a(t)(Â§6) where Jâ€”Red(Meh
where (t) will have the-behavior 2,prOVided it is not
t-M
e

destroyed by other factors in 6. This does happen occaSion-

ally. Starting with the Regge Form
7 _ I .
P ( xt)+Pa(xt))-(qtqt . )aÃ©t)
2 sinna M JM'M
a d

 

~(2n+1)(

where q ,q' are initial, final cm momenta for t-channel;
t t

 

 

 

using
s-u sâ€”l/2(M:+M:+M:+MÂ§)
Xtzcoset=4qtqÂ£ 6>>t 2qtqÂ£
and
d . .
T + _
Pa(xt)~ (a l/2)2 X:(t) _ Cx:(t) P (-Xt)=te lNaPa( t)
VFT(d+l)
we write
_ -iwa(t) .QAQ' d(t)
B(2d(t)+l)%:;:ï¬‚a(t) x: â€”E727â€”â€”] where have put
Ma/MbMd

cB=Â§ and T is the signature of the exchanged trajectory.

27

E is determined by-requiring this expression to coincide
with the field theory propagator at the-pole t=M:

-l+tefiï¬‚a(t) d(t)[]a(t)_
2 sinna(t) Xt â€™

 

2. -
6354: [B (2a (t) +1)

t-M
e

Near the pole (at to)
a(t):a(to)+a'(to)(t-t0)

sinna(t) : sinnd(t0)+nd (t0)cosnd(to)(tâ€”t0)

z sinnj+wa'(t0)cosnj(t-t0)2na'(t0)(t-t 'cosnj

0)
~ Nd (t0)(t t0)( )

 

. j . ,
t t0 2(-)3na'(t0)(tâ€”t0) t t"to

from which we readily obtain

 

 

"_ l -jâ€˜ _jâ€˜
B - Zï¬‚a (t0)xt /[(r+( ) )(23+l)]
2na'(t0) , s-l/2(M:+M:+M:+MÂ§)} -j
= , 1
_ J
(2j+1)[T+( ) ] 2Mav/MbMd
Example:

For J=l exchange the propagator is replaced by

 

_ i-eâ€™l"a(tâ€™ s-l/2(M:+M:+M:+MÂ§) â€œ(tâ€™
R(t)=B(2d(t)+l) A _ { mâ€œ }
251nna(t) ZMa/MbMd

where E is given by

 

s- l/2(M:+ +M2+M2+M2) d(t)
-_ TT l Mb C (21
2M aVMbMd

A 1â€” ._........_:_..r-â€”
Jâ€˜

28

The procedure is to calculate the cross-section and density
matrix elements with this form.for the-propagotor using

the experimentally determined Regge trajectory apprOpriate
to the exchanged state.â€˜ Since.the~Regge model is known

to give the correct asymptotic energy dependence for cross-
sections one hopes that this phenomenological ansatz

based on the Regge idea-will show similar soc1ally acceptable
behavior when it comes to energy dependence. The problem
of correct energy dependence is one of the chief diffi-
culties of the other methods-of-calculating peripheral
reactions. As a derivation of one-particleâ€”exchange Via
the Regge approach makes clear, the energy dependence is
intimately connected with the.spin of the exchanged quanta.
In the Regge approach, exchange of a spin j does not lead

to amplitudes which go as s3 but as de(t)

where dj(t) is'
the trajectory of the exchanged quanta.

We might wonder why the field-theoretic t-channel
amplitude is not used and explicitly continued to the
s-channel rather than making this prOpagator replacement.
Such calculations are being-done and they are more compli-
cated than the replacement procedure. Starting With the
t-channel exchange amplitude,.the kinematical Singularities

introduced by spin are explicitly removed and the amplitude

is continued analytically to the physical s-channel region.

 

29

Again, the residues are determined by requiring equality
with the Born term at the pole. Â»As an example of such
calculations, the reader is referred to the recent work
of Griffiths and JabburS on N*3.production, in KN
collisions. Recent calculations on np charge exchange
combining the absorption model with a-conSistent treat-
ment of spin in the Regge model will be outlined later in

this paper.

 

ONE-MESON-EXCHANGE CALCULATIONS

AND COMPARISON WITH EXPERIMENTS

Most of the basic approaches to theoretical analysis
of peripheral collisions have been discussed. To familiar-
ize the reader with the general picture before describing

present attempts to find improvements to OME models,

! .

consider two reactions. Â«One, nN+pN, is dominated by
pseudoscalar meson exchange, the other, ï¬n+wp, by veCtor
meson exchange. We discuss them in terms of the models
just outlined. This discuSSion will be qualitative. The
figures show general 5 and t-behavior only. Notation

and conventions are detailed in Appendix I.

The first of these reactions has been studied by
many high energy theorists 3 and summarizes the successes
of the models. This is both good and bad. It is good
that everyone can invent a model, but bad because there is
no resultant discrimination among ideas. The second
reaction has been chosen because none of the models already
discussed can explain all of the data. Some models give
correct angular distributions and others handle the energy-

dependence.

30

31

. do
Figure 6 shows dï¬

a linear scale of arbitrary normalization.

; %'
$ï¬?

ï¬‚â€”
Â»
a, _

 

 

 

 

o ' 't if '
on (a) , 25- â€˜ b) 4'.

Figure 6. Qualitative sketches of the t-angular
distributions for (a) wp+pp at 4.0 GeV/c and (b) nn+wp
at 3.25 GeV/c.

Notice the peaking, particularly strong for ï¬‚p+pp, in

the forward direction. This peaking is characteristic

of peripheral processes. The angular distribution for
n+n+wp, dominated by-p exchange, is less peripheral

than the n-exchange reaction, as is to be expected since
mp is much larger than mâ€. In each case the total crossâ€”
section drops quickly with increasing energy as

indicated in Figure 7.

015)

 

4
Â¥

S 3
Figure 7. Sketch of sâ€”dependence of the total
cross-section in a quasi two body high energy scattering
involving quantum number exchange.

4F

vs t for each of the reactions on

32

Data on the production density matrix are meager.
Because of invariance principles, there are only three
elements of interest and they are the same ones for each

reaction: The element 010 is complex but

poolplolpl'_lÂ°
we need only the real part for the decay distributions
W(cosa) and W(B). For wp+pp at 4.0 GeV/c the available
data are (000) = 0.53:0.12, (pl,-l> = 0.16:0.10,

> = -0.06i0.05. Similarly, for ï¬‚+n(p)wp at 3.25 GeV/c

â€˜ï¬' _

<p10
Jackson gives <p00> = 0.5 and at 1.7 GeV/c he gives

Jâ€œ ' Va; Pin._â€˜: â€™31 â€œ

> = 0.6:0.12, > = 0.0i0.712, > = -0.06i0.08.

<poo (91,-1 (910

In both cases the average is over cosG from 1.0 to about

I

0.80. This data is not very limiting to a model, but we

shall See that it does tell us something.

The Born Term Model (BTM)

:2
dt

experimental data as indicated in Figure 8 we see immed-

If we superimpose theoretical curves for on the
iately that the BTM is just not good enough. The crossâ€”
sections are too large and not nearly peripheral enough.
It should be noted that the vector-meson-baryon coupling
involves an arbitrary factor. There are two coupling
constants, the vector coupling GV and the tensor GT'
Even if one uses some symmetry scheme to fix Gv' he must
. Jackson gives GT/GV = 3.7

T/Gv
inferred from T=l electromagnetic form factors.

look elsewhere to get G

33

Jw :
m (a) ,/" ' ' ' â€˜
\\\\â€œsu'
EXP 1â€˜.

 

-t
Figure 8. Comparison sketches of the BTM and

experimental (EXP) for (a) np+pp and (b) nn+wp.
Comparison of theoretical and experimental total
cross-sections and density matrix elements is as unfavor-
able as for the angular distribution. The energy dependâ€”
ence is particularly bad for the vector meson exchange
reaction. But it is of practical value to carry through
an analysis to determine the BTM predictions for pmm"
It is not necessary to do any calculating to get answers.
For w-p(n)p-p, Figure 9 shows the nnp vertex in thegbrest

frame.

1â€˜ "Hâ€˜- â€œ â€˜3. 951â€˜

Figure 9. The wnp-vertex in the p-meson rest frame.

This case was considered in the discussion of
density matrix elements. The analysis was done in the
rest frame of the pâ€”meson and we concluded that the only
non-vanishing element is 000 and using the normalization
condition Trp :1 we obtained 000:1. The data tell us
that the BTM prediction is incorrect.

We ask if the fault is with the BTM or with the
quantum we have assumed to be exchanged. To answer this

we look at the decay angular distributions of the resonant

state, W(cosa) and W(p) and note that even though 91 â€”l

I
and p10 are doubtless non-zero, they are extremely small,
which is consistent with pseudoscalar exchange. The

experimental W's, shown in Figurelï¬‚ support this conclusion.

 

 

 

(a) Mn)
N. ,r
â€œâ€˜ 't" ........
-1 o 1 4t o W
(.031 ï¬

Figure 10. Sketches showing the experimental decay
distributions for the decay p+nn in the reaction np+ppennp
assuming 0 exchange (a) W(cosa) (b) W(B).

Additional support comes from the BTM distributions
assuming vector meson exchange which is also allowed in
this reaction. Figure 11 shows these distributions. (We
will see how to obtain these distributions shortly). It
is apparent that these distributions are not at all close
to the experimental data. The conclusion is that the basic
idea of pseudoscalar exchange dominance is a good one

but that it is too simple; the BTM neglects too much.

 

 

m 00â€˜
xâ€™ .5â€œ I. Nâ€˜ Iâ€™\\
â€™ \\ I \ I \
\ \ / N
- L O 4, -!â€˜ o 1!?
cos 9k P

Figure 11. Sketches showing the experimental decay_
distributions for the decay p+wn in the reaction np+pp+nnp
assuming 1 exchange (a) W(cosa) (b) W(B).

â€”â€”â€˜-â€”s-rrv -':â€˜?..3.- WP
. J

35

For N+n+wp the analySis procedes in the same manner
as for 0- exchange. In the rest frame of the w, we have,
from the angular momentum addition theorem that
0-+Jge+L(â€”)L=l-; Je=l implies L=0, l, or 2 and, since we
are in the w reSt frame, L220. The conservation of parity
in strong interactions tells us that (-)Pe(-)L=(-) or L
must be odd, Since Pe=-l. Therefore L=l. Conservation
of the z-component of J tells us that Jez=0,tl. So we
must have Ilm; coupled to |10>L to produce Ilm>w. Again
by the rules for adding momenta we see that the state
llmgilO>L contains only ilzl>w ie. the mw=0 state is not
produced. Therefore we can conclude that Dmo = pom = 0.
Substituting in the decay angular distribution formulas

we obtain

W(cosa,B) = 3/4 (l/2-p Cos2B)s1n2d
n l,-l

or

W(cosa) = 3/4sin2d W(B) = 1/4 :(l+20l _ c0528}.
I

1)'491,â€”1
(These formulas are valid for w decay even though it is

a 3n resonance if the angles are redefined such that d is

the angle between the incident n+ and the normal to the

nâ€”wÂ° decay pion plane; 8 is then the corresponding aZimuthal

angle).

v.2 m .ninâ€”â€”â€”

36

We see immediately that for vector meson production
in PB collisions the decay angular distributions are
distinctly different for pseudoscalar and vector exchange
dominance. (Actually the above analysis is valid for an
exchanged state Wlth Pe=(-)Je, Je>l; these are called
natural parity states. For unnatural parity, Pe=(â€”)Je+l

we get no restrictions).

