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The Reggeized U(6) ⊗ U(6) ⊗ O(3) absorptive peripheral model for O−12+ → O−12+ hypercharge-exchange reactions
~
Nuclear Physics B20 (1970) 381-396. North-Holland Publishing Company
THE REGGEIZED U(6) ® U(6) ® 0 ( 3 ) ABSORPTIVE PERIPHERAL MODEL FOR o - ½ + --,- 0 - ~ + H Y P E R C H A R G E - E X C H A N G E REACTIONS P . A. C O L L I N S , B . J . H A R T L E Y , R . W . MOORE a n d K. J . M. M O R I A R T Y
Physics Department, Imperial College, London SW. 7 Received 11 F e b r u a r y 1970 Abstract: The O-½ + ~ O-½ + hypercharge-exchange reactions: y - p --* K°A(5-0), ?r-p ~ K°A, K-n --* ~-A, K-n -- ~ - ~ o , ~+p --. K+~+ and K-p -- ?r-Z+ a r e studied within the f r a m e w o r k of the Reggeized U(6) ® U(6) ® 0(3) absorptive p e r i p h e r a l model. The differential c r o s s - s e c t i o n data for 7r-p ~ K°h(~ °) is used to determine the p a r a m e t e r s for the K*(890) and the KN(1420 ) exchanges. With these p a r a m e t e r s absolute predictions a r e made for the differential c r o s s sections for the other r e actions. Agreement with experiment is reasonable. The polarization p a r a m e t e r P(t) is presented. Good agreement is obtained for Itl < 03
1. INTRODUCTION In two r e c e n t p a p e r s [1] t h e O-½ + ~ O-½ + c h a r g e - e x c h a n g e r e a c t i o n s ~ - p -~ ~On, ~ - p ~ TOn, K - p ~ KOn a n d K+n ~ KOp w e r e w e l l e x p l a i n e d u s i n g R e g g e i z e d U(6) ® U(6) ® 0(3) a n d a b s o r p t i v e c o r r e c t i o n c u t s in t h e e n e r g y r a n g e 5.0 < P~ab "-< 1~.2 G e V / c . T h e c a l c u l a t i o n s h a v e now b e e n e x t e n d e d to O-½ +-~ O - ~ + h y p e r c h a r g e - e x c h a n g e r e a c t i o n s . In r e f . [1] t h e only s i g n i f i c a n t c o n s t r a i n t i m p o s e d by t h e h i g h e r - s y m m e t r y s c h e m e w a s in f i x i n g t h e r a t i o of t h e h e l i c i t y f l i p to t h e h e l i c i t y n o n flip amplitude. The baryon-baryon vertex involved both D- and F-couplings, t h e r a t i o of w h i c h w a s not f i x e d by SU(3) but h a d to b e f i x e d by s o m e h i g h e r s y m m e t r y s c h e m e . S i n c e t h e r e w a s only one t y p e of b a r y o n - b a r y o n v e r t e x , n a m e l y pn, t h e D / F r a t i o c o u l d b e e f f e c t i v e l y a b s o r b e d into t h e r e s i d u e s . T h u s , t h e r e a c t i o n s w e r e s i m p l y r e l a t e d by SU(3). S i n c e t h e r e a r e two t y p e s of b a r y o n - b a r y o n v e r t i c e s , n a m e l y n u c l e o n - A a n d n u c l e o n - ~ , f o r t h e hypercharge-exchange reactions considered here, the D / F ratios cannot s i m p l y b e a b s o r b e d into t h e r e s i d u e s . T h e U(6) ® U(6) ® O(3) s y m m e t r y scheme thus provides a significant constraint. In s e c t . 2 we o u t l i n e t h e f o r m a l i s m f o r O - -~1 + - ~ O - -~1 + r e a c t i o n s in t e r m s of t h e t - c h a n n e l M - f u n c t i o n s a n d t h e s - c h a n n e l h e l i c i t y a m p l i t u d e s a n d p r e $ Science R e s e a r c h Council R e s e a r c h Fellow.
