EXPLORING DYNAMICS THROUGH POLARIZATION

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EXPLORING DYNAMICS THROUGH POLARIZATION

G. Goldstein, M. Moravcsik

To cite this version:

G. Goldstein, M. Moravcsik. EXPLORING DYNAMICS THROUGH POLARIZATION. Journal de

Physique Colloques, 1985, 46 (C2), pp.C2-251-C2-260. �10.1051/jphyscol:1985228�. �jpa-00224539�

30URNAL DE PHYSIQUE

Colloque C2, supplément au n°2, Tome 46, février 1985 page C2-251

EXPLORING DYNAMICS THROUGH POLARIZATION

G.R. Goldstein and M.J. Moravcsik

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

+ I n s t i t u t e of Theoretical Soienoe, University of Oregon, Eugene, Oregon 97403, U.S.A.

Résumé - Nous présentons une revue du formalisme optimal qui est appliqué à la détermination de plusieurs amplitudes à partir des observables de la diffusion pp élastique. Un modèle de Regge élaboré ne réussit pas à expli- quer ces amplitudes, mais le schéma OPE reste valable. On trouve une éton- nante simplification dans le repère "de cSté". Nous considérons la perti- nence de QCD.

Abstract - The Optimal Formalism is reviewed and applied to determining vari- ous amplitudes from observables in pp elastic scattering. A sophisticated Regge model fails to explain these amplitudes, but OPE remains valid. Strik- ing simplicity is found in the "sidewise" frame. The relevance of QCD is considered.

I - INTRODUCTION

Polarization experiments provide important insights into the dynamical struc- ture of hadronic interactions. It has been demonstrated on many occasions that spin is not an "inessential complication" in understanding the underlying dynamics.

The activity and excitement generated in these High Energy Spin Symposia by new po- larization data attest to the importance of spin phenomena. Of particular interest has been the measurement and interpretation of two-body scattering. Such exclusive scattering provides the simplest examples of purely hadronic interactions while exhibiting sufficient complexity to test our theoretical understanding exten- sively.

A brief reminder of the history of one line of spin phenomenology will serve to orient the following discussion. Over 15 years ago it was realized that the single polarization asymmetry in irp charge exchange scattering was a clear signal of interfering exchanges(1) and therefore provided a probe of the then popular model, the Regge pole model(2). Experiments were performed in that and other sys- tems

(3)

with somewhat ambiguous results. The models had to be complicated with the addition of branch cuts and the structure of those cuts was the subject of consi- derable debate

(4).

In a perfect example of the interplay between theory and exp- eriment, more refined data led to more refined models which, in turn, led to more refined experiments until it was clear that basic changes in the theoretical struc- ture were necessary. In particular, the measurement of the A and R double polari- zation correlations in n^p elastic(5) dealt such a blow to the prevailing theoret- ical notions that the models never recovered their widespread credibility. The remarkable interplay between theory and polarization experiments led to further developments in experimental techniques. The most noteworthy program was undertaken at Argonne to measure a complete set of spin correlations from which the amplitudes for pp elastic scattering could be determined(6). As the results were gradually obtained over many years (long after the ZGS was shut down) many puzzles appeared. Some of the original motivation, however, was lost as notions about fun- damental processes changed. At this point there exists more than a complete set of observables at 6 GeV/c and several angles(7). It behooves us to obtain as much in- formation from these data as we can.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985228

The s p i n dependence of t h e pp system a t "medium" e n e r g i e s provides a t e s t i n g ground f o r many t h e o r e t i c a l i d e a s and s p e c i f i c models. I n t h i s t a l k we w i l l review t h e r e s u l t s of our own a n a l y s i s of t h e pp amplitudes(8) and present some s t r i k i n g new sYstematics(9). Using t h e very g e n e r a l framework of t h e "Optimal Formalism"

f o r p o l a r i z a t i o n phenomenology(lO) has enabled us t o explore hidden dynamical c l u e s ( 8 ~ 9 ) a? w l l as t o t e s t some model predictions(11,12) i n novel ways.

There i s one p a r t i c u l a r s-channel frame t h a t is simply r e l a t e d t o t h e t- channel h e l i c i t y frame. That i s t h e center-of-mass planar frame

-

t h e "Magic" frame

-

i n which t h e q u a n t i z a t i o n axes a r e i n t h e d i r e c t i o n s given by t h e c r o s s i n g angles.(18) Those a n g l e s depend on t h e energy, momentum t r a n s f e r and e x t e r n a l part- i c l e masses.(21) So f o r each kinematic point of a r e a c t i o n t h e r e i s a s e t of p l a n a r axes f o r which t h e corresponding Magic amplitudes a r e equal t o t-channel h e l i c i t y amplitudes. And f o r OPE of J t h e c o n s t r a i n t s of Eq.(l) apply t o t h e magic ampli- tudes d i r e c t l y g i v i n g many zero amplitudes, i n general. (See page 257).

