c

Volume 29B, number 7

REGGE

POLE

P H YS I CS L E T T E R S

EXCHANGE

ANALYSIS

OF

K + p -- K * ° ~ + p

23 June 1969

AT

7.3

GeV/c

C. Y. CHIEN, E . I . MALAMUD$, D . J . MELLEMA~$, F. D• RUDNICK #, P . E . SCHLEIN W. E. S L A T E R , D.H. STORK, H.K. TICHO and T. G. T R I P P E # # Physics Department, University of California, Los Angeles, California, USA Received 14 May 1969 A Regge pole exchange model is applied to the reaction K+p -, K *° (890)y+p at 7.3 GeV/c. The reaction is well desribed by exchanges of pion and Pomeranchuk trajectories with the Pomeranchuk exchange approximated by either a diffraction mechanism or on-mass shell y+p scattering. The model fits the production of the Q enhancement and also gives an adequate description of the virtual y+p scattering.

The m u l t i - R e g g e p o l e exchange m o d e l [1-3] h a s b e e n shown by r e c e n t w o r k [4-6] to be v e r y u s e f u l in i n t e r p r e t i n g v a r i o u s 3 body final s t a t e s in p - p , K - p , and ~ - p r e a c t i o n s . We have a d a p t e d t h i s m o d e l f o r u s e in the a n a l y s i s of K + + p -~ K*°(890) + ~+ + p

(1)

which incident K + b e a m m o m e n t u m at 7.3 G e V / c . Good a g r e e m e n t with the d a t a h a s obtained in a l l d i s t r i b u t i o n s i n v e s t i g a t e d . In p a r t i c u l a r the m o d e l s u c c e s s f u U y d e s c r i b e s the b r o a d Q e n h a n c e m e n t in the K*# s y s t e m n e a r 1.3 G e V / c z and p r o v i d e s an adequate d e s c r i p t i o n of the w r t u a l ~ p s c a t tering. E v e n t s u s e d in t h i s a n a l y s i s w e r e s e l e c t e d f r o m a s a m p l e of 3735 e v e n t s of the r e a c t i o n K + + p -~ K + + ~ - + ~ + + p

(2)

o b t a i n e d f r o m an e x p o s u r e of the BNL 80-inch h y d r o g e n bubble c h a m b e r to a r . f . s e p a r a t e d K + b e a m . R e a c t i o n (~2) i s c h a r a c t e r i z e d by c o p i ous p r o d u c t i o n of ATm(1236) an.d K*(890) and the b r o a d Q e n h a n c e m e n t in the K*~-~ + s y s t e m with i n v a r i a n t m a s s n e a r 1.3 G e V / c 2. T h i s Q enhanc e m e n t h a s s p i n - p a r i t y 1+ and c o n s i s t s of m a i n l y K*~, [7] but s o m e c o n t r o v e r s y e x i s t s a s to w b e t h e r it i s one o r s e v e r a l genuine r e s o n a n c e s o r a p u r e l y k i n e m a t i c e f f e c t [e. g. 8]. R e a c t i o n (1) w a s d e f i n e d a s t h o s e events f r o m r e a c t i o n (2) with 0.84 < M ( I C ~ -) < 0.94 G e V / c 2 *Work supported in part by the U• S. Atomic Energy Commission. ~Present address : National Accelerator Laboratory, Batavia, Ill. ~ N a t i o n a l Science Foundation l>redoctoral Fellow. #National Science Foundation Postdoctoral Fellow. ##Present address: CERN, Geneva, Switzerland.