382
t~.A. COLLINS et al.
sent the R e g g e i z e d S u p e r m u l t i p l e t a m p l i t u d e s . Sect. 3 contains a b r i e f d i s c u s s i o n of the r e s u l t s of the model as applied to the r e a c t i o n s
(i)
n - p ~ KOA(GO) ,
(ii)
u - p ~ KOA , K - n -~ n - A
(iii) (iv) (v) (vi)
,
K - n -~ ~ - F~° ,
~+p -~ K+E+
K-p ~ ~-E +
2. FORMALISM F o r r e a c t i o n s of the type -I+__ o
-I+
O~
,
the M-function [2] is
M=A+~B
,
w h e r e A and B a r e the i n v a r i a n t a m p l i t u d e s and Q is half the s u m of the initial and final m e s o n f o u r - m o m e n t a . We introduce the i n v a r i a n t a m p l i tude A' defined by
A'=A+
Ela b + t/4m 1
1- t / 4 m 2
B,
w h e r e E l a b is the lab. e n e r g y of the incoming m e s o n , t is the f o u r - m o m e n t u m t r a n s f e r , and m 1 is the m a s s of the t a r g e t nucleon. The independent s - c h a n n e l helicity a m p l i t u d e s [3] a r e defined by
1 c o s ½ 0 { m l A + ( E l W - m21)B}, 4~++ - 4nw 1 sin½0{E1A+ml(W - E1)B } , ~ +- = 4nw w h e r e 0 is the s - c h a n n e l c.m. s c a t t e r i n g angle, w is the c.m. e n e r g y and E1 is the c . m . e n e r g y of the t a r g e t nucleon. In the U(6) ® U(6) ® O(3) c l a s s i f i c a t i o n s c h e m e [4] the i n v a r i a n t a m p l i tudes A' and B a r e
A'= (1+ 4-~)gF
I f l _ h F r ( 1 - a_)(1- e - i n a - ) ( s - m
2-2mp/~2+½t) c~-
1
+ $+hDF(I_ 0%)(1+e-i~'o~+) (s-m ~2- i.t2+½t)a+
HYPERCHARGE-EXCHANGE REACTIONS
B= p-1 (1+ ~)(1-
× (s-
4-~)
383
gD+~FI~-hF F(1- a_)(1-e -i~a-)
m 2 - tl 2 + ½ t ) a - - 1 2m~+ fi+hDr(1 _ a+)(1 + e - ~ + )
( s - m2--2mp~2+½t)a+-l]'
w h e r e the g and h a r e b a r y o n - b a r y o n - m e s o n and m e s o n - m e s o n - m e s o n couplings, m and p a r e the m a s s e s a s s o c i a t e d with the (56, 1: 0) and (6, 8;0) m u l t i p l e t s and the ± r e f e r to the even and odd s i g n a t u r e t r a j e c t o r i e s , r e s p e c t i v e l y . The G e l l - M a n n ghost-killing m e c h a n i s m is employed. P a i r w i s e e q u a l - m a s s k i n e m a t i c s a r e a s s u m e d in this derivation. The helicity a m p l i t u d e s qS++ and qS+_ a r e expanded in p a r t i a l - w a v e s e r i e s , the a b s o r p t i v e c o r r e c t i o n s applied using the technique outlined in r e f . [ 1], and the p a r t i a l - w a v e s e r i e s then r e s u m m e d to give the modified helicity Y a m p l i t u d e s ¢ ++ and q~+_. We take a G a u s s i a n f o r m for the non-flip s c a t t e r i n g amplitude ( a s s u m e d the s a m e in the initial and final states) S J = 1- c e - J ( J + l ) / u 2 p 2 , w h e r e J is the angular m o m e n t u m , p is the c.m. t h r e e - m o m e n t u m of the initiai p a r t i c l e s , and u is the e l a s t i c r a d i u s of i n t e r a c t i o n of the p a r t i c l e s in the initial state. In t e r m s of the modified helicity a m p l i t u d e s , the differential c r o s s s e c tion and the p o l a r i z a t i o n a r e I ~gg d~ ~ , 2 I m [qS++~+_]
a~ = ~ [1~++12+ 1~--]2] ,
P(t) : 1¢):~+]2+ I~LI2 •