The second type of c o n s t r a i n t i s " f a c t o r i z a t i o n " . The couplings of t h e ex- changed p a r t i c l e of d e f i n i t e J t o t h e incoming and outgoing s t a t e s ( i n t h e t- channel) a r e independent s o t h a t the non-linear r e l a t i o n s of t h e form

hold, where the a , b , c , d r e f e r t o s p i n components along t h e Magic axes and the s i g n depends on t h e p a r i t y of the p a r t i c l e s . These r e l a t i o n s provide many c o n s t r a i n t s among the amplitudes but a r e of d i r e c t use only i n s p e c i a l circumstances a s we see i n t h e r e l e v a n t example of p-p e l a s t i c .

For p-p e l a s t i c amplitudes, when J=O exchange is dominant, Eq.(l) f o r c e s a l l but aJ=o and c j = o t o be zero (henceforth i n t h i s s e c t i o n t h e I b g i c frame f o r quan- t i z a t i o n w i l l be understood). Then f a c t o r i z a t i o n , Eq.(2), r e q u i r e s a 0 = +/-co, the s i g n depending on t h e p a r i t y of t h e exchange. F i n a l l y , because t h e angular f u n c t i o n s i n Eq.(l) a r e the same f o r the J=0 c o n t r i b u t i o n to a and c , t h e r e l a t i o n a = +/-c holds f o r the t o t a l amplitudes and b=d=e=O.

When J f o r OPE is 1 o r g r e a t e r t h e r e a r e no J - c o n s t r a i n t s f o r p-p e l a s t i c , but f a c t o r i z a t i o n g i v e s

., L

a j = +/-cj,d j = -/+ej, a j e j = +/-bJ

.

^{( 3 )}

Since a and c i n v o l v e the same angular f u n c t i o n i n E q . ( l ) , the remaining r e l a t i o n f o r t h e e n t i r e amplitudes from Eq. ( 3 ) i s j u s t

f o r n a t u r a l / u n n a t u r a l p a r i t y exchange of any d e f i n i t e J.

We have t e s t e d f o r OPE i n p-p e l a s t i c s c a t t e r i n g using t h e s e Magic amplitude t e s t s . ( l g ) To e s t a b l i s h some c r e d i b i l i t y f o r t h e r e s u l t s we f i r s t applied the t e s t s t o t h e " i n t e r m e d i a t e energy" amplitudes a t 300,580 800 MeV. For each of t h e s e e n e r g i e s t h e phase s h i f t a n a l y s i s of the SAID grouptz2) was used t o o b t a i n amplitudes i n the Magic frame. For t h e 580 MeV amplitudes, the amplitude a n a l y s i s of t h e complete s e t of p o l a r i z a t i o n d a t a from t h e SIN group(23) was used a l s o a s a check along with our own Optimal determination. I n no c a s e 'does t h e r e l a t i o n of Eq.(4) hold over t h e angular range of t h e analyses. Given t h e e x p e c t a t i o n t h a t many exchanges of d i f f e r e n t p a r i t i e s a r e important i n t h i s energy region and t h a t t h e u-channel poles must c o n t r i b u t e a s w e l l , i t i s not s u r p r i s i n g t o f i n d a n u l l r e s u l t here.

On t h e o t h e r hand, the 6 G e V / c Optimal amplitudes ( r o t a t e d t o the Magic frame) do s a t i s f y Eq.(4) w i t h t h e

+

s i g n a s Fig.6 shows. This i s t r u e , within uncertain- t i e s , f o r almost a l l f o u r s e t s of s o l u t i o n s . This i n d i c a t e s t h a t n a t u r a l p a r i t y exchange, with a t l e a s t some J

>

0 ( s i n c e b , d , e a r e non-zero), plays a s i g n i f i c a n t

r o l e i n t h e p-p dynamics a t 6 GeV/c. Why should t h i s be i f t h e standard i d e a s about Regge exchange seem t o f a i l a s we have seen? Perhaps t h e r e i s some new dy- namics t h a t s t i l l involves OPE i n an e s s e n t i a l way. Or perhaps the OPE dominance r e f l e c t s some QCD mechanism wherein one gluon exchange i s somehow dominant. We w i l l s p e c u l a t e on t h i s l a s t p o s s i b i l i t y l a t e r . But next we t r y t o look f o r o t h e r c l u e s t o t h e dynamical s t r u c t u r e using our a b i l i t y t o change p l a n a r frames.

