?s0^0/6JLO \ J

,\ COMITETUL DE STAT PENTRU ENERGIA NUCLEARA

ISTITUTUL C E N T R A L D E F IZ ICA

MieufWfTf-MAauricu

R O M A N I A

CENTRAL INSTITUTE OF PHYSICS

INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING

Buchaie.&t, P.0.B.S206, ROMANIA

FT-772-J979 KpUl

Dual diffractive resonances

new exotic states in hadron-nucleus

interactions

V.E.lon

Ab&tfiact : In this paper the experimental con­sequences of the spreading of the elementary hadron-nucleon resonances over all tlie hadron-nucleus partial waves are investigated. The dual d-l^KCLCtlve. i£Aonanc£A (DPR) as new pos­sible exotic states in the hadron-nucleus in­teractions, are discussed. The essential cha -raateristic features of the hadron - nucleus scattering in the DDE dominance limit are established.

T h e h a d r o n - n u c 1 e u s i n t e r a c t i o n i s a t p r e s t n r o n e o f

t h e m o s t i n t e r e s t i n g t o p i c s i n t h e r e a l m o f t h e c o n v e n t i o n a l

h a d r o n p h y s i c s . T h e r e v i v a l o f i n t e r e s t w a s c a u s e d by a c c u m u ­

l a t i o n o f m o r e a c c u r a t e e x p e r i m e n t a l d a t a \ I - 1 9 } a s w e l l as

by n e w t h e o r e t i c a l i d e a s o n t h i s s u b j e c t . One o f t h e m o s t a c ­

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

i s : i n w h a t f o r m c a n t h e e l e m e n t a r y h a d r o n - n u c 1 e o n r e s o n a n c e s

( s e e r e f e r e n c e [ 2 0 J ) be e x p e c t e d t o m a n i f e s t t h e m s e l v e s i n : h e

h a d r o n - n u c 1 e u s t o t s ! c r o s s s e c t i o n s ?

E x o t i c n u c l e i a r e t h o s e n u c l e i w h e r e o n e n u c l e o n i s

s u b s t i t u t e d b y a h e a v i e r b a r y o n . S i n c e t h e m o s t h e a v i e r

b a r y o n s a r e t h e h a d r o n - n u c k o n r e s o n a n c e s t h e n m o s t e x o t i c

n u c l e i a r e e x p e c t e d t o e x i s t o n l y a s r e s o n a n c e s . T h e s e e x c t i c

n u c l e i p o s e c o m p l e t e l y n e w p r o b l e m s o f n u c l e a r s t r u c t u r e s u c h

a s " e x o t i c g i a n t r e s o n a n c e s " [ 2 1 - 2 2 ] o r " e x o t i c [tt, 'd . Y .

e t c . ) - n u c l e u r d o o r w a y s t a t e s " f 2 3 j s i n c e , j u s t as t h e n u c l e o n -

h o l e a n d n u c I e o n - n u c I e o n i n t e r a c t i o n s i n t h e u s u a l n u c l e i g i v e

r i s e t o a l a r g e v a r i e t y o f c o l l e c t i v e s t a t e s , s i m i l a r l y t h e

h a d r o n - m j c l e u s s c a t t e r i n g i n t h e e l e m e n t a r y r e s o n a n c e r e g i o n

y i e l d t h e t r u e c o l l e c t i v e e x o t i c s t a t e s . A s u b s t a n t i a l amour , t

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

a c c u m u l a t e d [ ' ~ ' 9 J i n t h e r e g i o n c o r r e s p o n d i n g t o ^ ( ' 2 3 6 )

r e s o n a n c e i n t h e e l e m e n t a r y I ' M " i n t e r a c t i o n . T h e e s s e n t i a l r e ­

s u l t s c a n b e s u m m a r i z e d as f o l l o w s :

( a ) T h e t o t a l p i o n - n u c l e u s c r o s s s e c t i o n s e x i h i b i t a m a x i -

mum ( w h i c h s a t u r a t e s t r o n g a b s o r b t i o n l i m i t ) c l e a r l y

r e l a t e d t o Z i ( ' 2 36 ) r e s o n a n c e . The p o s i t i o n o f t h e

p e a k s h i f t s d o w n w a r d v ' i t h i n c r e a s i n g a t o m i c n u m b e r A .

