TESTS OF DYNAMICS THROUGH POLARIZATION EXPERIMENTS : THE CASE OF THE DIBARYON

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TESTS OF DYNAMICS THROUGH POLARIZATION EXPERIMENTS : THE CASE OF THE DIBARYON

G. Goldstein, M. Moravcsik

To cite this version:

G. Goldstein, M. Moravcsik. TESTS OF DYNAMICS THROUGH POLARIZATION EXPERI-

MENTS : THE CASE OF THE DIBARYON. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-

353-C2-362. �10.1051/jphyscol:1985241�. �jpa-00224555�

JOURNAL DE PHYSIQUE

Colloque C2, supplement au n

2, Tome ^6, fevrier 1985 page C2-353

TESTS OF DYNAMICS THROUGH POLARIZATION EXPERIMENTS : THE CASE OF THE DIBARYON

G.R. Goldstein and M.J. Moravcsik*

Department of Physios, Tufts University, Medford, Mass. 02155, U.S.A.

*Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, U.S.A.

Résumé - Nous présentons une revue du formalisme optimal pour relier les am- plitudes et les observables dans la diffusion à deux corps. Ce formalisme est appliqué au problème de la détermination complète d'ensembles non ambi- gus d'amplitudes ; il conduit au fait que le vecteur polarisation est su- perflu pour déterminer des amplitudes du deuteron et au fait que des états dibaryon singlet et triplet sont plausibles dans la diffusion élastique p-p.

Abstract - The Optimal Formalism for relating amplitudes and observables in two-body scattering is reviewed. It is applied to : the problem of determi- ning complete unambiguous sets of amplitudes ; the redundancy of vector po- larization for determining deuteron amplitudes ; the plausibility of singlet and triplet dibaryon states in p-p elastic scattering.

Introduction

The underlying dynamics of particle interactions is surely reflected in the spin dependence of scattering processes. The long history of particle physics confirms this as does the shorter history of this symposium. From the discovery of parity violation to the confirmation of the Electroweak Standard Model, polarization experiments have been crucial to our theoretical progress.

The enormous effort and expense involved in initiating and completing polarization studies has been rewarded with increasing understanding of dynamics. However, it is not always clear that our existing theoretical notions about dynamics provide adequate explanations of all polarization phenomena.

Experiments often leave us with more puzzles than we knew we had. For this reason it is important to be able to analyse polarization phenomena in a general way unprejudiced by particular dynamical schemes. For interpreting

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985241

e x i s t i n g d a t a , f o r f i n d i n g new tests of dynamical models and f o r d e s i g n i n g f u t u r e e x p e r i m e n t s i t i s c r u c i a l t o have a framework i n which t h e r e l a t i o n between o b s e r v a b l e q u a n t i t i e s and u n d e r l y i n g s p i n dependence can b e s p e c i f i e d , i n a model independent way.

Over t h e l a s t s e v e r a l y e a r s we have developed and a p p l i e d a formalism t h a t r e l a t e s t h e s p i n dependent a m p l i t u d e s f o r a two-body r e a c t i o n t o o b s e r v a b l e q u a n t i t i e s i n t h e s i m p l e s t p o s s i b l e ~ a y . ( l - ~ ) T h i s "Optimal Formalism" a l l o w s u s t o s t u d y g e n e r a l a b s t r a c t q u e s t i o n s a b o u t a m p l i t u d e s t r u c t u r e ( 1 ) u n d e r v a r i o u s symmetry c o n s t r a i n t s ( 2 ) and t h e c o r r e s p o n d i n g o b s e r v a b l e s t r u c t u r e . The freedom i n choosing d i f f e r e n t f r a m e s i n which t o s p e c i f y a m p l i t u d e s and o b s e r v a b l e s l e a d s t o a n e x p l o r a t i o n of t h e consequences o f t h o s e c h o i c e s f o r t e s t i n g d i f f e r e n t dynamical hypotheses. We have used t h e Optimal Formalism t o e x p l o r e many problems, b o t h a b s t r a c t and p r a c t i c a l . ( 4 )