I1

-

OPTI14AL FORMALISM

Several y e a r s ago we developed a formalism with which t o d e f i n e am l i t u d e s and observables i n two-body s c a t t e r i n g of p a r t i c l e s with a r b i t r a r y s p i n s ( l p ) . I n t h i s scheme the amplitudes a r e defined ( f o r each energy ana momentum t r a n s f e r ) i n terms of t h e s p i n p r o j e c t i o n s of each p a r t i c l e along q u a n t i z a t i o n axes defined s e p a r a t e l y f o r each p a r t i c l e . Because of v a r i o u s symmetry c o n s t r a i n t s t h e choices of axes a r e l i m i t e d , but i n f i n i t e n e v e r t h e l e s s .

For t h e pp system, p a r i t y , time r e v e r s a l and i d e n t i c a l p a r t i c l e con- s t r a i n t s ( 1 3 ) r e s t r i c t t h e axes t o be a l l normal t o the s c a t t e r i n g plane o r within t h a t plane. F u r t h e r , f o r t h e "planar" choice each p a r t i c l e ' s a x i s must m k e t h e same angle with t h a t p a r t i c l e ' s momentum a s every o t h e r p a r t i c l e ( i n t h e c e n t e r - of-mass frame). See Fig. 1 below. That i s , f o r example, i f the beam q u a n t i z a t i o n d i r e c t i o n i s 30' counterclockwise ( i n the s c a t t e r i n g plane) from t h e beam momentum, then the t a r g e t s p i n q u a n t i z a t i o n a x i s must be 30' from t h e t a r g e t momentum; and so on f o r t h e outgoing protons. Hence t h e r e i s one a r b i t r a r y angle t o choose f o r p l a n a r amplitudes t o be f u l l y s p e c i f i e d . Note t h a t t h e "planar angle" of 0" corre- sponds t o choosing h e l i c i t y amplitudes. For most models h e l i c i t i y or t r a n s v e r s i t y amplitudes a r e the simplest b u t , a p r i o r i , t h e r e i s no reason why some other planar amplitudes might not r e v e a l simpler dynamics. We w i l l make a c a s e f o r t h i s l a t t e r p o s s i b i l i t y l a t e r .

Once t h e axes a r e s p e c i f i e d , observables can be defined i n many d i f f e r e n t ways. The "Optimal"

choice involves d e f i n i n g the sim- p l e s t p o s s i b l e observables con- s i s t e n t with h e r m i t i c i t y ) l l O ) . Using each p a r t i c l e ' s d e n s i t y m a t r i x t o f i x t h a t p a r t i c l e ' s po- l a r i z a t i o n , t h e s i m p l e s t obser- vables a r e t h o s e f o r which the d e n s i t y m a t r i c e s have t h e s i m p l e s t forms :

Fig. 1: Planar axes i n p-p e l a s t i c s c a t t e r i n g . The "planar"

angle" i s 8.

For d e t a i l s , r e f e r t o the review i n t h i s Symposium and Refs. 10 and 13.

The optimal observables so defined a r e not u s u a l l y observed because they in- volve a l l p a r t i c l e s being polarized. Symmetries reduce t h a t requirement somewhat, e.g. i n pp e l a s t i c t h e f o u r t h p a r t i c l e ' s p o l a r i z a t i o n i s f i x e d once the o t h e r t h r e e a r e prepared and measured(l3). However, t o r e l a t e t o a c t u a l l y measured observables c e r t a i n averages and sums over s p i n s must be made. I n s p i t e of t h i s complication i t is s t i l l very i l l u m i n a t i n g t o formulate a n amplitude a n a l y s i s i r ? terms of the Optimal Formalism. What i s p a r t i c u l a r l y imporrant i s the g e n e r a l i t y of r e l a t i o n s between observables and amplitudes s o t h a t t h e frames can be chosen to f i t t h e app- l i c a t i o n s .

Since the pp e l a s t i c system i s our primary concern, herein we w i l l consider t h e t r a n s v e r s i c y , h e l i c i t y , and planar frames s p e c i f i e d by one planar angle, a s in- d i c a t e d above and d e f i n e d i n Table I.

-- -.

TABLE I. Amplitudes f o r p-p e l a s t i c s c a t t e r i n g General Form f o r A

+

B + C

+

^D

---

D(c,a;d,b) where c = <s(c).zc>

,.... .