3. DISCUSSION AND RESULTS 0 - -~1 + ~ 0 - - 1~+ h y p e r c h a r g e - e x c h a n g e r e a c t i o n s have been t r e a t e d in the conventional (no-cut contribution) Regge a p p r o a c h by Salin [5] and by R e e d e r and S a r m a [6] with r e a s o n a b l e s u c c e s s . As d i s c u s s e d in ref. [6] the only Regge poles which can be exchanged in the t - c h a n n e l a r e a s s o c i a t e d with the K*(890) ( J P = 1-) and KN(1420 ) ( J P = = 2 +) m e s o n s . The d i f f e r e n t i a l c r o s s - s e c t i o n data with the l a r g e s t v a r i a tions in s and t is for the r e a c t i o n ~ - p ~ K°A(~. °) (ref. [7]). We t h e r e f o r e fix the residue/3(t) and the t r a j e c t o r y function a(t) for both of t h e s e t r a j e c t o r i e s f r o m a X2 fit of the a b s o r b e d differential c r o s s section to this e x p e r imental data, and then p r e d i c t the 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 and p o l a r i z a t i o n s for the other r e a c t i o n s . Following ref. [ 1] we took the r e s i d u e s to be c o n s t a n t s and p a r a m e t r i z e d the t r a j e c t o r y functions for negative t by a(t) = a o+c~ l e ~ 2 t .
384
P . A . C O L L I N S et al.
The conventional linear parametrization e r e d in t h e p e r i p h e r a l r e g i o n
of a R e g g e t r a j e c t o r y i s r e c o v -
a(t) ~ (ao+ ~1) + ( c q a 2 ) t • U s i n g t h i s f o r m t h e R e g g e t r a j e c t o r i e s w e r e e x t r a p o l a t e d s o a s to p a s s through their respective poles, thus providing additional constraints. Alt h o u g h t h i s e o n s t r a i n t w a s not i m p o s e d f o r t h e P a n d A 2 t r a j e c t o r i e s in r e f . [1], t h e t r a j e c t o r i e s w e r e f o u n d to p a s s s u r p r i s i n g l y c l o s e t o t h e i r r e s p e c t i v e p o l e s . We t h e r e f o r e f e e l t h a t t h i s i s not a n u n n a t u r a l e o n s t r a i n t . H e n c e we h a v e t h r e e f r e e p a r a m e t e r s f o r e a c h p o l e , n a m e l y a c o n s t a n t r e s i d u e a n d t w o t r a j e c t o r y p a r a m e t e r s , g i v i n g a t o t a l of s i x f r e e p a r a m e ters. These were determined using MINUITS (CERN Program Library, n u m b e r D506). T h e D - a n d F - c o u p l i n g s f o r t h e p r o e e s s e s c o n s i d e r e d a r e s h o w n in t a b l e 1, a n d t h e e l a s t i c s c a t t e r i n g c o e f f i c i e n t s a r e s h o w n in t a b l e 2 ( s i n c e no d a t a i s a v a i l a b l e f o r t C n d i f f e r e n t i a l c r o s s s e c t i o n s we h a v e u s e d t h e K - p c o e f f i c i e n t s ) . T h e r e s u l t s of t h i s m i n i m i z a t i o n p r o c e d u r e a r e s h o w n in t a b l e 3. Table 1 D- and F - c o u p l i n g s . Baryon vertex Reaction
K*(890) and KN(1420 ) D + ~F
-p -~ KO A
?r-p -* K°~ K-n
°
--' rr-A
F
Meson vertex KN(1420 )
K*(890)
D-type
F-type
- V5
-45
~
-C5
1/3
-1
~/2
-'/-2
- ,/3
-if3
~/2
~/2
K-n ~ rr-E °
-1/3
1
~/2
~/2
#+p -* K+E +
ff-2/3
-V~
~-
-V~
K - p ~ 7r-E +
ff-2/3
- ~/2
~/2
~/2
A C h e w - F r a u t s c h i p l o t i s p r e s e n t e d in fig. 1. We n o t e t h a t t h e K*(tt90) a n d KN(1420 ) t r a j e c t o r i e s a r e a l m o s t p a r a l l e l a n d n e a r l y e x c h a n g e d e g e n e r a t e , a l t h o u g h t h i s w a s not i m p o s e d a s a c o n s t r a i n t in t h e d a t a f i t t i n g . W e now d i s c u s s e a c h r e a c t i o n s e p a r a t e l y : (i)
~ - p -~ K ° A ( E °) •
T h e m a s s r e s o l u t i o n of t h e r e c o i l h y p e r o n in t h i s e x p e r i m e n t w a s s u c h t h a t t h e A a n d E ° c o u l d not b e d i s t i n g u i s h e d . S i n c e t h e s e a r e in p r i n c i p l e distinguishable reactions d~ d(r dcr dr- (Tr-p --, K ° h ( E ° ) ) = d r - ( r r - p --' K°A) + ~ - ( r r - p --' K ° E °) .
