Experimental results on the (KK) and (Kp) systems as observed in the annihilations pp → KKπ at rest

Experimental results on the (KK) and (Kp) systems as observed in the annihilations pp → KKπ at rest

~ Nuclear PhysicsB3 (1967) 469-503. North-Holland Publ.Comp., Amsterdam EXPERIMENTAL RESULTS ON THE (KK) A N D (Kp) SYSTEMS AS OBSERVED IN THE A N N...
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Nuclear PhysicsB3 (1967) 469-503. North-Holland Publ.Comp., Amsterdam

EXPERIMENTAL RESULTS ON THE (KK) A N D (Kp) SYSTEMS AS OBSERVED IN THE A N N I H I L A T I O N S pp --~ K K z AT REST B. C O N F O R T O J , B. M A R E C H A L , L. M O N T A N E T a n d M. TOMAS ~ft" CERN, Geneva C. D ' A N L A U , A. A S T I E R , J. C O H E N - G A N O U N A a n d M. D E L L A N E G R A Laboratoire de Physique Nucl$aire, Coll}ge de France M. B A U B I L L I E R , J. DUBOC, F. J A M E S * a n d M. G O L D B E R G Institut de Physique Nucl$aire, Paris D. S P E N C E R Nuclear Physics R e s e a r c h Laboratory, University of Liverpool, U.K.

Received 31 August 1967

Abstract: In the course of an extensive study of ~p annihilations at rest involving KK pairs in the final state. 2000 events ~p .... K~K=LTr:F and 364 events ~p --~ro~o~o.~11,1. have been analyzed. This analysis shows that the K*(891) is strongly produced. In addition, we o b - _ serve a KK J P =2 + resonance at 1280 MeV and a significant enhancement in the KK system at threshold, in I = 1. We also need S-wave (Ky) amplitudes to reproduce the distribution of events on the Dalitz plot.

I. INTRODUCTION I n t h i s a r t i c l e , we a n a l y s e the e x p e r i m e n t a l r e s u l t s o b t a i n e d on the t h r e e body a n n i h i l a t i o n s ~p -~ K I ~ a t r e s t . T h e s e r e s u l t s c o r r e s p o n d to a s i g n i f i c a n t i n c r e a s e of the s t a t i s t i c s c o m p a r e d w i t h p r e v i o u s p u b l i c a t i o n s on the s a m e s u b j e c t ( r e f s . [1, 2]). T h e y c o r r e s p o n d to a t o t a l n u m b e r of a n n i h i l a t i o n s of 1.35 × 106 (the r e s u l t s we p r e s e n t h e r e i n c l u d e the d a t a d i s c u s s e d i n ref. [i]). T h e g e n e r a l a n a l y s i s of the t h r e e - b o d y ~p - . K~K+Tr:g w i l l be p r e s e n t e d i n s e c t . 3 w h i l e s e c t . 4 w i l l give the g e n e r a l r e s u l t s , w h i c h l e a d to the d i s c u s * P r e s e n t address: CERN. Geneva. J-Now at the Enrico F e r m i Institute, Chicago. ~'J Now at Junta de Energia Nuclear, Madrid.

470

B. CONFORTO et al.

sion of four main topics: evidence for a new KK effect at threshold, p r e s e n c e of a s m a l l s - w a v e s c a t t e r i n g length in the (Kn) s y s t e m s , p r e s e n c e of a (KK) JP =2+ r e s o n a n c e at 1280 MeV, p r o p e r t i e s of the K*(891) production. These four topics will be analysed in sects. 5, 6, 7 and 8 r e s p e c t i v e l y . In sect. 9. we shall c o m p a r e the r e s u l t s obtained for the annihilations ~p -~ K~lg~Tr6 to the p r o p e r t i e s of the annihilation ~p--~ K~K+T;~. Let us r e c a l l that there a r e at least three good r e a s o n s to study the (KK) and (K~) s y s t e m s in t h r e e - b o d y annihilations at r e s t ; the final state is f r e e f r o m any reflection due to the p r e s e n c e of a baryonic p a r t i c l e ; the initial state has well defined quantum n u m b e r s (since annihilations at r e s t o c c u r mainly in 1S0 and 3S 1 states); the limited n u m b e r of p a r t i c l e s in the final state allows a detailed a n a l y s i s , using for instance the f o r m a l i s m developed by Zemach (ref. [3]). In sect. 2, we d e s c r i b e the g e n e r a l p r o c e d u r e which has been followed for the identification of the events and their weighting.

2. TREATMENT OF THE DATA 1.35 × 106 antiprotons have been stopped in the 81 cm Saclay hydrogen bubble c h a m b e r , running at the CERN PS. The antiprotons, produced in a b e r y l l i u m t a r g e t at 700 M e V / c , have been m o m e n t u m and velocity a n a l y s e d by m e a n s of the beam k4 (ref. [4]) and subsequently degraded by a 10 cm copper a b s o r b e r down to 400 M e V / c just before entering the bubble c h a m b e r . The m o m e n t u m of 400 M e V / c has been chosen so that the antiproton stoppings would o c c u r near the middle of the bubble chamber. A r e s t r i c t e d fiducial volume has been chosen so as to r e d u c e to 12 + 2% the p e r c e n t a g e of annihilations in flight before m e a s u r e m e n t s and to e n s u r e a high detection efficiency for the types of events analysed. Subsequently, a p r e c i s e m e a s u r e m e n t of the incident t r a c k (the antiproton) (7 to 10 points m e a s u r e d on each of the three views on a length at least equal to 15 cm) lead to a f u r t h e r d i s c r i m i n a t i o n (prior to any k i n e m a t i c a l a n a l y s i s ) ; it a l lowed to r e d u c e the p e r c e n t a g e of annihilations "in flight" to 6 + 2%. This p e r c e n t a g e is still reduced when the k i n e m a t i c a l fit is c o n s i d e r e d : it is of the o r d e r of 2% for the f o u r - c o n s t r a i n t s fits (#p ~ K~IK~nV) and of the o r d e r of 4% for the o n e - c o n s t r a i n t fits (~p ~ K~lK~Tr°). In any c a s e , we have checked that our r g s u l t s a r e not significantly a l t e r e d by the inclusion of 5% of annihilations in flight which a r e f o r c e d to simulate t h r e e - b o d y annihilations at r e s t . We shall t h e r e f o r e take for granted that the s m a l l f r a c t i o n of annihilations "in flight" which may be left in our sample of annihilations "at r e s t " does not affect significantly our r e s u l t s . A double scan of the film shows that the scanning efficiency is of the o r d e r of 98% for events with V o p a r t i c l e s far enough (>~0.5 cm) f r o m the c o r r e s p o n d ing annihilations. To e s t i m a t e the l o s s of events with VO d e c a y s close to the annihilation, we have used the distribution of the decay lengths of the o b s e r v e d V o p a r t i c l e s . We have drawn this distribution for different m o m e n t u m r a n g e s of the V °, using all the events of the s a m e topology to i n c r e a s e our s t a t i s t i c s .

(KK) AND (KTr) SYSTEMS

471

M a k i n g u s e of the l i f e - t i m e of the K~I, we h a v e then b e e n a b l e to a s s i g n to e a c h e v e n t a w e i g h t d e p e n d i n g on the m o m e n t u m of the a s s o c i a t e d K~. T h i s w e i g h t t a k e s a l s o a c c o u n t of the l o s s e s due to the g e o m e t r i c a l l i m i t a t i o n s of the c h a m b e r . Fig. 1 s h o w s the d i s t r i b u t i o n of the w e i g h t s a s a f u n c t i o n of the m o m e n t u m of the K~I, a v e r a g e d o v e r all d i r e c t i o n s (this a v e r a g e a f f e c t s only the p a r t of the w~ight which d e p e n d s on the g e o m e t r i c a l l i m i t a t i o n s of the c h a m b e r . As it m a y be s e e n on fig. 1, this p a r t of the w e i g h t is l e s s i m p o r t a n t than the one due to the l o s s of V ° d e c a y s c l o s e to the a n n i h i l a t i o n : it i s n e g l i g e a b l e f o r m o m e n t a s m a l l e r than 400 M e V / c ) . We h a v e c h e c k e d t h a t the w e i g h t e d and u n w e i g h t e d d i s t r i b u t i o n s l e a d to the s a m e i n t e r p r e t a t i o n . T h e r e f o r e , in the following, we u s e the w e i g h t s only f o r the d e t e r m i n a t i o n of the a n n i h i l a t i o n r a t e s . ~p --' K~IK+n=~: T h e s e a n n i h i l a t i o n s a r e to be found in the " t w o - p r o n g - o n e - V ° " e v e n t s , 15000 s u c h e v e n t s h a v e b e e n s c a n n e d and m e a s u r e d , of w h i c h 2000 a r e identified a s pp -~ K~lK+~Wannihilations at r e s t . One s e e s that the n u m b e r of t h r e e - b o d y e v e n t s is s m a l l , c o m p a r e d to the total n u m b e r of e v e n t s with O 5:~ the s a m e t o p o l o g y , m o s t of t h e m b e i n g f o u r - b o d y a n n i h i l a t i o n s : ~p - _, K1K 7r 7rO and ~p ~ I~(K°)Tr+zr -. H o w e v e r , the f o u r - b o d y c o n t a m i n a t i o n is found to be l e s s than 17o a f t e r k i n e m a t i c a l and i o n i z a t i o n a n a l y s i s . On the o t h e r hand, 2% of the good t h r e e - b o d y a n n i h i l a t i o n s a r e a m b i g u o u s b e t w e e n the two s o l u t i o n s pp ~ KgK+~ - and pp ~ K~IK-V+. Since t h e s e e v e n t s r e p r e s e n t a s m a l l 1 f r a c t i o n of o u r s a m p l e , they a r e not i n c l u d e d in the a n a l y s e d data. ~p --~ K~lK~ln°: T h e s e a n n i h i l a t i o n s a r e to be found in the " z e r o - p r o n g - t w o - V ° ' ' e v e n t s . 800 s u c h e v e n t s have b e e n s c a n n e d and m e a s u r e d , of w h i c h 364 a r e l d e n t l h e d a s pp -~ K~K~g a r m l h i l a t l o n s at r e s t . In c o n t r a s t with what h a p p e n s f o r the c h a r g e d thre~-l~ody pp --~ K~K±TrT, the n e u t r a l t h r e e - b o d y ~p ~ K~lIX~.7r° r e p r e s e n t an i m p o r t a n t f r a c t i o n of :[he total n u m b e r of e v e n t s with the s a m e 1 topology. The c o n t a m i n a t i o n of the t h r e e - b o d y ~p -~ K~IK~Tr° into o t h e r a n n i h i l a t i o n c h a n n e l s (~p --~ K~K°nn ° with n > 1) is then, a p r i o r i , v e r y s m a l l . o o M o r e o v e r , it h a s b e e n show~1 that the c h a n n e l ~p ~ K1K 1 is f o r b i d d e n f o r a n n i h i l a t i o n s at r e s t . T h e s a m e q u a n t u m n u m b e r a r g u m e n t s m a k e it v e r y likely t h a t the c h a n n e l ~p -~ I~lK~Tr°Tr° is not open to a n n i h i l a t i o n s at r e s t (ref. [5]). T h e r e f o r e , the c o n t a m i n a t i o n of the t h r e e - b o d y ~p -~ K~K~Tr° could only c o m e , e v e n t u a l l y , f r o m the f i v e - b o d y c h a n n e l ~p .... I~lK~Tr°Tr°n% T h i s is v e r y unlikely. Fig. 2 s h o w s the m i s s i n g - m a s s s p e c t r u m f o r all e v e n t s of the type ~p K~IK°M ( w h e r e M i s f o r 0, 1, o r m o r e n e u t r a l pions). It c o n f i r m s the a r g u O men~s given a b o v e and s h o w s that in a d d i t i o n to the t h r e e - b o d y ~p K~IK~ITr a n n i h i l a t i o n s , t h e s e " z e r o - p r o n g - t w o - V ° " e v e n t s a r e m a i n l y due to the c h a n n e l s ~p ~ K~IK~IT?°and ~p -~ K~lK~l¢O°. .