and M(p~ +) > 1.33 GeV/c 2. The l a t t e r r e q u i r e ment r e m o v e d A++(1236) e v e n t s f r o m the s a m p l e . The r e m a i n i n g e v e n t s e x h i b i t e d p e r i p h e r a l p r o duction of both K* and p r o t o n which i s s u g g e s t i v e of the Regge p o l e exchange m e c h a n i s m shown in the i n s e r t in fig. l a ( d i a g r a m A). F r o m d i a g r a m A define t~he following q u a n t i t i e s : 2 S a = (p~ut + p ~ + ) 2 Sb = (PK* + P l r ~) , ta = ( p i n _ Pp _out~2 J , t b = (PK p K . ) 2. c~a a n d a b a r e the e x c h a n g e d Regge t r a j e c t o r i e s . The c o n t r i b u t i o n to r e a c t i o n (1) f r o m t h i s d i a g r a m should be d o m i n a t e d by the P o m e r a n c h u k (P) t r a j e c t o r y a a and the pion t r a j e c t o r y f o r c~b s i n c e the s t r o n g coupling of K* to K~ and the h i g h - l y i n g P t r a j e c t o r y g i v e s a l a r g e a m p l i t u d e [4]• We have t h e r e f o r e f i t t e d r e a c t i o n (1) with t h i s s e t of t r a j e c t o r i e s . To r e m a i n within the e x p e c t e d r a n ge of v a l i d i t y of the Regge p o l e m o d e l , we r e quire S a > 2.25 (GeV/c) 2

0.02 < Ital < 1.0 (GeV/c) 2 ~bl < 1.0 (GeV/c) 2

(3)

The lower Umit on Ital is introduced to elimate the effect of scanning inefficiencies for short recoil protons. This yields a sample of 445 events with very little contamination from other reactions t. t A possibly non-negligible contaxminati~n (~25 events) is present from the reaction KTp -* K pOp. A quntitative estimate is difficult because of the uncertain background subtraction. We have studied the events which satisfy all requirements for diagram A and in addition have 0.66~< M0r+~ -) ..<0.86 GeV/c2. They show no features appreciably different from the r e mainder of the sample. 433

Volume 29B, number 7

PHYSICS LETTERS

, , , , , , , , 80~ M(K,Op) GeV/Ct

~

.N

23 June 1969

= = Ifl(tb) ~ (tb)(Cosh ~b)ab 12 ~ ( t a, w, tb) 12 ~ ~/~p 121 w h e r e ~ is the r e s i d u e function and ~ is the s i g nature factor, [1 + exp(-i~c~b)]/sin ~f~b, of the plon t r a j e c t o r y . ~ is the v e r t e x function at the middle v e r t e x , and co is the T o l l e r angle [2, 3] defined by

(c)

f= t0 ; '=

(PK +× Pg*)" (p~n× p~Ut) cos co = ipK+ × PK* ] Inin× ,,out I rp t-p 1.4

1.8

2,2

1.6

60

2.0

2_4

2.8

2.~

5o

K

~.0

~.4

t8

with all m o m e n t u m v e c t o r s taken in the 7r+ r e s t f r a m e ~ 1.~1m[ 2 d e s c r i b e s the lower half of diag r a m A a n d l s substituted for by the e x p r e s s i o n given by a diffraction s c a t t e r i n g description and the optical t h e o r e m damp ~ I M ~ p 12 = 64~2 Sa d~ =

4(K*--T+) ~ / / c f (f)

1•

0

0.2 0.4 0.6 0.8 q~ 'c°s OK'K"

0

'

,,

(g)

-I~

0

0.2

l

I0

0.4 ~K,'

'

(h)

0'

5O'

06 '

Q5 1.0 1.5 2.0 160i ' Tol'ler ~

]I20F

~0'

~'

'

(i)

~

5O'

= 4k 2 Sa[aT(Sa)] 2 exp (k(Sa)ta) (7) w h e r e aT is the total ~-p c r o s s section and k is the ~-p c e n t e r of m a s s m o m e n t u m . In p r a c t i c e , we substitute constants f o r k (Sa) and aT(Sa). A value of 30 mb was used to p r e s e n t a v e r a g e s o v e r the Sa distribution in our data a s s u m i n g the exchanged pion to be on its m a s s shell. The function k has been a s s u m e d slowly v a r y i n g and hence r e p l a c e d by a constant. The e x p r e s s i o n f o r cosh ~b in t e r m s of Sb, t a and t b is given in ref. [3]. Finally, following B e r g e r [4], the entire m a t r i x e l e m e n t s q u a r e d is n o r m a l i z e d to the 2 equivalent one pion exchange model at th = m=. This fixes ~ (~b = mz) which is then take~ to have the s a m e vahie at all t b. The final r e s u l t is

~0'