T h e s h a p e o f t h e p e a k , i t s b r o a d e n i n g a n d a s y m m e t r y

i n c r e a s e w i t h i n c r e a s i n g A F ' « *•» 9 j •

- 2 -

( b ) T h e p i o n - n u c l e u s a n g u l a r d i s t r i b u t i o n s , i n t h e e n e r g y

r e g i o n o f £A O 2 36 ) r e s o n a n c e , d i s p l a y a t y p i c a l d i f f r o c -

t i o n p a t t e r n f 1 , 1 4 , 2 s » J w h i l e t h e e x c i t a t i o n f u n c t i o n s

h a v e a r e s o n a n c e b e h a v i o u r £ ' 9 j a t e n e r g i e s m u c h i o w e r

t h a n t h e f r e e £ ^ ( 1 2 3 6 ) r e s o n a n c e p o s i t i o n .

( c ) T h e p i o n - n u c l e u s p h a s e s h i f t s d i s p l a y e d i n A r g a n d

p l o t s h a v e p r o n o u n c e d r e s on a n c e - I i k e l o o p s £ 2 5 , 2 6 , 1 H ]

f o r a l l t h e p a r t i a l w a v e s ( e x c e p t i n g 5 - w a v e ) i n ' h e

r e s o n a n c e r e g i o n .

T h e r e e x i s ^ j a n i mp r r - M t n u m b e r o f t h e o r e t i c a l s t u d i e s

fib i ch h a v e b e e n d e v o t e d t o t h e s e p r o b l e m s . F o r a r e \ i e w o f t h e

p r e s e n t s i t u a t i o n s e e r e f s - £ 2 ^ , 2 7 , 2 9 ^ - T h e s p a t e o f t h e o r e -

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

m e n t b e t w e e n t h e t h e o r y a n d e x p e r i m e n t a l s c a t t e r i n g d a t a .

C a r r o l l e r a \ . [Sj h i v e f i t t e d t h e i r d a t a t o a s i n g l e a s y m m e ­

t r i c B r c i t - ti i n n e r r e s o n a n c e o b t a i n e d b y t h e a s s u m p t i o n o f a

s i n g l e r e s o n a n c e i n e a c h p a r t i a l c r o s s s e c t i o n s h i f t e d s y s t e ­

m a t i c a l l y , d u e t o t h e e f f e c t o f t h e c e n t r i f u g a l b a r r i e r I n d e ­

p e n d e n t a r g u m e n t " . , p r e s e n t e d b y f l c V o y £ 2 8 J a n d L a n d a u e t a l .

f"303 , i n d i c a t e t h a t r he o b s e r v e d oe a k i n t h e p i o n - n u c I e us t o ­

t a l i ' . m s ' j s e c t i o n s i s n o t f o be v i e v;e d . i s a s i n g l e r e s o n a n c e ,

b u t r a t h e r a s a n " e n e r g y b a n d " o t o v e r l a p p i n g r e s o n a n c e s , a l l

0 f w h i c h o c c u r i n e a c h p i on - n u c l e u s p a r t i a l v;a v e . D e t a i l e d i n ­

v e s t i q d t i o n s o f t h e c o n s e q u e n c e s o f s u c h a s p r e a d i n g o f t h e

e I . : MIe n t a r y r e s o n a n c e s ( S ţ R ) o v e r a l l h a i\ r o 11 - n u c I e u s p a r t i a l

i M » i ' S a r c n i _e s s j | * y s ' n e e t h e y mi q h t p r o v i d e an i m p o r t a n t c l u e

1 nwa r d a l i r t l i " 1111 .'n- r . t an d i n q o f t h e h ad r on - n u c I e us i n t e r j ' . -

1 i on.-. .