Figure 12 shows the experimental W's for n+n+wp.

Van-n moat-5 q

It is clear that the evidence is strong for the exchange
of a natural parity state. The simplest possible

exchange candidate is the p-meson, a member of the 1-

 

 

octet.
/ /â€˜ â€˜ \ ."\\ (f~â€˜
I I â€˜
\ I \JI \
l 2
â€˜1 o i -w 0 ï¬â€˜
$.20â€œ: ct E"
Figure 12. Sketches of the experimental decay distriâ€”

butions of w in the reaction nn+wp (a) W(cosa) (b) W(B).

Multiplicative Form Factors and Reggeized Propagators

Both of these approaches involve replacing the

EXF(s,t), where F(s,t) is a

exchange amplitude MEX by M
suitable function with as few aribtrary parameters as
possible. Figure 13a shows the effect of the Amaldiâ€”

Selleri form factor on the n-exchange BTM cross-section

for ï¬‚p+p-p. The effect is to damp the amplitude at

37

larger values of t. This model compares favorably with
experiment for incident pion momenta from 3.0 to 8.0 or 9.0
GeV/c in the lab. The same information is given in Figure 13b

for the Regge propagator model.

 

 

 

 

 

â€˜0) \ a)

Figure 13. Comparison sketches of (a) BTMF and (b)
BTMR with experiment of the reaction np+pp. The dashed
curves are the experimental results.

%%(t=0) is matched to the theoretical calculation at t=0.

One can see that the results are favorable. Again,

the effect of this modification is to damp the Born
amplitude at larger values of t, which is the desired
effect. There is some arbitrariness in the pion tra-
jectory; since it is difficult to isolate experimentally

one determines the form from general principles.

 

 

38

For vector exchange reactions, Jackson and Pilkuhn

improved the BTM angular distributions with an exponential

form E(0)e->â€˜t (usually normalize form factors to l at the
2
exchange pole: F(t=m:)=l; E(0)=le Ame).

Figure 14a shows the effect of such a factor with

)\=2.5(GeV/c)'-2 on the n+n+wp reaction. We see an improve-

ment in the angular distribution. There is, of course, Eâ€œ
not any improvement in the energy dependence. We could Z
always add a factor S_n and adjust n for a best fit. The
Regge propagator result is shown in 14b. The trajectory
used in the calculation is dp(t)=.57+l.08t; the parameters
are from eXperiment. The energy dependence also consider-

ably improved.

 

 

 

 

 

(m (b)

Figure 14. Comparison sketches of (a) BTMF and (b)
BTMR with experiment for the reaction ï¬n+wp. The dashed
curves are the experimental results. â€˜

%%(t=0) is matched to the theoretical calculation at t=0.

Theory and experiment are on the same scale.

39

Neither of these modifications to the BTM affect
the density matrix predictions; no spineindependent
modifications can do so. To see this we look closely at

the definition of pmm" assuming a resonant state d:

*
= N2<A mIMll A ><A m'IMIA l > Trp=l
c a b c . a

p b

mm'

where <IMI> are the production helicity amplitudes. Eh

If M denotes the BTM amplitude and F a function, independent

 

of spin, then the form factor modified or reggeized pro-
pagator amplitude is MF. Since F factors we have

p = N|F|22<IM1><|MI>* = NiFIZA'l

and we see immediately that Trp =1 gives N=AIF|â€”2 and

p , is unaffected.
mm
Absorptive Corrections

By the same kind of argument, it is apparent that
the absorptive corrections will_modify the denSity matrix
elements because the effective function F is different
for each spin amplitude. However, since absorption
usually damps down contributions to the lower partial waves
much more than the high ones for each amplitude one expects

the modifications to 03$? to be small for most of the

elements--exactly the experimental situation.

40

For the wâ€”exchange reaction it is difficult to
distinguish between the absorptive and Regge calculations.
The results are not identical but they are close for
scale and angular distribution. The density matrix predicâ€”
tion of the absorption model is good, while the BTMR
calculation gives the same prediction as the BTM. The
situation is somewhat different for the p-exchange reaction,

n+n+wp. Again the absorptive model does well with angular

 

distribution and density matrix elements but the absolute

 

magnitude of the total cross-section prediction is too
high. In this regard, the Regge propagator form does

much better. Because the meager data we cannot say just
how much better; it is pOSSible that the prediction is
low. As in the p-case, this Regge model gives no improve-7
ment in the production density matrix prediction over

that of the BTM.

The absorptive corrections for these calculations
were obtained from phenomenological fits to experiment
according to the prescription of Jackson and Gottfried4
(1964). This approach is quite successful and is used
by most everyone dOing absorptive calculations. The idea
is too obtain from the latest experimental data a best
fit to the elastic scattering amplitude and invert it to
obtain the elastic phase shift which is then used to

correct the inelastic amplitude.

41

Recall the DWBA formula for the elastic scattering

amplitude:

_i 'or
fEL = -AfdÂ£ e 9 â€”V(Â£)x(Â£)

Substituting the high energy approximate expression
for x(u) we obtain

2
_' U. ' . _'A U |
fEL = -AjdÂ£ e lg Â£V(Â£)ela E lqif: V(b,z )

_',Z\_ l | - . _ -'
-iqudbd6dz%Â§ e 1 [:2 V(b,z )elÃ© g Aâ€”q q

(for 6<<1 A a = 251 so 9'5: 9'2)

-iA/qf dz'v(b,z')_l=

â€”CD

The integral over 2 gives e
elx<b)-l. After the integration over 0 we obtain the

Glauber formula.

_ _. r ix(b)_
fEL - qubde0( b)[e 1]
using fbdbjn(Ab)Jn(A'b) = %6(Aâ€”A')
we obtain
ix(b) _ g
e â€” 1+qudAJO(Ab)fEL(A)

The standard parameterization of the experimental

data on elastic scattering is a Gaussian distribution in A:

 

. 2
f = ing e-CA 0
EL 4n _ T _ l
C- â€œT
1C 2 Y"C C
= 2Y9 e-A /4Y

Substitution in the Glauber formula readily gives

: _ 2 2
e1x(b) = 1_Ce YJ /q

 

42

The parameters C,Y are directly related to experiâ€”
mental quantities. Usually they are assumed independent
of energy. One fits them to the available data at one
incident energy and calculates the inelastic cross-
sections over a range of energies and compares with
experiment.' The results are quite good for many reactions,
but the very best fits to the data require energy dependent
C and y.

A very similar record of successes and failure
appears in other NN channel reactions and in KN and NN
reactions. The success or failure of a model depends
more on the tensor character of the exchanged quanta
than on the nature of the final state (eg. one resonance,
two resonances, etc.--). To summarize:

l. The Born Term Model (BTM) while exhibiting some
nice features is inadequate. Looking at the experimental
data we intuitively conclude that we should try to modify
it in some way rather than abandon it alltogether. One
makes this decision because one-particle exchange is
relatively simple to calculate. Also, the density matrix
data show quite well the effects expected from one-particle

exchange.

Ill: elllll â€˜1

 

 

43

2. Simple phenomenological form factors F(t) can
provide agreement with the production angular distribution
data over a respectable range of energies-~particularly
for p-exchange. Sometimes (pâ€”exchange) one also gets good
oT(s) vs. s; for some reactions an additional factor F(s)
is necessary. No improvement in density matrix elements
is obtained.

3. The Regge form of the propagator works well
for production angular distributions and does much better
on energy dependence than the BTM, but fails on density
matrix elements. More sophisticated Regge calculations
can give pmm' predictions. Basically this approach is
quite like using a form factor.

4. The absorptive model performs extremely well
whenever reactions go via pseudoscalar exchange. It gives

good OT(S) vs. 5, good gE-vs. t and usually good absolute

dt
magnitude. Also, predictions on pmm. are good. Whenever
V-exchange dominates, the angular shape is good and density
matrix predictions are acceptable, but absolute magnitude

of cross-sections is too large and the energy dependence

prediction incorrect.

ANOTHER APPROACH TO ABSORPTIVE CORRECTIONS

One can try to be more ambitious in applying
absorptive corrections to inelastichTM amplitudes.
Instead of purely phenomenological fits to the elastic
scattering data, one can construct models for the
elastic amplitude based on physical ideas, hopefully
containing as few parameters as possible and use them
to generate absorptive corrections. If the model is
good we expect a fine fit to the elastic scattering and
to the inelastic processes with the same values for the
parameters. We consider one such model.

Recently S. Frautschi and B. Margolisu)proposed
a model of high energy elastic scattering based on the
identification of the Born term with Regge exchange.
The t-channel Regge exchange is assumed to generate the
sâ€”channel potential rather than the amplitude. They

start with the Born approximation to Glauber's formula:

B _ _iq 2 . ibÂ°q
fEL(A,q)- 2n fa b216(b)e

The physics enters here.. fgL is equated to the

leading Regge trajectory pole contribution

44

45

A =ceâ€œa(t)
pole

p=ln.Â§â€”- -iï¬‚/2, signature is neglected, d(t) is the
0

Pomeranchuk trajectory a(t):a(to)+a'(t~t0)

Ap=ceua(t)= -%%-fd2b2i6(b)eib q
Frautschi and Margolis then solve for 5(b) and sum the
Born Series to obtain an expression for the elastic amplitude,
an amplitude consisting of the pole term plus a series of
multiple scattering terms which can be likened to Regge
cuts. The one free parameter (the slope of the Pomeranchuk
is obtained elsewhere) is fit to experiment at some incident
energy and calculations are compared with experiment over
â€˜a range of energies with much success.

This model can be used to generate absorptive
-corrections, which can be applied to inelastic processes
in the spirit of the Jackson-Gottfried approach. Inverting
'the above equation we have
21500) = â€” ï¬-e'bZ/4a'â€œ,
where we have assumed

a (t)=1+d't.
p .

Using this expression we obtain the corresponding absorp-

tive correction function
2

. . _ -b /4d'u

e1x(b)=6216(b)=e E/Ue

Notice that

iX(b)|+l

Ie as s+w.