T r a n s v e r s i t y

H e l i c i t y

Planar

D P ( ~ ' , a ' ; d ' , b ' ) ~ =

1

d(112)(-0)d~12)(-~)d(:'2)(-8)d~:~2)(-B)~h(c,a;d.b) a ' a 'b c c a,..d

where = ( B ) I

-

s i n $ {sinB[a+c+d-el

+

c o ~ 8 [ 4 b l )

~ ~ ( 8 ) ^t z(B

- Z) ^{+ g}

1 [a+c+s-e-4b]

111

-

AMPLITUDE DETERMINATION

As of two years ago t h e ZGS experimenters had measured and analyzed a com- p l e t e s e t of p o l a r i z a t i o n observables f o r pp e l a s t i c s c a t t e r i n g a t a momentum of 6 GeV/c and s e v e r a l s c a t t e r i n g angles(6,14). From those d a t a it is possible t o de- termine the f i v e complex amplitudes (up t o an o v e r a l l phase). There remain d i s - c r e t e ambiguities l e a v i n g f o u r p o s s i b l e s o l u t i o n s a t each angle(8).

The a c t u a l determination of t h e amplitudes can be done i n any frame. How- e v e r , an examination of the observable-amplitude b i l i n e a r r e l a t i o n s makes it c l e a r t h a t the t r a n s v e r s i t y frame provides the most d i r e c t connection and, t h u s , t h e l e a s t e r r o r *ropagation(8). The magnitudes of t h e f i v e t r a n s v e r s i t y amplitudes a r e f i x e d by 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 t h e f o u r p o l a r i z a t i o n q u a n t i t i e s P, C m ,

Dm

and

Km.

The phases a r e then f i x e d by the q u a n t i t i e s C11, Css, Cis. Kssr Hsns which have l a r g e r u n c e r t a i n t i e s . Although t h e r e a r e 10 observables i n t h i s s e t t h e r e is s t i l l a f o u r f o l d ambiguity. Furthermore t h e a d d i t i o n a l observable (beyond t h e min- imum of 9) is not q u i t e compatible with t h e o t h e r s a t some of t h e s c a t t e r i n g angles. The a c t u a l determination of t h e t r a n s v e r s i t y amplitudes was accomplished by doing a l e a s t s q u a r e s f i t of two of the phase a n g l e s t o s e v e r a l observables. At some s c a t t e r i n g a n g l e s t h i s gave r a t h e r l a r g e X2 v e r i f y i n g p o s s i b l e incon- s i s t e n c i e s .

When t h e f i n a l amplitudes were obtained and p l o t t e d i n t h e complex plane no s p e c i a l f e a t u r e s appeared. T r a n s v e r s i t y amplitudes do not appear t o be e s p e c i a l l y simple o r r e v e a l i n g of underlying dynamics. An example of two s o l u t i o n s a t one mo- mentum t r a n s f e r i s shown i n Figure 2. Note t h a t i n going from one momentum t r a n s - f e r ( o r s c a t t e r i n g a n g l e ) t o another i t was n o t always c l e a r how t h e 4 s o l u t i o n s

continued i n t o t h e 4 a t t h e neighboring momentum t r a n s f e r .

lrn"""1, Ampl,l"d.,

Some of t h e u n c e r t a i n t i e s and a m b i g u i t i e s a r e ex- pected t o be reduced when t h e a d d i t i o n a l observables re- c e n t l y analyzed and made a v a i l a b l e by t h e Argonne group(7) a r e incorporated i n t o our determination of tran- s v e r s i t y amplitudes. There i s s t i l l no e x p e c t a t i o n t h a t t r a n s v e r s i t y amplitudes w i l l have any p a r t i c u l a r l y note- worthy s t r u c t u r e .

I V

-

TESTING REGGE POLE MODEL

I

I

-L

As o u t l i n e d i n t h e i n t r o d u c t i o n , the Regge pqle model had t o become more and more complicated t o account f o r t h e considerable d a t a i n p o l a r i z e d two body s c a t - t e r i n g processes. Several r e f i n e d models were constructed t h a t could accomodate most of t h e r e l e v a n t d a t a by i n c o r p o r a t i n g many poles and c u t s with t h e i r accom- panying r e s i d u e pararneters(15,16). Before the Argonne program was completed, these models could be used t o make p r e d i c t i o n s f o r t h e pp amplitudes. We have t e s t e d one of t h e more s o p h i s t i c a t e d of the models; t h a t of Berger, e t a1. (15).