)

The a i m of t h e p r e s e n t p a p e r is to i n v e s t i q j t e the

main d y n a m i c a l - en s e q u e n c e s or the i E n o r f e e t i n the h a d r o n -

n u c l e u s s c a t t e r i n g i n ; .1 ._> j u d I J i • ' ' r . i t i i v o r e s o n a n c e ( D 0 R 1

l i m i t . In S e c t . 2 we d e f i n e a r. e .. t i a s ; o f , c J ' r f r i n g ţi h e - o -

me n a c a l l e d d u a l ;! : f r' r a c ' I . e ->r.a' : o r i , in D 0 S ; •. ,-h i I e ;ne s p e c i a

case o f t li e DDS s t a t e s , t h e d j j i - j i f f

r e s o n a n t d i f f r a c t i o n a r e 1 i ^ .. , ;sse. !

r 1 . e re s on an ce s or

S e c ; . 3.

2 . Dua I S i r f r a c t i v e S c a t t e r i n n P h e n o ̂ e n a

B e f o r e d i s c u s s i n g the a m p l i t udes o f the d u a l d i f f r a c -

t i v e s c a t t e r i n g (DDS) we w o u i d l i k e t o r e c a l l some e s s e n t i a l

f e a t u r e s o f t h e d i f f r a c t i o n and the r e s o n a n c e p h e n o m e n a .

S t a r t w i t h the 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 F ( E , 0 )

o f the s p i n z e r o p r o j e c t i l e on a s p i n z e r o t a r g e t w r i t t e n as

GO

\\)

where y is the c m . s c a t t e r i n g a n g l e , E and K are the c . m .

e n e r g y and momentum. T h e n , the e l e m e n t a r y d i f f r a c t i o n t n e o r i e s

can be d e r i v e d f rom the a s s u m p t i o n o f sharp c u t - o f f f o r s c a t ­

t e r i n g m a t r i x e l e m e n t s wh ich is e q u i v a l e n t to

f^-7l 4 or 31 it ^ ^ e o ~ Kit ( 2 a )

Ve*= ° ^ ir «I l > £ (2b )

A r e s o n a n c e is a i o l e on the second Riemann s h e e t of the

s c a t t e r i n g a m p l i t u d e . The p a r t i a l wave f o r a resonance w i t h " t * " C ,

is parametrized as

- •*

•St/2. a (£1= !S i± - -^ E0-E-iP/2.

wh i 1 e aii (\ f £ ) » for C "^ t » ,5 r e s m o o th , s 1 ow I y v a r y i n g w i th

e n e r g y , whore I n and | are the e l a s t i c and total w i d t h , £ is

the p o s i t i o n of the r e s o n a n c e on the real axis of the e n e r g y .

4 resonance nas ihe f o ! l o u i n q important p r o p e r t i e s :

(i) well d e f i n e d q u a n t u m n u m b e r s ( s p i n , i s o s p i n , p a r i t y ,

G - p a r i t y , b a r y o n i c q u a n t u m n u m b e r , e t c . ) ;

(ii) a mass (£„) and total w i d t h (") w h i c h are i n d e p e n d e n t

of the channel in w h i c h the re s o n a n c e is s e e n ;

(iii) coup ling, which can v a r y for channel to c h a n n e l ;

(iv) f a c t o r i z a t i o n for the c o u p l i n g to d i f f e r e n t c h a n n e l s .