46

This absorption function has several features
noticably different from the Jackson-Gottfried approach.

1. The correction is a complex function. We can
expect the density matrix elements to be slightly different.

2. The correction is energy dependent and leads to
a decrease in the amount of absorption as the energy
increases. The effect is to soften the s-dependence of the
BTM exchange.

We have compared this method of generating absorp-
tive corrections with the curve fit approach for the
reaction pp+nA3/2++. This reaction was chosen since
Frautschi and Margolis have fit the parameter (5) in their
hadron scattering model for p elastic scattering. As is
usual, approximate equality of the absorptive correction
function in the incident and final channel has been
assumed in the absence of definitive NN* elastic scatter-
ing data.

Recall that the procedure is to replace the appropriate
BTM partial wave amplitude Mj with

th

~j â€˜ = . . . .
<ACAdIM Ilalb>DW. Ai helic1ty of 1 particle

J

The Jackson-Gottfried correction function has been

parameterized as

-2/q2 2 2/q3]

2 -Y2 J + â€˜Y 3
(DWj) =[l-c+e + ][l-c_e -

i refer to incident and final channels, respectively.

47

The parameters Ci,yi are derived from elastic pp scatterâ€”
ing data to be:
Ci=1 Y+=-24 GeV/c
y_=.10~GeV/c
In the model based-approach, the absorptive function

used is
2

V _ 2 , 2 , _ _ 2 .
(DW.)2= exp(-Â£: e J /4a uq+)exp(-Â§-â€”Ie J /4a uq_)
J U n
We used the same value for the slope of the
Pomeranchuk trajectory as Frautschi and Margolis:

'=o,82(GeV/c)-2. Also, their fits to the elastic pp data

2

at 20 GeV/c with this value of the slope and s =lGeV

0
yield E=7. Starting with â‚¬Â£=7 this parameter was varied
over a wide range to get a good fit to the pp+nA;;2.

Also, some calculations were done to determine the effect
of different 0' on the result.

The initial appeal of Frautschiâ€˜s elastic scatter-
ing model for absorptive corrections is the logarithmic
'energy dependence which shows up in the expression for the
phase shift. Absorptive model calculations a' la'
Jackson-Gottfried at several energies show that the amount
of absorption necessary to get a good phenomonological
fit to the data decreases slowly with energy. It was hOped

that the Frautschi parameterization of elastic scattering

might allow for this decrease.

48

92
dt'

particular energy there is trivial difference between the

It is possible to get a good fit to but at a
different approaches as shown in Figure 15. Differences in
the predicted density matrix elements are easier to see--
as a function of energy; the Frautschi model corrections
give pmm' which change more rapidly. Unfortunately there
will not be enough good data until the summer, at which
time comparison from 6-30 GeV/c should be possible.

There are other objections to this approach. From
a phenomenological point of View it would be better to use
the energy dependent corrections. But values of the
absorptive parameters necessary to fit the inelastic data
do not provide a satisfactory fit to the elastic scatter-
ing. Figure 16 compares a theoretical curve of the Frautschi
model, using the parameters that provide a best fit to
pp+nA with the elastic scattering. It is not clear that
the model is internally consistent.- Sums of t-channel
exchange diagrams can show a Regge behavior; the Frautschi
model probably "double counts" contributions to the amplitude.
Also, one can show that resonance behavior at low energy

can be deduced from a Regge amplitude at high energy.

 

Resonance is a prOperty of amplitudes not potentials. The
Frautschi model says that Regge exchange in one channel
generates potentials in the crossed channel rather than

amplitudes. This seems inconsistent with the other result.

49

We must conclude that the Frautschi model is
probably not correct. It-is-possible~to-generate absorp-
tive corrections containing a small energy dependence
(anyRegge-model of-elastic scattering can work) and improve,
if only slightly, the fit to experiment. We are being
drawn to simultaneous use of the Regge model and absorp-
tive corrections. The only obstacle is the question of
double counting--a question that can_be answered in favor

of combination of the models.

50

 

20-0 l I 1 T

FIG. l5 â€˜
ppâ€”'-nA3/2

pmcf 55 GeV/c

I60 7 â€”â€”â€” ABSORPTIVE - '-
JACKSON-
GOTTFRIED

â€”-â€” ABSORPTIVE-
FRAUTSCHI {=25

 

 

 

 

 

I2.0 - â€”- wREGGE - "
do --; TRAJECTORY
dâ€”Q: â€œ g at=aâ€˜(t-m%)
\Ã©
mb/sr 1â€˜:
8.0 â€” (g i
â€˜3
i 1rREGGIE Tauscroav
_. V: ' _
4.0 .\ w M
L.\\ m {=25
I. \\r 1
t : ï¬qâ€˜g1 "1â€˜ --â€˜ r-..
00 l Kn; ' .â€˜ 1â€œâ€œ '~--J
' .95 9 .85 .8
cosecm

Figure 15. Comparison of theory and experiment for
the reaction pp+nA3/2.

 

â€˜1â€˜
U
\
>
8â€™
5a _ I I I 'j I I
5 FIG. I6
3|; 0 pp ELASTIC SCATTERING
I92 GeV/C
IOZS _ â€” FM MODEL {=7 â€”
--- FM MODEL {=20
I626 =aÂ°+a'1=l+082I T
â€˜x :p- 1 .
|(527__ \ o- -41

 

 

 

111111 21
1 2 3 4 5 6 IIIIGeV/C)

Figure 16. Comparison of pp elastic scattering
with theory giving best fit to pp+nA3/2.

OTHER MECHANISMS
With the successes and failures of'one-meson-exchange
in mind, we focus now on vector meson exchange and investigate
improvements to OMB models. As an example-of a vector

exchange dominated process we continue examination of nn+wp.
Vertex Correction Mechanisms

The use of phenomenological form factors to improve
simple OME was based on the assumption-that-other.more
complex diagrams contribute to the scattering amplitude,
and that such factors (functions of t) improve the angular
distribution over the BTM prediction for both pseudoscalar
and vector meson exchange dominated reactions. We ask.if
it is possible to get a better clue as to the functional
form of such factors by looking at other diagrams.which
can contribute. Figure-l7 shows the diagrams for.the BTM

and lowest order corrections for n+n+wp.

1r\\\ u â€˜\][,Iâ€˜
I / \
n Iâ€™ â€œmy

 

Figure 17. Diagrams showing (a) the BTM and (b)+(d)
some vertex corrections to the NpN vertex for nn+wp.

52

53

In addition there can be graphs where the virtual
nucleons are replaced by nucleon resonances.

We concentrate on diagrams 17b as the next simple
cases after the BTM. Using the Feynman rules for forming
the S-matrix, we obtain the transition amplitude assuming
this diagram to be the dominant contribution (detailed.

calculations can be found in Appendix III. We obtain
f

 

_ 2 vvp 1
(AcAdIMIAb>nâ€” 2cgNngpV Mc m2_tâ‚¬uvpca epâ‚¬o(câ€™xc)
U(d.Ad)Y5AuY5U(b.Ab)
where
__ (3) - (3) _ (3)
ysAuysâ€” 213anu (Mp)+21YA[LAu (Mp) LAu (Mn)]
4 1 Â£
(n) d l A u
L (M)=[dsf â€”-â€”~ A =M -M
Aâ€œ (211)Z 122-22pz+Az]n â€œ9 n P

All of the terms inllare finite, if one allows a
liberal interpretation of the rules for manipulating
infinite quantities. It is interesting to note that if
we neglect the neutron-proton mass difference for internal
lines the amplitude vanishes.

After very involved algebra, one obtains a compact

expression for the helicity amplitudes

 

<ACAdIM|Ab>n = iK M2 K[yz]
-t
p
Here Acâ€™Ad'Ab refer to the helicity quantum number

of the w,p,n respectively. {Ad'kb} is shorthand for the

spinor product 6(p,x )u(n,l ), K contains coupling constants
d b

lo

and miscellaneous factors, and B comes from the rpm

vertex. ConSider first the energy dependence. Refering

54

to the BTM calculation one easily finds that the calculated
gg~sz. We saw earlier that this result conflicts with

the experimental evidence. Turning now to the vertex
correction contribution, B and {Adâ€™xb} go as s3/2 and s
respectively (k~so). The energy dependence of K[YZ] is

the only unknown. Should this factor, which comes from

the closed loop in the diagram be independent of energy,

we will obtain the same-energy dependence of %% as the
BTM.' This will turn out to be the case. To see this we

look more closely at the algebra. K[yz] can be expressed

as a sum of three terms:

1 2M 2 2 2
K[yz],=>K [yz] + â€”9-VK'[y z ] + 2K"[yz ]
1 Anp l 1
Where the first term comes out of L(3)(Mn) and the others

from 2iY1[L(3)(Mn )- -L(3)(M )]. Rewriting in terms of the
fourth order Feynman parameter functions Fi(x,z) defined

in Appendix III we have
2 l-

 

 

K [yz] = dz F (l,z)M
l (2â€œ) I 3 n
2 2 2 l
Ki[y z ] = 73â€”;1-Â£ dz z[F5(l,z)Mn-F5(l,z)M ]
n
l
KinzZ] = 4 f dz 2 [F3(1,z)Mn-F3(l,z)M ]
(2N) Â°

This extra algebra proves very convenient; these
functions Fi(x,z) characteristic of fourth order diagrams
in the nN channel.

Again refering to Appendix V , examination of the
functioneri(X,Z) shows clearly that all s-dependence

vanishes at x=1: Fi(1,z) are independent of s.

54

.03+c= coï¬‚uommu ecu How u mo mmcmn H00ï¬m>nm onÂ» Hm>o HN>Hx
coï¬‚uocsw mnnuosuum mnu 00 when amen one .ma gunman

 

 

Eommoo
0 _- 0.0- 0.0 no 0.. .
mud
IIIIRIIIIII .I 0..
N. 11111111111111/1/11/
l/l/
Ilhï¬‚q
//
o .
1 Â¢Lov
5
s
1 gum , Qm
. a . n.2â€”
o\>u0 mmv mun- a i
9901:...Â»
I m. .9... 8
_ _ _ o.Â»

 

 

55

Figure 18 shows ReK[yz] for several values of
incident pion momentum over the physical range of t.

(ImK 10"3

ReK) Notice the similarity in the curves for
different energy and the forward peak. Since the scattering
amplitude is prOportional to wnK[yz] we can understand I
the peaking of %% at non-forward t. Assuming that these
vertex correction type graphs are the dominant production
'mechanism for wn+wp the cross section was calculated,
and is shown in-Figure 19. The angular distribution is
not bad and the-absolute magnitude is more in_line with
experiment.~ Keep in mind that no absorptive corrections
have been added. Addition of such corrections would shift
the peak more toward the forward direction, but also would
'damp the absolute magnitude far to much. For comparison
'Figure'l9 also shows the result of an absorptive BTM
calculation at the same energy.. Also, these graphs cannot
improve the fit to the w production density matrix elements;
indeed the prediction is the-same as the BTM.