385
HYPERCHARGE-EXCHANGE REACTIONS Table 2 Absorption coefficients. Channel
Plab
v-I(GeV-1)
c
7T+p
3.00 3.25 4.00 5.05 5.40 7.00
0.27 0.27 0.27 0.27 0.27 0.27
0.89 0.87 0.84 0.82 0.81 0.79
7r-p
6.00 7.91 8.00 10.00 11.20
0.26 0.26 0.26 0.26 0.26
0.79 0.76 0.76 0.74 0.73
K-p
4.07 4.25 5.47
0.26 0.26 0.26
0.78 0.77 0.73
Table 3 Regge p a r a m e t e r s . Trajectory
K* (890)
KN(1420)
oto
-0.829
-0.984
~1
1.098
1.177
or2 (GeV/c) -2
0.840
0.761
fl(GeV/c)-I
-7.162
4.712
No. of data points X2
76 65
A s d i s c u s s e d a b o v e t h i s r e a c t i o n w a s u s e d to d e t e r m i n e the Regge p a rameters. F i g . 2 s h o w s the e n e r g y d e p e n d e n c e of the 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 . T h e s - a n d t - d e p e n d e n c e of the d a t a a r e e x t r e m e l y well r e p r e s e n t e d . H a v i n g d e t e r m i n e d the R e g g e p a r a m e t e r s we c a l c u l a t e d t h e f o l l o w i n g r e a c t i o n s a n d c o m p a r e d the p r e d i c t i o n s with e x p e r i m e n t . (ii) ~ - p - * K°A. D a t a on t h i s e x p e r i m e n t e x i s t s at 7.91 G e V / c [8]. T h e 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 is s h o w n i n fig. 3. We s e e that the d a t a is w e l l r e p r e s e n t e d b y the t h e o r y f o r a l l v a l u e s of t. T h e r e is p o l a r i z a t i o n d a t a at 6.0 G e V / c [9]. A s s h o w n in fig. 4, f o r I tl < 0.3 ( G e V / c ) 2 t h e o r y a g r e e s with e x p e r i m e n t , but f o r > 0.3 the t h e o r y f a i l s t o fit the d a t a , i n t h a t we o b t a i n + 0 . 1 f o r t = - 0 . 3 5 ( G e V / c ) 2 w h e r e a s e x p e r i m e n t i s c o n s i s t e n t with p o l a r i z a t i o n b e t w e e n 0. a n d - 1.0.
Itl
386
P.A.COLLINS et al. 7.0 6.0
5.0 4.0 3.0 R e a l o=(t)
2.0 1.0
t"
0.0
-1.0 -2":3.o
'
-;.o
'
-,:o
'
o.o
'
I.'o
'
21o
'
3.0
t r(c.vlc~ 3
Fig. 1. Plot of ~(t) against t for the K*(890) trajectory ( ) and the KN(1420) tra. ' jectory ( . . . . ). Parameters from table 3. 100
. . . . . . . . . . . . . . 't["p~ KOA{,r o )
101
~
10'
rl
162 ~
11.2 GeV~:
1." . . . . . . 04)0
'
'
'
.50
t
. . . .
.
~
I.(30 . . . .