H e r e i n we w i l l d i s c u s s s e v e r a l f o r m a l a p p l i c a t i o n s of t h i s framework and a p a r t i c u l a r a p p l i c a t i o n t o t h e dynamics of p-p s c a t t e r i n g . We have r e c e n t l y answered t h e q u e s t i o n of what c o n s t i t u t e s a minimal s e t of e x p e r i m e n t s from which t o d e t e r m i n e t h e a m p l i t u d e s c o m p l e t e l y unambiguously.(3) We w i l l review t h i s work. A second i n t e r e s t i n g a p p l i c a t i o n i n v o l v e s t h e t e n s o r p o l a r i z a t i o n o f t h e deuteron.(5) F i n a l l y we d i s c u s s t h e a m p l i t u d e s t r u c t u r e f o r p-p s c a t t e r i n g i n t h e r e g i o n of t h e presumed dibaryon resonances.(6) Optimal Formalism

The Optimal Formalism h a s been d i s c u s s e d i n c o n s i d e r a b l e d e t a i l e l ~ e w h e r e ( 1 ' ~ ) a n d w i l l b e reviewed o n l y b r i e f l y here. Consider a two-body r e a c t i o n A+B+C+D f o r which t h e s p i n s a r e sA, sB, sC, and sD. Now t h e s p i n dependent a m p l i t u d e s a r e d e f i n e d i n terms of t h e s p i n q u a n t i z a t i o n a l o n g some p a r t i c u a r d i r e c t i o n s i n s p a c e f o r each of t h e f o u r p a r t i c l e s . L e t a , b , c , d b e t h e e x p e c t a t i o n v a l u e s of t h e s p i n s w i t h r e s p e c t t o t h e a x e s l a b e l l e d

- ^{A-B -C -D i . e .}

J = I ~ I ~ ~

The a m p l i t u d e s t h e n can b e w r i t t e n a s Fc,d;a,b (E,B) i n a n obvious n o t a t i o n . These a x e s a r e s p e c i f i e d i n d e p e n d e n t l y . For convenience t h e y a r e s p e c i f i e d i n t h e Center-of-Mass frame of t h e r e a c t i o n . There i s a continuum of p o s s i b i l i t i e s f o r t h e s e a x e s , a l t h o u g h symmetries r e s t r i c t t h o s e c h o i c e s somewhat ( 2 ) a s f o l l o w s .

P a r i t y c o n s e r v a t i o n r e q u i r e s t h a t t h e a x e s b e e i t h e r normal t o t h e s c a t t e r i n g p l a n e o r i n t h e p l a n e i n o r d e r t h a t t h e a m p l i t u d e s t r a n s f o r m i n t o one a n o t h e r under t h e p a r i t y o p e r a t i o n . Time r e v e r s a l i n v a r i a n c e r e q u i r e s

A A

t h a t t h e zA and zC b e p a r a l l e l a s w e l l a s t h e zB a n d zD a x e s . I d e n t i c a l

A

p a r t i c l e c o n s t r a i n t s r e q u i r e zA p a r a l l e l t o zB and zC p a r a l l e l t o zD. So f o r p a r i t y c o n s e r v a t i o n a l o n e , which w i l l b e assumed i n a l l a p p l i c a t i o n s

c o n s i d e r e d h e r e , e a c h p a r t i c l e ' s a x i s c a n b e s p e c i f i e d by t h e a n g l e t h a t a x i s makes w i t h t h e p a r t i c l e ' s momentum d i r e c t i o n when t h e a x i s l i e s i n t h e s c a t t e r i n g plane. I f o n l y p a r i t y i s r e l e v a n t then, f o u r a n g l e s parameterize t h e f o u r a x e s i n t h e plane. T h i s c h o i c e of a x e s i s c a l l e d a "Planar Frame"

a n d t h e c o r r e s p o n d i n g a m p l i t u d e s a r e " P l a n a r Amplitudes". A l t e r n a t i v e l y , of c o u r s e , any of t h e a x e s may b e t r a n s v e r s e t o t h e plane. When a l l a x e s a r e t r a n s v e r s e t h e a m p l i t u d e s a r e " T r a n s v e r s i t y Amplitudes".

With t h e s e r e s t r i c t i o n s t h e n u c l e o n a x e s i n pion-photoproduction, f o r example, can b e chosen e i t h e r i n t h e s c a t t e r i n g p l a n e w i t h two independent a n g l e s o r b o t h normal t o t h a t p l a n e . ( 7 ) On t h e o t h e r hand t h e a x e s f o r t h e p r o t o n s i n p-p e l a s t i c s c a t t e r i n g , i f t h e y a r e chosen p l a n a r , must a l l b e a t t h e same a n g l e w i t h r e s p e c t t o t h e i r momenta.(2) There i s o n l y on c h o i c e of o r i e n t a t i o n t o b e made and hence one a n g l e t o s p e c i f y .