,

H e l i c i t y amplitudes, however, a r e expected t o i n d i - c a t e underlying exchange processes. These amplitudes a r e , , simply obtained from l i n e a r combinations of t r a n s v e r s i t y

, amplitudes and e x h i b i t c e r t a i n s y s t e m a t i c s t h a t can be compared w i t h v a r i o u s models. We have done t h i s f o r a

The Regge model i n question uses Pomeron, p , A 2 , c, w, f , w ' , a s n a t u r a l p a r i t y exchanges and t h e "poor man's a b s o r p t i o n model" n, 8 , A l , "Z", a s u n n a t u r a l p a r i t y exhanges. With the dominance of the d i f f r a c t i v e component

-

t h e Pomeron

-

t h e h e l i c i t y non-flip n a t u r a l p a r i t y combination of amplitudes i s expected t o dom- i n a t e .

Regge pole model(l1) a s we w i l l review next.

Fig. 2: T r a n s v e r s i t y Amplitudes a t 6 GeV/c, t = 0.6 G ~ v ~ / c ~

I n Regge pole models t h e pole c o n t r i b u t i o n s a r e organized i n t o combinations of h e l i c i t y amplitudes t h a t have d e f i n i t e n a t u r a l i t y f o r high enough energies. For t h e pp system t h e s e a r e

R

No = a+c, N 1 = b, 1i2 = d-e, f o r n a t u r a l p a r i t y , and Uo = a-c, U2 = d+e, f o r u ~ a t u r a l p a r i t y .

So i t i s the No t h a t should be dominant; t h e o t h e r 0 d e f i n i t e n a t u r a l i t y amplitudes a r e p r e d i c t e d t o be about an o r d e r of magnitude smaller i n magnitude.

The phases have d e f i n i t e p r e d i c t i o n s a s well. The

d i f f r a c t i v e p a r t , and hence No, i s p r i m a r i l y imagin-

<

^{10% b MU&}

a r y , while t h e pion c o n t r i b u t i o n , Up, i s mostly r e a l . These a r e t h e f i r m e s t p r e d i c t i o n s of most Regge

Iqu2F7 models. The other d e t a i l s depend on p a r t i c u l a r para-

&#

m e t e r i z a t i o n s .

Regge (Berger) -

We have used t h e model(l5) t o c a l c u l a t e t h e 6

- ^----

^{Optimal (1)}

GeV/c amplitudes a t s e v e r a l momentum t r a n s f e r s o r t ''I v a l u e s . One of those s e t s , a t t = -0.2 GeV/c, i s

shown i n Fig. 3. ^t=-0.2

(?I2

Fig. 3: Comparison of Regge Model of Ref. 15 with Optimal Amplitudes a t t=-0. ~ ( G ~ v / c ) ~

To compare with the a c t u a l amplitudes t h a t we have obtained from t h e d a t a , t h e d e f i n i t e n a t u r a l i t y combinations must be formed. However, , t h e r e i s an o v e r a l l undetermined phase a t each t value. Such phases can only be measured by i n t e r - f e r e n c e with a known process, e.g. Coulomb s c a t t e r i n g . To f a c i l i t a t e the com- p a r i s o n with the d e f i n i t e phase p r e d i c t i o n of t h e Regge model we assume t h a t the phase of our No amplitude i s equal to the model p r e d i c t i o n . A l l o t h e r phases a r e then fixed.

Recall t h a t t h e r e i s a f o u r f o l d ambiguity i n t h e " a c t u a l " o r Optimal ampli- tudes. At each t value the four s e t s were compared with t h e predicted amplitudes.

Fig. 3 shows t h e one Optimal s e t t h a t i s c l o s e s t i n r e l a t i v e magnitudes t o t h e Regge p r e d i c t i o n a t t h a t s i n g l e t value. The agreement i s poor. Comparing o t h e r t values i n t h e same way does no b e t t e r . I n general the magnitude of No i s always r e l a t i v e l y smaller than the model would l i k e , although t h e r e is always a t l e a s t one Optimal s e t f o r which the No i s t h e l a r g e s t amplitude.

Another way t o compare t h e Optimal amplitudes with t h e model p r e d i c t i o n s i s t o look a t h e l i c i t y amplitudes d i r e c t l y

-

a s a f u n c t i o n of t. The agreement i s no b e t t e r but i s r e v e a l i n g of the r a t h e r s t r i k i n g f l u c t u a t i n g of t h e Optimal ampli- tudes while the model amplitudes a r e q u i t e smooth f u n c t i o n s of t. An example of t h i s i s shown i n Fig. 4 where Icl i s p l o t t e d . I n Fig. 5 t h e r e l a t i v e phase of e i s shown t o have s i m i l a r l a r g e excursions compared t o t h e model.

Fig. 4: Phase of amplitude e.

