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This is to certify that the

thesis entitled
FIXED-uFINITE-ENERGY SUM RULE

CALCULATIONS FOR TI’N SCATTERING
USING REALISTIC SPECTRAL FUNCTIONS
presented by

Laurence J. Sowash

has been accepted towards fulfillment
of the requirements for

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0-169

ABSTRACT

FIXED-u FINITE-ENERGY SUM RULE
CALCULATIONS FOR 11M SCATTERING
USING REALISTIC SPECTRAL FUNCTIONS

By

Laurence J. Sowash

Realistic spectral functions have been used to calculate fixed-u
Finite Energy Sum Rules (FESR) for va scattering. The FESR equate
positive integral moments of the Regge and low-energy scattering ampli-
tudes. The fixed-u FESR isolate baryon Regge amplitudes which dominate
high-energy backward scattering. The generation of realistic spectral
functions involves detailed consideration of t-channel threshold proper-
ties. Regge convergence properties implicit in the sum rules are used
to choose between available sets of direct channel phase shifts. The
FESR are used to fit coupling constants for 1=0, l mesons. The
calculations are extended to higher moments than possible in previous

calculations.

FIXED-u FlNlTE-ENERGY SUM RULE
CALCULATIONS FOR er SCATTERING
USING REALISTIC SPECTRAL FUNCTIONS

By

\
_..;.{\

%“\
Laurence J? Sowash

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

l97l

ACKNOWLEDGMENTS

My special thanks go to Professor Peter Signell for suggesting
this project and encouraging and guiding these efforts. Professors
Jules Kovacs, Joseph Kubis and Dr. Jonas Holdeman were especially

helpful.

TABLE OF CONTENTS

INTRODUCTION

THE FORMAL DESCRIPTION OF n’N SCATTERING

Notation and Kinematics

The Invariant T-Matrix

Cross Sections

Partial Wave Amplitudes

Crossing Symmetry

Kinematics of Cross-Channel Processes

The Mandelstam Plane

TCN DYNAMICS AND FINITE ENERGY SUM RULES

A.

The Mandelstam Representation

The Finite Energy Sum Rules

The Concept(s) of Duality
SPECTRAL FUNCTIONS

The Regge Amplitudes

The s-channel Spectral Function
The t-channel Spectral Function
The Preliminary Spectral Functions

Transforming the Partial Wave Series

RESULTS AND DISCUSSION

A.

B.

C.

The Finite Energy Sum Rules
The N“ and Np FESR

The A FESR

lO
lI

l2

19
I9
25
28
33
33
38
h]
1&9
59
7|
71
72
78

D. The Cutkosky-Deo Scheme
APPENDICES
A. Generalized Spinor Invariants for IZN Scattering
B. lsotopology of flN Scattering
C. Invariants in Terms of Partial Waves
D. Partial Wave Amplitudes for rtt-vNN

REFERENCES

78

8]

88

93
97

LIST OF TABLES

Table

l. The baryon trajectories and their quantum numbers.

2. Quantum numbers and coupling constants of the zero-width
mesons.

3. Quantum numbers and parameters of the finite-width mesons.

Page

36

A6

50

Figure
l. The
2. The
3. The
A. The
5. The
6. The
7. The
8. The
9. The

l0. The F: (X,u) Spectral function at u

LIST OF FIGURES

Mandelstam diagram for rtN scattering.
cut cosé Plane for “N scattering.
region of integration for the FESR.
FE (X,u) spectral function near X = X0.
Ff'(X,u) spectral function near X = X0.

FE (X,u) spectral function at u = 0.

Ff’(X,u) spectral function at u = 0.
FE (X,u) spectral function at u = l.O GeVZ.

F (X,u) spectral function at u = 1.0 GeVz.

1.0 GeVz.

II. The F: (X,u) spectral function at u = l.0 GeVZ.

lZ. Intersection of the Lehmann Ellipse and the Mandelstam
plane.

l3. Graph of x versus 2 at W = 1.] GeV. in the 5 channel.

l4. Graph of x versus 2 at W = l.8 GeV. in the 5 channel.

15. Graph of x versus 2 in the t channel at t =4./?5'.

lb. The F2 (X,z) spectral function at u = l.0 GeVZ. generated
from transformed partial wave amplitudes.

l7. The F2 (X,z) spectral function at u = l.0 GeV2 generated

by expanding x in terms of polynomials in 2.
l8. The n = l N“ and N, FESR.
19. The n =

3 Na. and N; FESR.

vi

Page

15
I7
37
42
A3
52
53
54
55
56
57

6]
66
66
66

68

70
73
7h

20.

2|.

22.

23.

The

The
the

The

The

n=5 Nm , and N, FESR.

contributions of each of Barger's zero-width mesons to
n=l N; FESR.

low-energy terms for the n=0, l, 2, A‘FESR.

low-energy terms for the n=3, 1+, 5, A‘FESR.

vii

75

77
79
79

INTRODUCTION

Finite energy sum rules are a development of the description of
strong interaction processes by integral equations for scattering ampli-
tudes. This description is derived from Quantum Field Theory as embodied
in perturbation theory and Feynman graph techniques.

Mandelstam demonstrated that perturbation theory as applied to two-
particle interactions was equivalent to a two-dimensional integral repre-
sentation of the scattering amplitude. This representation is not com-
plete since only a few basic properties of the spectral function (the
integrand of the two-dimensional integral) can be established apriori.
For this reason one cannot speak of 'solutions' to the Mandelstam repre-
sentation (nor, as we shall see, to finite energy sum rules). However,
these equations do serve as very strong constraints on the form the
scattering amplitude can take.

The Mandelstam representation exhibits the scattering amplitude in
the form of an integral which explicitly satisfies the Pauli principle
and Crossing symmetry. It is one of the basic features of S-matrix
theory that the scattering amplitude is viewed as an analytic function
of energy and momentum-transfer variables, defined by its analytic
structure and basic symmetry considerations rather than through any
connection with underlying 'fields'.

In the Mandelstam representation the scattering amplitude describing

some process a + b-+ c + d has singularities (poles or branch cuts) at

2
the allowed physical energies of any initial or final state in each of
its respective 'channels', as well as poles corresponding to intermediate
bound states. The separate channels of a reaction are generated by inter-
changing an initial and final state particle. lnterchanged particles
become antiparticles and vice versa. The respective channels are labeled
by the appropiate Mandelstam variables corresponding to the center of
mass energy squared for that channel.

In TtN scattering we consider three channels:

n'+ N —' 75+ N 's-channel'
N +N-"*-T-i+1t 't-channelI
75 + N—'"' '1? + N 'u-channel'

The self-conjugate property of the pion reduces the u-channel to a
reaction identical to the s-channel but having a branch cut in a differ-
ent variable. Crossing symmetry states that these three reactions are
described by the same analytic function continued to the appropriate
value of s, t, u.

Four scattering amplitudes are in fact required to describe (IN
scattering, since there are two allowed spin states and two allowed
isotopic spin states in each channel. Each of these amplitudes has its
separate Mandelstam representation. The details of these spin and
isospin complications will be dealt with in chapters II and III. For
the present we shall speak of a single 'scalar' amplitude F(S,t,u), neglect-
ing spin and isospin. The Mandelstam representation of F(S,t,u) has

the form

a
l c ’ 0""!
F1”, (.4) = fi: Iufi'“ (nits-(mf—

(buvpu‘ flyd‘ (1")
' , oql(tfcafll
+ 75: I“ P“ «WNW-u)
fin‘ hfirflny’

 

al.1(44' 6"]
-r ”Ida f3“ (.4 -.¢)(u’-o¢.)
(Ne/If (Mo’O
We have omitted the pole at the nucleon mass from F(s,t,u). Further,

assuming that F(s,t,u) has even rather than odd parity under Crossing,

(l.l) could be reduced to

a
I I
I I .._..—-—- __..___.
HA! ‘1 “)- = iii-IA" 54’ t) {4'14 * 4"“}JAI
rm Wyn)

+ ifq‘jflLflplfli’f,«dJ"7:t_

(I.2)

This is the so-called one-dimensional form of the Mandelstam representation,
the form used in most calculations.

As required, the cuts in s and u obviously extend from (M*/‘)2 to
infinity and the cut in t extends from #/2}' to infinity, even though the
process NN-vmt cannot physically occur on the low energy portion of the
cut 4,0- 5 t 4 you}. This unphysical cut has received a great deal of
attention in the context of the Mandelstam representation. This attention
was warranted by the connection between the process NN-flrfl' and the
electromagnetic structure of the nucleon. The nucleon isovector form
factors F¥’2 (t) (the 'structure constants' for electron-nucleon scatter-

ing) can be represented in the form of dispersion relations as

 

:f) —- I #53a7— a’t’ (L3)
—- —— (‘2.
z figu‘ *

The spectral functions lm Fv (t) contain linear combinations of

l ,2

i = I Nil-Nut partial wave amplitudes and are proportional to the pion

form factor. One of the earliest successes of the Mandelstam

I.

representation was to help establish the existence of the rho-meson by
combining equations similar to (I.2) and (l.3) with information on nucleon
resonances and data on the form factors. Similar calculations also yield
information on 1H1 elastic scattering phase shifts.

Lack of complete knowledge on the high energy behavior of scattering
amplitudes has limited attempts to apply the Mandelstam representation
over any very wide range of energies. This difficulty is enhanced by the
fact that the energy denominators in (I.2) suppress the effects of the
high-energy states. Further, cuts extending to infinity often make it
impossible to evaluate the contributions of terms in the amplitude which
do not converge rapidly at high energy.

Alternate forms of integral constraints on the scattering amplitude
have been proposed recently. One form which avoids the mathematical
difficulties inherent in the Mandelstam representation while still serv-
ing many of the same purposes is the so-called Finite Energy Sum Rule
(FESR). The FESRs we have investigated were first studied by Barger,
Michael, and Phillips, and take the form of moments of the scattering

amplitude

‘J; ;(’:4£,D<:;acx,as)aKX: =*:ZEXffltlgvaquCX3.¢)<{><

In this equation F (X,u) is the asymptotic form of the amplitude

Regge
F(X,u) beyond the energy at which the amplitude becomes Reggeized.

X0 is the value of the coordinate X at which this 'matching' takes
place. The coordinate X is defined such that high-energy states occur
at large positive and negative values of X. Thus the FESR enhances the

effects of high-energy terms (near1th). In chapter III we show that

these equations can at least be understood (if not derived) in the context

of the Mandelstam representation despite their radically different appear-
ance.

The Regge amplitudes which appear in these fixed-u FESR are generated
by baryon Regge exchange. Baryon exchange describes rtN scattering in
the backward direction at high energies just as meson Regge exchange des-
cribes high energy' er scattering in the forward direction as energy
increases along lines of fixed t.

We will not be concerned with the ambiguities which still remain
in determining the continuation of baryon Regge amplitudes to low energy,
but will simply use forms appropriate to high energy data analysis.

Part of our approach to implementing the FESR is to develop a para-
meterization of the meson resonances which is more realistic than the
simple zero-width pole approximation frequently encountered in dispersion
relations. We take resonance widths and threshold prOperties into
account in as realistic a fashion as possible so as to generate a con-
tinuous t-channel amplitude whose relation to the Regge amplitude near-Xo
can be examined in detail. This is an improvement over previous calcu-
lations which did not generate a continuous spectral function. In add-
ition we examine the behavior of the s-channel amplitude near X0 in
order to select the best set of phase shifts to describe the process.
This approach allows us to satisfy higher moments of the sum rules in
cases in which previous calculations were reasonable only for the lowest

(n=0 or I) moments.

II

THE FORMAL DESCRIPTION OF fl“N SCATTERING

A. Notation and Kinematics

Pion and nucleon mass will be labeled /¢(=.I396 GeV) andIMV(=.9382 GeV)
respectively(I ). Initial and final states will be subscripted l, 2
respectively. Pion and nucleon 4-momenta are denoted by q and p .
Momentum-energy conservation in the process If, + Nl-* 71} + N2 is

then expressed

qwrp. = 92 *P2 (2.1)

In general we will operate in the center of mass system, in which

are-n
T3 3-}32 1': k,
IE/=/P/5R

II
I In

a

Now, letting any 3-momentum IL define a coordinate axis, all h-momenta
are determined by specifying ll, the "C. M. momentum" and meal/{712.2
the ”scattering angle“.

If we wish to specify the scattering amplitude which describes the
interaction as a function F(ql q2 pI p2) it is sufficient to regard
it as a function of the two variables A ,me or any equivalent set.
In order to define a Lorentz-invariant scattering amplitude it is custom-

ary to construct the invariants( 2)

.-:= 4/7””? = +2/‘l'zn‘et-u. = W2 (2.2.)
t E - (ch-9.? = '7k2{/‘ma) (2.3)
u. E -(/>.-7z)a= (fr “4);- 2A’(/+c~9) (2.1+)
where: E. EW= initial and final nucleon energy,

aa‘ilfi;:€;z?

initial and final pion energy,

W

center of mass total energy.

