Quark Substructure and Isobar Effects on Deuteron Form-factors

7/28/2019 Quark Substructure and Isobar Effects on Deuteron Form-factors

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Hadron and Nuc lear Physics with ElectromagneticProbesK. Mam yamaand H. Okuno (Editors)2000 Elsevier Science B.V. All rights reserved. 193

Q u a r k s u b s t r u c t u r e a n d i s o b a r e f f e c t s o n d e u t e r o n f o r m - f a c t o r sE . L o m o n a *~ C e n t e r f o r T h e o r e t i c a l P h y s i c s a n d L a b o r a t o r y f o r N u c l e a r S c i e n c eM a s s a c h u s e t t s I n s t i t u t e o f T e c h n o lo g yC a m b r i d g e , M A 0 2 1 3 9

E l a s t i c e d s c a t t e r i n g , w i t h d e u t e r o n p o l a r i z a t i o n , u p t o h i g h m o m e n t u m t r a n s f e r p r o -v i d e s d e t a i l e d i n f o r m a t i o n o n t h e d e u t e r o n w a v e f u n c t i o n . T h i s d e t e r m i n e s t h e r a n g ed e p e n d e n c e o f t h e o r b i t a l a n d s p i n c o m p o n e n t s o f t h e o n e - a n d t w o - b o d y c u r r e n t s , r e -s t r i c t i n g c o n t r i b u t i o n s o f i s o b a r a n d m e s o n - e x c h a n g e c u r r e n t s a n d o f q u a r k / g l u o n d e g r e e so f f r e e d o m , a s w e l l a s t h e n u c l e o n c o m p o n e n t . T h e R - m a t r i x b o u n d a r y c o n d i t i o n m o d e lc o m b i n e s a l l t h e s e e f f e c t s , p r e d i c t i n g n u c l e o n - n u c l e o n r e a c t i o n s a n d t h e d e u t e r o n f o r m -f a c t o r s s i m u l t a n e o u s l y . A b r i e f d e s c r i p t i o n o f t h e m o d e l i s f o l l o w e d b y a c o m p a r i s o no f i t s r e s u l t s w i t h d a t a , e m p h a s i z i n g t h e r e s t r i c t i o n s p l a c e d o n t h e m o d e l b y e d e l a s t i cf o r m - f a c t o r s .

1 . I N T R O D U C T I O NT h e r e i s n o w e l a s t i c e l e c t r o n - d e u t e r o n s c a t t e r i n g d a t a d e t e r m i n i n g t h e e l e c t r i c f o r m -

f a c t o r A ( q 2 ) u p t o 6 ( G e V / c ) 2 [ l ] , t h e m a g n e t i c f o r m - f a c t o r B ( q 2 ) u p t o 2 . 8 ( G e V / c ) 2 [ 2 ] ,a n d t h e t e n s o r - p o l a r i z a t i o n f o r m - f a c t o r t 2 0 ( q ) u p t o 1 . 8 ( G e V / c ) 2 [ 3 ] . T h e s e d a t a r e s t r i c to r b i t a l a n d s p i n c o m p o n e n t s o f t h e d e u t e r o n w a v e f u n c t i o n a t s c a l e s a s s m a l l a s 0 . 2 f r o , a tw h i c h d i s t a n c e q u a r k d e g r e e s - o f - f r e e d o m ( d . o .f .) , i s o b a r c o m p o n e n t s a n d m e s o n - e x c h a n g ec u r r e n t s a l l h a v e a s i g n i f i c a n t r o l e .