I C I ( Y ~

Fig. 5: Magnitude of h e l i c i t y amplitude c a s a f u n c t i o n of t f o r Regge model vs f o u r s e t s of Optimal s o l u t i o n s .

What can be concluded from t h i s comparison? The s o p h i s t i c a t e d (Fourth gener- a t i o n ? ) Regge model f o r pp e l a s t i c s c a t t e r i n g amplitudes(15) does not agree with those amplitudes obtained from t h e complete s e t of p o l a r i z a t i o n d a t a a t 6 GeVfc.

The major disagreement involves t h e dominant amplitude, No. The Optimal r e s u l t does i n c l u d e a r e l a t i v e l y l a r g e r No, but not an order of magnitude l a r g e r than a l l o t h e r s . It i s t r u e t h a t t h e phases of a and c a r e n e a r l y e q u a l , so t h a t No i s an order of magnitude l a r g e r than U2 i n magnitude. This i s not t h e case f o r t h e re- maining t h r e e magnitudes, however, Perhaps such No dominance i s not i n d i c a t e d by t h e data.

The r e c e n t p u b l i c a t i o n of more p o l a r i z a t i o n observables from t h e ZGS experi- ments(7) b e a r s on t h i s q u e s t i o n of No dominance. The experimenters enhance the ob- s e r v a b l e s e t with t h e a d d i t i o n of KLS, DSS,

Ts,

HLSN, HNSS. I n p r i n c i p l e , t h i s gives an overdeterm'ned s e t . The amplitudes should be f i x e d uniquely- unambiguously. I n p r a c t i c e t h a t is not n e c e s s a r i l y the c a s e as we now r e p o r t .

The experimenters attempt t o determine t h e amplitudes from t h e i r d a t a ( 7 ) . They use a X2 minimization t o obtain those amplitudes t h a t give t h e best f i t t o t h e new observables. To make t h i s procedure t r a c t a b l e they assume t h a t No i s dominant.

I n f a c t , they a r e unable t o o b t a i n a reasonable X 2 f o r the f u l l observable s e t . The

RS

d a t a must be excluded t o o b t a i n a meaningful X2 f o r a f i t with No dominated amplitudes. Then the r e s u l t i n g d e f i n i t e p a r i t y h e l i c i t y amplitudes a r e i n f a i r but not close agreement with various Regge ~ t o d e l s ( ~ ~ , ~ ~ ) .

N

However, t h e r e i s no reason t o expect t h e DLS d a t a t o be incompatible w i t h t h e o t h e r o b s e r v a b l e s ( l 7 ) . E i t h e r the No dominance assumption i s unwarranted o r t h e r e a r e i n c o n s i s t e n c i e s i n the f u l l d a t a s e t . The former p o s s i b i l i t y i s under s t u d y s i n c e our determination from the smaller d a t a s e t c a s t s doubt on No dominance a t 6 GeV/c.

I f indeed No dominance i s i n c o r r e c t , even a t the s m a l l e s t t v a l u e s , a s we found p r e v i o u s l y , then what can be s a i d about t h e dynamics i n t h i s i n t e r m e d i a t e energy region? The Regge p i c t u r e i n i t s l a t e s t form(l5,16) c e r t a i n l y w i l l have t o be modified s i g n i f i c a n t l y . And t h a t l a t e s t form has so many complications t h a t t h e whole e n t e r p r i s e has few remaining proponents t o modify t h e scheme f u r t h e r . What o t h e r dynanical approaches would be f r u i t f u l ? We consider o t h e r p o i n t s of view next.

V

- ----

ONE PARTICLE EXCHANGE TESTS

While the p a r t i c u l a r Regge pole approach t o p a r t i c l e exchange may be inade- quate o r i n c o r r e c t , t h e notion of p a r t i c l e exchanges dominating hadronic i n t e r - a c t i o n s i s as old a s Yukawa's proposal t h a t l e d t o t h e pion. As complete explana- t i o n s of dynamics, s i n g l e p a r t i c l e exchange i s long known t o have t h e o r e t i c a l problems

-

r e a l i t y of amplitudes, u n i t a r i t y , exploding energy dependences, e t c . Those various problems have received considerable a t t e n t i o n through the years. A s a r e s u l t , many s o l u t i o n s have been proposed t h a t maintain some of t h e s t r u c t u r e of one p a r t i c l e exchange (OPE) while modifying t h a t s t r u c t u r e t o s a t i s f y some of t h e necessary t h e o r e t i c a l c o n s t r a i n t s . OPE has r e t u r n e d i n many d i f f e r e n t g u i s e s and has u s u a l l y provided a t l e a s t q u a l i t a t i v e understanding of s c a t t e r i n g phenomena.