Only two of these variables can be regarded as independent since

A+t+4c=2m2 *2/3' (2.5)

B. The Invariant T-matrix

To determine the most general scalar invariant form of the

scattering operator in nucleon spinor space we could write,

F = IE (I. a.) + .2. r...- i-Aeuw 4- 7. 310M“
7.1;;- "Pi l-y-j ‘Fl’(l,mG)

+ O 0 c O

forming an infinite set of spinor operators by constructing all possible

higher-ordered products. The Dirac equation:

(Y-p +m)u.r,a)=o

can be used to eliminate the terms linear in nucleon momentum. This

8

fact, together with Eq. (2.l), shows that the terms linear in pion
momentum are not independent. The quadratic and higher-ordered terms can
all be shown to be equivalent to a spinor-independent function plus a term

linear in either pion momentum (Appendix A). The conventional choice is:

Fa, m9 é-lffl. «9) + x: 1-6? Emu”
where Q '5 'd’.(?t ’7')

The invariant T-matrix can thus be written:

T = -Aa,cue) 4- zr-Qflmmue) (2.6)

T is related to the S-matrix by:

3!) = 5*; —- (21:)"1 {(7.7% - 7,-7.2 (7% ‘51}Tw, (2.7)
where: find. are nucleon spinors for initial and final nucleons.
ibis the 'no scattering' Delta-function, f and i denoting final
and initial states.
A and B are also matrices in isotopic-spin space. In the so-called

'numbered' pion representation,
to) C')
A 3 A (4, t,“) L! I. flat-us 2’] Afd,“,w) (2.8)
_. to) J- , f"

where: on, ,3 = l, 2, 3 are the final and initial pion labels.
'Z;,T; are the nucleon isospin operators.

The details of constructing this representation and relating A and
B to matrix elements corresponding to good isospin l = 1/2. 3/2 and the
physical scattering processes of 72" p, rr'p elastic scattering and

charge exchange (Ir’p on"! ) are presented in Appendix B.

The major results are:

(2-9)

(2.10)

The same relations held for the B amplitudes.

C. Cross Sections

The cross-section for any process is commonly written

21sz = /F(A,mO)/2 (2.”)

where F(k, cosa) is the 'scattering amplitude'. The connection between

F and the T-matrix for boson-fermion scattering is

Foam) = - 1%?!" T(h,m9) (2.12)

To compute the elastic cross-section for some isospin state it is necess-
ary to sum (2.ll) over initial and final spin states preperly normalized.

(A)

The details are presented in Gasiorowicz(3 ). The result is,

-1. I
0:0“. =- (Kl‘fl') loan. A (Mme) (2. I3)

I o 2
:5 = .513 o-AUx/A (Amer WAC- Wane”?
where:

A '0, can) ' A (A. «0) 4- (K 4 r/c—‘lfl- */I»‘).’B (A, m0)
K = (4 'M' 749/(2-M)

10

Here as in (2.ll) and (2.12) isospin labels have been suppressed.

D. Partial Wave Amplitudes
(if) (I) -
In order to relate A and B to amplitudes for good 4,] (14:1)

first define

{has} : gf'k)€(me) ‘zfrUPr-«w (2.1“)
£71., m0) = :(JA) - firkflf’ww
where
fans) 5 mi?) fme)
15:“ )fi;, -‘—{Zc"8 III are the partial wave amplitudes

for isospin T and #21172.

Then A, B are related to f,, fz by

£={';%[A*(W-M}B} (2-15)

{SEQ/:4 (WM-'25}

The details are presented in Appendix C.

Partial waves can be calculated in terms of invariants by inverting

(2.l4), giving,

£02) = fi{£?¢,malfa~a)

(2.l6)

,
+ fire, we) firmed «(rum

II

E. Crossing Symmetry
The basic idea of crossing symmetry as applied to pion-nucleon scatter-
ing is that the scattering amplitude possesses well-defined properties
under the interchange of initial and final nucleons or initial and final
pions. In particular, it is observed that under the interchange 71‘”"?z
the scattering amplitude( 5) has the property
F’VIPIVtFa) ‘3 F(‘sza'9iPJ
While this relation does not relate the amplitude for physical processes
(note that the pion energies are negative on the right-hand side), this
relation implies that the process
772’N4C -—’ 7zl'*/Mz
is not independent of the process
;z;1>4( -—+~715 +jAé
In the foregoing discussion the implicit assumption was made that
all quantum numbers of TE, and f5, were interchanged in addition to the
initial and final pion momenta. An exact statement of Crossing Symmetry
is that the T-matrix must be invariant under the transformation generated
by the product of Charge Conjugation and G-parity operators. This imposes

(‘5)

+
on A and BLT) the symmetry requirements,

I!) (3)
A (41%“): IA (“ltl‘a') (2.17)

(1) (a;
B (4.4....) = :5 ra,t,—o)
since the replacements 9,0—O—9, and P’673 each effect the transfor-
mation .40“, t being unchanged (see Eqs. (2.2) - (21+)).
The square of the Center of Mass total energy of the reaction
r5" * All "" 7T, f/Vl is just the Mandelstam variable u. This leads

to the designation 'u-channel process' for the reactions produced from

12

the direct interaction ('s-channel process') by applying the Crossing

relations.

F. Kinematics of Cross-Channel Processes

One could begin the analysis of pion-nucleon scattering by labeling
the direct channel energy-square 'u' and construct the s-channel amplitudes,
if necessary, from Eq. (2.17). Partial wave amplitudes as functions of
u-channel momentum It, could also be extracted as in equations (2.6)
through (2.16), with c059 replaced by cos 0“

However, other processes than those related by the Crossing relations
will also be of interest. In particular, consider the reaction generated
by interchanging q, and -f5 . Conservation of energy-momentum,

PIT/pa ' 9:"?!
now describes the reaction,
N+ 1V —-: m + n

It is possible to express the T-matrix in terms of t-channel
variables (for this reaction t is the C.M. energy square) as an altern-

ative to (2.lh) - (2.l6).

To do this first express 5, t, u in terms of t-channel variables

= 7"- 9’ + 2M «9‘
= 7(15 7093441.!) =l/(937ua)

II

“P3 -9: "2/37 cap¢
where: magnitude of Nucleon C.M. momentum

= magnitude of pion C.M. momentum

t-channel scattering angle.

(6)

“PVRNL
II

In terms of p, q, cos¢ we have

I3

(2) a);
l4 (1cm!) = 71);; (1'9 (197)7171"??::77 7": (f) coe¢ 432%?)
19:” (t) Pram}

g I {0/
Bag (0495) =Xfl;7;% (P9)? 7?: (flPrw 9‘)

(r)
The amplitudes ‘F: ’0') describe states of total angular momentum f;

(2.19)

the subscripts I! refer to NN states of identical, +, helicity or
opposite, -, helicity. The superscripts (i) label amplitudes proportional
to states of good t-channel isospin, since

(—) (1)

A“): 72" A“) A = %A (2.20)

3
These relations hold for B‘ ) amplitudes as well.
The Pauli principle requires that the (+) iSOSpin amplitudes contain
only states of even 3 and the (-) amplitudes only states of odd f.

Equation (2.19) can be inverted to give;

{.(Z’J‘: rk"; 9'57: A: )W W'("") ,(‘Jhdésé’fiajn
(flu *
{(1‘): fflWflT’-'{B-tf)-é: (1)]

where: 4,
IA?) (4) ,E. 518)] = JDI (t, mi) ; B(t,ca))]€(¢~‘) 4"“‘H
-I
8
It is convenient for later applications to express A‘ l, 8“, in terms

I
of t-channel partial wave amplitudes {1”(1‘) as well as the helicity
“”1 .
amplitudes f; 44:) . Here.£ denotes the orbital angular momentum of
the NN system (for the pions, 3'0 so that 9 3.0). The details of the

calculation are presented in Appendix D. The major result is;

(ili

J.
I I.) =,-,-(.,+,——)"‘ f7.) wage—2c 2..

1" (2.22)
3 .1 i It]
1C: Z)" :fi(21*l ){ {75.31%} 4724617457 )7( ’u’“)

Note that no singlet NN states contribute. This is due to parity con-

servation which in this cale requires that [-1 1/ for the NN states.

G. The Mandelstam Plane

The singularity structure of rtN scattering can be conveniently
represented in 2-dimensional graphic form by using the sides of an
equilateral triangle for s, t, u axes as in Figure l. The median is
scaled to be of length 2.0002129 so that the constraint 4+f4t¢8 lm'flpa
is satisfied.

The s, t, u-channel physical regions are labeled. The symmetry of
the diagram about the mid-line is just one consequence of Crossing
symmetry. Note that the t-channel threshold for two pions is 9’4.) , while
for two nucleons it is 95!! . Thus we can think of the t-channel singu-
larities as extending down to 4711 in the sense that an external state of
the t-channel exists in the region 7/4." t 5 ”It: even though the
process flit-ONN cannot physically occur in this region.

The only stable particle intermediate state available to any channel
is the nucleon itself, which can appear in the s- or u-channel. Such
fixed-energy singularities appear as lines in the Mandelstam plane as
seen in Figure 1. Note that each nucleon pole occurs below the threshold
of its respective channel.

Physical regions for scattering processes are limited not only by

energy considerations but also by the physical limits on scattering angles.

IS

 

 

 

 

 

 

 

    
          

   

7171:421sz M11011

\
A\ £31111--- . _ --
\K‘ \ ... -.--.-..... .,...
‘\ \|
it?

 

-‘l-AW - .m*.

\_g\ \ \ )\\\V \\\_\‘ \\\

. . ‘ \

 

- __- t 3 1(1)»)- ...".-.

" "'I” W 2220’ //,

 

 

  

 

   
 
 

 

. '\‘\ x\\\\‘ K .

. ' _ ‘- :\ ‘ ‘ ' . / //
Tim \ ~ ‘0. 2 ”is: ;l
I, w , \\ . (1/ a 2
K\ - “\\‘ \x\‘ / 2 /// ,‘I ,/'
Lj\ll‘\ \\:\:\‘\ V// / ./2’./:/
k\\\‘\ \\\\ ///J

 

Figure l. The Mandelstam plane showing the basic cut structure of 11N
scattering.

16

The line t=0 corresponds to forward scattering in both 5 and u-channels.
Backward scattering in these channels is limited by the curves ,4; :(m‘yd)?
This curve also limits the t-channel scattering angle in both the forward
and backward directions.

The delineation of the region of physical scattering angle for
lion’s titan} is complicated by the fact that it does not lie exclusively
in the Mandelstam plane. In this range of t, pion momentum q is real but
nucleon momentum p is pure imaginary. Then the t-channel scattering
angle

A 4- p‘4-93
“f7

flby

‘

cofl2¢’==

 

can be seen to be pure imaginary for real 5, u (i.e., points in the
Mandelstam plane), except for the locus 4+p’v’ao which is the line
s=u in the Mandelstam plane.

In the region ‘f’u‘ét‘fiuo‘ the condition --/5 «441/ is equivalent
to ,4.“ stub/£92. Further, since 5 and u are complex conjugates of one
another in this region, this latter inequality is equivalent to
IAI$M‘-fc‘. The physical region in cos+ then is limited to the
interior of a circle of radius [ne’jsz centered on the origin in the
complex s-plane as shown in Figure 2. The physical regions of s, t, u
channel processes project onto the real axis in the complex s-plane as
shown.

The point behind this detailed discussion of locating physical regions
in both energy and angle on the Mandelstam plane should be clear; the
evaluation of the scattering amplitude for some arbitrary choice of
kinematic variables is typically limited by the fact that no experimental

information regarding the process may exist at the corresponding location

17

.u box_w um m< newsman oc__ on“ mon_cummo

... ocm _+ c8503 #80 act/cg umfi 302 . né.\x~wure\) Low c039. tmou 339:3 of. .N 0.33...

 

AASCgV n... m. ...u
~k<\+ex>vuuuu...o

N§}m ...U

 

e30 \b§&§‘0 $~3

 

esmqmaaaflwuwIILJTI

 

 

 

_
at. in W

 

 

 

l8

in the Mandelstam plane except as an extrapolation from some (hOpefully
nearby) experimentally accessible region. Just as strong a caveat is to
be observed as regards extrapolation in angle as in energy. This is
particularly true in the case of input data parameterized as partial waves
in some channel. The partial wave series is necessarily truncated after
some infinite number of terms, typically when the error estimate on some
high angular momentum phases makes them consistent with zero. For
increasingly large values of cos¢ outside the physical region the
Legendre polynomials diverge like (cos#)’. Obviously, even a very

small coefficient of some high-L Legendre polynomial will dominate the
lower terms in the series if this extrapolation is carried too far in cos¢’.
This fact causes serious problems even before the limits of convergence
of the partial wave series (the Lehmann Ellipse) are reached, as we shall

see.

TEN DYNAMICS AND FINITE ENERGY SUM RULES

A. The Mandelstam Representation

Before the advent of Duality and Finite Energy Sum Rules the basic
language of dispersion relations was the Mandelstam representation. A
brief review will provide a context in which Finite Energy Sum Rules can
be discussed.

Assuming that the amplitude to be calculated possessed the singularity
structure of field theoretic perturbation theory, Mandelstam(7') was able
to write down a two-dimensional integral representation which the amplitude

in question was required to satisfy. For TEN scattering the Mandelstam

representation has the form:

 

m «yd-4,, a.)
24m,t.u)= Tz’JJ‘fOJO‘, (A’-—4)(u’- «2)
nonqd anyd

a(€)'(u't’
at I
+ 75f fl,“ fdt 7u'-u7t’-t§ (3.1)

fimvdz

o: '(t '41')
rfd‘fdj W
1%“ M~7l9

(x) 2 I -+ '
5(426“) = 9 (m ~m)
4- similar torus.
It can be seen that the integral over each integration variable
covers the range of that variable appropriate to physical values of the
energy-square in that channel. Of particular note is the t-channel

integration which begins at the 2-pion threshold {IV/4‘ rather than the

'9

20
Nii threshold tsl/u‘,

Crossing symmetry is then translated into a set of symmetry relations
on the spectral functions cg“, 0L4, , 0L”, so that the Mandelstam represent-
ation explicitly satisfies the conditions of Eq. (2.l7). Since an apriori
knowledge of the structure of the spectral functions is not contained in the
Mandelstam representation it does not serve as the sole basis for determin-
ing the amplitudes Act) (s,t,u), Bct)(s,t,u) although at the same time it
does determine their analyticity properties and also serves as a strong
constraint or consistency requirement on any proposed solution.