'- Another approach would be to interpret these fourth

order corrections in a strictly field theoretic sense as

next order corrections to the BTM and assume

\
\(
<AcAd|M|Ab>n = BTM + â€™\ + -

/â€”\ '

55

.ds+c: MOM gamma coï¬‚uomuuoo xmuum>

may H00 unmeï¬‚ummxw 0cm muomru mo QOmAHmmEou .ma musmam

Eommoo

 

 

 

:1Â»
m. 0.â€œ.

02.8% as

 

_ _ _ _ _

 

 

 

 

/ \ 4N.

 

 

ESE
at

56

Here we can have interference and reference to the
calculational details for the separate terms shows that
the corrections to the BTM helicity amplitudes are such
that we could obtain a prediction for the denity matrix
elements. However, absorptive BTM gives us this already
and there is still the problem of energy dependence.
There are, of course, still the other fourth order
contributions of this type, diagrams 176: for example, as
well as others. These other graphs are plagued by severe
divergence problems and the non-divergent terms show no
improvement in the energy dependence. The conclusion is
that vertex corrections are not the answer to the energy
dependence problem, so one must try other things.

There is currently much interest in the relation-
ship between absorptive corrections and Regge poles; most
of it devoted to justifying simultaneous use of the two
models. The idea is to write a Regge form for the inelastic-
interactions and apply absorptive corrections.

We will see that such calculations can be successful.

There is,however, one other alternative to be considered.
Two-Pion Exchange

We abandon the BTM and guess, perhaps, that only
pions can be exchanged in the interaction. Thetnresonance
is assumed to be produced by sequential pickup of two
virtual pions. The simplest relevant Feynman graphs are
shown in Figure 20. Note that this time we have 6

contributing graphs.

 

 

 

Figure 20. Simple two pion exchange diagrams
for the reaction ï¬‚n+wp.

As before, using the usual rules, one obtains expressions
for the-matrix elements:

<A11'|MI [11>n =2/2'G 2M:â‚¬ elvpo aAq a; (c, A)u(d,uâ€˜ )YSA: y5u(b,u) q=b-d

 

(All. IMIï¬lJ>n= _________________________________ Ag]:
* _

<Au'IMIII|u>n= ----------- axrvso(d,l)u(c,u')Y5A iIIY5u(b,u)r=b- c

with
4
I â€˜ d e e
o no I _ .
(2") [e2 +M 2+M][(a+e)2 :I[(q-e)2+Mi][(b-e)Â§+M2]

+ other terms and obtain II by the replacement a+-c in AI
and III by the replacement q+r in AI

It is necessary to be very careful when making
approximations to be sure the dominant contributions are
'retained. We assume, however, that the contribution from
III can be neglected relative tO'I and II.

Reference to Appendix IV shows that we.can write
(using the same notation as for the vertex-correction

â€˜calculation)

- AI -11 14> .- <4) _ (4)
Y5 st- nPLuv (Mp)+1YX[LW(Mn) LW (Mp)]

p o p___
L(n) (M)=fds"9Â§ 4 p u â€˜
(2â€œ) [e2 -Zepz+pz]n

 

58

A seemingly endless number of algebraic manipula-
tions leads to a compact form for the scattering amplitudes.

We obtain
. I A I < Wu'u A
<Au'IT Iu>n=iKB {u',u}I[yz]-1KB Eâ€”â€”â€”I3-KMu,uJ/Anp I=Il-I2
np

Similar expressions can be written for the TII and

TIII terms. Here K contains coupling constants, and
miscellaneous factors of- and has the same definition as
in the vertex correction amplitude.

The other factors are

"A i . .-
B =â€”qq' Slne 6 w . =1U(d,u')#U(b,u)
/2 cm A,il u u
_ A â€” , A
{min} = U(d,u')U(b,u) Mu'u=U(d'â€œ )N U(b,u)

N=â‚¬Avpo aAqvâ‚¬Â§(C'A)

Rewriting in terms of the fourth order parameter
functions Fi(x,z) for convenience in investigating the

energy dependence, we have.
I - "2
1.(2")4

 

[dzdx22(l-z)Fi(x,Z)Mp

M
4 fdzdx23(l-z){[Fl(x,z)+Zâ€”E F2(X,Z)]n '
(2N) M np

[F1<x.z>+Zâ€”B- F2(x,z)1Mp}
nP

_ n
I2â€”

 

2
2 2
13=73373 Idzdxz (l-z) (1â€˜X)[Fl(xrz)Mnâ€”F1(Xâ€™Z)Mp]

2
w
J +7;;TI f dzdx(1-z)z[F4(x,z)Mn-F4(x,z)Mp]

The energy dependence of the F's is discussed in

the Appendix, as is that of the other factors. We find that

F sâ€œ2 and F ~sâ€™1 so I, I I ~sâ€™2 and J~S-l.

Fl' 2 4 2' 3

59

Similarly, examination of the expressions for Wu.

and M3,â€œ gives S1 and 83/2

dependence. (Mu.u~S0 ). Therefore we can-write

,-respectively, for the energy

I
T = al+a2%Â§'
=Squaring, etc. leads to an expression for %%~-~

I
_ ' b C 0 ' 0 1
ENT'- a+â€”â€”' E const. (one can obtain Similar resuits

ï¬‚?
for-I and III).

This result is obviously much more pleasing that
the BTM and its-simple vertex corrections;w there is at
least some~reason to be optimistic about-ZW-exchange.
The t-behavior can be seen easily in Figure 21, which
shows ReIl(I) and ImIlâ€˜I) as a function of CM scattering
angle for incident pion momenta of 3.25GeV/c. Note the
predominant forward peak and absence of any backward
â€˜peak. It should be noted that-the peak in the cross section
will be shifted away from the forward direction because
of the influence of angular factors in the kinematics.

Figure 22 shows the same information with the T(II)

contribution added, eg. Re(Il-Tl) and Im(Il-f1)' The T(II)
contribution is about 1/30 of Tâ€˜I) in magnitude. It is

clear that T(II)

could have been neglected.

The term proportional to J (J looks like I but is
of Opposite-sign) provides a small correction which
additively effects the helicity amplitudes and can there-

fore give a non-BTM density matrix-prediction, but at-

these energies this correction is not noticable. Figure_23

60

.o\>mw mm.m um m3+c= coï¬‚uommu wnu How u

mo mmcmu Hmoï¬‚mxcm may um>oAHWH EH pew AHWH cm .HN musmï¬‚m

QT

 

 

 

 

60080
M? 9 i. NV no â€œw t a m. q_
_ _ _ _ _ _ _ _ _
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I ,_o
i
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i
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_
o\>momm.m .2 H
a3 :8 c +k
I am: Ind
_ _ _ p _ _ _ _ _

 

 

61

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mmcmu HMUHm>Â£m may Hm>o AHMIHHVEH can AHMIHvam .NN musmï¬‚m
E0080
0..- 0.. 0.. Â«V N.. 0.0 N0 Â«.0 0.0 0.0 0._
_ _ _ _ _ _ _ _ _
â€œHunnHHHthHHHHHMHHHhMhHHHHHHHHHHHHHHMHHHHHHMMHWlulfl:lll Ilnru
:III///._mm/,
// /
.5. x J
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$88.Â» 2 Z

aaakmv c+h
NN.0_.._

_ _ _ _ . _ _ _ _ _

 

.Emï¬cmnomï¬ ucmcHEOU may mm mmcmsoxmipm
#0

mcHESmmm ms+c: coï¬‚uoomu onâ€œ How u .m> op

. mm 83m:

800.80

 

62

 

 

 

 

535
8
l O.â€”
o\>mo mNM QSAIctF N._

mN .GE
_ _ _ _ _ L â€” _ _ r Â¢._

 

 

63

shows the angular distribution assuming 2â€ exchange

for the production mechanism. Absorptive corrections have
been included, the result is acceptable. The conclusion
isthat~2Tr exchange can work and give an acceptable energy
dependence.v Cursory examination of final states containing
higher spin resonant states shows that 2N exchange will
break down, ie. predict an incorrect energy dependence,

but that nTr exchange for some apprOpriate n will restore

the prediction to an acceptable value. It is interesting
that sums of exchange graphs can give Regge energy behaVior.â€˜
Also absorption is clearly a different physical effect

from multiparticle exchange. The argument that the physical
effects behind-Regge exchange and absorption are distinct
has-recently been-given in detail by Marc Ross, Frank Henyey,

and-GordonKane.ll

ABSORPTIVE REGGE MODEL

We accept the qualitative arguments against double
counting and apply absorptive corrections to the Regge
exchange amplitudes for the reaction np+pn assuming.
dominance of the 1r,p,A2 trajectories. This reaction is
interesting because of the availability of experimental
data for the cross section and an experiment now in
progress to measure the final neutron asymmetry, ie. a
polarization. If the cross section for this reaction
(and related reactions) can be fit in a consistent manner
then one obtains a prediction for the polarizations.

Regge poles have definite spin and paritylzand the
Regge model tells us that the trajectories with the largest
a(t) will dominate the amplitude at high energy. The
conservation laws tell us that the exchange quanta must
be members of non-strange isotopic Spin triplets of
mesons; 1r,p,A2 are acceptable candidates. We must determine
how these trajectories contribute to various t-channel
helicity amplitudes.

Expanding in t-channel partial waves we have

<A2 A 4t|T(t,e )|AA nt=2(2j+1)<A2A4|T1(t)|A1A3>ndi (6t)

13> j nt tut

64

65

At=A1'A3

utzxzâ€™M

n = â€”
At ut

t

In np charge exchange A =A2=A3=A4=l/2, giving a

1
total of 16 helicity amplitudes. Application of parity
conservation and time reversal invariance reduces this
number to 6. Further, strong interactions are I-spin
invariant so we neglect the n-p mass difference and
kinematically we have identical particle scattering which

gives one more relation and so reduces the number of

independent helicity amplitudes to these five:

Â¢Â§=<l/2 1/2lTll/2 i/2>O
Â¢Â§=<1/2 1/2|T|-1/2-1/2>O
Â¢Â§=<l/2-l/2|Tll/2-1/2>O
Â¢Â§=<1/2-1/2|T|-1/2 1/2>2

Â¢Â§=<1/2 1/2|T|1/2-1/2>l

To determine the Regge contribution to each amplitude
we write the amplitudes as linear combinations of parity
conserving amplitudes (Regge poles have definite parity),
expand these into partial waves (definite spin assoc.
with each trajectory) and perform the Sommerfield-Watson
transform to obtain the appropriate Regge forms. After
removing kinematic singularities we obtain (after long

and tedious algebra)

66

t_ 1+ i- ai
Â¢1â€”1/2E(BOO+BOO)Ci(S/SO)

t_ i+_ i- ai
Â¢2-1/2Â§(800 BOO)Si(s/so)

t_ Z i+ i- ai
Â¢3-l/2iqi(811+Bll)si(s/SO)

t__ 2 i+_ i- ai
Â¢4â€” l/ziai(Bll Bll)si(S/SO)

t- - 1+ ai-l
Â¢5â€”1/251netÂ§8108i(S/SO)

th

where Ci is the usual signature factor, ai(t) is the i

'+
Regge trajectory, and the 8%; are the residue functions,

natural

(the i refer to (unnatural)

parity).