Can the b a s i c n o t i o n of OPE dominance be t e s t e d , independent of p a r t i c u l a r models f o r such exchanges, e.g. Regge p o l e s , complex p o l e s , absorbed poles? We have r e c e n t l y shown how to accomplish t h i s given some complete s e t of amplitudes f o r a r e a c t i o n a t some energy and momentum t r a n s f e r ( l 8 ) . The t e s t i s obtained f a i r l y e a s i l y i n t h e Optimal Formalism.

Consider t h e r e a c t i o n A+B + C+D with s p i n s

sA.. .sD.

Suppose a s i n g l e "par- t i c l e " of d e f i n i t e J and p a r i t y i s exchanged i n t h e crossed t channel. Then f o r p l a n a r amplitudes defined i n t h e t channel, i . e . f o r the p r o c e s s AX: + BtD, two types of c o n s t r a i n t s operate. The f i r s t , the " J - c o n s t r a i n t " , r e q u i r e s t h a t t h e net s p i n p r o j e c t i o n , SZ, f o r t h e incoming s t a t e along some planar a x i s , z , not exceed J ; s i m i l a r l y f o r t h e outgoing s t a t e . Hence f o r t channel h e l i c i t y amplitudes f o r example, the h e l i c i t i e s a r e constrained by

I

a-c

I <

J,

I

^b-d (

<

J

( t ) 3

s i n c e ~ ( t ) ( c , a ; d , b ) =

1

^DJ ( c , a ; d , b ) dC-,;d-b ( O t ) , ( 1 ) J

where the a,...,d a r e the s p i n p r o j e c t i o n s along the momenta ( i . e . h e l i ~ i t i e s ) ( * ~ ) f o r p a r t i c l e s A,...,D i n the t-channel. For r e a c t i o n s with high spin p a r t i c l e s t h e s e J - c o n s t r a i n t s can f o r c e many amplitudes t o be zero. Of course when t h e phys- i c a l s-channel amplitudes a r e formed a s l i n e a r combinations of continued t-channel amplitudes(21) t h o s e zeroes give r i s e t o e q u a l i t i e s o r l i n e a r r e l a t i o n s among the s-channel amplitudes. The form of the l i n e a r r e l a t i o n s depends on the choice of frame again.

(a) (b)

Fig. 6: Amplitudes a and c i n the magic frame vs t; (a) magnitudes, (b) phases.

V I

-

SIDEWISE AMPLITUDES

Once the p-p e l a s t i c t r a n s v e r s i t y amplitudes were obtained a t 6 GeV/c we ex- p l o r e d d i f f e r e n t choices of planar amplitudes t o s e e whether o r not s i m p l i f i c a t i o n s i n the s t r u c t u r e of t h e amplitudes ap e a r f o r p a r t i c u l a r q u a n t i z a t i o n axes, o t h e r than t h e standard ones. Ue reported(54) i n t h e previous Symposium t h a t one propi- t i o u s choice of p l a n a r axes i s t r a n s v e r s e t o t h e momenta, but i n t h e s c a t t e r i n g plane. I n t h a t p l a n a r - t r a n s v e r s i t y ( 8 ) , o r "sidewise", frame the amplitudes become almost e x c l u s i v e l y pure r e a l or pure imaginary with r e s p e c t t o one another a s Fig.7 i l l u s t r a t e s . Since then we have looked a t t h e lower energy d a t a , 300 t o 800 rieV u s i n g phase s h i f t s and amplitudes t h e r e a s well.(19) The r e s u l t s a r e r a t h e r s t a r t - l i n g a s Fig. 8 shows. These lower e n e r g i e s e x h i b i t t h e same kind of s i m p l i c i t y i n t h e sidewise frame a s the 6 GeV/c case.

Why should t h e sidewise frame have p a r t i c u l a r l y simple phases f o r t h e ampli- t u d e s There seems t o be no simple explanation i n terms of exchanges s i n c e sidewise amplitudes a r e combinations of both n a t u r a l i t i e s . Furthermore t h e p e r s i s t e n c e over t h e l a r g e energy range considered s u g g e s t s none of t h e "standard" p i c t u r e s t h a t apply only t o t h e lower i n t e r m e d i a t e o r h i g h e r medium e n e r g i e s could be o p e r a t i n g h e r e i n . We continue t o puzzle over t h i s s t r i k i n g phenomenon. Is t h i s new dyn- amics?