To transform the Mandelstam representation into a useful calculation
tool it is convenient to reduce it to one-dimensional form. By formally

evaluating one integration in each of the double integrals and exploiting

(3.1) can be reduced( 8)

 

 

 

 

 

the symmetry of the spectral functions 07-,- ,
to
/
+
A(:,t,x)"fif;‘:: (‘4’ t) 4' :4. " 4"“)
(“0,03
(t) 2 (3'2)
at) , I ..
flatmfl'fOIATB, (‘1‘!) 4'—.—4 +45“)
(mt-30‘“
/
"' P? (m - ’tzmhu)
(1') (t) _ ,
In the equations one is guaranteed only that A; , Bi , the l-dumensuonal

spectral functions, are real. This fact allows us to separate out the

real and imaginary parts of each side of the equations. We will use

A (3:) :)

(s,t,u) as an example although 8( (s,t,u) also satisfies the same

2|

. . I
conditions. Let s,t,u have values such that only the term“n,¢_becomes

singular. Then Eq. (3.2) has real and imaginary parts

on (3)
kg, A12)" *2 u) = é 9 [d4 )4, (431)4-4q

(nu-«'74P
a. (:l) I I
I! if $541 (4. {23732
Cur/L” (3 - 3)

- I (’2 I I
«réfa’tA, (to—07:?
*3
(t)
unguaA/‘d/t, “-J = A,/‘4')t)

The second equation allows us to identify the spectral functions unambigu-

ously. Substituting this relation into the first of Eq. (3.3) gives

 

to "-51.24 (431:)
bA’4,*2“)='—Lf J41

“’u‘ JLL’ (3.“)

We have replaced the dummy variables 5' by u' to get the second term and
have implicitly carried out the procedure used in obtaining Eq. (3.3) for
each channel. Thus, one of the terms in this equation may be thought of
as a Principal value integral depending on the values of s,t,u selected.
Also, if s,t,u are all simultaneously outside their respective physical
regions none of the integrals in singular and hence the amplitude is pure
real (except for the Nucleon pole terms atAm.‘ ora-A’ in Bfl’) (s,t,u).
The imaginary part of any amplitude should thus be required to vanish
outside the physical energy region of any given channel.

The Mandelstam representation explicitly exhibits the real part of
the amplitude in terms of contributions from the imaginary part in each

channel, but in order to calculate the real part, the spectral functions

22

(imaginary parts) are assumed to be stable against cross-channel contri-
butions. This implies that the integrand in each term possesses the
threshold properties and high energy convergence (or divergence!) prop-
erties of the particular channel over which it is to be integrated. For
example, the imaginary part of A(i) (s,t,u) is assumed to contain only
contributions from non-strange isotopic spin l/2 or 3/2 Baryon intermediate
states for s- or u-channel integrations and the t-channel term is assumed to
be calculable in terms of non-strange Isotopic spin 0 or I meson intermed-
iate states.

One of the most persistent difficulties in the path of using the
Mandelstam representation, even on the modest scale outlined here, before
the advent of Regge theory, was the fact that the integration in Eq. (3.h)
extended to infinite energy in each channel while only a few low energy
resonances were known. This led to truncating the integrations at a point
above which little was known about the spectral functions( 9) as well as

(‘0)

the use of subtractions and the introduction of phenomenological

constants(ll) to account for 'higher-states', to mention only a few of
the more popular responses to an unsatisfactory situation.

While these approaches all relied on the obvious fact that nearby
singularities ought to influence the amplitude at a particular point
more than those farther away in the Mandelstam plane none is as intrin-
sically satisfying as the possibility of taking high-energy behavior
into account more or less analytically via Regge poles and/or cuts.

The subtraction approach provides a sufficient number of similarities
to the philosophy of Finite Energy Sum Rules that a few words about this

method are in order.

Suppose one was presented with the dispersion relation of the form

23

I £7I/47 'I ’
R.— ffab) = if] M"4é 3‘
Without detailed information about the high-energy behavior ofo/f) it
is advantageous to also write
{4' I
_ _L W
3.2%; — 22f .2 0’4

so that the difference of the two equations appears as a subtracted dis-

persion relation:

M Jami/“'2 0/ 2
Raf(49=/Q¢}(°) "’ fi/MYM’WP) 4‘
The integral appearing in this equation now is of smaller magnitude than
previously and also converges more rapidly. Of course the price paid for
this improved property is that a new piece of input data has been introduced,
or another phenomenological constant has been introduced.

Looked at in another way, this equation can be interpreted as having
replaced the problem of evaluating the function Re f(x) by the problem of
evaluating the difference AWN? Re(f(x)-f(0)).

One could extend the subtraction approach by performing more than
one subtraction or by evaluating the derivative of Re f(x), although the
price to be paid for minimizing the effect of the high-x behavior would
be the introduction of a successively greater amount of input data or a
host of phenomenological constants. However, if one can regard the high-
energy behavior of the spectral functions for a suitably('2) chosen
amplitude as essentially known, all of these problems could be avoided.
This is the basic point of view underlying Finite Energy Sum Rules.

Suppose, for example, that the Regge parameterization of A(t) (s,t,u)

satisfied the Mandelstam representation as expressed in Eq. (3.4). Then,

24

4 .
writing Eq. (3.Q) for A(’)(s,t,u) as well as for its Regge-asymptotic

(1' )

parameterization ARegge (w,t,u) and subtracting these two equations one

has
°' (0
(’ Jaezkzéiéfiilél
ext...» =22] -... w
(mu/U: .
’ on Awa’f 0/ 2 (3 S)
175/ a’—a¢ u.
(“79" (5)
£4 (#24) 2
*‘k/ w—t ”It
97¢"
(1.) (1’) _
where A (4,,tu)5 Au, t,’u.)-—A (45,»)

4.
Now, since the amplitude A( )(s,t,u) must of necessity approach the Regge
, (2)
amplitude as energy increases, the function A (4,1, “) must vanish
beyond that point. Therefore we are in a position to analytically truncate
the integration of Eq. (3.5) at some finite energy in each channel.

Equation (3.5) can then be cast in the form

13
2:2, ,
flaA ’42-'2 “2" ”afl'A—J” .4 J‘ / I1"-
(4'70" (Zip/0);” V/‘
In this equation so, to, uo are the values of the energy-squared in each

channel above which the difference Aflfih 19,“) is required to vanish
(the limits of the 'Regge' region in each channel). If we were to evaluate
(3.6) at some point well above the 'Regge limit' in each channel, the
left-hand side would be zero. One could then expand the denominators on

the right-hand side to obtain
"0

fl
2 - , (:2,
0:2 5 [“4“ ’fAM-Ava"2"’°I-4I
M'0 m (dayd‘
. (1) (3.7)
+ ‘4'“"fu’mAcA raffle/(24’
lousy)“

25

t.
. - (A. (3
+t"f{ cLyA?¢:‘)o/f’
fin‘

In this equation, Bn are the coefficients of the binomial expansion.
Equation (3.7) could naively be assumed to require that the coefficients
of each power of the energy-square s,t,u vanish identically. However,
since only two of the variables s,t,u are independent these coefficients
are likewise not independent. One could, however, expand one of the terms
(f'wflg) in terms of s,u and reduce Eq. (3.7) to the form
" r:
a 2 ,
0 =55... {...-Ml [am-(WA (4.0.222
and "W‘
*0
r
4. rkliQ-Au—A t?‘,’“)Jt:7
#‘

u.
,. “-4—! [ jétflravqkeA'fig #2 Ju’
(“or

+ f: (t 9.1.... A‘h/a) It? ]
r}

The weight functions )Af/t’) is a simple polynominal in terms of
positive powers of IK" . However, one could now require that the coeffic-
ients of powers of s,u vanish identically. Such a relation has the form

of a ”Finite Energy Sum Rule“.

We have deliberately been rather unspecific about a number of points
in this development of Finite Energy Sum Rules (FESR), precisely because
although the Mandelstam representation forms the background for intro-
ducing the concept, the FESR as actually used cannot be rigorously derived
from it. Our interest in these relations stems more from the fact that
they seem to be obeyed by 1rN scattering amplitudes than that they can be

derived in some rigorous fashion from the Mandelstam representation.

B. The Finite Energy Sum Rules

 

The FESR as we actually use them were first introduced by Barger,

26

Michael and Phillips(l3). They derive from the work of Dolen, Horn, and

(1“)

Schmid and subsequent investigations of Chiu and DerSarkissian(]5).
They can be extracted more or less from the Mandelstam representation as
expressed in Eq. (3.5) if the u-channel term on the right-hand side is

first eliminated. That is, let us rewrite (3.5) as

‘. t
/€¢A(4,t,u.) = 3" ‘u‘. "‘ 7i" ‘ t -
luv)‘ 9'

We will examine the motivation for neglecting the u-channel terms shortly.
Note that both terms on the right hand side have been written as fixed-u
integrals although they did not appear in quite that form in the Mandelstam
representation. Further, the amplitudes 13‘!) are to be considered as
written in terms of either A(2)(s,t,u), BC!) (s,t,u) or any of a fairly
wide range of linear combinations of these amplitudes. This is to allow
for the writing of FESR in terms of amplitudes which can be saturated by
some small number of Regge pole contributions at high energies. For the
same reason we will also write the FESR in terms of u-channel isospin,
rather than the t-channel isospin indicated by the superscripts (i) which
have been carried along to this point only because of their convenience
in expressing the Crossing relations (2.l7).

It is convenient to write (3.9) in terms of the variable X defined
as

X '3 ffd’t) =4'M‘—,¢‘*I"¢ = ”t+m‘+,u'-fw (3.9a)

In terms of X,u we can express (3.9) as

4:4. é) , ’ £31 a) ’
fleArj)xu)=%f‘mfi—“‘WJX *yé/de’ (3.10)
’ #:(avf)“ t'fi“

Now choose values for so, to so that the range of integration with respect

to X is symmetric about zero. This implies that only one of the quantities

27
so, to can be arbitrarily chosen and, as we shall see, the value of so
is rather well limited by available phase shifts to be so = h.8 GeV2
which fixes to to be h.8 GeV2 as well, considerable above the 2-nucleon
threshold hmz = 3.52 GeVZ. These values correspond to the limits
X =11(3.9 GeV2 + fiu): 3 X0. The fact that our choice of to has been
'forced' does not at the present time cause any glaring conflicts with
data available on the process flit-’N'N. In terms of these explicit limits

on X equation (3.lO) can be written

+

A'"
3422,“): 'L W‘Zé‘fiA—A xm‘a/x (3.11)

Expanding the denominator for some value of X such that [X] >>/x./

gives

ng x""fx "ea... Act (2: 'u) JX

which can be satisfied only if

f;2«.4 A‘JJ/X,60JX= o ; mac, an integer. (3.l2)

-x,

if the Regge and non-Regge terms are separated the FESR can be put in the

form
fx°“.!2ef Janna/x fz’X’ZéeFero’X (3.13)
.x0
('5)
This is the form of the FESR as written down by Barger et.al. which

we wish to examine in terms of much more realistic spectral functions
(I)
It should be noted that while the spectral functions which entered
into the Mandelstam representation vanished at points outside the physical

region of a given channel, the Regge term in Eq. (3.l3) will be seen to be

28

non-zero throughout the entire region - X. 5 x .1 X, ,

It should also be noted that while we write Eq. (3.l3) in terms of
the 'imaginary part' of the appropiate amplitude, the proper spectral
function is the so-called 'Discontinuity' defined for some arbitrary
amplitude f(x) as

'Dm'fm) 5:233... (fawn) -fr2¢-..-n)/2; (3.1L1)
Further, if f(x) is to be an analytic function of x, it must have the
property that f(x)* = f(x ) so that Disc f(x) = Im f(x). The fact that
the variable X is defined in terms of -t in Eq. (3.9a) interchanges the
two terms in (3.lh), introducing an over all (-) sign into the contribu-
tion of the t-channel singularities in the FESR. These considerations do
not affect the s-channel terms since X and s have the same sign. Nor is
the Regge term affected since, as we shall see shortly, it is constructed

in terms of u-channel singularities which are independent of X.

C. The Concept(s) of Duality

 

'Duality' refers to the idea that the separate channels of a given
reaction are not 'independent'. The word is (or has been) understood in
a variety of ways variously categorized as 'Strong duality', 'Weak
Duality', etc. The difference between the various uses of the word hinge
on the degree to which the separate channels are or are not thought of
as 'independent'.

In terms of the Mandelstam representation as expressed by Eq. (3.4),
the separate channels contribute to the amplitude in an identical fashion
(through a fixed momentum-transfer integral over the physical cut in
energy). In this sense the separate channels are completely independent.
Further, the spectral function on the energy cut in each channel is con-

structed solely from intermediate states occuring in the channel (i.e. the

29

spectral function is not constructed via exchange mechanisms). Exchanges
appear when the Mandelstam representation is evaluated only because it is
not possible to select values of s, t, u simultaneously in the physical
region of each channel. As evaluated in the Mandelstam representation
the real part of an amplitude appears as a superposition of direct-and
crossed-channel contributions.

Now consider the situation as viewed in Regge theory. In the forward
direction the s-channel amplitude is calculated in terms of t-channel
Regge pole exchange, in the backward direction u-channel Regge exchange
dominates the amplitude at high energies. It would be formally possible
to write down a direct channel Regge pole contribution, but in fact this
is never done. Admittedly a direct channel Regge pole would be highly
singular if it were represented by the simplest form of the trajectory
(real 0L). But, even if the imaginary part of space were included a
direct channel Reggeon would not contribute appreciably to the amplitude
at high energies. This is due to the fact that the recurrences at high
energy couple weakly to the elastic channel, being intercepted far from
the physical region (at large Just“).