Which of t h e i n f i n i t u d e of a x e s a r e chosen depends on t h e a p p l i c a t i o n

a n t i c i p a t e d . For e x t r a c t i n g a m p l i t u d e s from o b s e r v a b l e s t h e t r a n s v e r s i t y a x e s

a r e u s u a l l y most c o n v e n i e n t . S i n g l e p a r t i c l e p o l a r i z a t i o n asymmetry i s t h e n a

combination of s q u a r e s of magnitudes of t r a n s v e r s i t y a m p l i t u d e s . For making

model p r e d i c t i o n s t h e h e l i c i t y frame h a s been p r e f e r r e d b e c a u s e of t h e

s i m p l i c i t y of t h e h e l i c i t y dependence of most h y p o t h e t i c a l i n t e r a c t i o n s a t

h i g h e n e r g i e s .

The h e l i c i t y frame i s j u s t a s p e c i a l c a s e of t h e p l a n a r frame f o r which a l l a n g l e s a r e zero.

Given a s e t of t r a n s v e r s i t y o r h e l i c i t y amplitudes e x t r a c t e d from d a t a f o r some r e a c t i o n , i t i s a n e a s y m a t t e r t o o b t a i n p l a n a r a m p l i t u d e s f o r any p a r t i c u l a r c h o i c e of o r i e n t a t i o n angles. Choosing t h o s e a n g l e s p r o p i t i o u s l y may r e v e a l some h i t h e r t o unexpected s t r u c t u r e r e f l e c t i n g new dynamics. I n t h e p-p e l a s t i c system t h i s e x p l o r a t i o n h a s been done f o r t h o s e e n e r g i e s and s c a t t e r i n g a n g l e s a t which a complete set of p o l a r i z a t i o n d a t a e x i s t . ( 8 ) For t h e 6 GeV/c and t h e 300 t o 800 MeV data a remarkable s i m p l i c i t y r e s u l t s when t h e p l a n a r frame i s chosen w i t h e a c h a x i s a t 90' t o t h e corresponding momentum. I n t h i s Planar-Transversity o r "Sideways" frame t h e amplitudes a r e almost e n t i r e l y p u r e r e a l o r pure imaginary w i t h r e s p e c t t o one another. Why t h i s should b e i s n o t c l e a r b u t c e r t a i n l y s u g g e s t s some u n d e r l y i n g s i m p l i c i t y n o t inc'orporated i n any c o n v e n t i o n a l dynamical model. ( 9 )

Having d e f i n e d frames and amplitudes t h e o b s e r v a b l e s can now be s p e c i f i e d . ( l ) The s i m p l e s t s t r u c t u r e is o b t a i n e d by choosing i n i t i a l s t a t e p a r t i c l e d e n s i t y m a t r i c e s t o have s i n g l e e n t r i e s on t h e d i a g o n a l o r two off-diagonal elements t o i n s u r e h e r m i t i c i t y . S i m i l a r l y t h e f i n a l s t a t e p o l a r i z a t i o n s a r e t r a c e s of f i n a l s t a t e d e n s i t y m a t r i c e s w i t h s p i n o p e r a t o r s t h a t a r e chosen w i t h one d i a g o n a l e n t r y o r two off-diagonal e n t r i e s . For s p i n 1 / 2 t h e s e m a t r i c e s a r e

These correspond t o complete p o l a r i z a t i o n a l o n g t h e z-axis ( ~ u r e l y + o r -

1 / 2 ) , o r asymmetries a l o n g x- o r y-axes. For spin-1 t h e s i t u a t i o n i s more

c o m p l i c a t e d , ( l ~ 2 ~ 5 ) i n v o l v i n g b o t h v e c t o r and t e n s o r p o l a r i z a t i o n s a s

i n d i c a t e d here:

Note t h a t m a t r i c e s w i t h imaginary components correspond t o odd numbers of S y l s . Each m a t r i x i s a l i n e a r combination of s c a l a r a n d / o r v e c t o r w i t h t e n s o r p o l a r i z a t i o n s . A s o b s e r v a b l e s t h e y a r e c o n c e p t u a l l y a s s i m p l e a s v e c t o r a n d t e n s o r p o l a r i z a t i o n s . They correspond t o v a r i o u s combinations of pure s p i n components +1, 0, -1 a'long t h e z-axis, t h e x-axis, t h e y-axis, 45O a x i s i n t h e x-z p l a n e , e t c . I n p r a c t i c e , of c o u r s e , n o t a l l p a r t i c l e s a r e i n p r e p a r e d o r measured p o l a r i z a t i o n s t a t e s . A c t u a l o b s e r v a b l e s a r e sums o f t h e s e o p t i m a l c h o i c e s . N e v e r t h e l e s s c o n s i d e r a b l e i n s i g h t i n t o b o t h f o r m a l and p r a c t i c a l a p p l i c a t i o n s i s o b t a i n e d from s t u d y of t h e s e optimal o b s e r v a b l e s a s t h e examples below i n d i c a t e .

Formal a p p l i c a t i o n s

The Optimal Formalism a l l o w s u s t o answer a s i m p l e and very fundamental quantum maechanical q u e s t i o n . For a p a r t i c u l a r two body r e a c t i o n , what t y p e s of measurements are n e c e s s a r y for c o m p l e t e l y d e t e r m i n i n g * u n d e r l y i n g

-

a m p l i t u d e s unambiguously? Without any symmetry r e s t r i c t i o n s we showed p r e v i o u s l y t h a t measurements of a l l t h e s p i n p r o j e c t i o n s of e a c h p a r t i c l e ' s s p i n a l o n g two independent d i r e c t i o n s were n e c e s s a r y t o c o m p l e t e l y determine t h e amplitudes.(4) Of c o u r s e t h o s e measurements must i n c l u d e a t l e a s t double c o r r e l a t i o n s w i t h o t h e r p a r t i c l e ' s s p i n s . ( l O ) T h i s s p e c i f i c a t i o n , however, a l l o w s d i s c r e t e a m b i g u i t i e s t o remain i n t h e a m p l i t u d e s .

To remove d i s c r e t e a m b i g u i t i e s a s w e l l i t i s n e c e s s a r y t o measure a t

l e a s t one o b s e r v a b l e w i t h s p i n p o l a r i z a t i o n a l o n g t h r e e independent

d i r e c t i o n s . The proof i s s t r a i g h t f o r w a r d u s i n g t h e o p t i m a l formalism.(3)

Measuring o r p r e p a r i n g one p a r t i c l e ' s p o l a r i z a t i o n a l o n g a s i n g l e d i r e c t i o n

i n v o l v e s d i a g o n a l e l e m e n t s of t h e d e n s i t y m a t r i x f o r t h a t p a r t i c l e ( i n t h e

b a s i s f o r which t h a t d i r e c t i o n i s t h e q u a n t i z a t i o n d i r e c t i o n ) . Hence t h e

c o r r e s p o n d i n g o b s e r v a b l e s i n v o l v e sums of s q u a r e s of magnitudes. Such

o b s e r v a b l e s can y i e l d t h e magnitudes b u t n o t r e l a t i v e phases of t h o s e a m p l i t u d e s .

P o l a r i z a t i o n a l o n g a second o r t h o g o n a l d i r e c t i o n g i v e s o b s e r v a b l e s i n v o l v i n g t h e sum of r e a l ( o r imaginary, depending on c o n v e n t i o n ) p a r t s of b i l i n e a r p r o d u c t s of a m p l i t u d e s . From t h e s e o b s e r v a b l e s t h e c o s i n e s of r e l a t i v e phases between a m p l i t u d e s i s determined. But c o s i n e s l e a v e s i g n a m b i g u i t i e s i n t h e r e s u l t i n g phase a n g l e s .

I t i s f i n a l l y n e c e s s a r y t h e n t o measure o r p r e p a r e a t l e a s t some p o l a r i z a t i o n s a l o n g a t h i r d o r t h o g o n a l d i r e c t i o n . T h i s i n v o l v e s o b s e r v a b l e s t h a t a r e sums of imaginary p a r t s of b i l i n e a r p r o d u c t s of a m p l i t u d e s . Those o b s e r v a b l e s t h e r e f o r e d e t e r m i n e t h e s i n e s of some of t h e same phase a n g l e s and, t h e r e b y , remove t h e remaining q u a d r a n t a m b i g u i t i e s .

T h i s s k e t c h of t h e proof shows t h e u t i l i t y of t h e formalism and i n d i c a t e s why i t i s a v a l u a b l e t o o l f o r e x p l o r i n g s u c h g e n e r a l q u e s t i o n s .