Now, let us c o n s i d e r a c l a s s of limiting s c a t t e r i n g

p h e n o m e n a in which the c o n t r i b u t i o n of each partial wave

at full s c a t t e r i n g a m p l i t u d e is i n d e p e n d e n t of the a n g u l a r m o -

men turn (as in the case of d i f f r a c t i o n ) for al1

T h e r e f o r e , let us d e f i n e the dual d i f f r a c t i v e s c a t t e r i n g (DDS,

a m p l i t u d e s as f o l l o w s :

ae(E) = a (E)

aCE)= 0

c»«

C»b)

I -bOS,

L e t r ( £ > « ) > » ( E . , v ) , I ( l l , C r ) be t h e s c a t t e r i n g am­

p l i t u d e s c o r r e s p o n d i n g t o t h e r e s o n a n c e ( f t ) , t h e d i f f r a c t i v e (D)

and t h e d u a l d i f f r a c t i v e s c a t t e r i n g (DDS) p h e n o m e n a , r e s p e c t i v e l y

T h e n , u s i n g e q s . f ( ' ) " CO J we h a v e

F*(E,8)= f <&+*> P* - r?(co5e; ( 5 )

E R - E - 0 - C ^

- 5 -

F*(E,9)=-££(2UL)Pt(cos9) (6)

where r^— K~ "s t n e Compton wavelength of the projectile

C= I)• The cross sections of the DDS phenomena are given by

(I ) total cross section

(2 ) elastic cross section

(r6=finci(v^f|a(E.)r <•« (3 ° ) i n e l a s t i c c ross s e c t i o n

c i,v= W£((.+1)1 {3m a(E) - KE ) f ] Then, if U^K-K 3'' t n e s e cross sections have gigantic values

(proport iona1 to

The d i fferential cross sections are expressed by

( 3 c )

dcr (o°) 5*fiL (zt+i)*?^'©)

?tYt0+i)*faCE)|x

( 8 d )

( 8 c )

4%(uo°) = %*(*.+ if | a (e)r ( 8 f )

So, the differential cross sections of the DOS have the dif­

fract i ve patterns and they satisfy the relation

35>WV d o r ' **• (8g)

The e s s e n t i a l c h a r a c t e r i s t i c f e a t u r e s a re d i s c u s s e d in

d e t a i I in r e f . ^ 3 ^ -

3- The Dual D i f f r a c t i v e - Resonance Phenomena

We a r e now in p o s i t i o n t o s t u d y the dua I d i f f r a c t i ve -

resonance (DDR) phenomena wh ich can occur e s p e c i a l l y i n the

h a d r o n - n u c l e u s s c a t t e r i n g as an e f f e c t o f the " s p r e a d i n g " of the

e l e m e n t a r y h a d r o n - n u c ' e o n resonances over a l l h a d r o n - n u c l e u s

p a r t i a l waves. A p h y s i c a l u n d e r s t a n d i n g of such remarkab le

" s p r e a d i n g " i n v can be g i v e n in many f o r m s . For e x a m p l e , i n

the f i x e d nuc leon l i m i t , i n wh i ch the o p t i c a l p o t e n t i a l has a

resonance po le J_ VabL ^ -T" ' * - » " } « C ^ ) j ^ fa ) t l e n u c , f c u s

form f a c t o r _ J t h i s resonance spreads over a l l h a d r o n - n u c l e u s p a r ­

t i a l waves, bu t i n t h i s case each phase s h i f t has a s i n g l e p o l e

as a f u n c t i o n of e n e r g y and the p o s i t i o n o f the po le i s a t tne

sjme complex energy ( i . e . t u a t of the e l e m e n t a r y h a d r o n - n u c l e o n

resonance) in sach p a r t i a l wave. In t e res t i or; v i e w p o i n t s on t h i s

" s p r e a d i n g " in L of the e l e m e n t a r y resonances are t o be found

in the works of UcVoy [ 2 8 j and Landau e t a l . ţ 3 0 ] [ .