The conflict implied by the co-existence of these two views on the
structure of the scattering amplitude is heightened by the fact that in
the Mandelstam representation the close-by singularities are thought of
as being the greater contributors to the amplitude at some point, while
in Regge theory it is just these direct-channel contributors which are
eliminated.

The means by which this situation can be understood is the existence
of Finite Energy Sum Rules in the context of Duality.

(17)

The work of Dolen, Horn, Schmid is the first element in the

30
explanation of the relationship between Regge theory and the Mandelstam
representation. These authors were able to show that in the case of
charge-exchange the )0 trajectory extrapolated to low 5 was a good repre-
sentation of the average scattering amplitude as a function of energy
along lines of fixed t corresponding to forward TIN scattering. Such a
statement is equivalent to an FESR. A function and its average have
identical moments. This is the simplest sense in which Eq. (3.l3) can be
understood and the weakest statement of Duality. However, Dolen, Horn,
Schmid noticed an even more interesting character of the amplitudes. It
was unnecessary to integrate over the entire low-energy spectrum in order
to get the scattering amplitude to 'average out' to the Regge trajectory;
Una/o trajectory appeared to average the scattering amplitude over much
smaller regions of energy. Thus the Regge exchange contributions could be
said to represent the 'local' average of the scattering amplitude. Sub-
sequently Dolen et.al. attempted to ascertain the validity of the local
average concept in the backward direction. Although they found the concept
to be valid near the point at which the amplitude becomes 'pure'Regge,
this is to be expected. We shall show that the local average idea cannot
be maintained convincingly at lower energies.

(18),

The ideas of Dolen et.al. were further elaborated by Schmid who
observed that it was possible to extract 'resonant' partial waves from
the,;>-Regge charge exchange amplitude which agreed surprisingly well with
three established resonances. That is, the Regge amplitude corresponded
to a 900 phase shift at C.M. energies l920 MeV, 2l90 MeV, 2420 MeV, the
positions of three 'well-established' rtN resonances (The I920 and

2h20 are listed at I990 and 2650 (j=?) in the January I970 data tables).

This simple Regge parameterization coincided so well with available data

3|

that Schmid's work could be interpreted as suggesting that a sufficiently
well elaborated Regge model might be able to supplant the direct-channel
resonance model, even at low energies. This is Duality in its strongest
form.

While the successes of Schmid's calculation have not been achieved
in other cases and might be laid aside as coincidence, the equivalence of
Regge and resonance models can be demonstrated in the so-called 'Regge'
region even more convincingly than in the low or intermediate energy range.

(19)

This demonstration has been carried out by Lichtenberg et al. These
authors were able to show that the backward 11M differential cross section

in the energy range 2.24 GeVS Wc .5 3.2 GeV could be fitted in a

m

(20)

direct-channel resonance model as well as with u-channel Regge exchange
While not all of the resonances required can be said to be well established,
the authors demonstrated conclusively that the resonance model could be
extended a considerable distance into the region in which the Regge model
has been considered the only way to fit data. In this form Duality appears
in a way stronger than the requirement that the FESR be satisfied, but
weaker than the requirement of absolute interchangeability suggested by
Schmid's work.

In the FESR written down by Barger et.al. the Regge term is constructed
from u-channel Reggeons, the N and.A.trajectories.. The FESR are evaluated
for 06 a- t to GCV‘. Referring to Figure I, it can be seen that for X
approaching Xo these u values correspond to backward scattering in the
s-channel and for X approaching -Xo backward t-channel scattering. Meson
Regge Trajectories (~uc'"”") are suppressed because either 5 is small
or, for large s,<1(t)-/ is large and negative. Therefore, there is no

u-channel contribution to Eq. (3.9) and (3.l3) (a u-channel amplitude

32
would assymptotically approach a t-channel Regge amplitude in the forward
direction (along a line of fixed t) or an s-channel Regge amplitude in the

backward direction (along a line of fixed 5)).

IV

THE SPECTRAL FUNCTIONS

A. The Regge Amplitudes

The FESR are written down in terms of the amplitudes.
111,0 _. MA MA
Ema = + A (X,u) - (firm)5(x.w) w")

The subscripts I should not be confused with the superscripts used
to denote t-channel isotopic spin. The superscripts N,t3 refer to u-
channel isotopic spin l/2. 3/2 respectively which are related to s- and

(21)

t-channel isotopic spin by

(3/1) (,5)

at (V2) (‘9

F =-3’F +iF =/—‘ -— a—F (2.21
A (Ila) (I/z) (+2 6-)

r = 3F + if“ =F +F

Thus, for a given moment number,n, in Eq. (3.l3) there are four FESR,

N N

one for each of F+ , F , F+ , F

2A .
The amplitudes F:’ can be related to the s-channel spin-flip, no

spin-flip amplitudes by applying the crossing relations (2.l7) to the

2 2 (it) (i)
amplitudes F;(4,-t) alnw/(I—fl")-F,,,(4,f), related to A , B by Eq. (2.l5).

One finds

(t) (.2)
[ (.4,t, 09) = 1' Em, ea.)

(:2
/E:(%b'£¢€) =’it/€:£2u,t,4v)

Singh(22) has shown that the amplitudes fl and f2 are the appropiate

(5.3)

33

3A

ones to Reggeize. The factor W is introduced to avoid a kinematic sing-
ularity at s(or u)=O. Eliminating the factor (£itan) simplifies the cross-
ing relations (h.2) and avoids the possibility of suppressing low-lying
Regge recurrences.

The subscripts refer to the t’P values (signature X parity) of the
trajectories which contribute to each amplitude at high energies. The
eight possible trajectories, four for each isospin value corresponding to
the various choices (‘F,P) = (ififli), are listed in Table I along with the
labels customarily used to refer to them.

Regge analysis of the existing data are consistent with the assump-
tion of zero residue for the N t" N‘ , A“, A, trajectories(23). Thus
the number of trajectories we need consider are reduced by half.

The Regge amplitudes are parameterized as

5:;,.1= 22212
where no
R23 flim/(m2mmwm)
3’2"“? 3 87tlm-M'Ia. +(m+m)b)
7.7m = in: (a. max/2 22222,, 16"”

a2=a.,cc'“' , b-b,ec‘“'
oL:(It) = -o.as «2.9/22.

062(sz = 0.2! + .2“

An explicit parameterization is necessary for only one of the residues

XQIMEI for each isospin since the other can be determined by McDowell

(2A) which

Symmetry requires that

35

Y, ME!) = 4,222.)
This is simply an expression of the content of Eq. (2.l5)under the con-
dition of allowing A(:) (s,t,u) and Bct) (s,t,u) to be defined for neg-
ative energies (waz'is the C.M. energy in the u-channel).

The constanta a0, b0, c0, do, f0, 90 introduced above are assigned
the values 0.8, l.8, 0.5, 0.2, 0.09, -0.7 respectively (u is measured in
GeVZ). It can be seen from (h.h) that the Regge spectral function have
well defined X-parity for real u>0, namely

£19509“) = " T$~Er—x,w
The well-defined parity of the Reggeons allows them to be isolated in the
FESR. The N“_, N; trajectories contribute only to odd-n moments of the
FESR, the A ’A)‘ to even-n moments only. Thus the even-n 'N' FESR and
odd-n 'll' FESR have the form of finite energy superconvergence relations
for the total amplitude. However, we are not surprised to discover that
the amplitudes do not exactly satisfy these superconvergence conditions.
While the residues of the corresponding Reggeons are consistent with zero
residue, it is likely that they are non-zero although they are apparently
much smaller than the other Regge residues. This contention is supported
by the fact that the 'zero' moments of the FESR are always very much
smaller than the corresponding non-zero ones.

Our FESR calculations are limited by the above consideration to real
0‘ 2° . Further, since the Regge fits are done for real oLlu.) , we are
limited to values of u below the u~channel threshold (the Regge recurr-
ences have zero width and hence correspond to singularities in the u-
channel physical region). Thus we limit our calculations to the interval

as as IGeVZ. This region in the Mandelstam plane is shown in Figure 3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ISosPln TP (T,P) 1.8ch m
+ + Na: odd
.+_
I= 2i (N)
‘4' ‘- IV, 0J¢J
... N‘
+ + A“
+
‘- T- 3 even
I = 23- (A)
‘I’ “' A5
"’ + A; even

 

Table l. The Baryon trajectories and their quantum numbers. The
second column lists the subscripts of the u-channel helicity

amplitudes to which a given trajectory contributes.

The last

column lists the values of the moment number n for which the
corresponding Regge amplitude yields a non-zero contribution to

the FESR.

 

37

 

‘a¢ 1: r1: 4 N N

 

1_L_.,«=4 3

 

 

 

 

u - clan/rel
Phys/cal

reg/on

 

 

 

 

Figure 3. The region of integration associated with the FESR. The
cross-hatching corresponds to lines of fixed u and X.

38

B. The s-channel Spectral Functions

 

’

A
In order to determine F (X,u) on the cut Aalarf)’it is necess-
ary to specify the partial wave amplitudes fl,‘ over the range (b+q)2

2
smax = h.8 GeV . Then equations (2.lh), (2.l5) and (4.l) deter-

UK

5 s
mine the spectral functions $61.23,“). The partial wave amplitudes are
specified by the (real) phase shift, 5, and the (real) inelasticity,”(,
as in (l.ll+). Unitarity limits the inelasticity to be in the range 0171/.
Each partial wave state of good 1, l, I is specified by a separate phase
shift and inelasticity, thought of as functions of energy, although 1 is
identically one below the limit of pion production (I.2l8 GeV). In
practice only a relatively few lOWfi‘ partial waves are assumed to be significant
Phase shift analysis is commonly done by one or both of two methods.
In the first method a set of values are chosen for the phase shift (includ-
ing the inelasticity) for each state at each energy and these values are
used to 'predict' the measured cross sections and angular distributions
at various energies. Discrepancies between the predictions and measure-
ments are then minimized by varying the phases and performing subsequent
predictions. This process is iterated until some error criterion
(typically a Chi-square test) is satisfactorily minimized. A 'smooth'
curve connecting the final values of each phase shift and inelasticity
then represents the energy-dependence of that particular parameter. In
fact, the initial choice of values is not so arbitrary as it might appear
from this discussion. Available knowledge of the known resonances can be
used to fix rather accurately both the phase shift and inelasticity of
some states over a wide energy range. It would be difficult to ration-
alize arbitrary choices for the P33 phase in the range W 4 L3 GeV for

example. Some ambiguities can be resolved by imposing dispersion relation

39

constraints. The Chew-Low ambiguity was resolved in this fashion. Phase
shift analyses which extend to high energies also have the existing low
energy analyses available to minimize these initial choices. This approach
is often characterized as 'model-independent' or phenomenological phase
shift analysis.

The second approach is to predict cross sections and angular dis-
tributions as a means of fitting the phenomenological constants of some
model of the rFN interaction from which phase shift and inelasticity
information can be extracted. The resonance model is the major one of
these 'model dependent' phase shift analyses. The parameters to be fitted
in this approach are the masses, widths, and inelasticities of various
hypothetical Baryon resonances. The 'Roper resonance' (N (l520)) is the
earliest example of a resonance 'discovered' by phase shift analysis.

The use of dispersion relations as additional constraints can lead to the
simultaneous determination of meson resonance parameters.

While we have discussed model-dependent and phenomenological phase
shift analyses as separate approaches, they are frequently used as parts
of the same analysis. This is the case with the phase shifts we use - -

(25). This

the CERN PIP phases of Donnachie, Kirsopp, and Lovelace
analysis resulted in two separate sets of phases and inelasticities, the
'TH' and 'EXP' solutions respectively.

The EXP phases are the model-independent set. They provide a good
fit to rsN data in the range 1.078 GeV<W< 2.l90 GeV (chi-square of
~1000 for ~l+ooo data points) and provide the 'data' for the model-
dependent TH solution. The reasons for attempting a second solution in

the face of such a good fit to data are that the EXP phases do not possess

very smooth energy dependence and also do not provide a meaningful

A0

theoretical model for the flN interaction (it is virtually impossible to
extract resonance parameters, etc. from such phases, even though they do

fit the data). The TH solution is extracted from a resonance model by
fitting the EXP phases as data and imposing dispersion relation constraints;
i.e., the dispersion relations were saturated by the EXP phases. The
resonances which figure in the TH solution are supplied only if an accept-
able solution (the EXP phases are the guide here) cannot be found for a
particular partial wave state without one. This raises the problem of
distinguishing resonances from uncorrelated 'phase space' or 'background'
effects. The phase shift associated with the 'background' can not only

be quite large but can also vary quite rapidly with energy while its
associated inelasticity is a (presumably) slowly-varying function of energy.
Thus Lovelace et.al. identify a resonance by a minimum in the inelasticity
rather than a phase passing through 90°. By this criterion nine new

Baryon resonances were identified. Coupling constants were also deter-
mined for the f(1 = l) and 0’(1 = 0) mesons. Energy-dependent phase

shifts and inelasticity parameters were determined for 22 partial wave
states, 5,, through H3,11, throughout the energy range l.078 GeV. 5
W$2.l90 GeV.

The TH phases describe partial wave amplitudes which are smooth
functions of energy with a tabulated resonance structure. In this respect
these phases are an admittedly significant improvement over competing
phases produced at Berkeley and Saclay. However, the TH phases are
markedly inferior to the EXP set in reproducing experimental data, partic-
ularly for nQ’elastic scattering. This was to be expected since the TH
phases were fitted to the EXP phases as 'data' and therefore cannot be

expected to fit cross sections and angular distributions as well as a

A]

set fitted directly. The recent rediscovery of this fact has caused some
comment as to which set is 'better'(26). The result of this discussion is
that while the TH phases are superior in terms of interpreting nrN inter-
actions, the EXP phases are more desirable for purposes of calculation.