Fig. 1. Various invariant mass, f o u r - m o m e n t u m t r a n s f e r squared and angular distributions of the data c o m pared w i t h the curves predicted by the model. The an-

gles are defined in the text. The insert in fig. la, r e ferred to in the text as diagram A, is used in this analysis. The differential c r o s s section of r e a c t i o n (1) is given by d a = (2¢)-5(4FI)-1 ~ I M [ 2) d~b3

IMI2 =

(8) exp(Xta)

=

)21-cos

¢

~ 2 (r.b/So)2O Ir a

. where g 2k ' ~ * - is the K , K + ~r- coupling constant taken as"4~r ~ 1.65, ~ i s t h ~ p i o n t r a j e c t o r y a s s u m e d to be c~lr -- c ~ ( t b - M~). The other f a c t o r s a r e defined by

(4)

w h e r e d~b3 is the phase s p a c e e l e m e n t and F I is the invariant flux, equal to the product of the t a r g e t pr0~ton m a s s and the incident kaon lab m o mentum. ~ I M [ 2 is the absolute s q u a r e of the invariant amplitude, a v e r a g e d o v e r initial and s u m m e d o v e r final spins. The Regge pole hypothesis [2, 3] w a s adopted f o r ~ t MI2,

434

(6)

_

,)

2 = [Sa_(Mp÷M,r)2][Sa_(Mp_M,r)2]

~a

2

~b

= u

z

2 -r~

2

2 ~,

,,

~S~a-tM ' K~-'(M='*-MKh-ta(9) )(t+ht-M~h)t"

We note that p r e v i o u s a n a l y s e s [4, 5] have used a m o r e explicit f o r m of P o m e r o n exchange. We have v e r i f i e d that for our data such an ex-

Volume29B, number 7 OS.I;

6., I;8,2;0 ,2;2 ,24,26,

PHYSICS LETTERS )6

1,6 L8 2.0 22 24

26

0.0 ~)A 021~

02~-/

0,t ~ , . I .///"

02L)/~.'~./

"

-

-02

o.o~---:~

-.--

•

"

)e-

')':_021~ Y

'

0.0

': -02

' (' "

00

0.4

-02

O0 '

'

oo

>

-02

+ Y ~ ~

~

'

-

-0.2 00

oc

-02 00 -02 ,

Ie I'8" 2.0 2.2 24

1

M(-rr+p) GeVlcz

,

i

i

i

2'l0 2,2 '

~:4

)

r

~' 6

Fig. 2. Moments of the spherical harmonics of the angular distribution m the ~ p rest frame as functions of lr÷p mass. The angles and cut are defined in the text. The solid and dashed curves are the predictions from the Regge pole exchange and th~ dot-dash curves were calculated directly from real ~r'p elastic scattering data. p r e s s i o n gives n e a r l y i d e n t i c a l r e s u l t s to those of eq. (9)$ A n u m e r i c a l i n t e g r a t i o n of eq. (8) was p e r ' were f o r m e d in which the p a r a m e t e r s k, So, ot~r v a r i e d to give the b e s t r e p r e s e n t a t i o n of the data ~ . F o r a l l values, the r e s u l t i n g total c r o s s s e c t i o n was about 60% below the e x p e r i m e n t a l value of 0.31 ± 0.03 rob. T h e r e f o r e , all t h e o r e t i c a l d i s t r i b u t i o n s w e r e n o r m a l i z e d to the total n u m b e r of e v e n t s for c o m p a r i s o n to e x p e r i m e n t . Acceptable fits to the data w e r e obtained with 5 < k < 7 (GeV/c) -2, 0.6 < SO < 1.0 (GeV/c)2 and 1.0 < c ~ < 1.4 (GeV/c) "~'. T h e b e s t fit was obT, tained with k = 6 (GeV/c) -z, So = 0.8 (GeV/c) ~" and o~ = 1.2 (GeV/c) - 2 ~ . The fitting proc~d,~.-~ was r e p e , t e d by r e p l a c ing in lower half of d i a g r a m . A with o n - m a s s s h e l l e l a s t i c ~r+p s c a t t e r i n g , i.e. s u b s t i t u t i n g for d(r~/df~ in eq. (7) the e x p e r i m e n t a l ~+p e l a s t i c d i f f e r e n t i a l c r o s s s e c t i o n at the given S # The $ When ~aWaS replaced by

[Sa-tb-M2+~(ta+tb-M2)]2

as usedby Berger in ref. [4], it yi$1ded=the same fits with a cross section of 0.15 mb. $$A fixed K* mass of 0.8914 GeV/c 2 was used in this three-body calculation. The results were subsequently checked by a four-body Monte Carlo program which generated K*lr+p events with a BreitWigner shape for the K* mass and a width r = 0.05 c-ev/c 2. The results are consistent. ~ T h e s e values are consistent with those obtained for reactions ~-p -* pO~-p and K-p --' K*~-p in ref. [5].