A n a t u r a l way of i n t r o d u c i n g the e f f e c t of t l ie e l emen ­

t a r y hadron-nuc Icon rcson. incos on the had ron -nu c 1 eus a m p l i t u d e is

t f> u'*e j - . i r i g l c Re <| «|c po le .ipp rox im.i t i on f o r each ( s - c h a n n e l )

p j r t i . i l wavi' . J H , WO s i a r t w i t h the f o l l o * # i n g had ron -nuc leus

• c<) t l o i i n i | .imp I i i udo :

kpt K P(cos0) 19)

where

E=-Q*N- Kx]lh+ O J + Ka3 ' /X- <m + <*v ( 1 0 )

K a " J Q a r e the momentum and the c . m . s c a t t e r i n g a n g l e s . O n . ,

• • A , N , n . , a r e the masses of t h e n u c ' e u s , the n u c l e o n and tne

i n c i d e n t h a d r o n , r e s p e c t i v e l y . Ql ( jEjand / ^ , ( t ) a r e the Regge

( p o l e and r e s i d u e ) p a r a m e t e r s vthich d e s c r i b e the s i n g l e p o l e c o n ­

t r i b u t i o n o f each p a r t i a l wave (up to l—l =^ K K ) a t the f u ' I

s c a t t e r i n g a m p l i t u d e .

riow, in the n e i g h b o u r h o o d of the e n e r g y C © where

passes through i n t e g e r u , we may w r i t e : I

where we have d e f i n e d

w i th

<*.«)= [J veydEi,;.,.*. I j ' s 2. /LCE*/ft <*e(E>a /\(E<> Therefore, i f

is a real function on E, then, e*ach hadron-noc leus partial wave is

described by the amplitude

(13)

which y i e l d s the u s u a l pre i t-W'i gner e x p r e s s i o n f o r a resonance a t

wi th a t o t a I w i d t h

- 8 -

rl fe re -rl

Sow, when a l l the resonance parameters I , *~0» *-D . [0 ,

ore independent o f t, . using eqs- ( 7 ) , ( 9 ) . ( ' 3 ) we obta in tha

amp I i tude 9

( 1 5 )

where p- [ E z Q V * the s c a t t e r i n g a m p l i t u d e ( 6 ) . The s c a t t e r i n g

phenomena d e s c r i b e d by the a m p l i t u d e { ' 5 ) M I I I be c a l l e d c>'ie

d u a l d i f f r a c t i v e resonance (DDR) ohenomena.

A p h y s i c a l u n d e r s t a n d i n g of t h i s remarkab le s c a t t e r i n g

a m p l i t u d e (15) can be g i ven as f o l l o w s . The hadron i n c i d e n t on

n u c l e u s / at e n e r q y E near Cne p o s i t r o n E of an e l e m e n t a r y r e -

sonance 3 w i l l e x c i t e the " a n g u l a r momentum bands"

="KV\ | o f o v e r l a p p i n g resonances . Assuming t h a t a l l these r e s o n a n ­

ces coup le e q u a l l y t o the incoming channe I j ft (•^•/Ai i ndependen t of U

then they w i l l be e x c i t e d c o h e r e n t l y to form wave packe ts in an-

g u l a r momentum of magni tudes J & J j ( ? - R e o l \ +Q r .v iOc'-) V

of " 3 r e i t - W i g n e r " fo rm i n (J. , each peaked at - ^ • ^K . ^ .O^ v/i th

a width: £ f - 2>0 t f (E j y ^ r f o i (E ) ^KRso , i n the , i m ! t in w i l i c h

«>n the pa ramete rs P , P P * and Î V ? ^ a r e ' " d e p e n d e n t o f

<£, , we o b t a i n the DDR-ampI i tude (15) i n wh ich the t o t a l w i d t h

can s a t u r a t e the upper l i m i t

where we have used the r e l a t i o n R » RQ A . r l e x t , i t i s easy to

see t h a t , i n the dua l d i f f r a c t i v e resonance l i m i t , we o b t a i n the

p r e d i c t i o n s l i s t e d i •» t a b l e s 1-2, where

4 (17)

« - 0

Tha angular behaviour of the quantity K^\^l^OSQjy^ »

for different E . is shown in Fig- I. Moreover, for s»a I I

angles and high values of \J^ K r\, , we have

(18}

s the Bessel function of order i

and « o - f^- is the t r a n s f e r o o a t n t u i .