1.50
- t EfGWIc )z'l
Fig. 2. Differential cross section for ~-p ~ K°A(~°). Data from ref. [7].
HYPERCHARGE-EXCHANGE REACTIONS
1
387
ffp--K°A at 7.91 GeVIc
f T\
K~3/ . . . . . . . . . . . . . . 0.00 .50 1.00 -t r'( GeVic)2"l
1.50
Fig. 3. Differential cross section for ?r-p ~ K ° A . Data from r ef. [8].
~'p--,-K°^ at 6.0 C,eV/c
0.50
Z 0
N_ n,
-1.00 0.50 tOO -t C(GeVIc )2:]
Fig. 4. Polarization for 7r-p -~ K°A. Data from ref. [9]. (iii) K - n ~ v - A . T h e t h e o r e t i c a l p r e d i c t i o n f o r the d i f f e r e n t i a l c r o s s s e c t i o n for t h i s r e a c t i o n is p l o t t e d a g a i n s t the e x p e r i m e n t a l d a t a at 4.25 G e V / c [10] i n fig. 5. T h e n o r m a l i z a t i o n in the f o r w a r d d i r e c t i o n i s r e p r o d u c e d , b u t the t h e o r y p r e d i c t s too m u c h s c a t t e r i n g f o r l a r g e t. In fig. 6 the t h e o r e t i c a l a n d e x p e r i m e n t a l p o l a r i z a t i o n s [ 10] a r e c o m p a r e d . T h e t h e o r y d o e s not r e p r e s e n t the data.
388
P.A.COLLINS '
I
.
.
.
.
et al. i
,
,
•
,
[
K-n~"E-A at 4.25 GeV/c
101 I"I
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1152
.
i
,
F i g . 5. D i f f e r e n t i a l c r o s s
I
,
,
,
i
~0
I - t C(C_~//c)2 D
i
I i
.
I
1.50
,
s e c t i o n f o r K - n - * 7r-A. D a t a f r o m r e f . [ i 0 ] ~n..,,~
ot &.25 C~k:
Z O
N
0D0 F i g . 6. P o l a r i z a t i o n
0.50 1.00 1.50 -t E(6eWct 3 f o r K - n -~ 7r-A. D a t a f r o m r e f . [10].
HYPERCHARGE-EXCHANGE REACTIONS
10(
,
,
,
i
I
.
.
.
.
i
.
.
K-rt~
.
.
389
i
tr'~ ° Qt 4.25GeV/c
u
E U
10
16: . . . . . . . . 0.00 0.50
~ j 1.00
, , 1.5O
-t E (GeV/c)2~
F i g . 7. D i f f e r e n t i a l c r o s s s e c t i o n f o r K - n - * 7r-S ° . Data f r o m r e f . [10].
tOO
.
.
.
.
i
,
,
,
050
•
i
,
•
,
,
!
K-n - - tY'L"° at 4.25GeVIc
~oo N
-050
--1.0(
,
0.00
.
,
,
i
,
0.50
,
,
,
I
.
-t
E( GeV/c)2 ~
1,00
,
.
.
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,
1.50
F i g . 8. P o l a r i z a t i o n f o r K - n --* 7r-S ° .
•
390
P . A . COLLINS et al.