.

.

.

O

.

.

.

.

472

B. C O N F O R T O et al. I

120 I.I0

]

I

]

J

~

b J

1.0

I

I

0.05

02

a

0.6

014 Momentum

K,°

I 0.8

(GeV/c)

F i g . 1. W e i g h t i n g of the e v e n t s . The weight i s given, f o r e a c h e v e n t a s a f u n c t i o n of the m o m e n t u m of the a s s o c i a t e d K~. T h i s weight is the p r o d u c t of two t e r m s , given by the c u r v e s (a) and (b). C u r v e (aT c o r r e s p o n d s to the l o s s of K~ which d e c a y c l o s e ly to the a n n i h i l a t i o n : it a f f e c t s m a i n l y the s m a l l m o m e n t a . C u r v e (b) c o r r e s p o n d s to the l o s s of K ~ w h i c h e s c a p e f r o m the v i s i b l e v o l u m e of the c h a m b e r . T h i s l o s s d e p e n d s on the ~ i r e c t i o n of the K~. The c u r v e g i v e s an a v e r a g e l o s s , i n t e g r a t e d on all directions.

2501 N

>=

(.O

20G --

OJ

o 0

150 q) Q) I O0

E Z

50

o 0

0.2

0.4

(Missing Mass) =

-

O

O

0.6

GeV =

.

Fig. 2. Missing m a s s squared M for pp ~ KIKIM; M is for 0, 1 or more neutral pions. The three-body annihilations ]~p ~ K~K~"° are represented by the dark histogram. The remaining events are mainly concentrated around M ~?]o and M ~ ¢o°.

(KK) AND (K~') SYSTEMS

473

3. G E N E R A L M E T H O D USED FOR T H E ANALYSIS O F THE T H R E E - B O D Y Dp ~ KKTr 3.1. Isospin analysis F r o m the point of v i e w of the i s o s p i n I and c h a r g e c o n j u g a t i o n C, we have to c o n s i d e r f o u r p o s s i b l e final s t a t e s : Op-4K°K-~

+ ,

~)p-~ K+~°u - ,

~p ~ KO~'OTro ,

PP --. K+K-~O .

If the c h a r g e c o n j u g a t i o n is c o n s e r v e d , the f i r s t two r e a c t i o n s c o r r e s p o n d to the s a m e p r o c e s s (ref. [6]). As we only o b s e r v e the c h a r g e d d e c a y m o d e of the K~I, K~I ~ 7r+Tr-, we s h a l l only c o n s i d e r the f o u r r e a c t i o n s :

pp

K~K+zr~: ,

(i)

(2) pp ~ K+K-zr ° .

(4)

T h e r e a r e a l s o f o u r p o s s i b l e initial s t a t e s c o r r e s p o n d i n g to

I=0,- 1.

C =±1,

The i n v a r i a n t t r a n s i t i o n a m p l i t u d e s a r e o b t a i n e d by p r o j e c t i o n of the final s t a t e s with well d e f i n e d c h a r g e c o n j u g a t i o n C = ±1 on the f o u r p o s s i b l e initial s t a t e s . U s i n g the r e l e v a n t S - m a t r i x e l e m e n t s : S l~p+E~p> =

D A~ il4)~,/> ,

(5)

I, i ' with ¢ = +1 the c h a r g e c o n j u g a t i o n e i g e n v a l u e , I the i s o s p i n of the total s y s t e m , and i o r I ~ the i s o s p i n of the KK s y s t e m . M o r e e x p h c l t l y : AI, i = (4)I,i 1S lPP + Epp) : t r a n s i t i o n a m p l i t u d e •

•

*~**

E

E

-

--

SIp~+EDp> = AO,II4)O,I E E > +A~,o14)~,0>+ A E i,i14)~,i>' with 4)0,1 - _ _

+ ¢[K-K°Tr+ + ~OK+Tr- - ~ 2 ( K - K + + K'°K'°)u o] } ,

(6)

4)1,0 ¢ = ~1[(K+K - - KoK.o)zro + E(K-K + - K-OKO)%O] ,

(7)

4)El,1 = ½ [(K+Y(°~- - K°K-7r+) + E(K-K°Tr+ - K°K+~-)] .

(8)

A s s u m i n g the initial s t a t e is r e p r e s e n t e d by an i n c o h e r e n t m i x t u r e of C

474

B. CONFORTO et al.

eigenstates, and using the following notation for the transition r a t e s :

nl = ]12'

R2 = l 2,

R3 = I(~p]SII~IK~ o>[2 '

R4 = I<~plS]K+K- o) 2

we obtain R 1 = .~t ~/2A~, 1+ ~/-3A~,1] 2+ ~ i~/-2A6,1 + ~/3AI,1 i2

R2

IA+0,i

+

~,0

(9) (lO)

12 '

R 3 = ~'!A0,1 + ~/3"A1,012 ,

(11)

R 4 : ~ [ A +0,1 - 4-3A~,012 + ~IA0,1-~f3A1,012

(12)

In these r a t e s , we have introduced the branching ratio F(~°~ °) _ 1 r(~+_ i - ~ , for the K~, since we only observe the charged decay mode ~ -~ ~+~-. In this~formulation, we have used the decomposition (KK, y). It is of course possible to use another decomposition. The Racah recoupling coefficients allow to go f r o m one decomposition to the other one in a simple way: If @ and ~p r e f e r to the decomposition (KI~,~) and (Ky,K) respectively: 4,~),1 = ¢0,½ E '

(13)

e 1 e 4"1,0 = - ~ P l , ½ + ~

e

4'1,1

:_~-~3 z e

~i,½

1 - ~

E ~01,~ '

e

~1,~"

(14) (15)

In the appendix, we p r e s e n t the decomposition according to the KTr coupling scheme. It leads directly to the above relations (13), (14) and (15). For the following discussion, it is interesting to point out that 4,~, 1 is a superposition of I(K~) =½ states (~1,½) and I(K~) = ~ states (
(K'K) AND (KTDSYSTEMS

475

G(I~) = (-i) ICK~)+ l(KK)

(16)

On t h e o t h e r h a n d t h e G - p a r i t y of t h e ~p s y s t e m i s e q u a l to

(17)

G(~p) = e(-l) I = G(KK) × G(u) . T h a t l e a d s to the s y m m e t r y p r o p e r t i e s

g i v e n in t a b l e l a .

Table l a Symmetry p r o p e r t i e s of the amplitudes Amplitudes

G(~p)

G(KK)

l(KK)

A;1

+

-

even

A;1

-

+

odd

A+ 10 A 10

-

+

even

+

-

odd

A {1

-

+

odd

A 11

+

-

even

AI. i

T h e s e s y m m e t r y p r o p e r t i e s m a y a l s o b e found in t h e e x p r e s s i o n s of t h e w a v e f u n c t i o n s 1"I,i a s e x p e c t e d f r o m the s y m m e t r i z a t i o n p o s t u l a t e : by d e f i n i t i o n , c o n s e r v a t i o n of C a n d I l e a d s to G - c o n s e r v a t i o n . M o r e p r e c i s e l y , t h e s y m m e t r y p r o p e r t i e s of t h e ¢~,i s t a t e s in t h e e x c h a n g e of the i n t e r n a l v a r i a b l e s of t h e K, K p a r t i c l e s a r e g i v e n in t a b l e l b . •

Table lb Symmetry p r o p e r t i e s of the amplitudes 1"1, i +

Amplitude Symmetry in the internal v a r i a b l e s of the K , K

+

+

1"01 1"01

1"10 1"10 1"11 I ' l l

even

even

3.3. Partial wave decomposition (a) Definitions: L e t u s s t a r t w i t h

odd

odd

odd

even

a simple non-relativistic description. F i g . 3 g i v e s t h e d e f i n i t i o n of t h e v a r i a b l e s w h i c h w i l l be u s e d : i = 1, 2, 3, c o r r e s p o n d s to t h e p a r t i c l e s K, K, u; Pi i s t h e m o m e n t u m of t h e p a r t i c l e i ; Pjk = (m4pk-mkPj)/(m4+mk) t h e r e l a t i v e m o m e n t u m of p a r t i c l e s j a n d k; a n d c o s ~jk :fijk "fii, w h e r e fijk :Pjk/[Pjkl a n d P i = Pi/ IPi [. W e d e f i n e a l s o : J , t h e t o t a l a n g u l a r m o m e n t u m ; ljk, the r e l a t i v e o r b i t a l m o m e n t u m b e t w e e n p a r t i c l e s j a n d k; a n d Li, t h e r e l a t i v e o r b i t a l m o m e n t u m b e t w e e n p a r t i c l e i a n d t h e (jk) s y s t e m . S i n c e c = +1 c o r r e s p o n d s to t h e 1S 0 i n i t i a l s t a t e (C =+1, J = 0 ) a n d ¢ = - 1 c o r r e s p o n d s to the 3S 1 i n i t i a l s t a t e (C = - 1 , J = + l ) , w e c a n r e w r i t e A¢l,i a s AJI,i

476

B. CONFORTO et al.

K, 81,

8mm

(a)

cyclic

I

(b) Fig. 3. (a) Wick triangle showing the relations between the momenta Pi (i= 1, 2, 3) of the three p a r t i c l e s , in the total c e n t r e - o f - m a s s system, the relative momenta Pij and the angles Ojk used for the p a r t i a l - w a v e decomposition. (b) Diagram showing the two-body decomposition of the t h r e e - b o d y annihilations, as it is used in our model (b) Multiple final state interaction model: W e s h a l l a s s u m e t h a t the t h r e e body a n n i h i l a t i o n p r o c e s s ~p ~ KK~ c a n b e d e s c r i b e d by t h e t e r m s a p p e a r i n g in t h e d e v e l o p m e n t s h o w n on fig. 3b. W i t h t h i s m o d e l , w e a s s u m e t h a t t h e final state interactions are much stronger than the single production term ( r e f . [7]). F u r t h e r m o r e , f o r e a c h t e r m , w e s u p p o s e t h e e n e r g y d e p e n d e n c e of t h e t o t a l p h a s e of t h e a n n i h i l a t i o n a m p l i t u d e i s g i v e n by t h e p h a s e 6jk of the ( j , k) s c a t t e r i n g in the a n g u l a r m o m e n t u m s t a t e ljk a n d t h e i s o s p i n s t a t e

Ijk.

We c h o o s e a s i n d e p e n d e n t d y n a m i c a l v a r i a b l e s f o r t h e i n v a r i a n t m a s s s q u a r e d of t h e jk s y s t e m a n d 6jk. We o b t a i n :

jk

coupling

Sjk

the

(KK) AND (K~ SYSTEMS

AI,J I12 =

~

a d 3 112 (s12' 012)

L3,112 +

~

477

ei512 sin 512 P12

bL2 113 (s13, 013)

ei513 sin 513 P13

L2,/13, I13 e i523 sin 523

+

~ i23CJLll23(s23 , 023)

L 1, 123,

P23

(18)

(c) Seleclion rules: The negative parity of the initial state imposes Li= ljk, The G-parity imposes that each amplitude Ai, i12is even or odd in the exchange of the space variable of the K (cf. sect. 3.2). We must have:

AId,IK~ =

~

l even odd

e iSKI~ sin 5KK PKK

J

all (SK~' 0K~)

(19) e

+

iSK~

sin 5K7r e i5~7r sin 5 ~ ] . PKzr ± bxx(s~ , 0~) p~

X,IKTr~ [bkx(SKTr'OKzr)

3.4. Separation of lhe angular variables . . Making use of s y m m e t r i c , t r a c e l e s s c a r t e s i a n t e n s o r s T zl" " "ll (/~) il,. • •, il = 1, 2, 3 constructed on the unit t h r e e - v e c t o r s / ~ which define the directions of the relative impulsions corresponding to the decomposition lwJk, L i (ref. [3]) for the r e p r e s e n t a t i o n s of the angular momentum l, we r i t e all J or bll J :

a~l = pl(sjklf(Ojk ) , where ol(sjk ) is a s c a l a r function of sjk and f(Ojk ) is given by • .