T h e R - m a t r i x b o u n d a r y c o n d i t i o n m e t h o d [ 4 , 5 ] p r o v i d e s a h y b r i d q u a r k / g l u o n a n dh a d r o n m o d e l , i n c o r p o r a t i n g a l l t h e a b o v e c o n t r i b u t i o n s . O n l y a f e w o f t h e p a r a m e t e r s a r en o t p r e d e t e r m i n e d b y d a t a i n d e p e n d e n t o f t h e n u c l e o n - n u c l e o n ( N N ) i n t e r a c t i o n a n d s y m -m e t r y r e q u i r e m e n t s . T h e r e m a i n i n g f e w a r e a l m o s t a l l d e t e r m i n e d b y N N s c a t t e r i n g d a t a .E s s e n t i a l l y o n e p a r a m e t e r i s f r e e t o d e t e r m i n e t h e b e h a v i o r o f t h e t h r e e i n d e p e n d e n t e l a s -t i c e l e c t r o n - d e u t e r o n f o r m - f a c t o r s ( e d f f ) o v e r t h e l a r g e r a n g e o f m o m e n t u m - t r a n s f e r s , q .T h i s p a r a m e t e r d e t e r m i n e s t h e r e l a t iv e a m o u n t o f A A (7 D 1 ) a n d A A (3 D 1 ) i n t h e d e u t e r o n ,w h i c h p r o f o u n d l y a f f e c t s t h e q d e p e n d e n c e o f t h e s p i n a n d c o n v e c t i v e c u r r e n ts [ 6 ] . T h eN N s c a t t e r i n g i s n o t s e n s i t i v e t o t h e r a t i o , b u t o n l y t o t h e s u m .

F o l lo w i n g a r e v i e w of t h e R - m a t r i x m e t h o d a n d i t s a p p l ic a t i o n t o t h e N N s y s t e m , t h r e espec i f ic mo de l s f o r t he I = 0 , JV = I + sec t o r , o f d i f f e r en t l eve ls o f com ple t en ess , w i l l be*This wo rk is suppo rted in part by fund s provid ed by the U .S. Dep artmen t of Energy (D.O.E.) undercooperative research agreement #DF-FC02-94ER 40818 and in part by KEK fund s for Foreign VisitingScientists.Present address: Theory Group, KE K-Tan ashi branch, Tokyo


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c o m p a r e d w i t h t h e N N s c a t t e r i n g a n d e df f d a t a .m o d e l t h a t r e p r e s e n t s a l l t h e d a t a .

F r o m t h e s e o n e c a n e x t r a p o l a t e t o a

2 . T H E R - M A T R I X B O U N D A R Y C O N D I T I O N M O D E LA t h i g h m o m e n t u m - t r a n s f e r ( s h or t r an g e ) t h e r u n n i n g c o u p li n g c o n s t a n t o f Q C D i s

s m a ll , p e r m i t t i n g a p e r t u r b a t i v e d e s c r ip t i o n i n t e r m s o f c u r r e n t q u a r k s a n d g l u on s ( a s y m p -t o t i c f r e e d o m ) . A t l ow m o m e n t u m - t r a n s f e r ( lo n g r a n g e ) n o n p e r t u r b a t i v e e f fe c ts p r o d u c ec l u s t e r i n g i n t o c o lo r s i n g le t h a d r o n s ( c o n f i n e m e n t ) . T h e t r a n s i t i o n b e t w e e n t h e s e ex -t r e m e s h a s b e e n s h o w n t o o c c u r o v e r a s m a l l r a n g e o f t h e r u n n i n g c o u p l i n g c o n s t a n t[ 7 ] ,a n d t h e r e f o r e o v e r a s h o r t d i s t a n c e . T h e R - m a t r i x m e t h o d[ 8 ] i s w e ll s u i t e d t o t h i s s it u a -t i o n i n w h i c h t w o r e g i o n s , e a c h w e l l r e p r e s e n t e d b y a d i f f e r e n t a p p r o x i m a t e H a m i l t o n i a n ,h a v e t h e i r w a v e f u n c t i o n s c o n n e c t e d b y a b o u n d a r y c o n d i t i o n a t t h e s e p a r a t i n g s u r f a c e .