Nonetheless, the TH phases were commonly used to generate 1tN ampli-
tudes until quite recently. We have examined the TH and EXP amplitudes
and find that the EXP amplitudes possess better convergence pr0perties,
i.e., they more closely approximate the Regge amplitudes near X = X0
(5 = h.8 GeV2) than do the TH amplitudes. The TH, EXP, and Regge ampli-
tudes in the region near X = X0 are shown in Figure h and S. The contri-
butions of the TH and EXP amplitudes to the FESR are virtually identical,
as we shall see, despite the evident discrepancies.

In addition to the physical cut A7,!m9p)‘ one must also include the
Nucleon 'Born term'I which contributes a singularity of the form 92/ (s-mz)
whose imaginary part is just a delta function. We choose the value of the

2
coupling constant so that 92/4 = l4.6.( 7)

C. The t-channel Spectral Function

In the region 't 3»fin3 the spectral function is constructed through
Eq. (2.l9), which exhibits the amplitudes A(i) (s,t,u), Bet) (s,t,u) as a
sum of states of good (t-channel) total angular momentum and like or
unlike Ni helicity.

Contrary to the case of the s-channel cut, little direct information
is available about the t-channel amplitudes. The region “fut ‘ Val,"
is experimentally inaccessible. Little information is available above
the physical nucleon threshold. However, the relationship between the
process Ira-"Ii and elastic rtfl scattering provides something of a
guide to the determination of the amplitude near t'Y/a' . Further, Barger

and Cline(28) have shown that a Regge model using N. and A; trajectories

10‘

42

 

 

 

 

 

av"

-zal

 

 

/ s
I \
I \
I \
I \ o . 0
I l . . L o . . .0 ’
' f. \ 6 AW . 7———i
’ . o 0 I “ O I
I g .. \ .. I \. I
. . . . o ‘ .0 ’ 8? ~ -'
O . l O.
'0‘ .. I .0
‘ .° I .0
0.. . . I o.
I ...
I 0,
l , -.
l I ..
\-l
r T
3 )(°
X to 00V: (X. a 3. 9)
Figure #. The spectral function for the state described by

Fa (X,u) at u=0. The region of X near X is shown. The
solid curve is the Regge spectral function. The low energy
amplitudes generated by the TH and EXP phase shifts are
shown as dotted and dashed curves respectively.

 

6w"

1+3

 

O

 

fifl"

-{O-I

 

 

X M Gav: (X.' 3.?)

The spectral functions for the state described by
F: (X,u) at u=0. The region of X near X0 is shown. The
solid curve is the Regge spectral function. The low energy
amplitudes generated by the TH and EXP phase shifts are
shown as dotted and dashed curves respectively.

Figure 5.

 

Ah
is consistent with early data only/[5 vflfi'at backward angles (small u).
A reasonable t-channel spectral function should therefore be expected to
converge to the corresponding Regge amplitude near t=quv2, The spectral
functions developed in this chapter show rather good convergence properties,
especially at small u.

The t-channel spectral functions are constructed as follows. Begin-

ning with Eq. (2.l9)

(J) (t) a

. I 1 '
Alissa-f) 3.3;“ §)(f’7) [fi{(flm¢€(m¢)

mg
" +(é) Efm¢)]
(t) Z L [+722 "Ala” 1
Baud) =37! 177.77 (P?) -(t) 1 (can

1

. . w; . (I) (I)
one needs to specufy the set of amplitudes f1 60 to determine A , B
. ‘6‘
and hence the spectral functions f; (X,u),

The simplest approximation which is used to generate spectral
functions is to saturate the amplitudes with zero-width meson resonances,
parameterized by a mass-square and a coupling constant for each of the
NW helicity states. Notice that the system cannot couple to an inter-
mediate state Baryon.

In this approximation each resonance would be accounted for by a

'Born' pole of the form

C!
f36h)== 551:;;-'

corresponding to a massxn43175 and two free parameters C,. Spin and
isospin labels on f;(t) will be the quantum numbers of the resonance.
This is the approach used by Barger. He included four mesons; fya,f.,7.
Coupling constants for the/o and 6 were available from the work of

Lovelace. Barger fixed the coupling constants of the f0 and g by

45

requiring that the FESR which isolated the N,” Npfi. , and A, trajec-
tories be small compared the the No, FESR for low moment number n (see
Table l). A complete list of Barger's resonance parameters is presented
in Table 2. Notice that the I=O, }=0 member is here labelled '6' rather
than 'cr'. Also, by inspection of Eq. (2.l9) it is clear that the units
of the coupling constants C: must depend on 3. Also, the/‘7is listed at
an admittedly low mass. As Lovelace points out(z9) this is a consequence
of attempting to represent all the weight of the low energy J=l, I=l
state by a simple pole. This part of the cut is sufficiently near the
s-channel threshold that the distribution along the cut is as important
as the total weight. We will shortly adjust the masses of both theJ/D
and f'to more meaningful values.

The FESR calculations of Barger, et.al., using the zero-width mesons
listed in Table 2(along with the CERN-TH phases to generate the s-channel
spectral function) produced remarkably good agreement for the lowest
moment (n=l) No, and N,i FESR. However, since the imaginary part of a
Born pole is a delta-function the t-channel spectral function is not
continuous and therefore cannot converge to the corresponding Regge ampli-
tude in any meaningful fashion. Thus, without a continuous spectral
function the FESR are not as meaningful as they might be - - if Regge
convergence could be explicitly demonstrated. For this reason we deter-
mined to replace the zero-width spectral functions by continuous ones and
see if the FESR could still be satisfied. The continuous spectral
functions arrived at not only satisfied the FESR but also show good con-
vergence behavior although not constrained in any way in this respect.

The new spectral functions were constructed as follows. First, the

Born poles were replaced by Breit-Wigner poles of the form

us

 

M35 5 I‘Spl'n Spin C-o- C-

 

e £37m o o .chev’ """"

 

p .5‘Ho.v. 1 1 1.56m 3.1

 

f; (.2506!V. O 2

11.5’ Gav." IO. 7 Gw‘z

 

 

 

 

 

 

<3 LGGOGcV. 1 5 4,2 a.v'3 aflaev'“

 

 

Table 2. Quantum numbers and coupling constants for the zero-width
mesons used in Barger's FESR calculations. Note that there is no C-

for the 1=0, J=0 state since only like helicity NN pairs can occur in
this state.

 

1+7

_ 32
flit) ' t,—t ~znr'

The values of M and!“ corresponding to each meson in Barger's list were
selected from the data tables. This involved a considerable change in
the masses of the/p and 6 . To fix the coupling constants 9; it was
required that the overall weight of each t-channel amplitude remain un-

changed. That is, that

 

n’C Swan/t = k Mg:
.,.f . .,.l at

This reduces to

 

a: = C: (Ll-5)

’/2. v (Wt) Tan "fig-gav/Mrl

Notice that g, approaches C: in the limit of zero width. The singular-
ities associated with the zero width parameterization of the intermediate
states have now been eliminated. No new arbitrary parameters were intro-
duced in the process.

The spectral function associated with the Breit-Wigner resonance
form is still not completely adequate to allow the convergence properties
to be thoroughly examined since A(:) (t,cos¢) is singular at the NN
threshold. To see this observe that cosé diverges as (pq)"l at p=0.
Thus, while the terms (pq)‘P1’(m¢-) and (pq)’ P1 (cosf) are finite at
p=0, the over all factor l/p2 produces a divergence. To remove this
singularity we must take into account the threshold properties of the
amplitude F‘?’(*).

For the spinless pions, orbital and angular momentum are identical
so that Jmffl’g must behave like 7"" as q approaches zero.
Similarly the absorptive part of any amplitude corresponding to good

- 2.12"
orbital angular momentum of the NN system must behave like P as p

1+8

approaches zero. In Appendix D the amplitudesfgw (f) are decomposed into
contributions from states of good.L(=11l) for the NN system. The results,

expressed in Eq. (2.22) are

a):

-)1 ’1‘)!
.,ft} =f(21+1) ”if” -fi(21+l)h(£;' 15)
cm (1)! as "3.1
‘f. (t) '17; (23:5) {/0 +75%? 75““)
The subscripts 4:] denote the orbital angular momentuml. of the NN system.

(01
In each of the amplitudes £(f) we replace the fixed width Hf' by the

21¢! 11 4' I
q

t-dependent term ’i/f/ , defining I: such that

1 22 l I
Mr" = I; m '* (7‘3” /,,M.
The absorptive part of each amplitude f: ('1’) now has the form

a fig : 92. J: ”7""? 21.: 2 (1+ 6)
’3! (t_t.)1+(3&1/P/2.2+/q24+1) .

 

The coupling constants 31 are determined by inverting Eq. (2.22) at

tO=M2, which yields

91"12 (2314)”;3- _ (2.44.4) (14.7)

:- (214-2 ”2-
3’1: (21’2”!) +(2::I 3"
There is still another difficulty to be overcome at the NN threshold.

(t)!
The j=l, l=0 state 7:. (1’) behaves like q3p as p approaches zero, which

will not overcome the l/p2 term in A(t) (t,cos¢'), although all the other
states will be finite at p=0. This difficulty can easily be overcome by
momentarily reverting to the Mandelstam representation. In this case
Frazer and Fulco(30)have shown that the partial waves of the amplitudes

+ 1
Act) (t,cos¢), B(") (t,cos¢) converge at least as rapidly as (pq) +

49

an
aim) )*--etc. as either p or q approaches zero. Combining this require-

«9/

ment with the NN angular momentum barrier factor f>11 , it can be shown

that the 4=l amplitude must converge at least as rapidly as p2 as p

approaches zero.

To do this first invert Eq. (2.l9) giving

any

= .1. “£3. “’ 422...... . m a)
,(t) ]fl{ (,9)? A4“? 4' 23¢] (P1,)l" [H 0515,) +1 5—61’]] (14.8)

(1); - —L- iii—1.13.4 "J (r)
1‘7 - 3’" 2: * ’ 9012' "[5,.6” ‘ Brow}

The amplitude we are concerned with is the j=l,.l=0 state. In this case

the threshold properties deducted by Frazer and Fulco reduce to

(a) I

f, (:1 ~ Gut 4' 0’09)
However, this state is composed of 1F0,2 NN states which guarantees that
{:26 converges at least as rapidly as p+'. Thus the constant must be
zero and so the first non-zero term is at least quadratic in nucleon
momentum. This behavior is guaranteed by adding an additional factor of

(or
p in the t-dependence of the width of the j=l, 1=0 amplitude 1:13).

2

Thus no term in Eq. (2.l9) converges slower than p at the NN threshold

and the apparent singularity in A(i) (t,cos4i) at this point is removed.

D. The Preliminary Spectral Functions

The spectral functions constructed according to the specifications
of the preceding sections of this chapter are presented in Figures 6
through ll. Even or odd moments of these functions generate the corres-
ponding FESR.

The parameters of the mesons used to construct the t-channel ampli-
tude are listed in Table 3 along with their coupling constants. In order

to compare these values with those in Table 2 we quote the coupling

50

 

 

 

 

 

 

 

 

 

 

 

 

 

I 4 mfg... ”"2... 9+ 9..
E 0 0 .725 .‘100 /.39 cw"
,0 1 1 .765 .I25 2,57 e.v 14.42 mv"
l; O 2 L253 ./'/5’ 11.95 Gav" ”.13 GeV‘2
9 1 '5 / .660 ./20 4.28 GeV‘" 5,53 cw"
fl! 1 '3 Z./20 .250 2,3: cw“ 3.0, c.v“

 

Table 3.
t-channel amplitude.

Parameters for the meson states used to construct the

 

5]

Figure 6. The spectral functions for the state described by the amplitude
F! (X,u). Moments of this function generate the Na. FESR. The spectral
functions are shown for u=0. The solid curve is the Regge Spectral

function. The dashed curve is the low energy spectrum.

Eigggg_z. The Ff' (X,u) spectral functions u=0. As in Fig. 5 the solid
and dashed curves represent the Regge and low energy spectral functions

respectively.

Figure 8. The F9 (X,u) spectral functions at u=l.0 GeVz.

Figure 2. The F2 (X,u) spectral functions at u=l.0 GeVZ.

E13g£g_lg. The Ff (X,u) spectral functions at u=l.0 GeVZ. The Regge

term is essentially zero in this case.

Eiflg£g_ll. 'The F: (X,u) spectral functions at u=l.0 GeVZ. Again, the

Regge term is essentially zero.

52

o um:o_u

as .. x

 

 

.25

 

 

 

 

 

 

«use.

 

MfixN

 

 

53

 

 

 

 

I

“
-p—-———-"
-
’fl’-’-
d”
A'
/
“
u
~-
~--)
" '
’
’
’
’
\
."

 

 

FIGURE 7

Sh

 

 

 

 

 

 

 

 

 

 

 

23

-l

d

-21]

FIGURE 8

55

m 552.".

 

 

v.0}.

maths

 

 

 

$2.».

 

 

 

56

 

I‘
II

(I

I l/\

I

\

 

 

\
)
I
V
I
K
k,
”s
I’,
I
\
\\

 

 

 

-5' 1'le

-/

FIGURE l0

 

57

 

 

 

 

 

 

 

 

‘Fth
Mo’-

'2x/0’«

Xe

FIGURE ll

58

constants 9, rather than 9; . In addition we have added one more meson

in order to extend the t-channel spectrum to -X This state, labelled

0°
’0', is known to couple to both 271 and NN states and was selected from
the list of high mass suspected mesons for this reason. The coupling
constants of the/0’ were selected to enhance the agreement of the N.,
and N, n=l FESR near u=l.O GeVZ.

Including the/a ands at their proper masses produced rather large
changes in their coupling constants, although the other refinements we
have made affected them little as can be seen by comparing the f0 and 9
coupling constants in Tables 2 and 3.