.i l

f : Ttl "

f = Sm~mii'T

^

(Pjk)T

iil..,

il_ 1

i 1. .

.i l

^

(Pjk)T

(Pi)

i'i 1.

when J = 0 , "" ~l-l(/~i) when J = 1 ,

S m (rn = 1,2,3) r e p r e s e n t s the spin-1 of the initial state, cmii' is the completely a n t i s y m m e t r i c tensor with three indices (E 123 = +1). 3.5. Approximations introduced for the energy -dependent part of the amplitude The amplitudes a r e then of the f o r m

478

B. CONFORTO et al.

EiSJk s i n pl(sjk) Pjk

with/(0jk)

: Tl(

5jk

f(gk) ,

(20)

jk) : Tl (Pi) "

T h e s c a l a r function old c o n d i t i o n s : when

Pjk --* O,

when

Pi ~ O,

pl ei6jk

sin

5jk/Pj k

pl ei6jk

sin

6jk

Pjk

- ¢

m u s t s a t i s f y the following t h r e s h te

l P)k

pl eiSjk sin 5jk te. l Pjk ~ c Pi

'

(21)

and to d e s c r i b e a r e s o n a n c e , we u s e the B r e i t - W i g n e r f o r m u l a . F o r i n s t a n c e , f o r the K*(891) r e s o n a n c e p r o d u c e d on a p - w a v e , the e n e r g y d e p e n d e n c e of the a m p l i t u d e will be given by the f o r m u l a :

m2K

cte pKTrPK _ m2 + irm °

(22)

w h e r e F, the width, is taken a s a c o n s t a n t , H o w e v e r , f o r a r e s o n a n c e n e a r t h r e s h o l d , we s h a l l u s e the f o r m u l a (29). We s h a l l a l s o m a k e u s e of the s c a t t e r i n g length in the z e r o r a n g e a p p r o x i m a t i o n , to d e s c r i b e an i n t e r a c t i o n n e a r t h r e s h o l d : f o r i n s t a n c e , f o r the KTr s y s t e m n e a r t h r e s h o l d p r o d u c e d on an s - w a v e

ct e •

(23)

1 -/aPKTr With t h e s e d e f i n i t i o n s , e a c h a m p l i t u d e is d e t e r m i n e d but to a s t a n t w h i c h is left a s a f r e e c o m p l e x p a r a m e t e r in the fitting Although we have given, f o r s i m p l i c i t y , a n o n - r e l a t i v i s t i c the g e n e r a l m e t h o d of a n a l y s i s , o u r r e s u l t s a r e given, in the t e r m s of r e l a t i v i s t i c m a t r i x e l e m e n t s .

complex conprocedure. d e s c r i p t i o n of following, in

4. G E N E R A L E X P E R I M E N T A L R E S U L T S FOR pp ~ I~IK±~ Fig. 4 s h o w s the p r o d u c t i o n D a l i t z plot for the r e a c t i o n ~p ~ K°K+~ff. T h i s D a l i t z plot m a y be qualified by the following r e m a r k s : (i) A n o t i c e a b l e a s y m m e t r y , with r e s p e c t to the f i r s t d i a g o n a l , w h i c h is an e v i d e n c e f o r the p r e s e n c e of i n t e r f e r e n c e e f f e c t s b e t w e e n the a m p l i t u d e s of the two i s o s p i n s t a t e s I =0, ! = 1. M o r e p r e c i s e l y , if we define two r e g i o n s in the D a l i t z - p l o t by the i n e q u a l i t i e s : Region I

:

M2(KTr) ° > M2(KTr) + ,

R e g i o n II : M2(Kn) ° < M2(KTr) ± , the d e n s i t i e s a r e then:

(KK) AND (KTr)SYSTEMS

479

20

-> 0

~:'~.::,~ :, '.:i". :: ..., •:~:(:!!.~:i.-: ::"): :'% 15

~,:i-:

J!'*>; ~ --:-'" "".", :; " ':"'-. ::,.~ :~:.... ,':., - ;....: ." ::..", C-(:.'. :' ':.: : :!'.

: - : . . . : .~.~:::.: ... , ' . ;::::'.:.:.

.:

I0

• -:~'.,:,....

:...:.:

~:'. (' : ::'~,:':~:::: : '." ,

%: . :.'.,;~j-

*

;

,"

• ~; .. , /

/.

.

:,

~'/ ?':' :.:'~"

"

~.::::."/..

'~ " ~ ':'~ ,'.":,-: ( 2 . ~ . ; ; : ' ~ : : • ::~ .' -: ': ;"

L' .: ~,: !:. '. ':' .~: '. :;., .:; ,"',, -:' ,.t"~:~, : .':":.~":': '": .', :'".'.L'"~ .'."."~ " i.:~..::~..~;.:, : : / ! . : ~ .....

o 5

~I~ K* (,891)

I

o

05

Io Mass z

:

@{

'.

"

%

I

I

15

20

( K°~ -~)

GeM

2!5

z

F i g . 4. E f f e c t i v e - m a s s s q u a r e d D a l i t z plot f o r ~p ~ K ° K ~ F ~ (2000 e v e n t s ) . F o u r a c c u m u l a t i o n s of e v e n t s a r e v i s i b l e : the K*°(891), K*+{891), KK(1280) and KK(990). T h e y a r e i n d i c a t e d by the a r r o w s .

',

=

-

+~I-~

o,i+

1,1

"

On the folded Dalitz plot (fig. 5), the density is given by the sum of the

~>

1.5

,F¢

/

-

0

'~

,

1.0

.

. (- K" (891)

(D

0.5

K*(~I)

o5

,io Mass z ( K ~)

.i~

I

..o

GeM 2

Fig. 5. F o l d e d Dalitz plot for pp ~ K~K=E?T=F.

480

B. CONFORTO et al.

contributions of the four amplitudes:

AO,I

A~,I,

A{), 1 and A1, 1:

R(I+II) = :~4 A+ 0 , 1 1 2 + ~ I A + l , l I 2 + ~]I A2 0+, 1 ~ ! '

AI,- 112"

(26)

The i n t e r f e r e n c e effects disappear. (ii) An accumulation of events in the two K*(891) regions, the K *± being less abundant tMn the K *o (figs. 6 and 7). The decay angular distributions of the K* (figs. 3 and 9) do not c o r r e s p o n d to a pure sin20 or cos20 d i s t r i bution, as it would be the c a s e if all K* w e r e produced in a pure 3S 1 or 1S 0 annihilation state. (iii) A sharp peaking of events n e a r the KK threshold (fig. 10). This peaking is even m o r e striking when we c o n c e n t r a t e our attention on the events for which both (K~) effective m a s s squared a r e l a r g e r than 0.9 GeV 2 (the main p u r p o s e of this cut is to r e m o v e the K* contribution) (fig. 11). Fig. 12 shows the decay angular distribution of the KK s y s t e m for M2(KK) < 1.08 GeV2: it is compatible with isotropy. (iv) An accumulation of events in the (KK) s y s t e m around 1280 MeV. This accumulation is seen, both on the complete sample (fig. 10) and on the s e lected one (fig. 11). Fig. 13 shows the decay angular distribution of the KK s y s t e m in the 1280 MeV region: the p r e s e n c e of the charged K*(cos 0~ -0.6) and of the n e u t r a l K*(cos ~ ~0.6) contribute to its anisotropy. (v) In addition to the contributions of the K* and of the KK enhancements at threshold and at 1280 MeV, we still o b s e r v e a n o n - u n i f o r m density of events which may be c h a r a c t e r i z e d by a hole in the middle of the D a l i t z plot and a small but significant i n c r e a s e towards the (K~r) threshold e n e r g i e s . We have obtained a good i n t e r p r e t a t i o n of these o b s e r v a t i o n s with the i n t r o d u c tion of S-wave amplitudes r e p r e s e n t e d by a s c a t t e r i n g length, for the (KTr) systems. Taking into account these qualitative f a c t s , we have attempted to i n t e r p r e t the complete distribution of the Dalitz-plot, using the g e n e r a l method d e v e l oped in sect. 3. We use the following definitions: gt~:

m e t r i c t e n s o r 1, -1, -1, -1;

P K I ' pit p~: K2'

f o u r - m o m e n t u m v e c t o r s for K~I, K~, f i "

tt Et~ = PK1 + PK~2 + P~ : total f o u r - m o m e n t u m v e c t o r ; SKl~r = (PK1 + PTr)2 : invariant m a s s squared for ( K ~ +) ; SK27r = (PK2 + p~)2 : i n v a r i a n t m a s s squared for (K±~r=r) ; SK1K2 = (PK1 + PK2 )2 : invariant m a s s squared for (K~IK±) ;

PK1 = PK1 -Ett(Ev'PK.1/E2)' i.e. ( 0 , P g l ) in the tot~ 1 ~'.st f r a m e ; -r,.b'P'2= P~21,,-EtZ(E~' "PK2/E2)' i.e. (0,PK2) in the total r e s t f r a m e ;

(KK) AND (K~7)SYSTEMS

481

O ~

,

r

.2 ~ .~

~:=

o

.~=~ = ~

O

m

.

~

o

I

,+~-~-"- ~ ____J o

,Aa9 ~O0/SluaAa ~o JaqLunN

-H

-

-

F

. . . . .

~

3

4-1

T~O

L

_

_

I

I _ _ - -

,AB9 ~OO/s~,ua^a io JaqtunN

0 .,-.~'~

°

.~

0

~ ~

•

482

B . C O N F O R T O et al.

IJ J i

'

I

= [

T----

I

r ¢o

.•

g 60

60

/ 4O

40

E

r

z

2

-I

A

0

0

Decay cosine

Decay cosine

Fig. 8. Decay a n g u l a r d i s t r i b u t i o n of the K*~=(891) ~ K ~ "± in pp ~ K~K±zr~. The c u r v e r e p r e s e n t s the t h e o r e t i c a l d i s t r i bution a s given by the b e s t fit (solution (iii) of table 2).

~

150

i

]

I

F i g . 9. Decay a n g u l a r d i s t r i b u t i o n of the K*°(891) ~ K±~r~ in ~p ~ K°K±zr:':. The c u r v e r e p r e s e n t s the t h e o r e t i c a l d i s t r i bution a s given by the b e s t fit (solution (iii) of table 2).

I

I

I1

CD

1

O

O ¢o

100

~_ 50t~

E

z I

0 0.98

130

I

3

I

I

1.70

2.1

2.5

2.9

Mass = (K~K +)

GeV =

Fig. 10. ( Ko1 K•) e f f e c t i v e m a s s s q u a r e d d i s t r i b u t i o n f o r ~p -~ K°K±~":F. The c u r v e c o r r e s p o n d s to the t h e o r e t i c a l d i s t r i b u t i o n as* -given by the b e s t fit (solution (iii) of table 2). The a c c u m u l a t i o n s of e v e n t s at 990 MeV and at 1280 MeV (effective m a s s s q u a r e d of 0.98.and 1.64 GeV2) a r e c l e a r l y v i s i b l e .

(KK) AND (K70 SYSTEMS I o.)

I

I

[

I 1.3

t 1.7

483 E

6o

CO 0

40-¢t) q.)

"Z 20

E Z

0.98

2.l

M ass' (K°K±)GeV ' . 0 ± Fig. 11. ( K0I K±) effective m a s s s q u a r e d f o r -pp ~ K~K ~-:[: when M2(K/F) > 0.9 GeV 2. This cut is to r e m o v e the K*(891) contribution. The full curve c o r r e s p o n d s to the t h e o r e t i c a l d i s t r i b u t i o n as given by the best fit (solution (iii) of table 2). The dashed curve r e p r e s e n t s the phase s p a c e . The a c c u m u l a t i o n s of events at 990 MeV and at 1280 MeV a r e c l e a r l y visible.