F o r t h e Q C D a p p l i c a t i o n , i n w h i c h c o n f i n e m e n t r e q u i r e s t h e q u a r k w a v e f u n c t i o n t o b es m a l l a t t h e t r a n s i t i o n b o u n d a r y , t h e s u i t a b l e f o r m o f t h e R - m a t r i x e q u a t i o n i s[4 ,5 ]

w)Or ) to = ~ f~(W )r W ) (1)w i t h

pi ( 2 )= f o + w - w ,l

i n w h i c h ~ a i s t h e e x t e r i o r w a v e f u n c t i o n f o r h a d r o n - p a i r c h a n n e l a , W is t h e t o t a l e n e r g y ,the po le s Wi a re the ene rg ie s o f a comp le te s e t o f in te rna l s t a te s van i sh ing a t r0 , and t heres idues i~ a r e g iv e n b y a b s o l u t e s q u a r e s o f t h e i n t e r n a l w a v e f u n c t io n d e r i v a t i v e s a tt h e b o u n d a r i e s . T h u s th e f ~ ( W ) a r e m e r o m o r p h i c f u n c t i o n s w i t h r e a l p o le s o f p o s i t iv er e s i d u e . T h e r e s i d u e s c a n b e e x p r e s s e d a s

~ = - r 0 - ~ r ~ ~ ~ (3)wh ere the f rac t io na l pa r en t age coe f f i cien ts ~ a re geom et r ic coe f fi c ien t s exp res sed in t e rm so f C l e b s c h - G o r d o n c o e ff ic i en t s o f t h e s p i n / f l a v o r / c o l o r s p a c e o f t h e g i v e n q u a r k c o n fi g u-r a t i o n . T h e r e f o r e o n l y t h e f o , r e p r e s e n t i n g t h e ef fe c ti v e c o n s t a n t o f d i s t a n t p o l e s a n dthe po le a t in f in i ty , a re f ree pa ramete rs .

T h e h a d r o n s a t r > r 0 i n t e r a c t v i a h a d r o n e x c h a n g e p o t e n t i a l s , a s g i v e n b y k n o w nh a d r o n m a s s e s a n d c o u p l i n g c o n s t a n t s f i x e d b y i n d e p e n d e n t e x p e r i m e n t s o r s y m m e t r yc o n d i t i o n s .

T h e s e p a r a t i o n r a d i u s , r 0 , m u s t b e w i t h i n t h e r a n g e o f a s y m p t o t i c f r e e d o m ( < 0 .8 5o f t h e e q u i l i b r i u m r a d i u s o f t h e i n t e r i o r b a g m o d e l ) b e c a u s e o f t h e s e n s i t i v i t y o f i~ tothe d e r iva t ive s a t r0 o f the in te rna l wave func t ion . Eq . (1 ) a l so requ i re s th a t r0 s a t i s fyr Wi(ro) ) = 0 . Us ing the low ene rg y da ta , the l a t t e r co nd i t ion f ixes the va lue o f r0w i t h s o m e p r e c i s i o n , g i v i n g a v a l u e c o n s i s t e n t w i t h t h e f i r s t c o n d i t i o n f o r t h e C l o u d y b a gm o d e l , b u t n o t f o r t h e M I T b a g m o d e l [ 4 , 5 ] , r u l i n g o u t t h e l a t t e r f o r t h e m u l t i - h a d r o nd o m a i n . T h e C l o u d y b a g m o d e l d e t e r m i n e s r0 = 1 .0 5 f m . T h e m o d e l t h e n g i v e s a g o o dd e t a i l e d f i t t o N N d a t a f o r rLa b ~ 0 .8 GeV[9] and a l so i s cons i s ten t wi th some ev idence fo r


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the lowes t I = 1 exo t ic re sonance , JP = 0+ [10] a t 2 .70 GeV . These re sonan ces , p ro du cedn e a r t h e f - p o l e s a t W i c o r r e s p o n d t o m u l t i - q u a r k c o n f i g u r a t i o n s o t h e r t h a n t h e m i n i m a lq~ and q3 conf igu ra t ions . Th e lowes t in the N N sys te m i s the I = 0 , JP = 1+ a t 2 .63 GeV.3 . D E U T E R O N P R E D I C T I O N S3 .1 . I n t e r i o r w a v e f u n c t i o n