Another f o r m a l a p p l i c a t i o n t o d e u t e r o n p o l a r i z a t i o n l e a d s t o a d r a m a t i c a n d unexpected r e s u l t . We have r e c e n t l y shown t h a t i n r e a c t i o n s i n v o l v i n g d e u t e r o n s ( o r any s p i n - 1 p a r t i c l e s ) all amplitude i n f o r m a t i o n can b e o b t a i n e d by measuring p r e p a r i n g only t e n s o r p o l a r i z a t i o n s of the d e u t e r o n s . C5)

-

Whether o r n o t t h i s has some p r a c t i c a l s i g n i f i c a n c e remains t o b e s e e n , b u t i t d o e s . a l t e r o n e ' s i n t u i t i o n a b o u t spin-1. We w i l l p r e s e n t a p l a u s i b i l i t y argument h e r e ; t h e proof b e i n g t o o l e n g t h y .

Consider t h e s i m p l e r e a c t i o n i n v o l v i n g s p i n s 0+0+0+1 a n d assume only L o r e n t z i n v a r i a n c e . There a r e 3 complex a m p l i t u d e s , A,B,C c o r r e s p o n d i n g t o 3 v a l u e s of t h e s p i n p r o j e c t i o n s a l o n g some a x i s t o be chosen a s t h e z-axis.

With 3 a m p l i t u d e s , 5 o b s e r v a b l e s a r e needed from which t o s p e c i f y t h e magnitudes and 2 r e l a t i v e p h a s e s of t h e amplitudes.

Choose t h e o b s e r v a b l e s t o b e t h e d i f f e r e n t i a l c r o s s s e c t i o n and 4 of t h e

5 independent t e n s o r p o l a r i z a t i o n s d e f i n e d h e r e i n terms of t h e b i l i n e a r

p r o d u c t s of t h e a m p l i t u d e s :

p,,'~e (BC*-AB*) p y z " l m ( B ~ * - ~ B * )

P ~ ~ - ~ ~ " R ~ ( A c * ) pxy"lm(Ac*)

From

a n d P,, t h e q u a n t i t i e s 1 ~ 1 ~ + 1 ~ 1 ~ a n d ( B I a r e d e t e r m i n e d w h i l e Pxx-Pyy a n d Pxy t h e n f i x A a n d C ( c h o o s i n g A r e a l s a y ) . Then o n l y Pxz o r PyZ s u f f i c e s t o s p e c i f y B. When more s p i n n i n g p a r t i c l e s a r e i n v o l v e d i n t h e p r o c e s s t h e r e a r e c o m p l i c a t i o n s (which h a v e b e e n e l u c i d a t e d i n r e f . 5 ) b u t t h e

g e n e r a l i z a t i o n g o e s t h r o u g h by i n c l u d i n g t h o s e same t e n s o r measurements i n c o r r e l a t i o n w i t h o t h e r s p i n p o l a r i z a t i o n s .

T h i s t h e o r e m c o u l d h a v e s i g n i f i c a n c e i n t h e p u r s u i t of more e x p e r i m e n t a l i n f o r m a t i o n on t h e i n e l a s t i c c h a n n e l s i n p-p a n d n-d s c a t t e r i n g w h e r e i n d i b a r y o n s may b e p r o m i n a n t . T h i s l e a d s t o o u r l a s t a p p l i c a t i o n

here--dibaryons i n p-p e l a s t i c s c a t t e r i n g a m p l i t u d e s . The Case o f t h e D i b a r y o n

----

We a r e g o i n g t o c o n s i d e r how t h e e l a s t i c a m p l i t u d e s r e f l e c t t h e e x i s t e n c e o f r e s o n a n c e s w i t h v a r i o u s s p i n - a n g u l a r momentum quantum numbers. For t h i s a p p l i c a t i o n h e l i c i t y a m p l i t u d e s p r o v e most c o n v e n i e n t ; t h e y r e f l e c t t h e s y m m e t r i e s i n p-p e l a s t i c s c a t t e r i n g most d i r e c t l y when decomposed i n t o d e f i n i t e a n g u l a r momentum s t a t e s .