-I

0 Decoy

I cosine

F i g . e l 2 . Decay angular d i s t r i b u t i o n of the KK s y s t e m for M2(KK) < 1.08 GeV 2 in ~p K~K ?T*. The c u r v e r e p r e s e n t s the t h e o r e t i c a l d i s t r i b u t i o n as given by the best fit (solution (iii) of table 2).

!~ -- P~t~,:Eg(E u. p~/E2), P i ~ l n = (Pn - P K 1 )/~ - [( rn2 -

i . e . ( 0 , p n ) i n the t o t a l r e s t f r a m e ;

rn21)/SKl~](P ~ +PK1)p,

i . e . (0, p ~ - P K 1 ) i n t h e (K17r) r e s t f r a m e ;

P ~ 2 ~ = (PTT - P K 2 )/z - [ (rn2 -

rn2K2)/SK2n](PTr+PK2)1~, i . e .

(0, p n - P K 2 ) i n t h e

(K27r) r e s t f r a m e ;

/1 PK1K2 =

(PK2 - PK1 )j~ - [ ( m 2 2

m21)/SKIK2](PK1 +PK2 )/~,

i . e . (0, P K 2 - P K 1 )

in t h e (K1K2) r e s t f r a m e .

484

B. CONFORTO et al.

100

80

6C

4(]

L

J*=Q"

0

I -1

0

Decay

cosine

Fig. 13. Decay angular distribution of the KK(1280) in ~p ~ K~Kirr~:. The three curve, represent the theoretical distributions as given by the fit for different spin-parity a s signments. The best fit corresponds to spin-parity 2 + . T h e s e f o u r - v e c t o r s a r e the r e l a t i v i s t i c g e n e r a l i z a t i o n of the v e c t o r s Pij and P K c o n s i d e r e d in sect. 3. We now give the e x p l i c i t f o r m of the a m p l i t u d e s A~, 1' A+1, 1, A~, 1 and A 1 1 which g o v e r n the r e a c t i o n (1). '(i~ The a m p l i t u d e A~ 1 c o r r e s p o n d s to 1S0, I = 0 initial s t a t e . We have to c o n s i d e r the contributio]as of the K*(891), the KK(1280), the KK at t h r e s h o l d and the (Kg) S - w a v e in IK~ = ½. a) K*(891); J P = 1- , P I @-K 2 ~ PK

PK 2 a P K 1> + SK17r - SK. + iq SK. F K . SK2 ~ - S K , + i S4~K.FK * .

(27)

b) KK(1280); J P =2 + , T# uCPK1 K2 ) Tp ~(p=) SKIK2 - SA 2 + i S4~A2 FA 2 ' where T t-Lu P. u _ ~ ( g L t U _ (PK1 + PK2 )/~(PK1 + PK2 )u (PKIK2) = P K I K 2 P K I K 2 SK1K2

T

) =p~pTr- ~

- E2

/Prr"

2 PKIK2 '

(KK) AND (K?DSYSTEMS

485

c) KK(1000); J P = 0+

1

(29)

SK1K2 - s o + iPK1K2 ~ •

In the l i m i t of a n a r r o w r e s o n a n c e f a r f r o m t h r e s h o l d , this width ~ m a y be r e l a t e d to the m o r e c l a s s i c a l "constant width" F by the r e l a t i o n F :PK1K2~/v~ o .

d) KTT (S-wave; IK~r = ½) ,

1

1 +

(30)

(ii) The a m p l i t u d e A +1 1 c o r r e s p o n d s to 1 SO, I =1 initial state. We have to c o n s i d e r the c o n t r i b u ~ n of the K*(891), the KK(1280) and of the (Ky) Sw a v e s in IK~ = ½ and IKn = ~. a) K*(891); J P =1- , bt PKlzrPK2P

PK2~FK1/~

SKlzr - S K . + i svr~K, F K .

SK2 ~ - S K , + i4 S K , F K , "

(31)

b) Kn (S-wave, IK~T=½) , 1 1 -

(32)

~½J-P227 T .

C) K~ (S-wave, IKzr = ~) , 1

(33) i

-~ ~ J ~ 2 ~

"

(iii) The amplitude An,_1 corresponds to 3S1, I= 0 initial state; only K*(891) will contribute 1

1

SK27r - SK, + iq SK, F K ,

t' (34)

w h e r e E~vp~ is the c o m p l e t e l y a n t i s y m m e t r i c t e n s o r with four indices

(E0123 = +1) . (iv) The a m p l i t u d e A1, 1 c o r r e s p o n d s to 3Sl, I = 1 initial state. ~) K*(891); J P = I -

,

486

B. CONFORTO et al.

v p ai1 1 -! E#vPcrPKlPK2P~I SK1 ~ - SK, + i s~f~K, I ' K , + SK2 ~ - SK, + i~/SK, F K , ~"

(35)

b) KK(1280); J P = 2 + , u

p

cr

-

1

¢ # uPcYPKlPK2PTr (PK1K2 P=~") s K1K2- SA2 + i~/SA2 FA2

•

(36)

Each amplitude A~ 1' A+ 1, Aa 1 and AT 1 is a linear combination of the above contribution's, e a c ~ o f thorn being'i'n~roduced with a complex p a r a m eter PK exp(ic~K) to be fitted with the data. We a s s u m e no i n t e r f e r e n c e effects between the amplitude c o r r e s p o n d i n g to 3S 1 and 1S 0 initial states. T h e r e f o r e , a b s o r b i n g the C l e b s c h - G o r d a n c o e f ficients in the p a r a m e t e r s PK exp(iq~K), the e x p r e s s i o n to be fitted to the density distribution is of the f o r m :

n- lA + 0,1 + A ÷1, 1i 2 + IA;, 1 + A I , 1 !

(37)

The 10 contributions d e s c r i b e d above lead then to a fit of 17 r e a l p a r a m e t e r s (disposing of an o v e r a l l n o r m a l i z a t i o n f a c t o r and of a phase for each of the initial states 381, and 1S0). The Dalitz plot has been divided into 192 cells (0.1 GeV 2 × 0.1 GeV2). As it will be explained in sects. 5 and 6, we have been led to p r o c e e d to three different l e a s t s q u a r e fittings c o r r e s p o n d i n g to the following p a r a m etrizations: (i) a B r e i t - W i g n e r p a r a m e t r i z a t i o n for the KK(1000) and a~(K~) = -a±(K~) (ii) a B r e i t - W i g n e r p a r a m e t r i z a t i o n for the KK(1000) and a.3(KTr) = 0 2 (iii) a s c a t t e r i n g length p a r a m e t r i z a t i o n for the KK(1000) add a~(KTr) = -a±(K~r) = -0.3 fm. The r2esults a r e given in table 2. The distribution of the cells for which the contribution to the X2 is l a r g e does not show a s y s t e m a t i c a r r a n g e m e n t on the Dalitz plot, but for the i n t e r f e r e n c e r e g i o n s of the K* and the KK(1280) r e s o n a n c e s . It gives us confidence in the fits obtained for the g e n e r a l f e a t u r e s of the amplitudes introduced in these fits, in p a r t i c u l a r for the m a s s e s , widths and s c a t t e r i n g lengths, a s well as for the intensities p2, but indicates that the p h a s e s q~, quoted in table 2 may be subject to l a r g e e r r o r s . This table 2 gives, for each solution, the v a l u e s obtained for the p a r a m e t e r s and the c o r r e s p o n d i n g number of events to be attributed to each c o n t r i bution (given by the integration of the s q u a r e d modulus of the amplitudes). In this fitting p r o c e d u r e the m a s s e s , widths and s c a t t e r i n g lengths have been kept constant. In a second step, the best values for these quantities have been obtained with a fit where all the physical quantities w e r e kept constant, but one. One may notice that the total n u m b e r of events obtained when adding all the contributions is not equal to the total n u m b e r of events: this is due to s o m e of the i n t e r f e r e n c e s for which the i n t e g r a l over the Dalitz plot is not zero.

(KK) AND (K~ SYSTEMS

487

Table 2 The p a r a m e t e r s p and Solution (i), X2 = 2 6 0 , Channel

Amplitude

K* S - w a v e KY IK =½

+ A01


p

99

Number of e v e n t s

0.61

0.00 a)

216

K *°, M=890 MeV. 1"=45 MeV

0.96

0.60

534

K *±. M=895 MeV, F=45 MeV

KK(1280)

0.67

1.02

260

KK(1000)

0.27

2.68

42

KK(1280), M=1280 MeV, F=90 MeV

0.23

-0.80

30

KK(1000), M=1016 MeV. T=200 MeV

0.58

1.84

191

0.71

-1.74

290

a~_(Kn) = 0.3 fm

580 41

a~ (K;r) = -0.3 fm

K* S - w a v e Kgr

IK =½

+

All

S - w a v e Ky IK?T=~

K* KK(1280) K*

AI~I

1.00a) 0.26

0.00a) 3.07

A01

0.22

1.11

2

28

Solution (ii) X2 = 258,
Amplitude

K* S - w a v e Kff [ _± K~-2 Ki~(1280)

+ A01

KK(1000) K* S - w a v e K~7 IK __½ K* KI~(1280) K*

p

69

Number of e v e n t s

0.60

0.00 a)

190

K*O; M=887 MeV, F=40 MeV

I.Ii

1.79

650

K*~=; M=897MeV, F=40 MeV

0.68

1.00

247

0.28

2.91

40

KK(1280) ; M=1280 MeV, F=90 MeV

0.31

5.24

51

KK{1000) ;M=1016 MeV, T =200MeV

0.64

5.87

214

a&(K~) = -0.4 fm

1.00 a)

0.00 a)

526

a~(KTr)

0.30

3.04

48

0.16

0.98

14

A+ 11

-

A 11 A 01

a) One module set to +1, two p h a s e s set to z e r o .

2

= o

B. CONFORTO et al.

488

Table 2 (continued) Solution (iii), X2 = 262, = 175, KK(1000): scattering length p

cp

Number of events

0.63

0.00 a)

210

K*O;M =890 MeV, F=45 MeV

1.40

0.36

1033

K*±; M=895 MeV, F=45 MeV

0.71

1.27

264

KK(1000)

0.95

2.01

475

K*

0.23

-0.04

28

0.64

1.26

218

0.38

-1.78

75

Channel

Amplitude

K* S-wave K~r -2rK~=! +

KK(1280)

S-wave K?; IK =½

A01

A1l+

S -wave KTr K* KK(1280) K*

All

1.00 a) 0.31

0.00 a) 3.08

A01

0.37

1.28

530 52

KF,(1280) ;M:1280 MeV, F=90 MeV K~(1000); a0(Ki~)= 2 fm

a½(K?D = 0.3 fm a~(Kff) = -0.3 fm

72

a) One module set to +1, two phases set to zero.