23 June 1969

s a m e v a l u e s of S o and c~L w e r e obtainsd. In figs. 1-2, both m e ~ o d s for c a l c u l a t i n g the model a r e c o m p a r e d to the e x p e r i m e n t a l data. The p r e d i c t i o n s of e x p r e s s i o n (8) a r e displayed a s solid c u r v e s . The r e s u l t s of u s i n g o n - m a s s shell ~+p s c a t t e r i n g a r e given by the dashed c u r v e s . Figs. l a - f show v a r i o u s i n v a r i a n t m a s s and m o m e n t u m t r a n s f e r d i s t r i b u t i o n s . The two t h e o r e t i c a l p r e d i c t i o n s a r e seen to be in good a g r e e m e n t with the data and with each other. The model fits the Q e n h a n c e m e n t in the M(K*# +) d i s t r i b u t i o n c e n t e r i n g at 1.27 GeV/c 2 with a width of 0.36 GeV/c 2 (fig. la). Figs. l g and l h show the a n g u l a r d i s t r i b u t i o n of the K* in the K*# + r e s t f r a m e with cos eKK* =/~K.JbK* and ~bK defined by f o r m u l a (6) with all m o m e n t u m v e c t o r s taken in the K % + r e s t f r a m e . Both the data and the p r e d i c t i o n s have r e l a t i v e l y fiat cos eKK* d i s t r i b u t i o n s which is c o n s i s t e n t with p r e v i o u s r e s u l t that the s p i n - p a r i t y of the Q e n h a n c e m e n t i s 1+ [7]. The d i s t r i b u t i o n of the T o i l e r angle, ¢o, is shown in fig. l i . The data a r e adequately d e s c r i b e d by the model prediction, hence the approximation of neglecting the ¢~-dependence at the middle v e r t e x i s valid. A detailed d e s c r i p t i o n of the a n g u l a r d i s t r i b u tion of the v i r t u a l ~+p s c a t t e r i n g is m o s t c o m pletely given in t e r m s of the s p h e r i c a l h a r m o n i c m o m e n t s (Y~ (e, ~)). The angle e i s defined by cos 0 = / ~ i n ~. ~ o u t and ~b i s defined by e x p r e s s i o n (6) with all mor~entum v e c t o r s taken in the ~+p r e s t f r a m e . We show in fig. 2 the ~Y~) and (Re Y~) m o m e n t s for l ~ 6 as a function of M(~+p)## for our data s a m p l e . In addition to the two s e t s of p r e d i c t i o n s p r e v i o u s l y explained, we also show the m o m e n t s c a l c u l a t e d d i r e c t l y f r o m p u r e o n - m a s s - s h e l l ~+p s c a t t e r i n g (dot-dashed

curves). The(Y~) moments in our data are consistently larger than the pure on-mass-shell moments (dot-dashed curves)###. The use of on-mass-shell scattering within the framework of a Reggeized ~ exchange (dashed curves) pro-

~T~e experimental differential cross section of e l a s t i c scattering was extracted from the summary of the 7r-N data in the CERN phase shift analysis [9]. ~ I n calculating the moments (Y~/~, the same cut, 0.02 ~<- t o o ~<1.0, was applied to the present experimenfal data,+the Monte Carlo events and the + on-mass-shell ~. p scattering data. For given M(~" p). this cut in tpp corresponds to a well defined cut in cos 8. hence moments shown by dahsed curves are d~fferent from those obtained from on-mass-shell rr p scattering without any cut. ### This effect has been observed in a previous analysis []o].