T h e f i r st f - p o l e i n th e N N s y s t e m , c o r r e s p o n d i n g t o t h e [ q ( l S l ) ] a c o n f i g u r a t io n , is i nt h e I = 0 , JP = 1 + s t a t e , 0 . 7 6 G e V a b o v e t h e d e u t e r o n m a s s . A s t h e w i d t h o f t h e e x o t i cr e s o n a n c e i s o n l y 0. 03 G e V , t h e f ~ z a re n e a r l y c o n s t a n t . I t h a s b e e n s h o w n [ l l ] t h a t t h ei n t e r i o r w a v e f u n c t i o n v a n i s h e s fo r c o n s t a n t f , a n d t h e a c t u a l p r o b a b i l i t y o f b e i n g i n t h ein te r io r has bee n e s t i m a te d to be < 0 .004[6]. Th i s impl ie s a l a rge "ho le" in the d eu t e ro nw a v e f u n c t i o n , w h i c h h a s l o n g b e e n n o t e d a s a p r o p e r t y o f t h e e d f f a n d is al so e m b o d i e di n t h e e f f e ct iv e " h a r d c o re " o f N N s c a t t e r i n g . I n o u r m o d e l t h i s a p p a r e n t r e p u l s i o n a r i se sf rom the rap id chan ge in the e f fec tive d .o .f , a t r0 . However a t the ene rgy Wi the re i s" m a t c h i n g " o f t h e i n t e r io r a n d e x t e r i o r w a v e fu n c t i o n s , r e s u l t i n g in a s u b s t a n t i a l i n t e r io rp r o b a b i l i t y a n d a r e s o n a n c e .

T h e f a c t t h a t r 0 is 2 -3 t i m e s l a r g e r t h a n t h e c o r e s o f r e p u l s iv e c o r e m o d e l s i s c o m -p e n s a t e d b y t h e d i s c o n t i n u o u s i n c r e a se o f w a v e f u n c t i o n s a t r 0, s o t h a t t h e e x p e r i m e n t a le f fe c ti v e r a n g e p a r a m e t e r s c a n b e p r o d u c e d b y b o t h t y p e s o f c o r e e ff ec t. H o w e v e r t h emo de ls d i f fe r a t h ighe r N N sca t t e r ing ene rg ie s fo r la rge q edf f.3 .2 . E x t e r i o r w a v e f u n c t i o nIn the I = 0 , JP - 1+ sys tem, the NN (aS1) and NN (3D1) s t a te s a re coup led toe a c h o t h e r a n d t o i s o b a r c h a n n e l s b y m e s o n e x c h a n g e p o t e n t i a l s a s w e l l a s t h e f - m a t r i x .B e c a u s e o f t h e i r l o w t h r e s h o l d m a s s a n d s t r o n g t e n s o r c ou p l in g , t h e A A ( a S 1 ) , A A ( a D 1 )a n d A A ( a D 1 ) c h a n n e ls a r e m o s t i m p o r t a n t a n d a r e i n c lu d e d in a ll o u r m o d e l s. N N( 3 S1 ) a n d A A ( a S 1 ) s t a t e s h a v e n o n v a n i s h i n g ~ w i t h t h e [ q (l s8 9 6 q u a r k c o n f i g u r a t i o n ,c o n t r i b u t i n g t o t h e f - p o l e r e s i d u e . T h e N N * ( 14 4 0 )( 3 S 1 ) c h a n n e l a ls o h a s a l ow th r e s h o l d ,a n d w i t h i ts la r g e w i d t h i s o f n e x t i m p o r t a n c e t o t h e d e u t e r o n . B u t o v e r t h e e n e r g yr a n g e w h i c h i n c l u d e s t h e f i r st e x o t ic s , o t h e r c h a n n e l s a r e a ls o of s o m e i m p o r t a n c e a n d t h eN S l l ( 1 5 3 5 ) ( 1 P 1 ) , N S l l ( 1 6 5 0 ) ( 1 P 1 ) and AS31(1620) (3P1) a re inc luded in our recen t work .T h e s e c h a n n e l s m o d i f y t h e b e s t c h o i ce o f f ~ z f o r t h e l o w e r t h r e s h o l d c h a n n e l s , b u t h a v en e g l i g i b l e c o m p o n e n t s i n t h e d e u t e r o n .3 .3 . D e t e r m i n i n g t h e f o f or t h r e e m o d e l s