The s-channel spectral function were generated from the EXP phases
to take advantage of their better convergence properties (see Figures h
and 5).

Before examining the spectral function in detail note that at u=0
the amplitudes F, are simply the negative of one another. In this case
Eq. (th) reduces to F: = 7 (A + m8). Thus at u=0 we plot only F-u'b.
These amplitudes have units of inverse energy; we plot them in units of

l

GeV' . Figures 6 and 7 are the u=0 spectral functions,Figures 8 through

ll show the u=l.0 GeV2

spectral functions.

Some general features can be seen in Figure 6 showing the spectral
function for the state described by F-N (X,u). First, the divergence at
the NN threshold (near -Xo) has been eliminated. The large peak just to
the right of X=0 is the/O . The f0 and g are well resolved peaks on the
negative X axis. The c lies under theJP and can be seen only as a
shoulder to the left of the/0 . The N3: overlaps the} on the right

and appears as a somewhat attenuated dip. The Regge amplitude is singular

at X=0 since, for u=0, «‘3 = -.88 so that the amplitude diverges as X-'88.

59

The convergence properties of the low energy spectral function is
particularly good at u=0. The amplitude is dominated by the resonances
occurring at low X and the higher resonances damp out monotonically as
X approaches 1 X0. This behavior is also evident in F_ at u=0
(Figure 7). However, good convergence properties alone do not guarantee
that the FESR will be satisfied (In fact the AFESR are not nearly so
well satisfied as the N FESR).

The spectral functions at u=l.0 GeV2 are plotted in Figures 8 through
ll. Unlike the u=0 case the oscillations in the low energy amplitude do
not damp out as X approaches IIXO but in some cases even increase, although
a few of the amplitudes still show relatively good convergence.

This distortion is due to the fact that cos¢ and cosO* are larger

in magnitude along the line u=l.0 GeV2

than at the corresponding value

of X along u=0. The higher terms in each partial wave series (the high
energy contributors) are beginning to dominate the lower terms. This is
the situation mentioned previously in Chapter 2. The partial wave series
deteriorates, as any polynomial approximation does, if extrapolated too
far from the region in which the successive coefficients (in this case
the s- and t-channel partial wave amplitudes) were fitted to actual data.

In the next section we shall examine some attempts to deal with this

situation.

E. Transforming the Partial Wave Series

Before considering how the approximation of fitting a scattering
amplitude to a few partial waves can be improved we will first consider
some of the basic limitations of the Legendre series itself.

If we consider energy and angle as basic independent variables to

describe the scattering amplitude in some channel it is clear that cross-

60

channel singularities appear as'poles in the x = cose plane. At a fixed
energy the partial wave series has the form of a Taylor expansion about
x=0. The series converges for '4‘, r 41 4 4;, , where 44. is the magni-
tude of cosOi at the pole nearest to x=0.

In the s-channel the nearest singularity in the cosa’ plane is

encountered at the t-channel threshold z‘-‘za3 . To see this note that

urea/by," = /+ 2.03/10
If we now evaluate u at c059 ==«-(/*'%E:) we find

“v = (“.../‘3)7'4 4- qut

These values are only #fd‘ larger than the value of u in the backward
direction. Hence the region of convergence of the Legendre series falls
far short of the u-channel threshold. This can be seen in Figure l2.

In the t-channel the cross-channel thresholds are symmetric, since

c054 can be expressed as
«w = ”fiat"—
Thus the values of cosqi at the s and u-channel thresholds for fixed t
are the negative of one another and the t-channel Legendre series converges
throughout the region between the cross-channel cuts.

In the context of the Mandelstam representation it can be shown that
the Legendre expansion converges not only along ‘4. 5 4’ 54‘, but
throughout an ellipse in the x = cosO' complex plane. The semi-major
and semi-minor axes are N. and W with foci at x =11. This is
the so-called Lehmann ellipse(3'). Figure l2 indicates the intersection
of the Lehmann ellipse with the region throughout which the spectral

functions are calculated in the present case.

The finite extent of the region of convergence of the Legendre

6l

 

Figure 12. The intersection of the regions of convergence of the
s-channel and t-channel Legendre series with the region in which
the FESR are calculated. '

62
series is a serious obstacle to applying any technique which requires
extrapolating the scattering amplitudes outside the physical region. In
the present case the s-channel spectral function has to be extrapolated
beyond the Lehmann ellipse in the forward direction for small u- and in
the backward direction for large u. For “gel-“$915.9” the entire 5-
channel cut lies outside the region of convergence of the Legendre series.

Extrapolating a finite approximation to the Legendre series (construct-
ed from a few lownl partial waves) beyond the limits of convergence will
obviously produce a finite result. Further, although the procedure is
not mathematically justified, a comparison of corresponding u=0 and
u=l GeV2 spectral functions makes it clear that some of the information
content of the physical region is preserved in the process. One could
reasonably hope to improve this situation in two respects. In the s-
and u-channel the region of convergence could be extended to include the
entire region between the cross-channel cuts as in the t-channel case.
Also it is evident that one can never fit more than a finite number of
partial waves to data in the physical region. Such a finite approximation
to the full Legendre series cannot reproduce the analyticity properties
of the amplitudes in the cosO plane. One cure for this situation is to
replace x by a coordinate which itself is an analytic function in the
c059 plane with cuts corresponding to the cross-channel singularities.

A transformation possessing the desired properties has been constructed
by Cutkosky and Deo(32). In our problem this transformation has been
useful in reducing somewhat the unsatisfactory features of u = l.0 GeV2
spectral functions.

This transformation is effected in two stages. First the cuts

(‘°"."4-), (4o,°") in the c059 plane are mapped into the symmetric

63

cuts (‘0‘,‘W), (Ix/’00) in the w plane by the mapping

w" = (M-qd/(I-Mfii.) (“-9)

where

“do 5 (4-i/p*)/(/¢'+M- +Z+Z- '1 )
.EE+ E Il/fiv‘ "/

W E'W(”¢-) ‘ ”(M-r)

Note first that this transformation maps the physical region into itself.
In the t-channel x+ = x_ so that yo = 0. Then of course4v=4£ In the
s-channel however this transformation extends the region of convergence
of the Legendre series (in.wV) to include the entire region between the
t- and u-channel branch points. However a finite expansion in place of
x would not explicitly exhibit the analytic structure of a 1zN scatter-
ing amplitude since whas only a simple pole at x = l/yo.

The second stage of the Cutkosky-Dec transformation is to map w into
the z-plane by the transformation

I} = M(§(w',f)) (“0)

where
,6 3'/w

f(usfl) = (7W2)F'(.4.¢.J’w-, fl) /Kr;)

b Jag _

frfb,¢h)55 JL ’//—-é:§:zz:§4p'
”/1.
KM) =,[ y—v‘jfl. '5‘...“ =F/i’, a)
The functions K(a) and F(b,a) are respectively the complete and incomplete
3
Elliptic Integrals of the first kind . This transformation maps x -tl
into 2 =tl ( to see this substitute w =:tl into h.IO). The coordinate z

is an analytic function of w with branch cuts (way-W) and (M09) in the

64

w-plane. As a function of x, then 2 possesses the analytic structure of
the cross-channel cuts. To see this note that the denominator of F969;!)
will be singular forlur/>W , i.e., for x outside the range ”LG/$54,. If
we now consider a'nN scattering amplitude approximated by a finite power
series in 2, even this finite approximation contains information regarding
the cross-channel cuts which was absent from a finite Legendre series
in x.

At any energy one can calculate 2 as a function of x or x in terms of
2 very efficiently by means of the Landen transfromation(3h). The Landen
transformation is a set of recursion relations which allows the values
b0, ao of the arguments of F(b,a) to be replaced by a pair of values
bl, a] such that

F(meJ/Krwz) = F(bcaa’O)/K(a’o)

Writing the incomplete elliptic integral of the first kind as

up]
F/M’A,fl)=j___i&___— (LLH)
o '7./"19344bv5fi

the recursion relations take the form

P:=(/-W7,é:,7’)/(/+1/7=’E“) (£4.12)

A, = A, (I .../Wyn .W)
Since Boil/W is less than unity,flm,<fi,,. . If we apply these relations
repeatedly the parameter} will eventually be reduced to zero. From
(4.l0) and (h.ll) we see that 2 will be equal to the corresponding value
of A . Since the successive values of fl are independent of /\ one can
use them to recover x if z is known, simply by inverting (h.12). Details

of the procedure are presented in the work cited in reference 33.

65

This mapping also preserves the threshold properties of the partial
wave expansion. This is apparent since U approaches infinity as either
p, q, or k vanishes. Equation (4.l0) then gives 2 = x. This guarantees
that the t-channel amplitudes will be well-behaved as functions of 2.
Figures l3 and IA illustrate the behavior of 2 as a function of x in the
s-channel. Figure l5 shows 2 and x at a representative value of t.

The Cutkosky-Deo transformation does not specify the form of the
scattering amplitude as a function of 2 which is to replace the Legendre
series in x. One approach would be to replace x by z in the partial wave
expansions (2.lh) and (2.l9), replacing the partial wave amplitudes them-
selves by a set of 'transformed' partial waves.

We have investigated this approach in detail for the s-channel

amplitudes. In this case the 'transformed' partial waves are defined by

£(£,co‘9) =Z£mfijai ~£f5nffra
1:0 ' '1

l

£(fi, ...e) = ;’ {Em " an} fr,»

fl
it”) 5 S; {f (h,m9)f(t) -r fita,mo)fi'n;}de

The transformed partial waves ‘3‘“) no longer correspond to good 1-
and 1 except at threshold; however this nomenclature is more than simply
a convenience. One would expect that the improved analyticity prOperties
of the transformed partial wave series would allow an equally good
description of 1tN scattering with fewer terms in the series. Thus we
might hope to eliminate the ”small” G and H waves from the expansion.
This led us to attempt two expansions of the form of Eq. (h.l3), the

first containing only S, P, D, F transformed partial waves, a total of

 

 

 

 

 

 

Figure l3.
in the physical region just

above the s-channel threshold,
(w=l.l GeV).

Graphs of x and z

 

 

‘ /

.1 o f

 

.J

 

 

 

Figure 14. Graphs of x and z
in the physical region well

above the s-channel threshold,
(W=l.8 GeV).

 

66

 

 

 

 

 

 

1&5
.1
,..:
b‘5
:3- a ma
Figure l5.

Graphs of x and z
in the t-channel covering the
region (-x, x+). These gra hs
were plotted at t=h.l25 GeV .

67

IA terms, and the second containing S through H waves, 22 terms in all,
the same number of terms as the CERN phases used as 'data'.

The results of this calculation are presented in part in Figure l6
which shows the FE spectral function in the s-channel for u=l.0 GeVZ.
The l4 term and 22 term transformed partial wave expansions easily
reproduce the lower energy structure of the partial wave expansion.
However, the IA term transformed expansion fails completely to reproduce
the high energy behavior of the amplitude while the 22 term expansion
wildly exaggerates that structure. Further, neither transformed expansion
possesses acceptable Regge convergence properties. From Figure l6 it is
apparent that some transformed partial wave expansion of more than IA
terms but less than 22 terms might very well be able to reproduce the
content of the partial wave expansion but it would be unlikely to represent
any real improvement in general as a means of performing the extrapolation
from the physical region.

An alternative procedure is to preserve the form of equations (2.lh)
and (2.l9) in their respective physical regions,expanding x in orthogonal

polynomials in z. We have chosen to write

no =);a.,,,fi(2)

(A.lA)
v:
a; (mI/nf’vfimk
Since x =tl map into 2 =1] we have the constraints
ZebM ’/ ; Z(-)“a.,_'-I (lug)
Av fin

In order to preserve the information content of the partial wave expan-
sions in their respective physical regions we require that the constraints
be satisfied to l part in l06. The resultant spectral functions reproduce

the partial wave expansions exactly within their respective physical

3
I 1/0 ,

68

 

 

 

 

 

IO,

 

 

 

 

 

 

 

A /\ R
Wk A
v C "

 

X. 2 X.
Figure l6. The F2 spectral function at u=l.0 GeV near X=XO. A, B, C, R
label the amplitudes generated using 22 partial waves, 22 transformed
partial waves, IA transformed partial waves, and the Regge amplitude
respectively. The value of X at the s-channel threshold Xt is .78 GeVZ.

69

regions but will differ outside.

The criterion we have used to determine when to truncate the expansion
of x in terms of 2 means that the number of expansion coefficients
required in Eq. (h.l4) differs at various energies. In general the higher
the energy the greater the number of coefficients required. Thus this
technique does not produce a spectral function of fixed order in 2 at all
energies. The higher-ordered partial waves are small at low energies but
dominate the amplitude at high energies and large unphysical angles. Hence
at high energies for large u the expansion in z generates significant
contributions from very high ordered polynomials in z - - much higher
than the original expansion in terms of x. Yet, the resultant expansion
preserves the information content of the partial wave expansion well out-
side the physical region. Figure l7 shows the F: spectral function

generated at u=l.0 GeV2

by this method. The expansion in z differs
appreciably from the partial wave Legendre series in x only at high energies.
As we shall see in the next section these small changes will have notice-

able effects on the FESR, although they have only a slight effect on

Regge convergence properties.