QCD BASED MODELS

VI1

- - - -

It i s not a t a l l obvious t h a t QCD should have anything t o say about 6 GeV/c p-p e l a s t i c s c a t t e r i n g . The energy may be too low; t h e momentum t r a n s f e r i s too s o f t ; t h e process i s an e x c l u s i v e one. Yet i t is worth asking whether t h e OPE dom- inance found above can r e f l e c t a gluonic sub-process.

There a r e many v e r s i o n s of QCD i n s p i r e d models f o r e x c l u s i v e s c a t t e r i n g i n t h e hard r e g i o n , i.e. s / Z = It

I>>

rnz(quark) o r m2(hadron) o r ~~(QcD). I n such models t h e hadrons d i s s o c i a t e i n t o c o n s t i t u e n t s and the c o n s t i t u e n t s i n t e r a c t v i a funda- mental QCD p e r t u r b a t i v e diagrams l i k e one-gluon-exchange, quark i n t e r c h a n g e , m l t i - p l e gluon exchange, m u l t i p l e quark exchange, etc.(25) Because t h e h e l i c i t y s t r u c t - u r e of t h e fundamental quark-gluon v e c t o r coupling is so simple f o r l a r g e momentum t r a n s f e r , t h e r e a r e simple p r e d i c t i o n s f o r t h e h e l i c i t y s t r u c t u r e of t h e hadronic process i t s e l f . ( 2 6 ) I n p a r t i c u l a r we know t h a t s i n g l e o v e r a l l h e l i c i t y f l i p i s d i f f i c u l t t o produce by hard s c a t t e r i n g ( y e t t h a t is c o n t r a d i c t e d by t h e l a t e s t BNL d a t a reported a t t h e Symposium). (27)

What becomes of t h i s p i c t u r e as t h e It

1

v a l u e s t a r t s decreasing? For one t h i n g the running s t r o n g coupling c o n s t a n t s t a r t s i n c r e a s i n g , although only l o g a r i - thmically. Hence higher o r d e r s become r e l a t i v e l y more important and m u l t i p l e gluon exchange should s t a r t t o dominate. The quark interchanges should be l e s s important a s t h e u-channel becomes f u r t h e r away. Now i n t h e m l t i p l e gluon exchanges t h e loop i n t e g r a t i o n s always include a region i n which one of t h e gluons c a r r i e s most of the momentum t r a n s f e r . The o t h e r exchanged gluons w i l l then be s o f t and serve only t o " f i x t h e color". Suppose these regions dominate t h e i n t e g r a l s a t t h e s e in- t e r m e d i a t e It1 values. Then t h e process w i l l have t h e kinematic s t r u c t u r e of "one gluon exchange" even though m u l t i p l e s o f t gluons a r e involved i n i n i t i a l and f i n a l s t a t e i n t e r a c t i o n s .

Adopting t h i s s p e c u l a t i v e p i c t u r e ( 1 2 ) of "one gluon exchange" a t moderate mo- mentum t r a n s f e r s , we have a simple conclusion. The amplitudes f o r the p-p e l a s t i c process, i n which t h e "OGE" is embedded, w i l l have t h e s t r u c t u r e of n a t u r a l p a r i t y OPE. The simple h e l i c i t y p r e d i c t i o n s f o r the hard region need not hold i n t h i s r e g i o n , but t h e b a s i c dominance of a s i n g l e s p i n exchange w i l l apply. So t h e near e q u a l i t y of a and c i n t h e magic frame could confirm t h i s point of view.

C e r t a i n l y t h e r e is nothing conclusive or compelling about t h e argument pre- sented h e r e exce t t h e phenomenological n e c e s s i t y . OPE has been seen t o dominate

P

t h e 6 GeV/c d a t a ( 9 ) but Regge models f a i l t o f i t t h e amplitudes.(11) What can be t h e source of the OPE s t r u c t u r e ? We simply p r e s e n t a p o s s i b l e i n t e r p r e t a t i o n i n terms of QCD based(12) notions. Whether the proposed "OGE" mechanism can a c t u a l l y dominate such r e l a t i v e l y s o f t kinematic r e g i o n s remains t o be seen.

We g r a t e f u l l y acknowledge very u s e f u l d i s c u s s i o n s with A. Yokosawa concerning t h e ANL d a t a , t h e work of our c o l l a b o r a t o r s N. Ghahramany and F. Arash, and t h e h o s p i t a l i t y of J. Soffer and the Symposium o r g a n i z e r s . This work was supported i n p a r t by g r a n t s from t h e U.S. Department of Energy.

Fig. 7: R e l a t i v e phases of sidewise Fig. 8: Relative phases of sidewise

amplitudes a t 6 GeV/c. amplitudes a t 579 and 800 MeV.

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