The t o t a l a n g u l a r momentum d e c o m p o s i t i o n o f h e l i c i t y a m p l i t u d e s ( l 1 )

c a n b e s e p a r a t e d i n t o s p i n s i n g l e t a n d t r i p l e t c o m b i n a t i o n s f o r t h e p-p

system. R e c a l l t h a t s i n g l e t s must h a v e e v e n L a n d t r i p l e t s odd L ( a n d J = L + l ,

L,L-1) due t o i d e n t i c a l p a r t i c l e symmetry. Then t h e h e l i c i t y a m p l i t u d e

d e c o m p o s i t i o n s become

b=D(+,+;+,-)

~ d $ ~ ( 8 ) < ~ ( ~ + 1 ) j , f l l ( ~ f l ) j , O > , J

w i t h n o t a t i o n S ( ~ ) J f o r t h e s t a t e s . ( 1 2 )

Note t h a t f o r a lJJ o r 3 ( ~ i l ) j r e s o n a n c e t h e a m p l i t u d e s a=-or+d.

However, f o r

o r 3 ( ~ + 1 ) J cf-or+e b e c a u s e of t h e d i f f e r e n t a n g u l a r f u n c t i o n s i n v o l v e d . T h e r e a r e r e l a t i o n s i n t h e l a t t e r c a s e n e v e r t h e l e s s ; t h e y a r e d e p e n d e n t on a n g l e a s we s h a l l s e e below. The a m p l i t u d e s t h u s p r o v i d e a means t o t e s t f o r t h e e x i s t e n c e of r e s o n a n c e s by o b s e r v i n g t h e a m p l i t u d e r e l a t i o n s a s e n e r g y v a r i e s . ( b )

Complete s e t s of a m p l i t u d e s do n o t e x i s t f o r a l l e n e r g i e s a n d a n g l e s i n t h e presumed d i b a r y o n r e g i o n s i n c e c o m p l e t e s e t s o f e x p e r i m e n t s have been performed o n l y a t s e l e c t k i n e m a t i c v a l u e s . Phase s h i f t a n a l y s e s h a v e been performed i n t h e r a n g e of i n t e r e s t - - h e r e 700 t o 900 MeV. To t h e e x t e n t t h a t p h a s e s h i f t s a r e b e l i e v a b l e t h e y i n t e r p o l a t e e x p e r i m e n t a l d a t a of v a r i o u s k i n d s by u s i n g u n i t a r i t y a n d t h e a s s u m p t i o n of a f i n i t e r a n g e i n t e r a c t i o n ( i . e . Lmax). U s i n g one of t h e " s t a n d a r d " p h a s e s h i f t a n a l y s e s , t h e S A I D s e t ( l 3 1 , we c a n d e t e r m i n e t h e a m p l i t u d e s a , . .. ^{, e .}

The r e s u l t i n g a m p l i t u d e s c a n b e s t u d i e d f o r t h e i r r e s o n a n c e s t r u c t u r e a s o u t l i n e d . The m a g n i t u d e s of a and d a r e q u i t e d i f f e r e n t and show no

i n d i c a t i o n of a p p r o a c h i n g one a n o t h e r a s t h e energy goes t h r o u g h t h e presumed mass of t h e s t a t e . The p h a s e d i f f e r e n c e between a a n d d e x h i b i t s n o s p e c i a l changes a s t h e same r e g i o n i s s t u d i e d . There i s no s i g n of r e s o n a n c e a c t i v i t y i n t h e s e a m p l i t u d e s i n d i c a t i n g e i t h e r a dominant background

c o n t r i b u t i o n o r n o r e a l e f f e c t .

The 3 ~ 3 s t a t e e s c a p e s n o t i c e i n t h e above t e s t . Examining t h e a n g u l a r dependence of I c ] and 1 el , however, shows a s t r i k i n g b e h a v i o r a s t h e f i g u r e below e x h i b i t s . The magnitudes a r e e q u a l a t 90°, where t h e y a r e s o

c o n s t r a i n e d . But t h e y a r e a l s o e q u a l a t somewhat s m a l l e r a n g l e s ; t h e r e i s a

c r o s s o v e r . And t h a t c r o s s o v e r p o i n t moves t o s m a l l e r a n g l e s a s t h e energy

i n c r e a s e s . Remember t h a t ~ " d : ~ (8) a n d e-d:-l (8). I f J=3 d o m i n a t e s t h e s e a m p l i t u d e s t h e n a t t h o s e a n g l e s f o r which d:1(8)=-d:-(p) t h e v a l u e s of cE-e.