5. KK S T R U C T U R E AROUND 1000 MeV The a c c u m u l a t i o n of e v e n t s o b s e r v e d f o r the (K~IK+) s y s t e m n e a r 1000 MeV (figs. 10 and 11) d o e s not c o r r e s p o n d to a w e l l - k n o w n p h e n o m e n o n . H o w e v e r , the s a m e o b s e r v a t i o n is m a d e f o r o t h e r a n n i h i l a t i o n c h a n n e l s (pp ~ .K~IK+~:~+~ - at r e s t , ( r e f . [8]), ~p ~ K ° K + ~ : ~ ° at 1.2 GeV/c (ref. [9]), pp a n n i h i l a t i o n s into 4, 5 and 6 p a r t i c l e s at 3".7 GeV/c (ref.[10])). In the t h r e e - b o d y pp ~ K~IK±Tr=7we a r e now a n a l y s i n g , the effect is m o r e i m p o r t a n t and d e s e r v e s a d e t a i l e d study. F i r s t of all, we h a v e m a d e s u r e this a c c u m u l a t i o n at the edge of the D a l i t z - p l o t is not due to m i s i d e n t i f i e d o r badly m e a s u r e d e v e n t s which could be f o r c e d , by k i n e m a t i c a l c o n s t r a i n t s to s i m u l a t e a t h r e e - b o d y a n n i h i l a t i o n with m i n i m u m m a s s f o r the (KI~) s y s t e m . I n d e e d , the D a l i t z plot of the t h r e e body e v e n t s still s h o w s the (KI~) t h r e s h o l d a c c u m u l a t i o n when the m e a s u r e d q u a n t i t i e s a r e u s e d i n s t e a d of the fitted ones. In a f i r s t a p p r o a c h , we have t r i e d to i n t e r p r e t this a c c u m u l a t i o n by the p r e s e n c e of s t r o n g i n t e r f e r e n c e e f f e c t s in the (K~) S - w a v e a m p l i t u d e s IK~ =½ and IK~ =~. As can be s e e n in t a b l e 3, none of the a t t e m p t e d s o l u t i o n s give an a c c e p t a b l e i n t e r p r e t a t i o n of the data. When the s c a t t e r i n g l e n g t h s a½ and a~ have c o m p a r a b l e v a l u e s , the (K~)IK~T =½ and IKy = ~ a m p l i t u d e s c o r r e s p o n d i n g to the IS0, I= 1 initial s t a t e have the s a m e e n e r g y d e p e n d e n c e , and a r e t h e r e f o r e i n d i s t i r g u i s h a b l e .

(KF~) AND (K7D SYSTEMS

489

Table 3 Results of the first approach

< X2>

=

a ~1

a~

X2

a .1_2

a ~_

X2

0.0 0.0 0.0 0.0 0.2

-0.4 -0.2 0.2 0.4 -0,4

525 469 444 426 478

-0.2 -0.2 -0.2 -0.4 -0.4

0.0 0.2 0.4 0.0 0.2

395 459 418 477 420

0.2 0.2 0.4 0.4 0.4

-0.2 0.0 -0.4 -0.2 0.0

476 434 428 453 477

-0.4

0.4

429

177.

T a b l e 3 gives the X2 obtained when a ! and a ] a r e sufficiently different. F o r s c a t t e r i n g lengths s m a l l e r than 0.4"fm or 1}trger than -0.4 f m , t h e r e is no solution. As expected, the m a i n contribution to the b a d n e s s of the fit c o m e s f r o m the e x c e s s of events in the (K~) t h r e s h o l d r e g i o n (fig. 14).

I

i

0.98

1.30

i

i

1.70

2. I

i

i

2.5

2.9

-> 150 co

o. o I00 ~p

$

50

E z

Moss' (K°K+)

GeV'

o • ~ :~. The c u r v e s F i g . 14. (K~K ±) e f f e c t i v e m a s s s q u a r e d s p e c t r u m f o r ~p ~ K1K r e p r e s e n t the t h e o r e t i c a l d i s t r i b u t i o n s a s given by the fit when no KK t h r e s h o l d e f f e c t i s i n t r o d u c e d . The c u r v e s a r e in good a g r e e m e n t with the e x p e r i m e n t a l data f o r e f f e c tive m a s s s q u a r e d s u f f i c i e n t l y f a r f r o m the t h r e s h o l d r e g i o n . N e a r t h r e s h o l d , the h a t c h e d r e g i o n c o r r e s p o n d s to the d o m a i n of e x p e c t e d d i s t r i b u t i o n s f o r d i f f e r e n t v a l u e s of a½ and a~ (the S - w a v e s c a t t e r i n g length of the (K~r) s y s t e m s ) . It is c l e a r that none of t h e s e c u r v e s is in good a g r e e m e n t with the data n e a r t h r e s h o l d .

490

B. CONFORTO et al.

T h e r e f o r e , we m u s t i n t r o d u c e a n a d d i t i o n a l a m p l i t u d e to t a k e into a c c o u n t of the (KK---')t h r e s h o l d effect. We p r e s e n t f i r s t the r e s u l t s o b t a i n e d when t h i s a m p l i t u d e i s p a r a m e t r i z e d i n t e r m s of one r e a l p a r a m e t e r , n a m e l y a s c a t t e r i n g l e n g t h a o i n the z e r o e f f e c t i v e r a n g e a p p r o x i m a t i o n . We s h a l l t h e n d i s c u s s the r e s u l t s o b t a i n e d when the p a r a m e t r i z a t i o n of the (KK) a m p l i t u d e i s d o n e i n t e r m s of two r e a l p a r a m e t e r s , s o a n d y, a s i t i s the c a s e for the Breit-Wigner formulation. We h a v e fixed the s c a t t e r i n g l e n g t h s a± a n d a3 to a l = 0.3 f m , a3 = -0.3 f m , a l t h o u g h the p a r a m e t e r s a o , (So,~) c h a r a c t e r i z i n g the (KK) eff~ect n e a r t h r e s h o l d a r e c o r r e l a t e d to the n u m e r i c a l v a l u e s c h o s e n f o r a± a n d a~; t h e s e 2 2 v a l u e s for a_~ a n d a a gave u s the b e s t i n t e r p r e t a t i o n of the D a l i t z plot d i s t r i • 2 2 . . b u t l o n a n d a r e t h e r e f o r e p r e f e r r e d to a n y o t h e r s e t , for a d t s c u s m o n of the KK s t r u c t u r e a t t h r e s h o l d . (i) Introduction of a real scattering lenglh. The (KK) a m p l i t u d e i s a p p r o x i m a t e d , n e a r t h r e s h o l d , by 1 1 - iaoPK1K2 " T a b l e 4. s h o w s the r e s u l t s o b t a i n e d f o r s e v e r a l v a l u e s of a o.

Table 4 Values of X2 for different a o X~I3

2 XKI~(1)

2 XKK(2)

ao (fm)

2 XD

0.0 0.4 1.0 2.0 3.0

389 286 266 262 264

134 39.6 26.5 21.0 19.9

125 44 24.8 17.0 15.9

-0.4 -1.0 -2.0 -3.0

323 287 266 266

65.8 37.0 27.0 24.6

47.6 34.0 15.0 13.5

4.0 6.0 10.0 20.0

266 274 274 278

19.4 22.0 21.0 23.0

15.7 16.7 17.3 18.9

-4.0 -6.0 -10.0

267 268 270

23.6 23.0 22.8

13.5 13.7 14.0

a o

(fm)

2 2 XKK,(1) XKK(2)

where X2 means X2 on the Dalitz plot ((X2 ) = 175), XKR(1) 2 means X2 on the (KK) el2 2 2 fective mass squared spectrum ((XKI~(1)) = 25), XK,X{2) means X on the (KK) effective mass squared spectrum when K* contributions are excluded (M2(K~) >/0.9GeV 2) 2 {(XK~,(2)) = 16).

T h e c u r v e SL on fig. 15 r e p r e s e n t s the d i s t r i b u t i o n e x p e c t e d for the s o l u t i o n a o = 2 f m (×2/4×2) = 1 7 / 1 6 ) . A s i t c a n be s e e n on t a b l e 4 a n d fig. 16, the

(KT~) AND (K~T) SYSTEMS

1 N

491

[

T

>~

(.9 0D O 50 C5

t

BW

E

Q) >

25 .Q

E

z

1.30

0.98

1.70

Mass 2 (K~K -+1

2.1 GeV z

Fig. 15. (K°K ±) e f f e c t i v e m a s s s q u a r e d d i s t r i b u t i o n f o r pp -~ K°Ka:~":F when M2(K;r) > 0.9 GeV 2. C u r v e SL r e p r e s e n t s the t h e o r e t i c a l d i s t r i b u t i o n a s g~iven by the fit when an S - w a v e s c a t t e r i n g l e n g t h of 2 f m is i n t r o d u c e d f o r the KK t h r e s h o l d effect.

4ooi550 --

"2 500260-- --[

"I

-I0

=~, -5

a=Z . ~ , 0 Qo

I

5

10

Fermi

2

F i g 16 Xr~ = f ( a . ) a,~ b e i n g a r e a l s c a t t e r i n g l e n g t h i n t r o d u c e d to i n t e r r e t the KK • " ~ . ' v + 1 0 P e n h a n c e m e n t at t h r e s h o l d . Two s o l u t i o n s a r e a c c e p t a b l e : a o = 2.0 n ' = f m which c o r r e s p o n d s to a m a r k e d m i n i m u m f o r t h e X~) v a l u e and a o = -2.3 + 0 : ~ f ~ n ~ h i c h h a s no well d e f i n e d l o w e r l i m i t ( A a o = 4 fm) but l e a d s to a n i n t e r e s t i n g i n t e r p r e t a t i o n of the KK e n h a n c e m e n t (see text).

scattering length formulation leads to two possible solutions, same absolute value for ao, but with opposite signs: a o =2.0

+ 1.0fm, -

0.5

a ° =-2.3

+ 0.3 fm -

4.0

"

with rougly the

492

B. CONFORTO et al.

A l t h o u g h a o ~ 2 f m i s n o t a v e r y l a r g e s c a t t e r i n g l e n g t h , the c u r v e SL r e p r o d u c e s q u i t e w e l l the s h a r p v a r i a t i o n of the d e n s i t y n e a r t h r e s h o l d o w i n g to the s t r o n g d e s t r u c t i v e i n t e r f e r e n c e e f f e c t s b e t w e e n the KI~ a m p l i t u d e a n d the S - w a v e K~, I = ½ a m p l i t u d e . (ii) I n t r o d u c t i o n o f a B r e i t - W i ~ o ~ e r a m p l i t u d e . To r e p r e s e n t the KK a m p l i t u d e n e a r t h r e s h o l d in t e r m s of two r e a l p a r a m e t e r s , w e h a v e i n t r o d u c e d the complex Breit-Wigner amplitude: 1 SK~ - s o + iPK~Y "

The following mass and width are obtained: 4-Fo=

m O

= 1016 + 6 M e V =

200±40MeV.

T h i s w i d t h y c o r r e s p o n d s to a " c l a s s i c a l w i d t h " F ~ 25 M e V , F b e i n g d e f i n e d by the r e l a t i o n F-

PKR ~ O

We h a v e d i s c u s s e d in s e c t . 4 t h e v a l i d i t y of s u c h a r e l a t i o n . T h e c u r v e BW on fig. 15 r e p r e s e n t s the d i s t r i b u t i o n e x p e c t e d f o r t h i s solution. T h e t h r e e s o l u t i o n s p r e s e n t e d in t a b l e 2 c o r r e s p o n d to a p r o d u c t i o n of 42, 40 a n d 475 e v e n t s , r e s p e c t i v e l y , f o r t h e K ~ t h r e s h o l d e n h a n c e m e n t . T h e s e f i g u r e s c o r r e s p o n d to p r o d u c t i o n r a t e s of t h e o r d e r of 10 -5 to 10 - 4 , d i f f e r i n g by a f a c t o r 10, b u t a s we h a v e s e e n s t r o n g i n t e r f e r e n c e s w i t h S w a v e KTr a m p l i t u d e s m a y b e p r e s e n t , l e a d i n g to l a r g e e r r o r s on the e s t i m a t i o n of the p r o d u c t i o n r a t e of t h e K K t h r e s h o l d e n h a n c e m e n t . A l t h o u g h , f r o m t h e a b o v e d i s c u s s i o n , the n a t u r e of the KK e n h a n c e m e n t o b s e r v e d a t t h r e s h o l d i s n o t c l e a r , i t m a y be i n t e r e s t i n g to c o n s i d e r , a t l e a s t , the f o l l o w i n g p o s s i b l e i n t e r p r e t a t i o n s " a) T h e I = 1 K K c h a n n e l i s d o m i n a t e d , in t h i s e n e r g y regior~, b y a n a m p l i tude r e p r e s e n t e d by a p o s i t i v e r e a l s c a t t e r i n g l e n g t h of 2 f m w h i c h m a y c o r r e s p o n d to a v i r t u a l s t a t e . b) T h e KI~ e n h a n c e m e n t i s d u e to t h e p r e s e n c e of a r e s o n a n c e w i t h m 1016 MeV a n d I G j P = 1-0+. T h e s p i n - p a r i t y a s s i g n m e n t i s s u g g e s t e d by°the p o s i t i o n of t h i s r e s o n a n c e n e a r t h r e s h o l d a n d the d e c a y a n g u l a r d i s t r i b u t i o n of the KK s y s t e m (fig. 12). W i t h t h e s e a s s i g n m e n t s , i t i s i m p o s s i b l e to i d e n t i f y t h i s r e s o n a n c e w i t h the ~ , S*, H ° o r 6 + m e s o n s . We m u s t t h e r e f o r e a s s u m e in t h i s c a s e t h a t t h i s i s a n e w o b j e c t , w h i c h c a n a l s o d e c a y into ~yr. No s u c h o b j e c t h a s y e t been observed. (c) T h e I = 1 KK c h a n n e l i s d o m i n a t e d , a t t h r e s h o l d , by a v i r t u a l bound s t a t e r e s o n a n c e ( r e f . [11]). T h i s r e s o n a n c e m a y b e c o u p l e d to the ~?Trc h a n n e l . To t a k e into a c c o u n t t h i s p o s s i b i l i t y , we h a v e t r i e d a f i t w i t h a c o m p l e x s c a t t e r i n g l e n g t h a o + ibo. The r e s u l t , bo = 0 +_8" 5 fm s u g g e s t s t h a t t h e p a r t i a l w i d t h of t h i s r e s o n a n c e in t h e ~ r c h a n n e l i s p r o b a b l y q u i t e n a r r o w . In t h e s e