(iv) K - n ~ ~ - ~ o . Fig. 7 s h o w s a c o m p a r i s o n of the r e c e n t d a t a [10] on the d i f f e r e n t i a l c r o s s s e c t i o n f o r t h i s r e a c t i o n at 4.25 G e V / c with the p r e d i c t i o n of this m o d e l . T h e a g r e e m e n t is m o s t e n c o u r a g i n g . Both the f o r w a r d n o r m a l i z a tion and the t - d e p e n d e n c e a r e well r e p r o d u c e d . T h e p o l a r i z a t i o n d i s t r i b u t i o n f o r which no d a t a e x i s t s at p r e s e n t is shown in fig. 8. (v) ~+p ~ K+~ +. T h e h i g h - e n e r g y e x p e r i m e n t a l d a t a on dg/dt f o r t h i s r e a c t i o n h a s b e e n m e a s u r e d by two e x p e r i m e n t a l g r o u p s [11, 12]. In fig. 9 the r e s u l t s of o u r c a l c u l a t i o n s a r e c o m p a r e d with the d a t a of ref. [ 11]. Both the n o r m a l i z a t i o n and t - d e p e n d e n c e of the m o d e l a r e c o n s i s t e n t with the data. A plot of the 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 and the c o r r e s p o n d i n g e x p e r i m e n t a l d a t a of ref. [12] is shown in fig. 10. We o b s e r v e that in this c a s e we do not obt a i n the c o r r e c t n o r m a l i z a t i o n . T h e r i g h t t - d e p e n d e n c e is o b t a i n e d out to about t = - 0 . 4 ( G e V / c ) 2, but b e y o n d this t h e r e a p p e a r s to be s o m e s t r u c t u r e which is not r e p r e d u c e d by our m o d e l . Since m u c h of this d a t a is at low ene r g y , and t h i s s t r u c t u r e is not p r e s e n t at the h i g h e s t a v a i l a b l e e n e r g y , we do not take this d i s a g r e e m e n t in t - d e p e n d e n c e too s e r i o u s l y . In figs. 11-13 the p o l a r i z a t i o n p r e d i c t i o n s a r e c o m p a r e d with the a v a i l able d a t a [11-13]. ~Figs. 11 and 13 s h o w that the p o l a r i z a t i o n is s m a l l and n e g a t i v e f o r it I < 0.3. H o w e v e r , b e t w e e n t = - 0 . 3 and t = - 0 . 5 the data
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HYPERCHARGE-EXCHANGE REACTIONS
391
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Fig. I0. Differential cross section for ~+p -~ K+~+. Data from ref. [12]. shows a d r a m a t i c change to a l a r g e positive value of about 0.7 at all e n e r gies, and this p e r s i s t s to l a r g e r values of t. Our model c o r r e c t l y r e p r o duces the p o l a r i z a t i o n for I tt < 0.3, which c o r r e s p o n d s to the f o r w a r d peak region in the differential c r o s s section, but beyond this the model f a i l s (vi) K-p -~ ~-~+. Our prediction of the differential cross section for this reaction is compared with the available experimental data [14, 15] in fig. 14. As in the previous reaction (v), we fail to obtain the correct normalization, although the t-dependence is reasonable. Since reactions (v) and (vi) are related by charge conjugation at the meson vertex, the K*(890) contribution simply changes sign from one reaction to the other. However, the K*(890) residue is approximately twice the KN(1420) residue and therefore this change in
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Fig.
11.
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ref.
[13].
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5.4 GeVl
ifi EOOOT 5’ 8
-0.50 -
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1 0.50 -t C(GcVk
Fig.
12.
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for ?‘r+p + K%+.
Data from
ref.
[ll].
HYPERCHARGE-EXCHANGE REACTIONS 1.2
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Fig. 13. Polarization for ~'+p -~ K+~,+. Data from ref. [12]. sign p r o d u c e s little difference in the f o r w a r d n o r m a l i z a t i o n s of t h e s e two reactions. Fig. 15 shows the p r e d i c t i o n for the p o l a r i z a t i o n for r e a c t i o n (vi). No h i g h - e n e r g y data is available at p r e s e n t . We note that in the c h a r g e - e x c h a n g e r e a c t i o n s of ref. [ 1] the data indic a t e s the helicity flip amplitude dominates in the f o r w a r d direction giving a t u r n o v e r , w h e r e a s in the h y p e r c h a r g e c a s e the helicity non-flip amplitude dominates giving a f o r w a r d peak. This f e a t u r e is c o r r e c t l y r e p r e s e n t e d by the U(6) ® U(6) ® O(3) s y m m e t r y scheme. Within the confines of our model, we feel that the a g r e e m e n t between theory and e x p e r i m e n t is s a t i s f a c t o r y . The conventional Regge a p p r o a c h of R e e d e r and S a r m a [6] contained many m o r e p a r a m e t e r s than the p r e s e n t work. In p a r t i c u l a r , they had r e s i d u e s c o r r e s p o n d i n g to v e c t o r and t e n s o r exchange, helicity flip and non-flip and A and ~, r e a c t i o n s , a total of eight
394
P. A. COLLINS et al. 100
.