 

7O

.on H Loo: XV mo_mcmco ;m_; Low >_co m>u=o

n__om 6:6 sou» mmmco>_o m>cau m_;u umzu ouoz .m>c:o 665mm6 m >6 czozm m_ N mo mcozoa mo mEcou

c_ >_uooe_p x m:_ocmaxu >6 pmumcocom co_uoc:m _mcuuoqm an och .m mc:m_u 50cm mco_uoc:m .mcuuoqm
>mcmcmuzo_ 6cm mmmmm ecu mum mo>cao 6__om och . >oo o._u: pm co_uoc:$ .mcuouam am och .m_ ou:m_m

N
P , . ,P

 

 

 

 

 

 

V

 

 

 

 

 

 

 

V

RESULTS AND DISCUSSION

A. The Finite Energy Sum Rules

The form of the FESR is given by Eq. (3.l3). For the sake of
comparing with Barger's results we introduce his normalization factor
l/l6fl . The FESR now have the form

,TI-gfmt {mm = ,fi/XU... Fonda/X
51' (S.l)

Isotopic spin and signative-parity indices have been suppressed for
simplicity. Since n is unrestricted except that it be a non-negative
integer Eq. (5.l) lists an infinite number of relations between the low-
energy and Regge spectral functions for each of the four choices of
isospin and signature-parity labels. The limited state of present know-
ledge of the singularities occuring on the t-channel cut make it unreason-
able to expect that more than a few of the lower moments be meaningful.
Further, our experience suggests that it is unnecessary to examine a
large number of moments since any improvement in the lowest moment is
always apparent in the higher moments. We have calculated moments up to
n=5 for each of the spectral functions using the standard Legendre series
in cosa and partial wave amplitudes. Only n=l Na, and N5 FESR were
calculated using the expansion of x in terms of Legendre polynomials in
2 according to Eq. (h.lh). Limited availability of computer time made

it impractical to calculate any more moments in this case since approx-

71

72
imately AS minutes of 6500 CP time were required in each case. Calculat-
ing only a few such moments is not a serious limitation since it can be
seen from Figure l7 that the spectral functions generated by the Cutkosky-
Deo continuation do not differ significantly in either qualitative or
quantitative aspects from those generated by standard partial wave expan-
sion since it can be seen from Figure l6 that these spectral functions
possess markedly inferior Regge convergence properties and do not represent

a meaningful way of satisfying the FESR without further elaboration.

B. The Na. and N; FESR

The N.‘ FESR are more well satisfied than any other. The n = l, 3, 5
FESR are shown in Figures l8, l9 and 20 respectively. The real signific-
ance of these results is apparent only with reference to Figures 6 and 9
which demonstrate that the corresponding spectral functions possess good
Regge convergence properties in addition to rather good agreement in the
sum rules. As expected the FESR are progressively less well satisfied
with increasing moment number. This is caused by not weighting the higher-
mass states heavily enough. The confusion regarding the high-mass meson
states made it impossible to include many of them. One would also have
to include some higher partial waves in the s-channel and extend phase
shift analysis to a somewhat higher energy before any marked improvement
could be seen in the higher moments.

Note that the rather small high-energy variations between the spectral
functions continued to u = I.0 GeV2 in x and 2 (Figure l7) produces a
noticeable difference in their respective contributions to the n = l N“,
FESR (Figure l8).

The N,. FESR is not so well satisfied as the N“, sum rules, at

least qualitatively in that the low energy term does not reproduce the

73

 

 

   

O OROO‘OO;OQO-OO:OO-OOOO-OO-

 
 

 

 

 

 

t I
(K flIIGiV‘ LC)

0-

Figure l8. The n=l N.. and N, FESR. The Regge terms
are indicated by solid lines. The low energy terms
generated by the Legendre series in x are shown as dashed
lines. Terms generated by the Cutkosky-Dec scheme are
shown as dotted lines. If either sum rule were exactly
satisfied the solid curve and the dashed or dotted curves
would coincide.

74

 

_ J'IIO’

- Io’

 

 

 

 

 

 

a U I
at .hv dhfl” 04,
Figure l9. The n=3 N. and N, FESR, Indicated as in the
preceding graphs.

7S

 

__ {via ‘V

 

 

 

 

 

 

 

'
a a 1; Gel” 1.0

Figure 20, The n=S N“ and N, FESR.

76

zero at u = m2 in the Regge term. This node is required in order to

eliminate an Np Regge recurrence at u = m2, corresponding to a negative-
parity nucleon. No such 'parity partner' to the N nucleon has been
observed. Since the need for including parity partners in Regge parameter-
izations is not a well established matter of necessity we do not consider
it a serious deficiency to be unable to reproduce this feature in the

low energy term.

It was mentioned earlier that the coupling constants of the/0’ were
chosen so as to enhance the agreement of the n = l Na and N, FESR near
u = l.0 GeVZ. In fact these coupling constants were chosen so as to
require that the n = l Nu, and N, FESR be exactly satisfied at u = l.0
GeV2 using a less sophisticated t-channel spectral function than that
used to generate the final results. Since this procedure masks the
effects of other high-mass mesons and high s-channel partial waves we
did not impose such a rigid criterion on the final calculations. Close
inspection of Figures l8, l9 and 20 shows that the Regge term diverges
more rapidly than the low energy term in each case as u increases. This
is further evidence that higher partial waves are required in both the
s- and t-channels to fit the FESR over any very wide range of u.

The connection between the rate of increase of the FESR as a function
of u and the mass and angular momentum of the states used to generate
the spectral functions is illustrated in Figure 2] which shows the
contribution of each of Barger's zero width mesons to the low energy
term in the n = l N“, FESR. Note that the high-mass, high-angular
momentum states f0 and 9 completely dominate the low-mass, low-angular
momentum states a’and/a for large u. This is due to the fact that

higher-ordered Legendre polynomials diverge faster as functions of u.

77

 

_ ctr/0'

I- ’0’

 

 

 

 

 

 

 

 

I

I
9 H. In OCV’ 1.0

d

Figure 2i. The contributions of each of Barger's zero-width mesons
to the n=l N.. FESR. The contribution of the o'(not shown) is
essentially zero.

78

Also, higher mass states are intercepted at large values of x than lower-
lying ones. Thus the rate of change of the low-energy terms with n is an
indication of the weight of higher-mass states while the rate of increase
as a function of u measures the effect of higher partial waves.

The fact that high-mass, high angular momentum states dominate the
FESR is important in that it offers the possibility of discriminating
between these states, a difficult task in the typical approach via
dispersion relations which tend to be sensitive only to low-mass, low-

angular momentum states.

C. The A FESR

 

Figures 22 and 23 Show the low-energy contributions to the A; FESR
for n = O, l, 2, 3, A, and 5. Note that these terms are of the same order
of magnitude as the errors in the corresponding N FESR. As a consequence
the 'zero' and 'nOn-zero' FESR are not well resolved. The Regge terms are
extremely small in this case as well and can all be regarded as zero.

Thus it is impossible to regard these sum rules as particularly significant.
One could attempt meaningful solutions in these cases only to resolve
very minute features of otherwise very well determined spectral functions.

Such a task is far beyond the scope of the present work.

0. The Cutkosky-Dec Scheme

While we have chosen to calculate the FESR in the context of the
Cutkosky-Dec scheme by the method of expanding x directly in terms of
polynomial in 2 it is clear that the transformed partial wave approach

is the definitive test of this idea. The expansion of x in powers of

2 produces a u l.0 GeV2 spectral function which differs from the

standard continuation only as a function of the criterion used to truncate

79

 

 

, if
e
nae
an!
_-i!
at:

 

 

 

 

1

a u A. o.v3 1.0

Figure 22. The low-energy terms for the n=0, l, 2.6;
FESR. The Regge terms, vanishingly small for both 'zero'
and 'non-zero' moments, are not shown.

 

., Jar/0'

 

 

 

 

o
\ Aw
p -‘.’°’
1W'3
a a. In GcV‘ 1.0

Figure 23. The low-energy terms for the n=3, A, 5 A;
FESR. The n=5 term have been scaled down by a factor of
IO. Again the Regge terms are all essentially zero.

80

the expansion. Had we been in a position to double or triple the number
of expansion coefficients for example the spectral functions in x and 2
could obviously have been made to agree to within any desired accuracy at
u = l.0 GeVZ. Hence this way of using the Cutkosky-Dec scheme has little
to add to any attempt at improving the qualitative features of spectral
functions.

However, the lack of good Regge convergence properties in the case
of the transformed partial wave expansion can be interpreted as suggest-
ing that the G and H waves of the CERN phases are not well determined.
This applies particularly to the case of the 22 term transformed partial
wave expansion. Imposing good Regge convergence on the transformed
partial wave spectral functions might be a means of determining the
higher phases much more exactly than can be done in the context of disper-
sion relations. If this calculation were to prove feasible, the FESR could
then be used solely to determine the higher-mass t-channel Spectrum. In
connection with improved data on cross-sections, etc. this could be a
powerful tool in determining the detailed features of the meson spectrum

at high energies.

APPENDICES

APPENDIX A

GENERALIZED SPINOR INVARIANTS FOR ITN SCATTERING

APPENDIX A

GENERALIZED SPINOR INVARIANTS FOR TIN SCATTERING

To construct the most general spinor invariant to describe the
interaction of a spin zero particle with a Spin-fi particle described
by the Dirac Equation
11 )‘-/‘5 + mum’s) = 0
we could begin by adding all possible invariants which can be constructed
from the f-matri‘ces and the four A-momentum vectors p],p2,ql,q2. Such a

function would have the form;

F = Emma) + 13 1,2,. Emma)
til-q.- G‘..(/¢,co¢9)}
1-;RY'PJXC“ x'lpk €1,5(k1we/

where F , F.
o

I, Gi’ F F are simply functions of k and c059 or

ik’ ijk

s, t, u. First, we observe that a matrix element

2.71;.) r-P qu1)x/':-(A, we) can be reduced by the Dirac Equation
to 1m iif+)“-(P¢)E'(fi,¢~9) by operating to the right or left on
11th or :4.an whichever is appropiate. Thus, terms linear in
Nucleon momenta are equivalent to terms containing no X-matrices at
all (i.e., F0), and can be eliminated. Second, the terms r'q, and

r'flz are not independent since

8l

82

CM WP: -,>.

and then

"it ’ ”'1'“ "1"" "P:
= ’1’, + ’Coagt,

the last term again following from the Dirac equation.
Now we can replace all terms up to those linear in any A-momentum
by the sum of a spinor-independent term plus a term linear in Pion A-

momentum.

Terms quadratic in 1"nmtrices can be shown to be spinor-independent
(35)

directly from the anti-commutation and normalization properties of

the I-matrices;

H Juli-2 {,.. (AI)

yzzt (A2)

where gh" are the elements of the AXA unit matrix. Write the quadratic

term, (32’ , as;

02’ = 42’. x'Mq' [1.156, We)
.: A 11 2 4 ,r}
+,;/:‘.a.~91£,;.cv.,r.1.m. +5.2; 1». 11
In the bracketed factor substitute (Al) and (A2) in the first and second

term respectively. Then;

2 F ° + J A
6" 2:151 (“'“9)[,;..2 S1‘“'/"‘II ;P171
or, Simplifying the f-function:

(9; :2 51111,...9); [DJ-‘7:

4"

83

an explicit spinor-independent.

To reduce any higher-ordered term with an odd number of Xl-matrix
indices one can first perform the simplification above on any pair of
indices as many times as necessary to reduce it to a linear form. That
linear form will then be equivalent to a spinor-independent plus a term
linear in Pion A-momentum as we have shown.

Any higher-ordered terms even in XLmatrix indices can be reduced
to a spinor-independent form simply by repetition of the operation
performed on the quadratic term for arbitrary pairs of indices.

(36,37)

The conventional choice of spinor-independent plus Pion

A-momentum is,
F = fiumm + i—zfl/fmwe)

where Q E 3561‘ +r,q2

APPENDIX B

ISOTOPOLOGY 0F n'N SCATTERING

APPENDIX B
ISOTOPOLOGY 0F rZN SCATTERING

The possible isotopic spins of the ITCI = I): N(1 = %) state are
I = l/2. 3/2. In terms of a set of projection operators_/szthe most
general form of any operator in rv'N isotopic Spin space is:

(9 A. (9* +A,ez (3.)
where (3%,(Eab are the 2respective eigzenamplitudes. The projection
operators can be constructed using the identity,

f=if+T (82>
where; Tiis the 2 X 2 Nucleon isospin 'Pauli matrix', identical in form

to the familiar Eflmatrix.
:r’is the corresponding Pion isotopic spin operator.

The Nucleon isotopic-spin operator is well known. In the charged

Pion basis the Pion isospin-matrix(38)
010 020 100
t_L/o/ ,Tzzi" O". T399000 (B3)
2 {2' . 15 Z
<7 [<9 0 1 ° C>C>-I

The Pion isospinors are,

0

1 - o
If)“; HEX-”(’9 1W"): ’ (3“)

Forming LIZ we have, from (82);

Iz= f-T + #732 +‘T‘

But;
rz= 3
T‘=Z
Or: T7“:- IZ‘H/‘I-

- ...—w.

 

 

85
Thus,J/\,i,_/q_3 can be written( ) as;
2

_-. .1 _ "‘f‘]
A; 3[/ zT
.. -f .
./\1. 3[2 Z‘T]
I.
We have now made explicit the content of equation (Bl). The next step

(39).

//r,> =f';_(/rz‘> + IIZ‘>)) 1722) =%(ln‘> - Irv?)

is to define the numbered Pion basis

m3) = / 7t°>

To show that (82) has the form (1.8) with./4L;‘,J/\.§ defined as
Z

in (85) note that in the numbered Pion basis

T _ .
E'fi “ ’ZCSXd-E (87)

Where; 'T'x‘ is any component of the Pion isospin matrix.
05,8 are labels of charged pions.

EVO‘A is the customary Levi-Civita symbol.

Then we can write; I
(T'T)up :; If Twp
1‘
where Fr is still a Nucleon isospin operator while T‘s indicates
the matrix element of the J‘-component of the Pion isospin between states

ot,fl.

Substituting (87) into (88) gives;

;- ‘1 7;» [tap
:'-4§Z}\

(2"- 7')...