(KK) AND (KY) SYSTEMS

493

conditions, the value a o = -2.3 fm c o r r e s p o n d s to a n a r r o w r/~ r e s o n a n c e with m a s s m = 872 MeV. If this i n t e r p r e t a t i o n is c o r r e c t , it is tempting to identify this n a r r o w 7/7r r e s o n a n c e , at 972 MeV, to the r e l a t i v e l y well e s t a b l i s h e d 6-meson. This a p p r o a c h of the 5 - m e s o n with a s t r u c t u r e in the I = l KK s y s t e m was f i r s t suggested by Dalitz (ref. [12]), who took the opposite a p p r o a c h and asked what I = 1 KK t h r e s h o l d s c a t t e r i n g length (a+ ib) would c o r r e s p o n d to the m a s s and width r e p o r t e d for the 5 - m e s o n and found a = -1.6 fm and b < 0.1 fm, in good a g r e e m e n t with our d i r e c t a n a l y s i s of the KK s t r u c t u r e . Then, our r e s u l t s lead to the following quantum n u m b e r a s s i g n m e n t s for the 6 - m e s o n : I G j P = 1-0 +. However, this i n t e r p r e t a t i o n s e e m s to p r e s e n t a difficulty if one takes s e r i o u s l y the indication that the width of this s c a l a r r e s o n a n c e should be r e l a t i v e l y s m a l l while the 7/~r channel is kinematically opened. If, for s o m e unexplained r e a s o n , the s t r o n g 7/~ decay mode was d e p r e s s e d , the e l e c t r o m a g n e t i c d e c a y s would be in competition.

6. THE S-WAVE (K~) AMPLITUDES In addition to the c o n c e n t r a t i o n s of events in the r e g i o n s : KK(1000), KK(1280), and K~(891), we o b s e r v e a distribution of events which suggest that the 3S 1 initial state does not c o n t r i b u t e to the b a c k g r o u n d , since the c e n t r a l r e g i o n of the Dalitz plot is empty (for 3SI, the density should be m a x i m u m in this region). Indeed, we o b s e r v e a n o n - u n i f o r m distribution c h a r a c t e r i z e d by a slow d e c r e a s e of the density when we go f r o m the edges of the D a l i t z - p l o t to the centre. A good i n t e r p r e t a t i o n of this effect is given by the introduction of S-wave s c a t t e r i n g lengths in the (K~) s y s t e m s . Since the 3S 1 initial state does not lead to S - w a v e (K~) amplitudes, we have just to c o n s i d e r the two p o s s i b l e initial s t a t e s IS 0 ( I = 0 , I= I). F o r the final s t a t e s , we c o n s i d e r the ! =½ and I~-~ (K~) state. The a p p r o p r i a t e a m p l i t u d e s have been d e s c r i b e d in sect. 4. The two a m p l i t u d e s c o r r e s p o n d i n g to the initial state 1S0, l = I , for I(K~) : ½ and ! (K~) = ~2, a r e r e p r e s e n t e d by the s a m e function if a ! ~ a ~. We t h e r e f o r e c o n s i d e r two p o s s i b i l i t i e s : 2 2 (i) The two a m p l i t u d e s a r e a l m o s t undistinguishable (a_* ~ a ~) and we int r o d u c e a single t e r m for these a m p l i t u d e s , in the fitting ~roce~dure, with a p h e n o m e n o l o g i c a l s c a t t e r i n g length a. The r e s u l t s a r e not v e r y sensitive to the value chosen f o r a. The best r e s u l t s a r e obtained for a ~ -0.4 fm, they c o r r e s p o n d to the solution (ii) of table 2. (ii) The two a m p l i t u d e s have a different e n e r g y dependence (a½ = -a~). The r e s u l t s a r e , again, not v e r y sensitive to the value chosen for a±. The b e s t r e s u l t s a r e obtained for 2 a l = 0.3 fm, 2

a ~ = -0.3 f m .

They c o r r e s p o n d to the solution (i) of table 2.

494

B. CONFORTO et al.

Whichever solution is chosen for the S-wave (K~) a m p l i t u d e s , the whole r e s u l t s r e m a i n e s s e n t i a l l y the s a m e with r e s p e c t to the KI<(1280) and K*(891) r e s o n a n c e s . We a r e t h e r e f o r e able, with our model, to investigate in m o r e detail the production of the KK(1280) and K*(891) r e s o n a n c e s .

7. THE (KK---)STRUCTURE AROUND 1280 MeV Figs. 10 and 11 show an a c c u m u l a t i o n of events in the ( I ~ ) s y s t e m around 1280 MeV. Fig. 13 shows the decay angular distribution of the (KI() s y s t e m a r o u n d 1280 MeV; its s t r o n g a n i s o t r o p y does not n e c e s s a r i l y mean that the spin of the (KK) r e s o n a n c e is different f r o m z e r o , since s t r o n g i n t e r f e r e n c e effects may be p r e s e n t , in p a r t i c u l a r with the K*(891) and the S-wave (K~) amplitudes. Table 5 gives the r e s u l t s of the fit obtained for different s p i n - p a r i t y a s signments. Table 5 Results of the fit for different spin-parity of KF~(1280)

JP(KK,)

X2(1)

X2(2)

0+

355

33

1-

340

26

2+

260

19

In table 5, ×2(1) is the ×2 of the fit of the Dalitz plot ((×2(1)) = 177), ×2(2) is the X2 of the fit of the angular distribution in the KI~(1280) region as shown on fig. 13 ((×2(2) > = 10). The three curves given on fig. 13 represent the angular distributions expected for the spin-parity 0+, 1-, 2+; J P = 2 + is favoured over 1- and 0 +. The central value for the mass has been fitted as described in sect. 4, with the following r e s u l t : M = 1280 ~ 12 MeV. The best fit is obtained for a width of 90 MeV. In fig. 11, the KI~(1280) enhancement is significantly displaced towards the low m a s s values. The fit gives a good i n t e r p r e t a t i o n of this phenomenon, which is due to i n t e r f e r e n c e effects. The two initial s t a t e s 3S1 and 1S 0 contribute to the production of the KI~(1280) r e s o n a n c e : taking an a v e r a g e value between the t h r e e solutions p r e s e n t e d in table 2 (the d i f f e r e n c e between the t h r e e solutions, f r o m this point of view, is s m a l l c o m p a r e d to s t a t i s t i c a l e r r o r s ) , we find that 257 +40 events a r e produced by annihilations in 1S0 state and 47 ± 20 events a r e p r o duced by annihilations in 3S 1 state. Taking into account the K~2K+ decay mode of the (KK) + r e s o n a n c e , the

495

(KK) AND (KTD SYSTEMS

n o n - o b s e r v a t i o n of the K~I -~ 2~ ° decay mode and an a v e r a g e weight of 1.14 for scanning and g e o m e t r i c a l l o s s e s , the r a t e of annihilation into the c h a n nel pp -- (KK)+~:~ is then 304 × 2 × 1.5 x 1.14 x

1 = (9+3)×10_ 4 . 1.15×106

The r e s u l t s obtained for the m a s s , the width and the spin make it p r o b able that this object is a manifestation of the s o - c a l l e d A2(1300) meson.

8. THE K*(891) PRODUCTION

The D a l i t z - p l o t (fig. 4) shows an abundant production of K"(891). Both 1Sn and 3S 1 may contribute to this production: indeed, the decay angular d i s t r i b u t i o n s of the K*(891) show, at l e a s t qualitatively, that both initial state contribute. M o r e o v e r , the i n t e r f e r e n c e between the two K* a p p e a r s to be s t r o n g l y d e s t r u c t i v e and indicates that the I = 1 initial state is p r e ponderant in the K* production. We have given in sect. 4 the explicit e x p r e s s i o n s for the four a m p l i t u d e s c o r r e s p o n d i n g to the initial s t a t e s 1S0 (I=O, 1=1) and 3S 1 (I=O, I=l). Table 6 gives the production r a t e s obtained for a v e r a g e over the t h r e e solutions given in table 2. Table 6 Production rates for pp ~ K*K(K*K) Initial state

Observed events

Events produced

1S01=0

205+30

1S0 I=1

36±7

150±35

f + l antisym = 1.3 x 10-4 ± 0.3 × 10 -4 JK*K

3S1 [=0

384-25

200±125

f K-0, K = 1.7x10 -4:~ 1.0Xl0 -4

3S1 I=1

546+50

2 6 5 0~-250

f - i sym = 23.4 ± 2.0 x i0 -4 JK*K

Rates

+0 950+150 fK*K = 8"4x10-4 ± 1"4x10-4

The 3S1, I = 1 initial state is p r e p o n d e r a n t for the production of the K*(891). However, the contribution of 1S0, I = 0 is not negligible. ½. One n o t i c e s that the r a t i o [~p -~ K*K(1S0)]/[~ p --" K*K(3S1) ] is c l o s e to Then the t h e o r e t i c a l d e c a y a n g u l a r d i s t r i b u t i o n s of the K* a r e expected to be 3 x 0.5 sin 2 8 + cos 2 ~ which is a l m o s t uniform. The t h e o r e t i c a l c u r v e s c o r r e s p o n d i n g to the b e s t fit drawn on figs. 8 and 9 a r e effectively u n i f o r m a p a r t f r o m i n t e r f e r e n c e effects between the two K* and in good a g r e e m e n t with the e x p e r i m e n t a l distributions.

496

B. CONFORTO et al.

It is possible to e x t r a c t f r o m these r e s u l t s (making use of the r e l a t i v e phase between the I = 0 and I = 1 K* production amplitudes) the n u m b e r of events for the two p r o c e s s e s PP "-* K ~ * °

,K+~'t:

500 ± 50 events ,

,K~I~:F

300 + 30 events .