We see immediately that the p and A2, both natural
parity particles, can contribute to all of the amplitudes.â€™
The n, unnatural parity, contributes only to 836 and

twpole_

therefore only to Â¢Â§ and Â¢Â§ such that Â¢l --Â¢;ï¬‚pÂ°1e.

With faith that our Â¢Â§ contain only dynamical singularities

13
we apply crossing to obtain the s-channel amplitudes:

s_ 1/2 1/2 _ 1/2 ,_
(Acxle'AaAb> â€”XdA' A (Xt)dA' A (" Xt)dA' A (" Xt)
a a b b c c
1/2 I I I | t
dA' A (Xt)<*c*a|T'Ad*b>
d d
where cosxt=[s/(s-4M2)]]'/2[t/(t--4M2)]1/2

After much calculating we obtain

67

2
s_ . 2 t t t t t _l+cos
Â¢1â€”l/25in xt[Â¢l+Â¢2+Â¢3+aÂ¢4+4bÂ¢5] aâ€” .
Sln x
s_ . 2 t_ t t_ t t
Â¢2â€˜1/251n Xt[Â¢l aÂ¢2+Â¢3 Â¢4+4bÂ¢sl b=cotx

t
s

Â¢Â§=l/Zsin2xt[Â¢Â§+Â¢Â§-aÂ¢Â§-Â¢:+4bÂ¢ ]

t t t t

Â¢j=1/2sin2xttaÂ¢1-Â¢2-Â¢3+Â¢4-4bÂ¢Â§1

t t' t t 2 t
l+Â¢2+Â¢3-Â¢4)+2(l-b )Â¢S]

s_ . 2

Â¢5-l/281n xtl-b(Â¢
Reference to the expressions for Â¢Â§ shows that;

only Â¢Â§ and Â¢2 contains the-n pole. One readily obtains

snpole= supole

Â¢2 Â¢4 =1/2'Ã©'"sï¬‚(s/so>Â°â€˜Tr

Now, Â¢Z is an n=2 amplitude, ie. net helicity transfer
from the incident to the-final state-is 2 units of angular
momentum, and must therefore vanish like (-t) as t+0.

Therefore we conclude that
Bâ€=<-t)8"

and the n pole cannot possible contribute to the experi-
mentally observed forward peak.' In Figure 24a we see the

â€pole amplitude as a function of t.

68

190â€œ do.
Â¢ 5;
\mb T

 

 

: ea _
N no; -t 00â€˜1 t

Figure 24. Sketches of (a)n pole amplitude and (b)
experimental t angular distribution for np pn at 8 GeV/c.
Since the n pole is nearest to the physical region it.is
expected-to be the dominant contribution to the cross;i-
section. The conclusion is that the Regge pole~model;-
predicts a dip in the forward direction in np charge
exchange. (Figure 24b shows the shape of the experimental
do/dt.) .

rThere is-a way around thisvdifficulty}4

Until-now_rw
the solution has been to assume the existence of a parity
doublet for the n. We require a second n trajectory,wd,

of positive parity which-conspires with the n such that

 

d
Â¢2+ï¬‚ =0, as required by conservation of angular-momentum,
without the necessity for Â¢g+ï¬‚ =0.

The Nd contributes to Â¢Â§ and $2 with opposite sign
so we obtain
5 + d _ ï¬‚ _ d d
Â¢2" " =1/ZBNSN(S/SO)Q +1/2eTr swim/so)OUT

d n d d
Sï¬+ï¬‚ _ -N a _ â€”ï¬ an -
Â¢4 â€”1/28 sï¬(S/SO) 1/28 Snd(s/so) s â€”Snd

69

Now angular momentum conservation gives

d
Â¢iï¬‚+ï¬‚ (t=0)=0 (the p and A2 contributions
vanish at t=0)

+ Â§"<t=0)=Â§"d(t=o>

dn(t=0)=dï¬‚d(t=0)

and we see that Â¢Â§ï¬‚+ï¬‚d need not vanish at t=0 and the
existence of the n pole at t=MÂ§ZO insures a large contri-
bution in the forward direction.

It is true that the p, A2 contributions to Â¢f,
and Â¢Â§ do not have to vanish in the forward direction as
indicated in Figure 25. However, the amplitudes are rather
smooth and flat, slowly decreasing, functions of t. When

â€˜the w pole is added and the other amplitudes

 

 

 

 

um (a
0â€˜1. 9' A7.
â€˜23 wmro
Ln: 6}
.5; 't .63 â€˜t

Figure 25. Sketches of p and A contributions to
(a) n=0 and (b) n+0 amplitudes in the geaction np+pn.

Etre considered the combined effect is to produce a dip

iri the forward cross section; the presence of the Nd

tJrajectory remedies this defect.

70

One problem with the conspiracy solution is

that the Nd is not observed so its introduction becomes
a non-physical extension of the model. We will seecthat
application of absorptive corrections introduces-cuts,
associated with each pole, each containing positive:andâ€™
negative parity parts, and that the behavior of the;n,
cut in the forward direction is effectively that of:the-
conspirator so we explain the data in a physical-way.;

We look at the absorptive formula-derived earlier:

A _ B
(ACAdIT |)â€˜a)â€˜b>n-<)\c>â€˜le l)â€˜a>â€˜b>n

. z ,, j
+iNj(23+l)dAu(6){<AcA L|Al A 2><A 1A2|T1j3 |AA

dlï¬ a b>

+<ACAd|TÂ§ |A1A2><A1A2|TÂ§L|Aakb>} -----
where, as is usual, the absorptive interactionsin the

incident and final channels are-assumed equal. Using of the

Jacob-Wick expansion and the elastic scattering parameterâ€”

ization
L 2 . At/Z -
<A A IT |A A >=-4q (i+p)0 p a 6 where
c d a b T ACAa Adlb
0T=total cross section and p=ReTEL/ImTEL are

experimentally determined as is A.

one obtains

At/ZIO At'/2

ip _

00

A . .
<AaAb|T |AaAb>n =<AC A lAaA >n+K dt 9 In(A/tt )

d|"B b

 

B . __ OT _.
<ACAd|T |AaAb>n With Klâ€” 4nA(l ip).

71

The Born pole term or Regge pole term TB enters.
in the second term which is a superposition of several.
poles, ie. a cut.- Thus to each pole there is associated
a cut generated by the absorptive correction to the ;.
amplitude. The cut is not a parity eigenstate and so.
contains both natural and unnatural parity parts.. This.
means that we can have interesting polarization effects,-
"self-conspiracy" etc. The important point is-that the
cut term could, and so we-shall see does, have the same.
- effect as the introduction of conspiring-trajectories. eg.,

np charge-exchange: the Â¢2Â°ut

need not vanish at t=0.

We fix the n-residue by equating, in the high energy
limit, the Born term to the Regge contribution at the n
pole. With n pole + n-cut most of the very forward.

cross section is reproduced. Figure 26 shows %%

vs t at 8 GeV/c with the experimental data}5 The same
procedure of equating Regge form and Born term at the pole
for the p meson is used. First we look-at nN charge-
exchange and use factorization of residues to determine

the ratio of non-flip to flip p residues. Extrapolation.

to the Born term of np charge exchange then fixes the.

, magnitudes and signs of the residues in a consistent manner
in terms of the vector and tensor coupling constants of
vector mesons to baryons. (np~charge exchange Born term

is enough by itself but nN charge exchange allows separation

'of residues into vertex parts and provides a consistency

check).

72

Atacamcm .Hamzhmm

 

 

 

.m.m.m.4 an no mcï¬‚ccmz .0 EOHM mum mummy .o\>w0 m um chmc How u .m> mm .mm musmï¬‚m
:80 moo
9v. OÂ¢. mm. Om. ON. ON. 0.. 0â€”. no. _.
_ _ IL _ i i _ _ _
+ / 1N.
/
I I /
/ 1?.
/ no.
/ um.
// IÂ».
/ 1m.
/ lm.
/
xx 10..
//J l _._
o m , .
Rapedâ€ ell/.1. N _
8.0... IÂ».
1V;
In;

 

 

 

To:

 

liaalllâ€˜! Ilv..l"l|l I I! III III- Iâ€˜lâ€˜llllllâ€˜ â€˜1

I'll-I. â€™llahlul Illu|u Iii! II III

73

We set the relative sign of the p and A2 by requiring
the (p+A2) contribution to an amplitude by predominantly .,

real. (The p and A are treated as exchange degenerate._

2
Experimentally %%(np+pn) holds its shape to low energies,
ie. no resonance formation). Since the signs of the n
and p are fixed by the Born terms, all signs are

determined absolutely.

[The procedure was to fix n pole + n cut~to reproduce
99.
dt 2
the overall t-distribution by varying the values of G

the break in and add p and A ~to get a best fit.to~

V .

and GT for the p and A within physically acceptable...

2

limits. The procedure uniquely fixes the polarizations. .
The experiment now in progress measures the recoil

neutron asymmetry by using a polarized proton target.

~A simple calculation determines this polarization in terms

of the Â¢i to be

do

| 2 â€”
dt

_ s-* s s s_ s â€”l_ s s 2 s 2 s 2
Aâ€”2NImÂ¢ 5<Â¢l+Â¢2+Â¢3 $4) N â€”[IÂ¢l +IÂ¢2| +|Â¢4| +|Â¢51 ]a
A close look at the contributions of the various poles

and cuts to the expression shows
s~ _ s
10 Â¢5â€”/-EÂ¢5
2. Â¢i+Â¢Â§+Â¢Â§â€”Â¢i does not contain the n pole

We expect the w cut to dominate at small t because.
of the proximity of the n pole to the physical region,
and therefore attribute the sharp forward peak to this
contribution. We suspect that gE-A/Jtï¬‚will be relatively
flat. Calculations clearly support these conjectures.

Figure 27 shows the results of the calculations.

74

There is one last point to be examined.- Analysis Of
the reaction, pp+nn shows that it has the same t-channel
as np charge exchange, and therefore must be dominated-
by the same poles. Application of the G-parity operator
to the two reactions shows that we obtain the pp+nn.
amplitudes by changing the sign-of the p contribution to.
the charge exchange. Consistency-requires a good fit
to pÂ§+nn with the same parameters. - Figure 28-compares
the theoretical fit to the experimental data for ppenï¬
at 7 GeV/c176 The fit is quite good so we conclude that
this model is a good one.