T h i s e q u a l i t y o c c u r s f o r 63' which i s c l o s e t o t h e c r o s s o v e r a n g l e a t 800-850 MeV a s t h e f i g u r e shows. F u r t h e r m o r e n o t e t h a t d:1(8)=~ a t 8 5 O s o t h a t c s h o u l d b e z e r o t h e r e . It i s f a r from z e r o showing t h a t t h e r e i s s i z e a b l e background c o n t r i b u t i n g . I n s p i t e of t h a t background t h e r e i s t h e s t r i k i n g a p p e a r a n c e and e n e r g y dependent b e h a v i o r of t h a t c r o s s o v e r t o v e r i f y a r e s o n a n t c o n t r i b u t i o n . I n a d d i t i o n t h e r e l a t i v e p h a s e between c and e v a r i e s r a p i d l y t h r o u g h 180° n e a r t h e 6 3 O s c a t t e r i n g a n g l e . The c a s e f o r t h e 3 ~ 3 i s q u i t e c o n v i n c i n g .

7 5 0 MeV

0

- - - - - - -

k ' *

_{- - -} ^{8 0 0} _{8 5 0} ^MeV _MeV

0-

.- .- ^{9 5 0 MeV}

0

To summarize t h i s a m p l i t u d e t e s t we emphasize t h a t w& h a v e u s e d t h e phase s h i f t s o n l y to r e p r e s e n t i n t e r p o l a t i o n a n d e x t r a p o l a t i o n of c r o s s s e c t i o n and p o l a r i z a t i o n data t o v a r i o u s a n g l e s a n d e n e r g i e s . The a m p l i t u d e s t h a t emerge a r e a n e q u i v a l e n t i n t e r p o l a t i o n - e x t r a p o l a t i o n of d a t a . Those a m p l i t u d e s show smooth " u n i n t e r e s t i n g " b e h a v i o r f o r t h e s i n g l e t s p i n a m p l i t u d e s . T h e r e f o r e o b s e r v a b l e 6 w i l l n o t show any " i n t e r e s t i n g " b e h a v i o r w i t h e n e r g y (even though t h e s i n g l e t phase s h i f t , ID^, l o o p s on t h e Argand diagram13). The q u e s t i o n of w h e t h e r o r n o t t h i s l o o p i n g p a r t i a l wave i s t r u e l y u n i q u e a r i s e s . Where i s

t h e e n e r g y v a r i a t i o n i n a n o b s e r v a b l e q u a n t i t y o r a m p l i t u d e t h a t shows s u c h a

s i n g l e t r e s o n a n c e ? Of c o u r s e t h e r e n e e d n ' t b e s u c h a v a r i a t i o n i n a n y

p a r t i c u l a r s e t of o b s e r v a b l e s t o s t i l l g i v e a r e s o n a t i n g p a r t i a l wave s o l o n g

a s s u b t l e c a n c e l a t i o n s occur. And l a r g e i n e l a s t i c i t i e s muddy t h e s i t u a t i o n . I n c o n t r a s t however, t h e 3~~ s t a t e i s f a r more p l a u s i b l e because of t h e c r o s s o v e r phenomenum observed above.

To conclude, we have demonstrated t h e u t i l i t y of t h e o p t i m a l formalism by c o n s i d e r i n g t h r e e v e r y d i s t i n c t a p p l i c a t i o n s . In each c a s e somewhat

s u r p r i s i n g r e s u l t s were p r e s e n t e d t o i l l u s t r a t e t h e improved i n t u i t i o n t h i s a m p l i t u d e formalism produces. The f o r m a l i s i m has many more a p p l i c a t i o n s , some of which were summarized i n t h e High Energy Workshop of t h i s Sumposium. It i s a n t i c i p a t e d t h a t t h e i n s i g h t s g a i n e d t h r o u g h t h i s approach w i l l c o n t i n u e t o prove advantageous i n b o t h t h e d e s i g n of e x p e r i m e n t a l programs f o r

p o l a r i z a t i o n measurements and t h e i n t e r p r e t a t i o n of t h e r e s u l t i n g d a t a . We acknowledge t h e s u p p o r t p r o v i d e d by g r a n t s from t h e U.S. Department of Energy. We a r e g r a t e f u l t o t h e o r g a n i z e r s of t h e Symposium f o r t h e i r

e x c e p t i o n a l h e l p ; p a r t i c u l a r l y Jacques S o f f e r . R e f e r e n c e s

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(1984)

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