~p ---.K+K*¥

]

So, with isospin i n t e r f e r e n c e effects, we a r e able to r e p r o d u c e the d i s s y m m e t r y between the production of the two K*, the K *:L being l e s s abundant than the K *o (figs. 6 and 7). It is not s t r a i g h t f o r w a r d to deduce f r o m these r e s u l t s , obtained for the annihilations ~p ~ K~K+~~, the total t r a n s i t i o n r a t e for the K*(891) p r o d u c tion in the three-bod~ annihilations in g e n e r a l pp ~ KK~. To the four possible initial s t a t e s , • = +1, I=O or 1 c o r r e s p o n d the four isospin states and t r a n s i t i o n r a t e s for the production of K*(891): ¢•0,½ = ½[(K+K*- - K°K*°) + •(K-K*+ - K°K*°)]' / ~•,0 K*K

(38)

~e

(39)

1,½ :

½[(K+K *- + K°K *o) + •(K-K *+ + K'°K*O)], f•K'.1

Let us f i r s t c o n s i d e r the two channels c o r r e s p o n d i n g to the s t a t e s (38), i.e. when 1=0. In this c a s e , as shown in the appendix (eq. A.5) the wave function ~ , ½ is identical to @~, 1, being s y m m e t r i c for • = +1 and a n t i s y m m e t r i c for • = -1 in the exchange of the two p a r t i c l e s K,K. Of c o u r s e , the t h e r e f o r e it is p o s s i b l e , in s t a t e s with I(Ki~) = 0 do not contribute to q~0.½; • this c a s e , to deduce the total annihilation ra~es f r o m the o b s e r v a t i o n s made on the r e a c t i o n ~p ~ K~K±~=F where I(KK) = 1. Taking into account the v i s i bility of the K~I, the weight of the events and the C l e b s c h - G o r d a n coefficients ~xplicited in eq. (A.2), one deduces f r o m the o b s e r v e d n u m b e r of events for 1S0, I=0 and 3S1, I=0 initial s t a t e s the p r o d u c e d n u m b e r of events and the total annihilation r a t e s f K . K +0 and f K . K -0 given in table 6. Let us now consider the two channels c o r r e s p o n d i n g to the states (39), l e when I = l In thls c a s e , a s l s shown by t h e r e l a t l o n ( A . 6 ) , ~1 - i s related, not only to @~, 1, but also the @~, 0" The r e a c t i o n (1): ~p "2K~IK±~=~, for which I(K~) ~ 1, p r o v i d e s i n f o r m a t i o n only on the p a r t of the state c o r responding to @1, 1, i.e. on the s y m m e t r i c p a r t of the state when • = -1 and the a n t i s y m m e t r i c p a r t when • = +1. To r e a c h the other p a r t s of the K* p r o duction, we have to o b s e r v e other channels: the explicit e x p r e s s i o n of q~• 1 given in (A.3) shows that the a n t i s y m m e t r i c p a r t of the wave function, w ~ • = -1, will be given by the o b s e r v a t i o n of the channels ~p ~ K+K-~ ° and ~wip.~ K~.K~ °. w h e r e a s the s y m m e t r i c p a r t of the wave function, when • =+1, 11 be ~iv~en by the o b s e r v a t i o n of the c-hannels PD ~ K+K-~° and ~p ~ K~ K~ 7r°. . • • • +, 1 a n f i s y m -, 1 s y m • In table 6, we give the anmhilation r a t e s f K * K and f K * K corre".

.

•

"

"

"

•

I

"

497

(KK) AND (KY)SYSTEMS

sponding to the o b s e r v e d n u m b e r of events for the channel ~p ~ .K~IK+~. Since the total i n t e r f e r e n c e effects between the s y m m e t r i c and a n t i s y m m e t r i c a m p l i t u d e s a m o u n t s to z e r o when i n t e g r a t e d o v e r the all Dalitz-plot, the total annihilation r a t e s f + , 1 and f ~ . , 1 may be written:

f+, 1 =f+, 1 antisym ~+, 1 sym K*K ~K*K +JK*K

'

-,1 = f - , l a n t i s y m ~ - , l s y m /K*K ~K*K +YK*K

(40) (41)

In sect. 9, we shall be able to estimate/~'lKsym from the obseTvation of the channel ~p --" I~1I~1 .~ ~.~°, t h e r e f o r e the total annihilation r a t e / k ' , l K will be -,1 given. But f o r / K * K we give only a lower limit, ( c o r r e s p o n d i n g to fI~-' 1 antisym = 0), since the channels ~p --" K~iI~2~° and ~p --, K+K-~O have *K not been analyzed. The values obtained for the masses and widths of the K*o and K*± are the following: M(K*O) = 889 ± 5 MeV ,

F(K*o) = 43 MeV ,

M(K *±) = 896 ± 5 MeV ,

F(K *~-) = 43 MeV .

The complexity of the t h r e e - b o d y annihilations ~p --" K~lK±~+, with the p r e s e n c e of at l e a s t ten amplitudes, some of them giving Else to s t r o n g i n t e r f e r e n c e effects, m a k e s the i n t e r p r e t a t i o n of the s m a l l m a s s d i f f e r e n c e obs e r v e d between the K *o and K *± v e r y difficult. We p r e f e r not to attach any significance to this difference. =~O.~O O 9. STUDY OF THE REACTION ~p --" i'~ll',l~ (364 events)

The relation (10)of sect. 3 shows that two amplitudes A ~ I and A i 0 m a y contribute to the annihilation ~p --* K~K~Tr °. The first of these amplitudes, A~I , has been already discussed when analysing the reaction Dp --. I~iK+n:~,-with the results (table 2) that A~I leads to the presence of K*(891), K K/+1000), KK(1280) and S-wave I=½ KTr background. These properties of A 0" must be reproduced in the annihilation Dp --*K°K°~ ° if our analysis of ~he annihilation ~p --*K°K±TrT is correct. In addition, the annihilation Dp --*I~iK~ir° m a y exhibit the properties of A'[O , which leads eventually to the presence of the known I = 0 K K resonances like the S*(1050) and the fo(1250), but can also contribute to the presence of some of the K*(891) and S-wave Kzr background. Our experimental information on the K~lI~/~rosystem is too limited to attempt a general fit on the Dalitz plot (fig.17)-, with all the contributions mentioned above, as it has been done for the K~IK~Tr~: system. Therefore, in a first step, we simply verify that our experimental information on the K~iI~l~r° system is in agreement with the properties of the A~I amplitude

498

B. CONFORTO et al.

~ a s deduced f r o m the study of the ~Tz-OTz--I" 1~1,, system. The Dalitz plot (fig. 17) shows, at l e a s t qualitatively, that this a g r e e m e n t is p o s s i b l e : we o b s e r v e an a c c u m u l a t i o n of e v e n t s in the K*(891) region, leading to a significant p e a k on the M2(Ku) p r o j e c t i o n (fig. 18). The c o r r e sponding decay a n g u l a r d i s t r i b u t i o n (fig. 19) s e e m s to be c o m p a t i b l e with cos2~ a p a r t f r o m background contributions. M o r e o v e r , we o b s e r v e an a c cumulation of events in the KK(1280) r e g i o n , with an a n g u l a r d i s t r i b u t i o n c h a r a c t e r i s t i c of J = 2 (figs. 20-22). The background is not u n i f o r m : it is m o r e i m p o r t a n t n e a r the edges of the Dalitz plot, a s expected for an S-wave Ku s c a t t e r i n g length effect. However, t h e r e is a p r i o r i no c l e a r evidence of an a c c u m u l a t i o n of events in the KI{(1000) region. H o w e v e r , a quantztative" v e r i f i c a t i o n of this agreemento shows that A~I,,, is not sufficient to r e p r o d u c e all the f e a t u r e s of the K~IK~lU s y s t e m . We have then introduced the A~0 contributions to which the quality of the fit is the m o r e s e n s i t i v e : the K*(891), and the S-wave Ku s c a t t e r i n g length, both with IK~ = ½ and IK~ = ~. In this fit, we have kept the A ~ l c o n t r i b u t i o n s constant, using the v a l u e s deduced f r o m the study of the K~K~:~:~ s y s t e m (the K~ v i s i b i l i t y and C l e b s c h Gordan c o e f f i c i e n t s lead to a f a c t o r i for the r a t e s o ~ s e r v e d in the K~IK~v° s y s t e m c o m p a r e d to the K~IK±~ ). M o r e o v e r , the s c a t t e r i n g lengths widths and m a s s v a l u e s of the B r 6 i t - W i g n e r a r e a l s o taken f r o m the r e s u l t s o b tained for the ~ --K~K±~ ~ system. J . T a b l e 7 g i v e s the r e s u l t s obtained for the t h r e e p o s s i b l e solutions p r e sented in sect. 4. Solutions (iii) and (i) give the b e s t r e s u l t s . They show that the p r o p e r t i e s of the K~IK~lY° s y s t e m a r e c o m p a t i b l e with the i n t e r p r e t a t i o n given f o r the annihilation pp ~ K ~ I K ~ .

I

i

i

I

I

]

/

/

.. .,**

1.0-

•

;;.:.

0

, . . . . I

• ,.-.-:

05

..'

,':ii.

.

•

. -..

%..

;. '- :.i-

,,":

":\

:.'."

• :';:.2

W

I

80

1.5~

i=

]

.

:":":

..

.

60

"

t ....

,

: :'""""

...~..

: ..i ;....,

"

.

.

40

.......,

,"

-

3

"

20 .,¢.

Z

K'1891)

1

I

0.5

1.0

0 0.40

I 1.5

2.0

I

0.80

P

1.20

I

1.60

Mass' ( K~"rr°) G e V '

Mass~ {K°,~o)

GeV

2

Fig. 18. (K~Tr°) effective mass squared distribution for pp -~ K~K~Tr° (two eomFig. 17. Folded Dalitz-plot for pp ~ ~lr~l/r binations per event). The-curve repre(364 events). The region of the K (891) is in- sents the theoretical distribution as given dicated by the arrows, by the best fit (solution (iii) of table 7). _

..o..o

o

(KK) AND (KTO SYSTEMS

499

Table 7 The p a r a m e t e r s p and ¢P

Solution (i). X2 = 60. (X2)= 39. a3¢0 2

Channel

Amplitude

p

K* S - w a v e K?T IKTT=½ KK(1280)

+ A01

KI~(1000) K* S - w a v e Ky IK~.=½ S - w a v e K~" IK =~

+ A10

Number of events

¢

0.53 a)

0.00 a)

25

1.19 a)

0.78 a)

130

0.68 a)

0.86 a)

42

KK(1280) ; M :1280 MeV. 1"=90 MeV

0.81 a) -0.61 a)

60

K~(1000) : M : 1 0 1 6 MeV. T:200 MeV

0.48

-0.85

21

1.52

-2.00

210

a½(KY) =0.3 fm

2.30

2.09

450

a~(K~) : - 0 . 3 fm

K*°; M=895MeV, 1" 45MeV

Solution (ii), X2 = 87. (X2> = 41. a ~ = 0 Channel

Amplitude

K* S - w a v e Kn II~=~ KK(1280)

A* 01

KI~(1000) K* S - w a v e K~"

IKy=½

A+ 10

Solution (iii). Channel

Amplitude

K* S - w a v e K~"

IK~=½

+

A 01

p

(P

Number of e v e n t s

0.62 a)

0.00 a)

33

1.16 a)

1.71 a)

117

0.70 a)

1.06 a)

42

K]~(1280); M:1280 MeV. r=90 MeV

0.48 a)

3.19 a)

20

K~(1000); M=1016MeV. 7=200MeV

0.22

-0.69

4

0.82

2.36

58

k2 = 55.

a~(Klr) = -0.4 fm 2 a~(Ky) == 0

(X 2) = 39. KK(1000) s c a t t e r i n g length

p

(P

Number of events

0.63 a)

0.00 a)

35

1.40 a)

0.36 a)

172

KK

0.71 a)

1.27 a)

44

K]~(1000)

0.95 a)

2.01 a)

79

0.66

5.19

37

2.08

2.90

350

0.69

1.04

40

K*

K*°; /I//--895 MeV, 1"=40 MeV

K*;M=895 MeV, F=40 MeV KK(1280); M=1280 MeV. 1"=90 MeV KK(1000), a o = 2 fm

a ! ( K y ) = +0.3 fm 2

S - w a v e Ky

ZK,=½

S-wave Kg IK~=~

A+

1o

a) Taken f r o m the a n a l y s i s of r e a c t i o n (1).

a~2(K?T) = -0.3 frn

500

B. CONFORTO et al.

I --

40~ . . . .