While other approaches to this problem of np-charge-
exchange can be successful,-we believe-that-absorptive
corrections to Regge pole amplitudes has strong appeal
from a physical point of view. -It is certainly a more
satisfying explanation than introduction of spurious
trajectories and arbitrary vanishing of residues to
satisfy experiment. We have faith that attempts to
fit other high-energy-inelastic-reactions with this model

will prove successful. Only time will tell.

75

.Ammm immmac mm .upmq

.mmzm .Hm um husngmd .m EOHM mum mummy .0\>m0 n um MG+MQ How p .m>

vo. meo. 90. 600.

Pm .
00 mm muswï¬‚m

 

:80 moo
mw_. e;. mWï¬ ï¬ve Mwnv mwnv
_ _ _ _ _ _
II: .
Â«III, .

+li+rll+lthII*I.+ +

o\>oom

amloc

NMw guru

 

â€” Â¢\

_

_

 

l
QQNâ€˜QOTON-to

 

 

10.

ll.

12.

REFERENCES

M. Jacob and G. C. Wick, Annals of Phys. l,(l959)404.

R. J. Glauber, Lect. in Theo. Phys. Vol. 1, Boulder

1958, Interscience Publishers.

J. D. Jackson, J. T. Donohue, K. Gottfried, R. Keyser,

B. E. Y. Svensson, Phys. Rev. 139(1965)B428.
J. D. Jackson, Rev. Mod. Phys. Â§1f1965)484.

J. D. Jackson and H. Pilkuhn, Nuovo Cimento, Series X,

33(1964)906.

A. Amaldi and F. Selleri, Nuovo Cimento ï¬l,(l964)360.

S. C. Frautschi, M. Gell-Mann, F. Zachariasen, Phys.

Rev. l26(l962)2204.
B. Margolis and A. Rotsstein, Preprint McGill Univ. 1968.
D. Griffiths, R. J. Jabbur, Preprint Wayne State Univ. 1968.
S. Frautschi, B. Margolis, CERN Preprint Th.909, 1968.

F. Henyey, G. L. Kane, Jon Pumplin, M. H. Ross, Univ.

of Mich. Preprint, 1969.

M. Gell-Mann, M. L. Goldberger, F. B. Low, E. Marx,

F. Zachariasen, Phys. Rev. l33(l964)Bl45.

76

13.

14.

15.

16.

77

References (continued)

T. L. Trueman and G. C. Wick, Annals of Physics

26(1964) 322.

L. Bertocchi, Heidelberg International Conference on

Elementary Particles, North-Holland Publishers, 1968.

J. V. Allaby, F. Binon, A. N. Diddens, et al., Submitted

to XIII Int. High Energy Conf., Vienna 1968.

E. W. Anderson, E. J. Bleser, et. al., Submitted to

Phys. Rev. Letts.

APPENDIX

APPENDIX I

Notation and Kinematics

1. For the diagram shown in Figure 28,a4b,c,d,e are
the 4-momenta of the particles of mass Ma' Mbâ€™ Mcâ€™ Mdâ€™
Me' respectively. The metric is such that scalar products

are written ab= -a b +a-b = a4b +aÃ©b.

 

 

 

o o â€”-â€” 4 â€” â€”

a c
e

b d.

Figure 28. Feynman diagram of OPE contribution

to ab+cd scattering.

2. Whenever convenient we use the Mandelstam variables

s,t,u given by

s = -(a+b)2 = -(c+d)2
t = -(b-d)2 = -e2 = -(c-a)2
u = -(aâ€”d)2 = -(c-b)2.

These variables are not independent; energy-momentum

conservation leads to s+t+u = M2+Mg+M2+M2
a c d

3. Figure 29 shows the orientations of the 4-momenta
relative to the coordinate axes in the s-channel center-

of-mass system

78

79

 

h
II.)
II
I70
3â€˜
IÂ»)

I;
,c
I2

Figure 29. The s-channel center-of-mass system for

ab+cd scattering

With this orientation, the 4-vectors for the momenta take

the form

a = (a0,o,o,-q) c (c0,-q'sin8,o,-q'cose)

b = (b0,o,o,q) d = (d0,q'sin6,o,q'cos6)

4. It is convenient to define the complete symmetric

function A(x,y,z):
A(x,y,z) = x2+y2+22â€”2xy-2xz-2yz.

Using this function we obtain
1

q= 27$ Al/2(s,M:, Mg) = incident channel CM momentum

l
Q'= 278 Al/2(slM:,MÂ§) = final channel CM momentum

1
ac: 2MC Al/2(t,M:,M:) = magnitude of 3-momentum of a in

0 rest frame

â€”lâ€”- 1/2 2 2 ;

bd= ZMd A (t,Mb,Md) = magnitude of 3-momentum of b in

d rest frame

Other useful kinematic quantities are easily cast into

this compact form.

 

 

I

80

5. We write the Dirac equation for spin-l/2 particles
as (iy-x+M)U=0 and normalize the spinors as UU=2M.- The
y-matrices are defined such that y+=y and y5=yly2y3y4=
1Y17273Y0â€˜ The quantity cu =-1[Yu,Y ] is also used
frequently. Using these 7's and referring to the CM-

coordinates we obtain the following positive energy

 

 

 

 

 

 

 

spinors: (U(X,Ax)=pos. E. spinor of 4-momentum x and
. . = _ 1/2
heliCity Ax) N(x) (x0 Mx)
l l l l
U(b,l/2) = N(d}
0 ENUD) g =N(b'bÂ£+â€”â€”xl 2
b + 0 O â€™b /
Jï¬b t 0 Mb
0 Mb
0
l 0 l
U(b.-l/2) = N(b _q = N(b)_ q x_1/2
b0+Mb 1 Lb +Mb
l
U(d,l/2) = N(d) _q (xl/zcose/Z + x_l/zsin6/2)
dO+Md

U(d,-l/2) = N(d - (X_l/2C059/2â€˜X1/251n9/2)

 

l
U(a 1/2) = N(a X

81

l
U(a.-l/2) = N(a)- X
a +M 1/2
0 a
l
U(c,l/Z) = N(c â€” ' (x cosB/Zâ€”x Sine/2)
c +M â€”l/2 1/2
0 c
l -
U(cl-l/Z) = N(c - ' (xl/Zcose/2+x_l/2sin8/2)
CO+M

We use these spinors to calculate useful scalar products:

(a) (Ad.Ab) = U(d,Ad)y5U(b.Ab)

(l/2,l/2) -(-l/2,-l/2) = -c-cosa/2

(1/271/2)

(-1/2,l/2) = C+Sln9/2 = Q+w/2

(b) {Ad'kb} = U(drxd)U(br>â€˜b)

{1/2,1/2} =I-1/2,â€”1/2} =w_cose/2

{l/2,-1/2} =-{-l/2,l/2} = Q+Sln6/2=w+U/2

where

 

1/2( 3 q
bO+Mb d0+Md

Â§:=[(bO+Mb)(dO+M )

dII

_ 1/2 r q Q'
Â¢ '-[(b +' .)(d 444 )1 (l. )
t 0 Mb 0 d b0+Mb dO+Md

 

Similar quantities c1,Â¢' can be defined for scalar

products involving spin 1/2 particles a,c

82

(c) The following products are also useful:

1) wxdxb= iU(d,Ad)Â¢U(b,Ab) W++=W__=(a0w+-qg+)
W+_=-W_+=(-a0w_+qc_)w/2

AC _ A A *
2) M =U(d A )N CU(b A ) with N C=â‚¬ a e e (c A )
Adxb ' d â€™ b n Avuo A o â€™ c

The 4-vector Eu is a spin 1 polarization vector.
It is discussed in Section 6.

l l
_. __ I _ : I " l,
M++â€”M__â€”/Â§-C+(qc0+q a0)w 72 C+an w7Â§w+qq w

0 O 2
M++â€” M__ -Â§Â§+qMcw
l
-l_ l _ _l â€” ,
M..-M-- - /3Â¢.qcowvzw.qq w
1 -l l 2 l 2 -l
M =-M =/2C ( c - 'a )--â€”â€”-C a q'w +â€”â€”â€”w qq'w =-M
+â€˜ â€˜+ â€˜q0q02f2"0 2J2" â€˜+
O _ 0 __
M+_-M_+â€” qMO;_w
MIE=-ME+=-i- c-qcow2+-lâ€”I_qq'w2
2/2â€˜ 2/2

All of these expressions are small angle approximations;

only the lowest powers of ware retained.

6. Specification of spinâ€”l states is in terms of polar-
ization 4â€”vectors â‚¬U(c,AC) subject to the subsidiary

condition CÂ°â‚¬=0 and defined such that

â‚¬u(C,X)â‚¬u(C,X )zÃ©AA' â‚¬u(c,A)ev(c,A)=6uV+â€”Â§â€”â€”

83

Refering to the CM coordinates, if
(a) particle c is spin 1

e (c,+l)=â€”];[o,coÂ§,-i,-Sin6]

â‚¬(c,o) =

ozle h

[q',-cosin6,o,â€”cose]

e(c,-l)=â€”l[o,cose,â€”i,-sin8]
.72â€”

(b) particle d is spin 1

E(d,+l)=â€”l[O,-COSG,â€˜l,Sln8]
J2

â‚¬(d,o) =ï¬i[q',dosin6,o,d0cose]

Â£1.

e(d,-l)=â€”l[o,cose,-i,â€”sin8]
f2â€”

7. For particles of spin greater than 1, we use the Rarita-
Schwinger formalism and construct high spin wave functions
by the angular momentum addition theorem subject to
subsidiary conditions:

Example: spin 3/2

Up(d,Ad)=<1A1/2AI3/2Ad>â‚¬u(d,A)U(d,A)

7 U =0 d U_=0
IJIJ UH

Expanding, we obtain
Uu(dâ€™3/2)=â‚¬u(d'+l)U(dll/2)

Uu(d,i/2)=â€”_l_e d,+l)U(d,-l/2)+/Â§eu(d,o)U(d,l/2)

Â«3 â€œ(

The other two spinors can be obtained by changing the Signs

of the helicities.

84

 

8. We rewrite the propagator % -to make the dependence

M -t
e

on w=25ine/2 more transparent:

2 _ 2 _ 2__ _ 2 _ . 2 , 2 2
Me tâ€”Me+(b d) â€” (bO do) +(q q ) +qq w +Me

now
S=(a+b)2=(a0+b0)2
so write

2
(bO-d0)2= [2(b0-d0)(a0+b0)]

Ere brd 2T:

so[(b -dO )(a H+b0+c0+d )]2 using a0+b0=co+d0

S-[(b d) (b+d)+(c- a) (c+a)]2

2 2 2 2 2
s[(Md-Mb)+(MC-Ma)]

.3).

which leads to

M2- -t= M2 e+(q- q ')2 MMÂ§)+(M -M: )] 2+qq' w2=qq' â‚¬2+qq' 'w2

which defines Â£2

â€”1_ 1 1
I

qq â‚¬2+w2
absorption calculations of OPE diagrams.