~'p

435

Volume 29B, n u m b e r 7

PHYSICS

vides reasonable agreement with the data. With the diffraction scattering approximation, the smooth solid curves are obtained. These deviate f r o m t h e d a s h e d c u r v e s o n l y a t low M 0 r + p ) w h e r e some resonance formation is present; therefore t h e d a s h e d c u r v e s b e t t e r r e p r e s e n t t h e t r e n d of t h e d a t a . T h e d a t a s h o w a s t r o n g ~b d e p e n d e n c e which is mainly given by the m = 1 moments as has been observed in a similar reaction, p p -~ A++Tr-p [11]. T h e ¢ d e p e n d e n c e i n t h i s m o d e l m a i n l y c o m e s f r o m t h e f a c t o r ~.2~Ir $ w h i c h i s i n d e p e n d e n t of t h e a p p r o x i m a t i o n - s u s e d f o r t h e P o m e r a n c h u k e x c h a n g e . H ~ n c e t h e two s e t s of p r e d i c t e d c u r v e s f o r a r e i d e n t i c a l a n d only the solid curves are shown. The Regge pole exchange model used here is seen to account in s o m e d e t a i l f o r b o t h t h e l a r g e a n d t h e q~ dependence and yields an adequate representation of t h e e x p e r i m e n t a l m o m e n t s . W e a l s o f i t t h e m o d e l to o u r d a t a w i t h a t i g h t e r c u t o n M 0 r + p ) ( > l . 8 G e V / c 2) t o r e m o v e t h e e f f e c t s d u e t o r e s o n a n c e s i n t h e 7r+p s y s t e m . T h e f i t s a r e s l i g h t l y i m p r o v e d (not s h o w n ) . A s m a l l f r a c t i o n of t h e e v e n t s s e l e c t e d w a s p r o d u c e d v i a a d i a g r a m i n w h i c h t h e K* a n d lr+ p o s i t i o n s i n d i a gram A are interchanged. Such a diagram is dominated by K*-P exchange$$. We therefore constructed amplitudes for this diagram, diagram A , a n d t h e i r i n t e r f e r e n c e t o f i t t h e d a t a $$~. T h e f i t s w e r e s l i g h t l y i m p r o v e d (not s h o w n ) , b u t to maintain agreement with the data only a small a m o u n t of K* e x c h a n g e c o u l d b e t o l e r a t e d . In conclusion, the Regge pole exchange model w i t h d i a g r a m A u s i n g e i t h e r o n - m a s s - s h e U 7r+p scattering or diffraction scattering approximation adequately describes the reaction K+p -~ K*Olr+p i n t h e k i n e m a t i c r e g i o n d e f i n e d b y eq. (3). I t p r o v i d e s a r e a s o n b l e d e s r i p t i o n to t h e v i r t u a l ~+p s c a t t e r i n g w i t h M 0 r + p ) > 1.5 G e V / c 2. I t a l s o p r e d i c t s t h e p r o d u c t i o n of t h e Q e n h a n c e ment. However, according to the Dolen-HornS c h m i d d u a l i t y h y p o t h e s i s [12], t h i s a g r e e m e n t SFor s m a l l t ' s , S h is the dominant t e r m inF. b. It can be shown that Sb~= F 1 - F 2 cos ~b w h e r e F 1 ~/nd F 2 a r e p o s i t i v e functions independent of S b and 4T h e r e f o r e a low m a s s e n h a n c e m e n t in the K*y s y s t e m is r e l a t e d to the peaking around 0 ° in the q~ d i s tribution. :~:~~Ve use the o r d e r i n g of the longitudinal m o m e n t a of the t h r e e outgoing p a r t i c l e s in the o v e r a l l cm s y s t e m to s e p a r a t e events into different d i a g r a m s [2]. We found that with cut (3), ~ of the events belong to the K*-exchange d i a g r a m while the r e s t belong to (A). The c u r v e s shown in figs. 1, 2 were all c a l culated with d i a g r a m (A) alone. $$:~The amplitude for the K*-exchange d i a g r a m is s i m i l a r to that for d i a g r a m (A). The phase between the two amplitudes is fixed by t h e i r s i g n a t u r e f a c t o r s , while t h e i r r e l a t i v e intensity was v a r i e d to fit the data. 436