TABLE I

°ă DIFFRACTION

ZY(R+1$

RESONANCE OIFFRACTIVC *ES<MA;<CE

Ил* ft г. ( Е Г Е ) Ь + ГЦ I, Wi) cn/D^t. rtfCr-TtCEi-^)1

<Г, HA T(R^x)1 W^+l\i7ifer T^VKy,Vr.g^ fc иг

IvA ftr(R+^

i/-2| + л — I t ( V E1 fe+rtf- tb r g.-&3 г* (e-E^-lpCr-r.Ce.-^

dft4 49C £ад te (Şr£)V Г гЛ

(g-fXf,

^»tf)<ţM* «Ю И * г (£<r Е-У+ j IT - VoCfc«T S ^

3&AW С ^ 6 ) ] ' den ^ ( o - X ^ C c o s ^ 1

- J) -

TABLE 2

CTE(E)

ycirr

n-Pt

(*) (*)

W(R *tf.f -£ Wft+K) C ««

i • i /

\, -fy P 2 -^ ( R - v - ^

CR^K- re? z% r o

J < ^ *t , n , i

We nor? that \f/j , ] and / - v i I I be taken at the corresponding

values of ER and E , r e s p e c t i v e l y

- 12 -

h. CONCLUS I OMS

I n t h i s pape r we l i nve i n u c i t i n a l e d t h e e x p e r i m e n t a l c o n ­

s e q u e n c e " o f the s p r e a d i n g o f t ! w • • l i . 'mon ta r y r e s o n a n c e s (SER) o v e r

a l l h a d r o n - n u c l e u s p a r t i a l waves i n t h e l i m i t in w h i c h | , Igp ,

.1 >0 o

T h e n , we have o b t a i n e d t h e d u a l d i f f r a c t i v e r e s o n a n c e s (QDR) tjs

.£. v 9

T and E o f t h e i n d u c e d r e s o n a n c e s a r e i n d e p e n d e n t o f v. .

o o

new p o s s i b l e e x o t i c s t a t e s i n h a d r o n - n u c l e u s i n t e r a c t i o n s . I f t h e

DDR s t a t e s a r e d o m i n a n t in t h e e n e r g y r e g i o n o f an e l e m e n t a r y

h a d r o n - n u c l eon r e s o n a n c e , t h e n t h e h a d r o n - n u c l e u s s c a t t e r i n g i s

c h a r a c t e r i z e d by the f o l l o w i n g e s s e n t i a l f e a t u r e s :

( t ) The h a d r o n - n u c l e u s t o t a l c r o s s s e c t i o n s ( s e e t a oIe I )

h a v e a B r e i t - W i q n e r a s y m m e t r i c f o r m w i t h an a s y m m e t r y

g i v e n by :

Tor / v ^ K » C b e h a v e s as A and the a s y m m e t r y i n c r e a s e

w i t h i n c r e a s i n g A . I f we n e g l e c t : t h e e n e r g y d e p e n d e n c e o f \

and \,p f a s w e l l as t h e s h i f t due t o the t e r m ( t ^ + % ) _2 t h e n we

f i n d t h a t t he p o s i t i o n o f peak i s d e s c r i b e d by

w e . - £ £ * - < > AT''"] S o , i f 0 o *> 0 t h e n t h e peak p o s i t i o n i s s h i f t e d d o w n w a r d s ,

w i t h i n c r e a s i n g ( » / » o ) as a f u n c t i o n of A, w h i l e i f Oo ^ 3

t h e n , t h e peak p o s i t i o n is s h i f t e d upwa rds w i t h i n c r e a s i n g ( V J t ^ .