 

and gives us [Mi-t] which we use frequently in

One can procede in the same way for u-channel exchanges

to obtain

 

 

2_ , 2 2
Me U qq â‚¬u+wu
with

, 2_ 2 _ , 2_l 2_ 2 2_ 2 2
qq â‚¬u-Me+(q q ) EÂ§[(Md Ma)+(Mb MCI]

85

9. For cross section calculations we use
0 4 - 1 I
S=l-i(2n) 6(pf-p.)[n(2E.)] M as our S-matrix
l. i l

and obtain

do_ 1 g; l 2

 

z |<Acxdlmlxaxb>|
(28a+1)(28b+l) spin

and

 

APPENDIX II

Calculation Details: OMB and Modifications for n+n+wp

In the center-of-mass we arrange the coordinate system

as shown in Figure 30.

91")

 

Figure 30- Coordinate system for OME calculation
of n n+wp. â€˜

Using this geometry we easily write the energy-

momentum 4â€”vectors of the particles.

a=(a OIOIâ€”q)

OI
b=(b0l)l)lq)
c=(cO,-qâ€˜Sin6,O,-q'cos6)

d=(d0,q'sin6,O,q'cose)

Application of the Feynmann rules leads to an expression
for the helicity amplitudes M:
<Acxdlmlxb>n n=|(Acâ€”Ad)+xb|

- "Ow

_ A
C '.
w GVZU(d,Ad)AuNU Ukb,lb)

86

87

where Z=[MÂ§-t]_1, fï¬‚pw is the PVV coupling constant and

GV is the vector BVB coupling constant.

_. . _ â€”1
Au-1Y0[nYu+21Andu] Aâ€”(Mb+Md)
N =8 a 5* (c A ) '=l+El =l+ï¬
u quc V00 0 ' c â€ GV â€

GT=tensor BVB coupling

â‚¬(c,AC) is the w-meson polarization 4-vector
Expanding and using the center-ofâ€”mass vectors a,b,c,d,
we obtain
A * _ * *
N â€”-A a c e +Al(a a C163

*
u u o 3 1 2 3Â°oâ€™aoc3)â‚¬2+A2[(aoc3'a3co)â‚¬1 o

* *
+a3cle ]+A3a0cle2

So one can write

A

' I I I
<Au'|M|u>~ {-Au â€œEX+AU uFA+AU â€œGX+Aâ€œ â€œH ]

0 l 2 3
where Au'â€œ-ï¬(d ')A U(b )
n " Iâ€œ n Iâ€œ
It is necessary to calculate only half of the amplitudes
because <â€”A-u'|M|-u> =-(-)n<lu'lM|u> which comes from
invariance principles. (Inversion of the y-axis in

connection with parity conservation gives this relation)

The factors A are easily evaluated:

AS+ : A6-=i[-n'Â¢++2Anw_d0]cose/2
A3- =-A6+=i[-nâ€˜w_+2Anw+dOJSin%

AI+ = A1-=i[4Anqâ€˜w_coszgâ€”n'C+]sin%
A:_ =-A1+=i[4Anq'w+sin2%+n'C_]cos%

88

++__ --_ , . 3
A2 - A2 â€”n ;+Sin2
+-_ â€”+_ , 3
A2 - A2 â€”n 2;_cos2
++.. --=Â° I _ I 3
A3 â€” A3 i[2Anq w_cose n c+]cos2
+"__ -+= l _ I Â° 9
A3 - A3â€˜ i[2Anq w+cose n z;__]Sin2
also: E =5â€”qq'sine=-Â§â€”Ht E0=FO=HO=0 G0=-quSln9
5 o

_L' _I i=l- _ I
F â€”/Â§ch0 q aocose] G i/Â§[qc0cose aoq ]
It is convenient to rewrite all of the angular
factors in terms of w=25inÂ§. Then we form the amplitudes

2
as

<A ' M > =6 A =
u I In n Z uNu BZÂ§_ x
â€”n
where the index a numbers amplitudes of a given helicity

'transfer n and

It is now a simple matter to obtain the scattering
amplitudes in a particular model.
1. The Born Term Model (BTM)
Calculate the cross section using the amplitudes
as they stand.
2. Born Term with Form Factor (BTMF)
Let Z=(MÂ§-t)-1F(s,t) and calculate using the BTM

expression for the amplitudes.

89

3. Reggeized PrOpagator Model (BTMR)

Follow the procedure discussed in the Regge

 

 

Model section; let
, _ -ina(t)
Z: 27â€). (t0) j (2a(t)+l)]2-Seinna(t)
(2j+l)[T+(-) 1
2 2 2 a(t)-l

S-l/2(M2+M +M +M
a C

b

(1)}
1/2
2Ma(MbMd)

 

and use the BTM expression.
4. The Absorptive Model (BTMA)

Let DW(X) be the absorptive function. Then,

neglecting terms in w higher than wn+2,

Xu
Ba=â€”â€”â€”[(X -â‚¬2Â§ )enf xde (wx)k (ex)DW(x)
n . an . n n
qq . 30

nâ€”
+w XanDW(JO)]
n

where we have used the identity â€”%â€”â€”5-
E +w

oo

=â‚¬n f xdx J (wx)K (ex).
J n n
O
In an actual calculation the upper limit will have
to finite--one simply chooses xu large enough to get most
of the amplitude. J is the lowest allowed partial wave:

0

J0=Max[|AC-Ad|,lla_lb|]. The error in this change is

small. This short cut as mentioned already is only good

for modifications to the plain BTM.

APPENDIX III

BVB Vertex Corrections

We consider only the two Simplest graphs as

shown in Figure 31.

r\\\ U) *T W
\ \â€˜s./
l
9 he
'5' \r â€œI
n ///r_F_\\\\
n P h ?

Figure 31. Diagrams giving vertex corrections to
NpN point vertex.

Using the Feynman Rules we obtain an expreSSion for the

helicity amplitudes:

 

f
_ 2 VVP 1 ï¬‚ *.
<)â€˜c>â€˜cilM|)â€˜ b>_/Â§CNNPgPPV MC M2_tÂ°uvpoavepâ‚¬o(c'xc)
p
U(d,Ad)75Auy5U(b,Ab)
where
4 22
A =f d 2 ii, u { l _ l }

 

â€œ (2n>z<22+MÂ§)<(e-2)2+MÂ§> 1(b-Z)+Mp 1<r+z)+Mn

9O

91

the following parameterization facilitates simplification

of the integrand;

 

 

 

 

 

l
l 1 l = 1 I dy l
23+M2 22-2e2+A 22-2b2+A 2Â§+M2 [22-2ep +A ]2
ï¬‚ 2 3 n 0 y y
l
l
=f 2(1- Z)dzdy 2
0 ds [2 -2epZ+AZ ]3
_ 2 2_ 2_ _
AZâ€”e +Mï¬‚â€”M1T t Pyâ€”e+dy
A =A (1-y)+A y A =b2+M2
y 2 3 3
Pz=Py(l-Z) =by(l-Z)+e(l-y)(l-Z)
A =A (1-Z)+M2
2 y n
Rearranging and using (ip+M)u(P)=O
we obtain
__ (3) (3) (3) _ _
Au 2Aan (M p)+2iYA[LM (Mn ) =LA (Mp )] Ampâ€”Mb Md
=-M
n P
with
M4 2 2
L(3) u v
(M)=fds Id 4 2 3
(2n) [2 -2ePZ+AZ]
L53) contains one finite and one divergent term.
Write 2=2'+PZ and obtain
4 . 2'2' +2' P +2'P +P P
V(M)=fdsfd 2 4 uAv u Z v Zu3 Zu Zv
(2n) [2' 2+AZ-PZ]3

92

The term odd in 2' vanishes and the quadratic term
is divergent (or, more correct, the integral does not
exist). Allowing a liberal interpretation of the rules

d

for manipulating infinities we write for the 2n term in Au

4 f .
. r d 2' 1 _ 1 \
Zlfdsï¬szJ â€˜ 4{ ,2. . 2 3 .2 2 3 }
(2n) [2 +Az(Mp)-Pz] [2 +Alen)-Pz]

recall
pz=byz+e(l-y)z (have let 2*1'2)

so we can write

ï¬<d>iyzU(b> -MbyzU(d)U(b)-(Mbâ€”Md)(l-Y)zU(d)U(b)

= -Mdyz -Ansz(d)U(b)

Using this expression we obtain

 

 

M
â€”. ' _(_i_ I 2 2 - II _ 2 __ -
Auâ€” ZlbUAnP{K1[yZ]+2Anp Klly z ]+2Kl[yz ]}- ZibuAnpK[yz]
with
1T2 A . .
Kl[yz]=â€”â€”â€”â€”IJdZF3(l,z)M
(2â€) p
2 2 n2
I _ .- _ . ,
Kl[y z ]â€”_ 4fdz zLF5(l,z)M F5(l,z)M 1
(2n) n p
K"[ 22]â€” â€2 {dz z[F (l 2â€˜ â€”F â€˜l 2) ]
Y â€˜ 4) 3* ' â€™M 3â€˜ ' 'M
1 (2n) n P

(the y integrations have been performed analytically.
Please refer to the discussion of the parameter functions

Fi).

93

Substituting, we obtain

A e=27Â§CZgÂ§â€”Anp
<ACAd|T|Ab>n=iBZB {Ad'xb}K[yz] c
2 -l
Z=M-t
[ p 1
x b b * A
B N=e11va navcpâ‚¬o(c' c)
*
=-qq'/Â§s2(c,k) in CM syStem
which gives
<0 l/2|T| 1/2>O=0
l w2
<1 1/2|T|-1/2>O=â€”â€”â€”ew+/ â€”Â§â€”â€”7 K[yz]
2/2 a +w

<oe1/2|T|1/2>l =0

<1 1/2]T| 1/2> =lâ€”ew_/ -Â§9â€”Â§K[yz]
l /2 e +w

<l-l/2lTI-l/2>l=<l 1/2|T| 1/2>l

<1-1/2|T| 1/2>2=-<1 l/2|T|-l/2>0

These amplitudes can be used to calculate cross sections

in the usual way.

APPENDIX IV

Two Meson Exchange in PB+VB

The relevant diagrams (the simplest) are shown in Figure 32.

 

 

 

 

 

1t â€œ"4, e I w -- V l.
I l \ /
11 Iw ;X\\
n :4, l ' P / 1
â€œW
1:. I

Figure 32, Two pion exchange graphs for n+n+wp.