LETTERS

23 June 1969

d o e s n o t n e c e s s a r i l y i m p l y t h e a b s e n c e of o n e o r m o r e K*Ir r e s o n a n c e s i n t h e Q r e g i o n . We are gratefully acknowledge the generous h e l p f r o m t h e B N L B u b b l e C h a m b e r a n d AGS G r o u p s a n d t h e c o n s c i e n t i e s w o r k of t h e U C L A scanning staff. 1. K.A. T e r - M a r t i r o s y a n , Nuclear P h y s i c s 68 (1964) 591; T . W . B . Kibble, Phys. Rev. 131 (1963) 2282; F. Z a c h a r i a s e n and G. Zweig, Phys. Rev. 160 (1967) 1326. 2. H. M. Chan, K. Kajantie and G. Ranft, Nuovo Cimento 49A (1967) 1159; 51A (1967) 696. 3. N. F. Bali, G.T. Chew and A. Pignotti, Phys. Rev. L e t t e r s 19 (1967) 614; Phys. Rev. 163 (1967) 1572. 4. E. L. Berger, E. G e l l e r t , G. Smith, E. Colton and P . E . S c h l e i n , P h y s . Rev. L e t t e r s 20 (1968) 964; E. L. B e r g e r , Phys. Rev. L e t t e r s 21 (1968) 701; Phys. Rev. 166 (1968) 1525. 5. M. L. Ioffredo, G.W. Brandenburg, A. E. B r e n n e r , B. Eisenstein, L. E i s e n s t e i n , W.H. Johnson Jr., J. K. Kim, M.E. Law, B.M. Salzberg, J.H. S c h a r e n guivel, L.K. S i s t e r s o n and J. J. Szymanski, Phys. Rev. L e t t e r s 21 (1968) 1212; S. U. Chung, R. L. E i s n e r , N. Bali and D. Liiers, Bull. Am. Phys. Soc. 14 (1969) 41 and BNL P r e p r int; M. S. F a r b e r , J . C . B e r l i n g h i e r i , B. Forman, T. F e r bel, P. F. Slattery, H. Yuta and D. Griffiths, Bull. Am. Phys. Soc. 14 (1969) 120; J. Andrews, J. Lach, T. Ludlam, J. Sandweiss and H. D. Taft, Yale University P r e p r i n t . 6. S. Ratti P r o c . 1968 Topical Conf. on High energy collisions of hadrons, CERN 68-7, Vol. I, p.61; J. G. Rushbrooke and J. R. Williams, Phys. Rev. L e t t e r s 22 (1969) 248; D. F. G r e t h e r and R. D. Sard, University of Illinois P r e p r i n t COO-1195-151; G. Alexandr, A. F i r e s t o n e , G. Goldhaber and A. Pignotti, Phys. Rev, to be published. 7. C.Y. Chien, P. M. Dauber, E.I. Malamud, D.J. Mellema, P . E . Schlein, P . A . S c h r e i n e r , W.E. Slater, D. H. Stork, H.K. Ticho and T. G. Trippe, Phys. Lett e r s 28B (1968) 143 and the r e f e r e n c e s cited. 8. J. B e r l i n g h i e r i et al., Phys. Rev. L e t t e r s 18 (1967) 1087, G. Goldhaber et al., Phys. Rev. L e t t e r s 19 (1967) 972. D. J. Crennel et a l . , Phys. Rev. L e t t e r s 19 (1967) 44; G. B a s s o m p i e r e et al. , Phys. L e t t e r s 26B (1967) 30 and U. Maor et al., P h y s . Rev. L e t t e r s 15 (1965) 281. 9. A. Donnachie, R. G. Kirsopp and C. Lovelace, CERN Th. 838 Addendum (phase shifts), (1967); C. Coffin, N. Dikmen, L. Ettlinger, D. Meyer, A. Sanlys, K. Terwillinger, D. Williams, Phys. Rev. L e t t e r s 17 (1966) 458; P h y s . Rev. 159 (1967) 1169. 10. T.G. Trippe, C.Y. Chien, E. Malamud, J. Mellema, P. E. Schlein,, W.E. Slater, D.H. Stork and H. K. Ticho, Phys. L e t t e r s 28B (1968) 203. 11. E. Colton, et al., UCLA p r e p r i n t No. 1023. 12. R. Dolen, D. Horn and C.Schmid, Phys. Rev. 166 (1968) 1768; G. F. Chew and A. Pignotti, Phys. Rev. L e t t e r s 20 (1968) 1078.