T h e f 0 z a r e s h a r p l y r e s t r i c t e d b y f i t t i n g t h e N N s c a t t e r i n g d a t a , w h i c h i n t h e d e u t e r o nsec to r requ i re s a f i t to the n p ( 3 S 1 _ 3 0 1 ) p h a s e p a r a m e t e r s 5 a n d ~ ( a S 1 ) , 5 a n d ~ ( 3 0 1 )and c1 fo r TLaD _< 0 .8 G eV. For the mo de ls C ' and D ' [12] , which have on ly NN and A Ac h a n n e l s , t h e l o w e r e n e r g y b e h a v i o r o f t h e d 's a n d e l d e t e r m i n e t h e N N s e c t o r f ~ z , w h i l ethe ene rgy de pen den ce fo r 0 .4 GeV < TLaD < 0 .8 GeV f ixes those co up l ing the NN an dA A ( 3 S 1 ) s e c t o r s , a n d t h e N N to th e s u m o f A A ( 3 D 1 ) a n d A A ( V D 1 ) s e c to r s . T h e l a stt w o h a v e t h e s a m e t h r e s h o l d b e h a v i o r , a f f e c t i n g t h e e l a s t i c s c a t t e r i n g i n t h e s a m e w a y .O n l y d e t a i l e d A - p r o d u c t i o n d a t a c o u l d s e p a r a t e t h e m , s o t h e r a t i o is f re e t o a d j u s t t o t h ee df f. T h e m a g n e t i c f o r m f a c t o r q - d e p e n d e n c e is v e r y s e n s it iv e t o t h e r a t i o b e c a u s e o f t h eo p p o s i t e s p i n a n d c o n v e c t i o n c u r r e n t s o f t h e s e s t at e s[ 6 ].


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T h e m o d e l C ' d i d n o t c o n si d e r q u a r k c o n f ig u ra ti on s [1 3 ]. W i t h o u t f - p o l e s t h e b e s tc h o ic e o f s e p a r a t i o n r a d i u s w a s r 0 = 0 .7 4 f m . M o d e l D ~ i n c l u d e d t h e l o w e s t f - p o l e w i t hr0 = 1 .05 fm as d i scussed above . A s show n p rev ious ly [12 ] , C ' and D ' g ive equ iva len t f i t sto the ~ ' s w h i l e case D ' i s a be t t e r f i t t o e l .

t2 .5,, 10.0

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0.960.96 ~0. 94 ~"0.92

. . . . I , , , , I , , , , I 0 . 9 00 250 500 750T L (MeV)

W h e n t h e o t h e r i s o b a r c h a n n e l sa r e i n c lu d e d , t h e N N * ( 1 4 4 0 ) , b e -c a u s e o f i t s la r g e r w i d t h , m o d i -t i e s t h e e n e r g y d e p e n d e n c e a n d i n -c reases the ine l as t ic i ty . Th i s r e -d u c e s t h e r e q u i r e d c o u p l i n g t o t h eA A c h a n n e l s. T h i s m o d e l E r e s u lt sin a be t t e r f it t o ~1 , (~ (3D 1) and toth e 77 's (Fig. 1).

F i g u r e 1 . T h e p h a s e p a r a m e t e r s f o r/ = 0 , J P ---- 1 +, np s c a t t e r i n g .M o d e l E (sol id curves); S A I D S P 0 0p h a s e p a r a m e t e r s (dashed curves ) ;B u g g 1 9 9 0 - 1 9 9 1 p h a s e p a r a m e t e r s(squares) .

3 .4 . T h e E D F F p r e d i c t i o n sI n p r e v i o u s w o r k [ 12 ], t h e e d ff f o r m o d e l s C ' a n d D ' w e r e c a l c u l a t e d w i t h t h e n o n r e l -