Using the Feynman rules and proceding in the usual manner

we obtain

f
I_ 2 new *, - I . _ _
T â€”2/2 GNNngnanc Elvpo axqveo(c)U(d)ysAuy5U(b) qâ€”b d
TII=-2/2 <32 Sipâ€”â€œls a 6*(c)t-J(d) AII U(b)
NngnanC Avpo Aqv 0 Y5 u Y5
TIII=-2/2 <32 fâ€”Tâ€˜we a r 6*(d)U(c) AIII U(b) r=b-c
NNngnand Avpo A v 0 Y5 u Y5

94

95

4 e
AI=A d e u

â€œ â€œP (2n)2f [e2+M:][(a+e)2+M:][(q-e)2+M:][(b-e)2+M:]

 

 

+iy d4e euex -{ 1
4 (2n)4 [e2+M:][(a+e)2+Mï¬][(qâ€”e)2+M:] b-e)2+Mi
_ l }
(b-e)2+M2
P

one obtains Ai by the replacement aÂ»-c in Ai

and AiII by the replacement q*r in Ai

We rewrite Au and compress it by the use of parameter

. . l_r 1
integrations Ebbï¬dfaf+b l-f)
to obtain

(4)(M )- Lâ€˜4â€™(Mp )1

_ (4)
YSAu Y5- A an (M p>+iYA[Lm n

where

4 ere l
L(4)_st I d e4 2 H v 2 4 ds'=â€”â€”â€”â€”zdxdydz 2(1-2)
(2n) [e -2ePZ+PZJ (2w)

 

iPz=iqZ+idyZâ€”ia(lâ€”x)(l-Z)

performing the loop integration we have (using the Dirac

equation to simplify)

L(4)(Mp )= -idurds â€”â€”Xâ€”â€”â€”

C2 (Mp )
o (4) 9 z o Ill. 2 '
iyu L (M) =id Ids â€”â€”Xâ€”â€” i? +lâ€”lY g ersâ€” ds=n ds
u C2(M) z 2 A Au C(M)

96

One can write

U(d,Ad)iPzU(b,A8=-AanUU-MpyzUU-W A (l-x)(l-z)

4d b

w =iï¬<d>Â¢U<b>
4d4b

Putting all of the terms together we finally obtain

6(d, A iZ):5AIYSU(b, A b):

 

 

 

 

[Ids-3xâ€”â€”â€” -fdsyz2 <1+YAP)(12 â€” 21 )
â€œp c (Mp ) np c (Mn ) c2 (Mp )
w
A A
- db bfdayz<1~ x)<1~z)( 21 â€” â€˜21 )1
np C(an)C(M)

 

-â€”U(d, A du)y U(b, A b)fds(CTM;7_ C(Mp)

which we rewrite as

dexb
idpAnp{Iï¬‚ 2"Aâ€œâ€˜ I3 }- %U(d, A du)y U(b, A b)J

where the definition of Ii and J is clearo

Substituting into the expression for TI we finally obtain

 

W
A A A
I _. Ac _. k d b . kM C
<ACAd|T |Ab>â€”iKB {Ad'Ab}I lkB I3- Zâ€”â€”AdAbJ
nP
AC_ 1 ' - ' '
where B -â€”;qq Vs SlnebdÃ©xc,il (in the CM)
{A Ab }â€” â€”U(d, A d)U(b, A b) MAc =6<a A )NACU(A A )
dâ€™ A A â€™ d â€™ b
d b
w = U(a A ) U(b A ) N4c=e ;(c, A C)
AdAb ' d â€™ b p uvpoa â€œq

K: 2/2G ng â€”Anp

97

The integrel J is small than I1, 12, I3 for moderate s
but will eventually dominate as s+w Since J~sâ€”l and
I ~S-2.

Performing the y-integration (parameter integral)

analytically we get

 

 

 

 

2
ï¬‚ 2
I = fdzdxz (l-z)F (x,z)M
1 (2n)4 1 p
1T2 I 3 M
I = dzdxz (l-z){[F (x,z)+â€”Eâ€”F (x,z)]M }
2 (2n)4 2 Anp 2 p
I = w fdzdx22(l-z)2(l-x)[F (x z)M -F (x z)M ]
3 4- l â€™ â€™ n l ' p
(2â€œ)
n2 _ m
J=(2n)4fdzdx(l-z)z[F4(x,z)Mnâ€”F4(x,z)Mp]

Again, there are 12 amplitudes, 6 of them obtainable from
the other six via symmetry considerations; the six

independent amplitudes are

 

1 . __ o 0
B0 <01/2|T|1/2>O â€” KMl/Z l/ZJ/Anp Vs
2 K ,- W1/2 1/2
B <ll/2ITIl/2> =-â€”+qq'/sw w[I-â€”â€”â€”â€”â€”â€”â€”â€” I ]
1 1 g~ - w A 3
/2 - np
- 1 1/2
KMl/Z l/ZJ/Anp ~S
(a w -q )
=' K Q'/Â§Â¢_w[I-. 2 + + I3] ~sl/2
J? wâ€” np
_ l , 1/2
KMl/Z 1/2J/Anp â€5
1 , __ 0 0
Bl <0 l/2|T|l/2>lâ€” KM_l/2 l/ZJ/Anp ~s

APPENDIX V

th

4 Order Parameter Functions for Pb+VB

h

1. All of the 4t order virtual momentum integrations

can be reduced to the form

1

o Cn(M)
where
ds = dzdxdy z(l-z)
n = positive integer
N(x,y,z)= polynomial in kinematical variables and x,y,z
and E(M) = A(x,z)y2+B(x,z)y+C(x,z)
M=mf

The 4th order functions Fi(x,z) are the result of

performing the y-integration analytically:

 

 

l
= w = .1; 2A+B â€” . :
F1(Xâ€™Z)M g 52 Q[A+B+C BF4(X'Z)M] G12(X'z)
F (x z) â€”111391 â€” l[- B+2C + 2CF (x z) 1 - G (x z)
2 ' M â€˜0 62 â€˜ Q A+B+C 4 ' M â€˜ 22 '

F3(x,z)M =

98

I â€˜ |Il|u| llxl

99

 

 

 

 

 

l _ _
F4(x,z) = I 91 = â€”3{tan 1 2A+B - tan 1 Eâ€”
M .â€” __
o c /6 /Q /5
F (X z) _ 2Ily2dy __ 3:23 10 A+B+C + Bz-ZAC F (X z) ]
5'M'zo-â€˜2AA9C A 4'M
2 -1 ma
Q = 4ACâ€”B G (x,z) = j Xâ€”â€”Xâ€”
nm n
O C (M)
N(x y 2) can always be written in the form 2 A xï¬ymzp
' â€™ Â£mp imp '
Therefore we write
1 .1 Km P ,
f dS N(XQY'Z) 3 I ds'dyï¬m aimpx n z = EmpjÃ©lis.X ZPG (x'z)M
0 c o p c (M) 0
It is usually necessary to perform the x,z integrations
with help of a computer.
2. The denominators C(M)
There are three topological diagrams to consider
as shown in Figure 33.
k
a -- I k ' C a.-W-.__c Qâ€”â€”- ' C.
I I
| 3\/C &! ,1
ti 3 j/ â€œ - K
b a b 4. â€˜ wt a ----â€” A
I â€˜E. EL

Figure_33. Configurations for twoâ€”meson exchange
graphs for n+n+wp.

100

(a) EI(X.z) = AI(x,2)y2+BI(x,z)y+CI(x,2)
AI(x,z) - Mgz2
BI(x,z) = (Mgâ€”Mg-t)zz+(Mg+M:-s-t)z(l-z)(l-x)-(MÂ§-t)z

CI(x,z) = tzz+(M:-M:+t)z(l-z(l-x)+(MÂ§-t)z

+[(Mï¬-M:)(l-x)+M:x](l-z)+M:(l-x)2(l-z)

(b) EII(x,z). Detailed examination of diagram II shows
that exchange of a<->-c in I will give II
This leads to

AII(x,z)

I
3
N

II _ 2_ 2_ 2 _ 2_ _ _ _ 2_
B (x,z) â€” (Mb Md t)z +(s Mc M )z(l z)(l x) (Mg t)z

(LN

II 2 2 2 2
C (x,z) tz +(MC-Ma+t)z(l-z)(l-x)+(Mg-t)z

+[(MÂ§â€”M:)(l-x)+M:x](l-z)+M:(l-x)2(l-z)

(c) EIII(x,z). Similarly we get III from I via t+u and
keeping labeling defined as in I (remember c,d masses

interchanged here)

This gives (alternatively: use I with
6=6bc instead of ebd
AIII(X,Z) = M222
d
III 2 2 2 2 2 2
B (x,z) â€” (Mb Md u)z +(Mb+MC s u)z(l z) (1 x) (Mg u)z

101

III 2

C (x,z) = uz +(Mi-M +u)z(l-z) (lâ€”x)+(MÂ§-u)

2
d
+[(MÂ§-M:)(iâ€”x)+M:x1(i-z)+M:(1-x)2(1-z)2

Remember that only two of s,t,u are independent variables
because of the relation

2 2 2 2 2
= Z =
s+t+u lMl Ma+Mb+Mc+Md

we use 5 and t

3. Energy Dependence
We use s and t as the independent variables.

Examination of the denominator polynomials CI(M), CII(M),

CIII(M) gives us the s-dependence of the functions FEâ€™IIâ€™III
at fixed t.
(a) Graphs I.
AI(x,z)~sO BI(x,z)~s CI(x,z)~sO +Q=4ACâ€”B2~sz
So we conclude
I -l
F4(x,z)~s
(b) Graphs II. Same as Graphs I.
(c) Graphs III. AIII(x,z)~s0 BIII(x,z)~s CIII(x,z)~s
+Q~asz+bs
and therefore
FIII(X,2)~s--l so FIII(x,z)~sâ€”2 FIII(x,z)~s0
4 l 3
F:11(x,z)~as_ +bs-3 FEII(x,z)~s

 

102

(d) Vertex Corrections (BVB) The Diagram for this correction

is shown in Figure 34.

&.\\\ 0
â€Gig;
hr \%
w
t
5/ \CL

Figure 34. Configuration of NeN-vertex correction
graphs for nn+wp.

0 O 0

Since AI(l,z)~s , BI(l,z)~s , CI(l,z)~s ,
we conclude that any energy dependence vanishes from the
F's at x=l:

I 0
Fi(l,z)~s ,

and therefore, that the triangle 100p cannot change the

energy dependence relative to the simple BTM.

IE

â€œnu'ryuï¬‚gï¬‚mï¬‚m'

(I)?

3 1293 0

 

Vl
Nm
U"
m
H
m
H
â€I