S i n c e t h e t o t a l w i d t h 1 s a t u r a t e s t h e l i m i t I 5 \ L l l n

\ I ft i s of t he f o r m

K N " a n d

ER - Eo " "• (22)

1+ if the b r o a d e n i n g i n c r e a s e s w i t h i n c r e a s i n g A.

( 2 ° ) The t o t a l e l a s t i c c r o s s s e c t i o n a l s o behaves as A s

O t ^ R ) bu t t h e y a r e d e s c r i b e d by an e x a c t 3 r e i t - W i g n e r

r e l a t i on peaked a t the e n e r g y

T, P l(i+ If)

and a w i d t h d e s c r i b e d by e q . ( 2 1 ) .

( 3 ° ) The h a d r o n - n u c l e u s a n g u l a r d i s t r i b u t i o n d i s p l a y s t y p i c a l

d i f f r a c t i o n p a t t e r n s : a pronounced f o r w a r d peak and

p e r i o d i c s i d e maxima and m i n i m a . I f j f o ^ O t h e e x c i t a t i o n

f u n c t i o n s have a r e s o n a n t b e h a v i o u r a t e n e r g i e s much

lower compared t o t h e f r e e h a d r o n - n u c l e o n resonance

pos i t i on E .

(*» ) The r e a l p a r t of the f o r w a r d h a d r o n - n u c I eus s c a t t e r i n g

amp 1i t u d e , in the DDR l i m i t , has a r e s o n a n t b e h a v i o u r

c h a r a c t e r i z e d by R f t l T O ^ l ^ ) 5 8 0 f ° r E * EQ :

, ^ e ^ t V E , 0 ) > 0 f o r E A < f c and R * f = ^ E , 0 ) < O f T

E > E o -

( 5 ° ) The D P S - s t a t e s , a t E =• E ) , s a t u r a t e the a x i o m a t i c bounds

C y T ^ . W t f ( k t l ) * 4 t f ( B . + tff f o r ^ 4 ? and

P - P , r e s p e c t i v e l y ( s e e r e f . ^ 3 1 ^ ) .

Mow, i t is easy to see t h a t a l l the e s s e n t i a l c h a r a c t e ­

r i s t i c f e a t u r e s o f the D D R - s t a t e s a r e in oood agreement w i t h the

r e s u l t s on the p i o n - n u c l e u s s c a t t e r i n g in the r e g i o n c o r r e s p o n d i n g

t o ^ ( 1 2 3 & ) r e s o n a n c e . In o r d e r to c o n f i r m the j . r e s e n c e of the

- 14 -

D D R - s t a t e s in t h e p i o n - n u c l e u s s c a t t e r i n g i t i s f i r s t n e c e s s a r y

a q u a n t i t a t i v e a n a l y s i s of the e x p e r i m e n t a l d a t a . Such an a n a l y ­

s i s is in p r o g r e s s 3"A w i l l be p r e s e n t e d in a w o r t h c o m i n g p a p e r .

F i n a l l y , <; n o t e t h a t a l l the r e s u l t s o b t a i n e d in t h i s

paper can be e x t e n d e d t o the cases when the i n c i d e n t p r o j e c t i l e

and t h e n u c l e i have a r b i t r a r y s p i n s .