a t i v i s t i c , c o u p l e d c h a n n e l i m p u l s e a p p r o x i m a t i o n ( I A ) a n d t h e m e s o n - e x c h a n g e c u r r e n t(M EC ) t e r m s o f ;r , p , an d w , "pa i r " c o r rec t ions an d the pTr7 t e r m to f i r s t r e l a t iv i s t i c o rder .F o r t h e I A t h e i s o b a r f o r m f a c t o r s a r e a s s u m e d p r o p o r t i o n a l t o t h e n u c l e o n e l e c t r o m a g -n e t i c f o r m - f a c t o r s . I n a l l c a s e s t h e M E C c o r r e c t i o n s t o t h e i s o b a r c h a n n e l s a r e n e g l e c t e d .B o t h H 6 h l e r e t a l.[1 4] ( H O ) a n d G a r i - K r i i m p e l m a n n [ 1 5 ] ( G K ) n u c l e o n f o r m f a c t o r s w e r eused . T he r esu l t s (F ig s . 3 -6 o f [12 ]) a re seen to be a good to A(q 2 ) fo r t he H O cho icea n d t o B ( q 2 ) fo r t he G K cho ice . Th i s is no t i ncon s i s t en t as A(q 2 ) i s d o m i n a t e d b y t h en u c l e o n e l e c t r i c f o r m - f a c t o r a n d B ( q 2 ) b y t h e n u c l e o n m a g n e t i c f o r m - f a c t o r . T h e t2o(q2)p r e d i c t i o n s w e r e c o n s i s t e n t w i t h t h e v e r y lo w q e x p e r i m e n t a l r e s u l t s a v a i l a b le a t t h e t i m e .

H e r e w e p r e s e n t , v e r s u s t h e e x t e n d e d d a t a r a n g e o f t h e e d ff , th e r e s u l t s o f m o d e l sC ' , D ' a n d E w h e r e t h e f i r s t o r d e r r e l a t i v i s t i c c o r r e c t i o n h a s b e e n a d d e d t o t h e i m p u l s ea p p r o x i m a t i o n a n d t h e s e c o n d o r d e r r e l a t i v i s t i c c o r r e c t i o n s h a v e b e e n i n c l u d e d i n t h eM E C [ 1 6 , 1 7 ]. F o r m o d e l E t h e r a t i o o f A A ( 7 D 1 ) t o A A ( 3 D 1 ) c o u p l i n g t o t h e N N s e c t o rw a s g u i d e d b y t h e C ~ a n d D ~ m o d e l r a t i o s . I t h a s n o t y e t b e e n o p t i m i z e d t o t h e d a t a .


5/6

1 9 7

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, I . . . . i . . . . I . . . . i . . . . I . . . . F25 50 75 100 125 150q 2 ( f e r m i - 2 )

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mf 1. . . I . . . . I . . . . I . . . . I . . . . 1 . . .1 2 3 4 5 6 7q ( f e r m i - 1 )Figur e 2 . A(q2): D at a points are de-scribed in Ref. [1]. M odel C ' (HO ) (solidline); m o d e l C ' (GK) (da sh -dash ) ; m o d e lD ' ( H O ) ( d a s h - d o t ) ; m o d e l D ' ( G K ) ( d o t -dot); m o d e l E (HO) ( long da shes ) ; m o d e lE ( G K ) ( d a s h - d o t - d o t ) .

Figure 3 . t20(q): Data points described inRef. [3]. Model C' (sol id l ine); m o d e l D '(dash-dot); m o d e l E ( long da shes ). T h edependence on nuc leon emff (HO or GK)is negligible.

Also the ba lance o f A A and NN *(1440) coup l ing to NN and the va lue o f the N *(1440)magnet ic moment , unknown f rom independen t da ta , have no t been var ied .

Tab le 1 shows the resu l t s o f model E for the s ta t i c p roper t i es o f the deu te ron . Formod els C ' an d D ' the resul ts are as s ta te d in [12] except for a smal l re la t iv is t ic chang e inQ d e u t .

Tab le 1Sta t i c deu te ron p roper t i es o f Model EB E ( M e V ) P D ( % ) P A h ( % ) PAd(%) PzxT(%) P N * ( % ) Q ( f m2) /~D(/~?)

M ode l E 2.2247 5.21Ex p g 2 .2246

.006 3.24 1.79 0.71 .273 .860.286 .857aThe higher mass channels have neglibible probability.