Where X does not equal d- orIB. Z} can be expressed in terms of

Zdand I; by the conmutation relation;
Ltd, 2"] s 21 Zr

86

Where (u,,8,2‘) is a circular permutation on (I, 2, 3). Then,

(f-T).,=-a‘:[z’.,z,3

Then from (85) we have;

Then an operator of the form;

can be written;

((9).; 3 (91134” * 1-11.13}
2 .
+91{%S~P -f[ anti-J}
: éfeé “29.84::
I 2 .L ,
+3{‘(9‘ *G‘fi[[~. IA]
Comparing this equation with (l.8) yields

('0

= é-(Ai +2A‘)

(-)_. J- g
A -3(A‘ -A)
The same relations hold for B.

Inverting these relations yields,

(4

A‘: = A ’+A"’
Ag : A“, *- ZAP,

which is just equation (1.9).
To relate good isotopic spin amplitudes to the amplitudes for physical

processes we first expand:

87

[If/0) =9',"/:'.§7 ’l/I/i'd’)
175'”) =)/§/3;f) Vii/{'3
’u'/9) = /a’$‘>

Hz?» -— .3 J

a , 1

Then simply taking matrix elements of (Bi)

Cu:

(9 = om Ag 0" ‘Aijgln'fD

“To a
(9 ’ = <n}2//l;8‘+/ig'35/rc’/o>
‘: (3;? {/3

(9“? .—. (rile/A;O£ +4.} <9 in)»

.L 3 z ‘
3(9‘+ 3'93

(0) , c-)
0+6»

These relations hold for both A and B, as well as any other amplitude

with isospin lables.

APPENDIX C

INVARIANTS IN TERMS OF PARTIAL WAVES

APPENDIX C
INVARIANTS IN TERMS OF PARTIAL WAVES

The T-matrix operator has the form

T = «A + éziB (0')
(LID)

Also, the scattering amplitude can be written in terms of projection

operatorspd:1+;L,P1:L—'/g as,

F(t,m9> =Z (21+')Ifii,..,£sh> + R.-. mu} (c2)
1:0 ‘ 3 ’Q 1

TO determine F1=L$’/2 write,

Then 2
T L = J21 «II—0"
:a(1+l)—L(l+ /)- 3/4,
So that
53L ”IQ-(EH) {... fi‘ITiésfl‘fi
Then

 

 

AF, _ (Tl: +13+/'
351+;- - 22"

P _ L_/ ‘0‘: I:
.}=£-—é 212*/

Now evaluate

.. --AA A, [A
“10‘9“! x (74’)
33,15" fr‘tfqua)

So that (C2) has the form;

Fa, m9) = {:Ium) in) *3 {PM >I if“ 9) (c3)

+ 4' a‘vfiljflnr £90} 5 ism)

The operator C7'is the spin operator in the space of Pauli spinors.

To reduce (CI) note that the Nucleon spinors LL06?) can be expressed

in Pauli form as;

'I " o I C“
u. (IP) = (2M(m+£)) '(sz‘mLhc 70) ( )
where
I O
...)... g 3]
o O

 

A convenient form of the r-matrices is

fez/5011' ELL/0295:

The 2 X 2 matrices‘jfiu'are formally identical to the Pauli matrices

(22. In

this representation,

-I1(;>")A up) = —(zm(M*£)7’/I Wolf: (any), i

+xf13-3’Xm«-j,[ *1/01 yqaflu’ (o)

B E‘F)f¢ u’p) -‘-' (4M(‘*5)).Bu?0)[f3 (In. «of. - if, 5".

90

(q‘+§")m +25... ‘Afwfi-(§'*§’)
+2wx/a,/,5‘-o'q" + 33,534»)
(m gain}, 6’- c‘j‘fluw;

The operators standing between 2-component spinors reduce to

 

 

m+£ -—(£-m~)m9 _ -.;.A [-m 1 (C5)
{ 2A0» .tafllt 2/nm
and
{ 154.03% :11). + lftegzgzflfl m 9 (C6)

,1???) (5—mg/Iana-zzw}5

 

respectively.

Comparing (C5), (C6) and (C3) yields,

m+£ +(E‘mv)~49
2m»,

in ) 2’(I 33 a (c7)
(f :2.” ‘na'w D3

1“an Z { (In) £110 +£ 5-64)}4’zm a)

The recurrence relations

Lfrme) = mafrmv) ‘i/Lue)

(flvyzdffimcavh'jfigkaHMQ) ' ¢.¢49.£?(hn42)
can be used to put the sum on the right-hand side of (C7) in the form
49

Z fife-t9 110;»: 9) “fi’aua))

[no

9]

Mme...» +4», m»

= mega-£159”)

+-

jwf glue) Tia/ff“ 9)

I

I {Ilka-46) 4' “49 éfk,“¢9)
Or, from (C7)

(ft-'0); i; [/4 “‘N‘/~05} (cs)
, W

fame-I9): 1;,” [A+(W+M)B}

These relations could also have been extracted by comparing coefficients
of the operator 55% in (C3), (C5) and (C6).

Note that [(4,0940 has the form;

HA, «9) = 7,660,049) + m6 7?, (Aqua)

‘ig-flme 1g (Laue)

In the forward direction “a .1 and,

Fame-z) = fa, mew) +- £(5, mew)

so that the optical theorum has the form

0;" = $1,151, ate-l)
-' ng {if/I’mau) ‘* t: (k,u¢9'/)]

_ g T
- pguuxpmng) +12 (I“’/,,¢N351-)

92
(in)

Below the inelastic limit the expression has the familiar form

..E * -3 _z
Ut’ot" ‘32:,“ 0.44mi}, *flm J...

APPENDIX D

PARTIAL WAVE AMPLITUDES FOR ’23 "'" NN

APPENDIX D
PARTIAL WAVE AMPLITUDES FOR rut—arm

The partial wave amplitudes we wish to define can be conveniently

extracted from the helicity formalism. Jacob and WickU‘Q) express the

helicity amplitude flu)” AcAd which describes a two particle

reaction dub

 

>c +d in terms of T-matrix elements as

, zA— N
(0]) figAbAcAJ: PL;(6*5)<JM’I¢NTI1MA«:@>€ ( I“ 01;.(9)

where A‘- = helicity of particles i.
I =I,—A,
PL z Ac,” Ad
IM-‘Ags O for spinless mesons
A; II); = 2pffor spin-éfermeons

4‘;(l)are the standard reduced rotation operators.

In terms of "FA‘A‘JcAJ o,

2.
3% zéj’fi‘Jb ACAJI
A

CIAJ

For the process Err—9 NN there are only two non-trivial helicity

labels -- those of the N and N -- which have values of Ac, )4 = 2‘5.

93

 

9A

The helicity amplitudes and Cross-section relations then take on the form

‘6‘)“: ‘LZ (1*1‘)<1MAJI Tumoo> e ‘Ffidzf‘ (9)
3

a = H24“ Ime- hm may“

Noting that the states I1M),A3> for 5,: SJ =f have the symmetry

A
property( 3)

E. IgmA.A.> =<-)’"‘/1mv a. ~A.>

one can write

<1W’Ic AIII T/1“"a/L> = <1“').'j/ TQM-Aral.)

So that,
<1ma-aITI1WOO)Z<3u-;’1ITI74wao):TJ
<1”? iITI1“°°>- Show flTI1M00>§ 7'1

i.e., the subscripts 1 refer to states of like or opposite helicity.
i
If we also note that the reduced rotation operators dg“,(9)

have the symmetry

4 __ 1
we) « atom

the helicity amplitudes and cross-sections can be written

'5‘

am. ‘ Zlfi/IQ/fi/
where the ¢-dependance, which cancels out in 3% has been neglected.
We can now construct partial wave amplitudes whose relation to
measurable quantities is clear by expanding the matrix elements 7:?

and 7:1 in terms of (1"“1‘4ITI1M102

Since only NN states withggetl contributes we have:

 

 

9S

«Me/W 77.1“ °°> =

to]

(ratio/1400 0),:‘510'4: AJ/j“£’> (1“! I/ 772“] O)

.3 1
In this case, the coupling coefficients take the form

(03) . '4
(imaI/I7MAIA1> :($%7> C(le'io’k) Off 1' /,').,‘/I,)

It is clear that I‘m’o) =11M00> so that

(14".10 I1MOO> = /

Then
a " 1
DA -
( ) I ’42-; {QMAtA5/7M11> 7;
9:1
‘ 2
Where 7; are just the partial wave amplitudes we seek.

~ (1+5)

The general expression for C(1tl'l,g"o‘$l) appear in Rose

The required results are;

ol ya
CH", 1,15 co) =[31*’]

1/2

C(1—I,I,1360) 5 [#TJ
Vi

C(1+I,I,j}0[ ) 1"- [3537—3)]
4- ll
CIj'MJMI a = Hit—'77] I

The Clebsch-Gordon constants C(‘gt/ 3%") are trivial.

Substituting these results in Equation (DA) yields

+ y:
(imiff limijufl > = ’f’zL'I‘TTET]

I Th
<1Mj-ff/1mjj—I’l) -' fi[21+l

 

 

96

__I,
t Z
(j’ij‘gL “'2’ Ijmjjd’] > 3 7%(2144)

<1”) 'aL'z' I]M,‘1"I/) :773L{27:'I )2

So that,

—-

(,5) T, =éP;——)‘7;-, “AL/{fi-I‘T‘
ml :

144

L‘f—I—j. T 512.1” (T .,

which serves to allow us to shuttle between the helicity formalism and

partial waves as needed.

. {
Further, since $331538} 7;: (there is no elastic scattering in

this case)we can apply the above results to S-matrix elements. In

particular we will use these results to expand the amplitudes

a
f; It) 5 (P/q),;€7. 77
{’10 .=. (”mg—,1 77

which appears in equation (l.l9).

 

REFERENCES

ll.

12.

I3.
14.

IS.
l6.
I7.
l8.

REFERENCES
These values, taken from the work of J. Durso,correspond to the charged
member of each multiplet.

G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu, Phys. Rev.,

Stephen Gasiorowicz, Elementary Particle Physics, John Wiley 5 Sons,
New York (I966), Chap. 23.

v. Singh, Phys. R_e_v_., E9, p. l889 (1963).

M. L. Goldberger and K. M. Watson, Collision Theory, John Wiley
8 Sons, New York (I964) p. 6A7.

 

 

J. Bowcock, W. N. Cottingham, and D. Lurie, Nuovo Cimento, X 16,
p. 9l8 (I960).

S. Mandelstam, Phys. Rev., ll5, p. I74l, p. I752 (I959).
C. Lovelace, CERN Report TH-838.
J. Durso, Private communication.

J. Hamilton and T. D. Spearman, Annals of Physics, 12, p. I72
(I96I).

Op. Cit. fl 6.

 

. . (1*)
One can obvnously erte FESR not only for A and B but also
for linear combinations of these amplitudes provided the coefficients
are not too pathological.

V. Barger, et.al., Phys. 523,, 185, p. I852 (I969).

R. Dolen, D. Horn, C. Schmid, Phys. 52!, Lett., 19, p. 402
(I967).

C. B. Chiu 8 M. DerSarkissian, Nuovo Cimento, 55, p. 396 (I968).

 

Op. Cit. # l3. See Eq. (7).
R. Dolen, et.al., Phys. Rev., I66, p. I768 (I968).

C. Schmid, Phys. Rev. Lett., 29, p. 689 (I968).

97

 

I9.

20.
2l.
22.
23.
24.

25.

26.

27.

28.
29.
30.

3I.

32.

33.

34.

35.
36.
37.
38.

39.

40.

D. R. Crittenden, R. M. Heinz, D. B. Lichtenberg, E. Predazzi,
3125; Ex- 9.. l. P- 169 (I970).

Op. Cit. # l3.
P. A. Carruthers 8 J. Krisch, Annals of Physics, 33, p. I (I965).

Op. Cit. # 4.

 

V. Barger and D. Cline, Phys. Rev. Lett., 21, p. 392 (I968).

S. W. McDowell, Phys. figy,, lI6, p. 774 (I960).

A. Donnachie, R. G. Kirsopp, and C. Lovelace, CERN Report TH-838
and Addendum (I967).

A. D. Brody et.al., SLAC-PUB-709 (I970). See especially the
bibliography.

This is the value quoted by Barger et.al. We use it for purposes
of comparison.

V. Barger and D. Cline, Physics Letters, 258, p. 4I5 (I967).

 

C. Lovelace, CERN Report TH-839 (I968).
W. Frazer and J. Fulco, Phys. Rev., Il7, p. I603 (I960).

A complete discussion of the convergence properties of the Legendre
expansion can be found in J. Hamilton and W. S. Woolcock, Rev. Mod.

Phys., 35, p. 737 (I963).
R. E. Cutkosky and B. B. Deo, Phys. figy,, 113, p. I859 (I968).

M. Abramowitz and l. A. Stegun, I'Handbook of Mathematical Functions“,
Nat. Bureau of Standards AMS 55 (I964).

I am indebted to Dr. Y. A. Chao for supplying me a copy of his
FORTRAN program for generating the Landen Transformation.

Op. Cit. # 3. p. 32.
Op. Cit. # 6.
Op. Cit. # l0.

R. Levi-Setti, Elementary Particles, Univ. of Chicago Press
(I963) pp. 26f.

 

H. Muirhead, The Physics of Elementarngarticles, Oxford,
Pergamon Press (I965) p. 388.

 

Op. Cit. # 3. p. 379.

98

 

41. Ibid. p. 373.

42. M. Jacob and G. C. Wick, Annals of Physics, 2, p. 404 (I959).

 

43. Ibid, p. 417.
44. Ibid, p. 427.

45. M. E. Rose, Elementary Theory of Angular Momentum, New York,
John Wiley 8 Sons (I957) p. 225.

99