R E F E R E N C E S

1 F . B inon e t a l . , N u c l . P h y s . B1_7_, ( 1 9 7 0 ) 168

N u c l . P h y s . B J 2 , ( 1 9 7 1 ) ':2

N u c l . P h y s . Bjȣ, ( 1 9 7 2 ) 6 0 8

2 N . O . G a b i t z s c h e t a l - , P h y s . l e t t . V7_£, ( 1 9 7 3 ) 23*»

3 K. G a b a t h u l e r e t a l . , N u c l . P h y s . B_55_, ( 1 9 7 3 ) 23*»

h C . W i l k i n e t a l . , N u c l . P h y s . B62_, ( 1 9 7 3 ) 6 1

5 B.W. A l l a r d y c e e t a l . , N u c l . P h y s . A 2 0 9 , ( 1 9 7 3 ) 1

6 A . S . C lough e t a l . , N u c l . P h y s . B_7_6_, 0 9 7 1 » ) 15

7 F . B a l e s t r a e t a l . , Nuovo C i m e n t o , AJ2_, ( 1 9 7 5 ) 351

Nuovo C i m e n t o , AJ_3_t ( 1 9 7 5 ) 673

8 Y u . A . S h e h e r b a k o v e t a l . , Nuovo C i m e n t o , A3_[ ( 1 9 7 6 ) 2^2

Nuovo C i m e n t o , A3J. ( 1 9 7 6 ) 2*»9

9 A . S . C a r r o l l e t a l . , P h y s . R e v . Vt£, ( 1 9 7 6 ) 635

10 H. D o l l a r d e t a l . , P h y s . l e t t . Bb_3, ( 1 9 7 6 ) M 6

11 S . A . Dytman e t a l . , P h y s . R e v . 3JÎ, ( 1 9 7 7 ) 1059

12 J . P i f f a r e t t i e t a l . , P h y s . l e t t . , B £ 7 . ( 1 9 7 7 ) 2 8 9

13 I . V . F a l m o k i n e t a l . , Nuovo C i m e n t o A*»3, ( 1 9 7 8 ) ^99

- 15 -

\k F . S inon e t a ) . , M u d . Phys. A 2 9 8 , ( 1 9 7 8 ) ^39

15 F . B a i e s t r a e t a l . , F r a s c a t i R e p o r t . L N F - 7 8 / 2 0

16 R . H . Cole e t a l . , P h y s . R e v . _T7_£. ( ' 3 7 3 ) 631

17 O . J . H a l b r o u g h e t a l . , P h y s . R e v . _1_7£' ( 1 9 7 3 ) 1395

13 J . P . A l b a n e s e e t a i . , P h y s . L e t t . Ş7J., ( 1 9 7 8 ) 113

19 F . B a l e s t a e t a l . , F r a s c a t i R e p o r t L N F - 7 8 / 3 b ( P )

20 T . G . T r i p p e e t a l . , P a r t i c l e Da ta G r o u p . . Rev. Mod .Phys .kă_, ( 1 9 7 6 )

21 K. K l i n g e n b e c k e t a i . , p r e p r i n t ( 1 9 7 8 )

22 R. Handel e t a l . , P h y s . L e t t . 73_B, 0 9 7 8 } *

23 L . S . K i s s l i n g e r e t a l . , Phys . Rev . Le t : . 30., ( 1 9 7 3 ) 1071

A n n . P h y s . ( N . Y . ) 9_9_, 0 9 7 6 ) 371»

2k J . H i i f n e r , P h y s . R e p o r t s 2_K, (J 975)

2-j I . V . F a l m o k i n e t a l . , Nuovo Cimento L e t t . Jj., ( I S 72) 1125

26 M . H . Hoen ig e t a l . , P h y s . R e v . J_0£, ( 1 9 7 * 0 2102

27 R.R. S i l b a r e t a l . , Ann . R e v . Nuci .Se i . 2k_, ( 1 9 7 M 2^9

28 K. McVoy, N u c i . P h y s . A 2 7 6 , ( 1977) «i9 1

29 B . L . F r i m a n , M u c l . P h y s . A29*», ( 1 9 7 3 ) k$0

30 R . H . Landau e t a l . , Ann. of P h y s . ( N . Y . ) 7_8, ( 1 9 7 3 ) 299

31 O . B . I o n , I . P . N . E . p r e p r i n t ( 1 9 7 9 )

A (9) \ i 11 u i i j 111 i 1111 n I I I I | i r 1 I { i I ! f | I I I I

A(91-[f(2lf1)P[(cos9)fl

A(W)-(i-*-ry

Fig.l. Angular behaviour of K P ^ C O S 9)> for different l0.

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Bwbtmt, P.O.B. AIM

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