!

I

I /

50

c.9 CO

~\

o

/t

"

1

I

o c:

"5

Z

o~ -I

20

E=

~, 0 Decoy

5O

I

Z

u

/

cosine

Fig. 19. Decay a n g u l a r distribution of the K*(891) in'~p -" K ~ K ~ "°. The curve r e p r e s e n t s the theoretical distribution as given by the best fit (solution (iii) of table 7).

OL__

t

1.0

15

I

i 2.0

Mass • ( K ~ )

I 25

50

GeV'

Fig. 20. (KTK°) e f f e c t i v e - m a s s squared d i s t r i b u t i o n fo~ pp ~ K~K ° 7r°. The curve r e p r e s e n t s the t h e o r e t i S a l ' d i s t r i b u t i o n as given by the best fit (solution (iii) of table 7).

2O

>~ (D 15

I

CO

o

\~1/

I0 "6

z

\

t

5

o IJ, 0..98

E Z

1.58 •

1.78 o

5

~

~,

2.18

o

Mass ( K, K, ) GeV'

I 0

Fig. 21. (KTK°) effective m a s s s q u a r e d

I O O O distribution for ~p ~ KIK~'Tr when MZ(K )>0.9GeV2. This cult is to remove the K*(891) reflection. A clear accumu- ^ lation of events is visible around 1.5 GeVz whereas no clear enhancement is visible at threshold. However, the theoretical curve, as given by the best fit (solution (iii) of table 7), gives a reasonable interpretation of the data.

Decay

cosine

Fig. 22. Decay angular distribution of the KK(1280) for ~p -~ K~K°?r°. The curve represents the theoretical d i s t r i bution as given by the best fit ~olution (iii) of table 7), and for spin JZ-=2+ for the KK(1280) resonance.

(KK) AND (K?r) SYSTEMS

501

Moreover, they are consistent with a non-negligible production of K*(891) related to the AI0 amplitude: 20 ± 10 events, leading to a production rate l sym of (5.35 ± 2.6)×10 -4 as given in table 8. *K Table 8 K* production Initial state

Observed events

Produced events

Rates

1S0 I = 0

31± 5 a)

9 5 0 i 150a)

fK*K+0 = (8.4 • 1.4) x 10 -4

1S0 I= 1

20 ± 10

616 =~308

fK+l sym = (5.35 ±2.6) × 10 -4 ,K"

a) Taken from reaction (1).

Then fK*K+I =JK*K~+I a n t i s y m +JK*Kf+I s y m = (6.65+2.7) x 10 -4.

(42)

In addition to the r e s u l t s p r e s e n t e d in table 8, the study of the K~IK~ s y s tem b r i n g s forth the following r e m a r k s : Our data do not show any evidence for the S*(1050) r e s o n a n c e . On the other hand, the K~K~ a c c u m u l a t i o n o b s e r v e d around 1250 MeV is partially explained by the p r e s e n c e of ~40 KI~(1280) events and the r e f l e c t i o n of the K*(891) (the cos * ~ angular distribution of the K*(891) leads to two a c c u m u lations of events in the M2(K~K~) s p e c t r u m , around 1.5 and 2.6 GeV2). However, we cannot completely exclude a s m a l l production of fo, an upp e r l i m i t for this production with the decay fo ~ KO~O being of the o r d e r of 10-4.

i0. C O N C L U S I O N A detailed study of the annihilations ~p ~ I~K+Tr; and ~p ~ I ~ K~ n ° at .t .t r e s t shows that s e v e r a l i n t e r f e r i n g amplitudes play an i m p o r t a n t role. To give a s a t i s f a c t o r y i n t e r p r e t a t i o n of the e x p e r i m e n t a l data, we have been led to the introduction of a model, as explained in sect. 3. T h r e e types of i n t e r f e r e n c e effects a r e n e c e s s a r y : the i n t e r f e r e n c e of two amplitudes not r e l a t e d by the Bose s t a t i s t i c s , a good example of which being the A 2 and the K* r e s o n a n c e s . The i n t e r f e r e n c e between two amplitudes r e l a t e d by the Bose s t a t i s t i c s , like the K* and K* r e s o n a n c e s . The i n t e r f e r e n c e between two a m p l i t u d e s of different total isospin, leading to the d i s y m m e t r y obs e r v e d on the Dalitz plot. We have also shown that it is i m p o s s i b l e to r e p r o d u c e the a c c u m u l a t i o n o b s e r v e d in the (Ki~ s y s t e m at threshold without the introduction of a s p e c i fic amplitude. Several i n t e r p r e t a t i o n s have been p r o p o s e d , in sect. 5, for

502

B. CONFORTO et al.

this I= 1 KK e n h a n c e m e n t at t h r e s h o l d . One of t h e m is to r e l a t e this e n h a n c e m e n t to the p r e s e n c e of a n a r r o w I G j P = 1-0 + r e s o n a n c e a r o u n d 972 MeV, w h i c h could be identified to the 5 - m e s o n . H o w e v e r , one c a n n o t e x c l u d e o t h e r i n t e r p r e t a t i o n s , like the e x i s t e n c e of a new r e s o n a n c e I G J P = 1-0 + at 1016 MeV with F ~ 25 MeV, o r , m o r e s i m p l y , the p r e s e n c e of a K ~ a m p l i t u d e r e p r e s e n t e d by a s c a t t e r i n g length of 2 fm with no c l e a r r e l a t i o n to a r e s o n a t i n g channel. The K*(891) p r o d u c t i o n is n e a r l y c o m p l e t e l y d e t e r m i n e d , the 3S1, I b e i n g p r e p o n d e r a n t , a s a l r e a d y shown by t3arash et al. (ref. [2]. The a n n i h i l a t i o n r a t e s f ~ l . K (6 = +1, I =0, 1) a r e : fK*K+0 = (8.4 ± 1.4) x 10 -4 '

fK*K-0 = (1.7 ± 1.0) x 10 -4 ,

f K+1 * K = (6.6 + 2.7) x 10 -4 '

f K-1* K >~ (23.4 ± 2 . 0 ) x 10 -4 m

We have a l s o o b s e r v e d the p r o d u c t i o n of a KK r e s o n a n c e with a m a s s M = 1280 ± 12 MeV and a width F ~ 90 MeV. Its m o s t p r o b a b l e q u a n t u m n u m b e r s a r e I G j P = 1-2 +. T h e s e c h a r a c t e r i s t i c s m a k e its i d e n t i f i c a t i o n with the A~ m e s o n highly p r o b a b l e . I t s p r o d u c t i o n r a t e s f~A0 and f - 1 ~ c o r r e s p o n d i n g r e s p e c t i v e l y to 1S0,1 = 0 and 3S1, I = 1 initial s t a t e s a r e 2 f+A02n = (6.5 ± 1.8) x 10 -4 ,

f -A 1 2 ~ = ( 2 . 2 • 0 , 9 ) x 10 -4

O

I

F i n a l l y , an i m p o r t a n t S - w a v e b a c k g r o u n d (annihilation r a t e of the o r d e r of 40 x 10 "4) m a y be r e p r e s e n t e d by a s m a l l S - w a v e K~ s c a t t e r i n g length IC/K~ < 1 fm. We should like to e x p r e s s o u r t h a n k s to P r o f e s s o r s t3. G r e g o r y , L. L e p r i n c e - R i n g u e t and J. T e i l l a c f o r t h e i r i n t e r e s t and s u p p o r t , and to P r o f e s s o r s R. A r m e n t e r o s , Ch. P e y r o u and J. P r e n t k i f o r m a n y s t i m u l a t i n g d i s c u s sions.

A P P E N D I X . ISOSPIN D E C O M P O S I T I O N A C C O R D I N G TO THE KTr C O U P LING SCHEME (A. 1) I ,I K Tr

w h e r e the n o t a t i o n s a r e e a s i l y d e d u c e d f r o m t h o s e of sect. 3, G96

' 0 , ½ - ~1

{(K+ -)FIo + (KOTr+)K- _ ~2_2((K+ O)K- +

(K%°)~o)

+ ¢ [(K-~+)K ° + (K°Tr-)K + - ~ 2 2 ( ( K - ~ ° ) K + + (K'°~°)K°)]} ,

(A.2)

(KK) AND (K7D SYSTEMS

503

¢p~ 1,½ = ~' lr,K+ ( u o,) K - _ ( K ° u ° ) ~ O + ~ ( ( K - ~ ° ) K + - ( K ' ° v ° ) K ° ) } ,

(A.3)

~ , ~ = ~l(1 r,K+u-,~o~ - (K°u+)K - + E ( ( K - u + ) K ° - (i~°Tr-)K+) } ,

(A.4)

??

£

w h e n w e c o m p a r e t h e s e e x p r e s s i o n s w i t h t h o s e o b t a i n e d f o r ~ I , I K K in s e c t . 3, one v e r i f i e s d i r e c t l y t h a t q~ ½

= ~E 0,1'

(A.5)

q~,½

-1 ~c - ~/-~ ~ ¢ =~=3- 1 , 0 l,l _-

-

(A.6)

'

(AT)

1

1,0

- ~f3

1,1

"

T h e i n v e r s e r e l a t i o n s w r i t t e n in s e c t . 3 a r e e a s i l y d e d u c e d . T h e s a m e • . ~ ~ . r e l a t m n s h o l d f o r t h e a m p h t u d e s A I I,r~ a n d Bl, i~r _. We c a n wr~te t h e t r a n s i t i o n r a t e s f o r t h e r e a c t m n s ( I ) - ( 4 ) i n ~ e r m s of t ~ a m p h t u d e s B:

Rl=~I',/2Bo,½-~'2 B1 1- - Bl,~21

+ t ¢ B5,½ -

_

,

R2 = ~ l BO, + ½- B +1,~ s ' + ~]-2~1,~21 _+s ,2 ,

=

Bo,½+ g l , ½ - ~

1,~

+ ~1 go,-~ +

,2

(A.8)

w h e r e R1, R2, R 3 a n d R 4 d e p e n d only on t h e s y m m e t r i c o r a n t i s y m m e t r i c p a r t of t h e B I IK~ a m p l i t u d e s (B s a n d B a) s i n c e the f i n a l s t a t e s (1)-(4) c o r r e s p o n d only ~0 t[~e s y m m e t r i c o r a n t i s y m m e t r i c p a r t of the s t a t e s q ~ I - - • T h e s y m m e t r y p r o p e r t i e s of t h e s t a t e s ~ I ~ a r e g i v e n by t h e G-~a~lUty c o n s e r v a t i o n ( s e e s e c t . 3.2) a n d t h o s e of the ~ta~es q3ie,iKu a r e d e d u c e d f r o m t h e e x p r e s s i o n s in t e r m s of t h e ¢ ¢I ,IK ~ s t a t e s . [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

R . A r m e n t e r o s et al., Phys. L e t t e r s 17 (1965) 170,344. N. B a r a s h et al., Phys. Rev. 139 (1965) B1659. Ch. Zemach, P h y s . R e v . 133 (1964) B1201. J. Duboc, A. Minten and S. Wojcicki, CERN r e p o r t 65-2. R . A r m e n t e r o s et al., P r o c . Geneva Intern. Conf. on High energy physics, 1962, p. 90. R . A r m e n t e r o s et al., P r e s e n t e d at Oxford Intern. Conf., 1965. Gillespie, Final State Interactions (Holden-Day advanced physics monographs). P . B a i l l o n et al., to be published in Nuovo Cimento. J. Barlow et al, to be published in Nuovo Cimento. C . B a l t a y e t a l . , P h y s . R e v . 142 (1966) 932. R. H. Dalitz, Strange p a r t i c l e s and strong interactions (Oxford University P r e s s , 1962) p.59. [12] R . H . D a l i t z . P r o c . 1965 Oxford Conf. on High energy physics, p. 162.