A (q 2 ) i s shown in Fig . 2. I t i s seen th at mod el E with ei ther choice of nucleon form-factors f i ts the da ta reas ona bly wel l for q2 < 2 .5 (G eV /c) 2, bu t is only large enough forla rger q2 whe n the GK cho ice i s made . Models C ~ and D ~ on the o ther han d a re b e t te rw ith the HO choice for q2 < 2 .5 (G eV /c2), b ut a t 6 (G eV /c 2) only model C with GK islarge enough.t2o(q2) i s shown in Fig . 3 . For the momentum transfer range there is negl ig ib le sensi-t iv i ty to th e choice of nucleon form -factors , as these cancel in the ra t io of qua dru pole to


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198

monopole electr ic ampli tudes which dominates t20 . The resul t i s however very sensi t ive tothe model used . The s imple cons tan t f -mat r ix model , C ' , g ives a good f i t over the who lerange o f q. M odel D ' pu t s the ma x im um of t20 a t m uch too smal l a m om entu m t rans fe r .Th is i s re la ted to the l a rge ampl i tude o f the L = 2 , A A s ta tes in th i s model. Model E ,wi th in te rm ed ia te A A comp onen ts , has the max im um of t20 be tween tha t o f models C 'a n d D ' .

T h e B ( q 2 ) for mo dels C ' an d D ' is s imilar to that of [12] with m inim a at s l ight lysmal le r q . However the min imu m of B ( q 2 ) for model E is a t much too smal l a value( q 2 1 .3 (Ge V/ c )2 ) .4 . C O N C L U S I O N S

The R-mat r ix boundary cond i t ion model E , wi th a l l the re levan t i sobar channe ls , rep ro -duces th e np(3S1 _3 D1) sca t ter ing phases w el l up to TLab < 0 .8 GeV . I t a lso reprod ucesvery well the s ta t i c p roper t i es o f the deu te ro n and A ( q 2 ) for q2 < 6 (G eV /c ) 2. It do esnot fi t t 2 o ( q2 ) as wel l as the s impler model C ' (a l though i t i s be t t e r than model D ' ) , andhas the f i r s t min imum of B ( q 2 ) at much too smal l a value. To correct the fu l l model (E)fo r B(q2) , the ra t io o f the A A (rD 1) to AA (aD1 ) coup l ings to the N N sec to r needs to bevar ied . Th a t , and perha ps a fu r ther subs t i tu t ion o f N*(1440) coup l ing fo r A A coup l ingto the NN sec to r may a l so co r rec t the f i t to the max imum of t 2 o ( q 2 ) .R E F E R E N C E S

.

2.3.4.5.6.7.8.9.10 .11.12 .13 .14.15.16.17.

L.C. A lexa et a l . , Phys . Rev. L et t . 82 (1999) 1 374, and references therein .R.G . A rnold et a l. , Phys . Rev. L et t . 58 (1987) 1723, and references therein .D. Abbot e t a l . , arXive:nucl-ex/0001006, and references therein .P. LaFrance and E.L. Lomon, Phys . Rev. D34 (1986) 1341.P. Gonzglez, P. LaFrance and E.L. Lomon, Phys . Rev. D35 (1987) 2142.W.P. Si tarski , P .G. Blunden and E.L. Lomon, Phys . Rev. C36 (1987) 2479.M. Creutz , Phys . Rev. Let t . 45 (1980) 313.E.P. Wigner and L. Eisenbud, Phys . Rev. 72 (1947) 29 .P . LaFrance , E .L . Lomon and M. Aw, nuc l - th /9306026 .J. Ball et al . , Phys. Lett . 320 (1994) 206.H. Feshbach and E.L. Lomon, Ann. Phys . (NY) 29 (1964) 19 .P.G. Blunden, W.R. Greenberg and E.L. Lomon, Phys . Rev. C40 (1989) 1541.E.L. Lomon and H. Feshbach, Ann. Phys . (NY) 48 (1968) 94 .G. HShler , E. Pier tar inen and I . Sabba-Stefanescu, Nucl . Phys . B144 (1976) 505.M. Gari and W. Kri impelmann, Phys . Let t . B173 (1986) 10 .P .G. B lunden , p r iva te comm unica t ion . T he fo rmulas o f B lunden and Ian Tow ner a reused here .H. ArenhSvel , F . Ri tz and T. Wilbois , to be publ ished Phys . Rev. C. This workincludes boo st corre ct ions not included in the work of P.G. B lunde n an d I . Towner.

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