.
.
.
i
.
.
.
.
i
.
.
.
.
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1.00
1.50
-t E(GeV/c)23 Fig. 14. Differential c r o s s section for K-p --o ?r-E +. Data from refs. [14, 15]. residue parameters. The 'so-called' scale parameters for vector and tensor exchange and helicity flip and non-flip were also included as parameters, adding four more. A 'turnover' parameter was included to obtain the d r a m a t i c c h a n g e in t h e ?r-p -o K+E+ p o l a r i z a t i o n at ! = - 0 . 4 ( G e V / c ) 2. H o w e v e r t h e i r t r a j e c t o r i e s w e r e not f r e e in t h a t t h e i r K*(890) t r a j e c t o r y w a s o b t a i n e d f r o m t h e i r p - t r a j e c t o r y a n d t h e i r KN(1420 ) t r a j e c t o r y f r o m t h e i r A 2 t r a j e c t o r y b y a d i s p l a c e m e n t e q u a l to t h e d i f f e r e n c e of t h e s q u a r e s of t h e r e s p e c t i v e m a s s e s . T h e F/D r a t i o f o r t h e v e c t o r a n d t e n s o r e x c h a n g e s and helicity flip and non-flip were effectively parameters, adding another f o u r f r e e p a r a m e t e r s . T h i s w a s a t o t a l of s i x t e e n p a r a m e t e r s a n d g a v e a g o o d r e p r e s e n t a t i o n of t h e d a t a . C o n s i d e r i n g t h e f a c t t h a t we h a v e o n l y s i x f r e e p a r a m e t e r s w e c o n s i d e r o u r f i t to b e a n a c c e p t a b l e r e p r e s e n t a t i o n of the data. Discrepancies between our model and the data could possibly be improved by different final- state absorption coefficients. However, these c a n n o t b e d e t e r m i n e d , a n d we f e e l t h a t l i t t l e i n s i g h t into t h e p h y s i c s i s to
HYPERCHARGE-EXCHANGE REACTIONS
395
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,
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,
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i
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F i g . 15. P o l a r i z a t i o n
1
.
•
•
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*
1.50
f o r K - p -~ ~ - ~ + .
be gained by making these p a r a m e t e r s . Another possibility is that exact SU(3) s y m m e t r y does not hold for the couplings. We o b s e r v e that in the work of Salin [5] and R e e d e r and S a r m a [6] the D/F r a t i o obtained is significantly different f r o m the SU(6) prediction of 2. This may be another s o u r c e of trouble. P r e l i m i n a r y application of this model to the r e a c t i o n yp ~ ~-A++ has shown that exact k i n e m a t i c s is important. P a i r - w i s e equal m a s s - k i n e m a t ics with the U(6) ® U(6) ® O(3) p r e d i c t e d r a t i o of helicity amplitudes leads to a pion f o r w a r d peak r a t h e r than the t u r n o v e r exhibited by the data. However, exact k i n e m a t i c s r e p r o d u c e s this f e a t u r e of the data. We t h e r e f o r e feel that the effect of exact k i n e m a t i c s may be significant for these h y p e r charge reactions. It m u s t also be b o r n e in mind that t h e r e a r e n o r m a l i s a t i o n e r r o r s in the e x p e r i m e n t a l data, typically of the o r d e r of 20-30%, which could account for some of our n o r m a l i z a t i o n difficulties. However, we see no point in r e adjusting the data within the stated n o r m a l i z a t i o n e r r o r s in o r d e r to improve the a p p e a r a n c e of our fits. We a r e generally able to account for the p o l a r i z a t i o n data for It l < 0.3 (GeV/c) 2, but beyond this the e x p e r i m e n t a l data undergoes a d r a m a t i c change which we cannot explain. Since p o l a r i z a t i o n is a subtle effect this m a y be explainable in t e r m s of the h i g h e r - o r d e r c o r r e c t i o n s to the amplitudes [ It has been suggested [1'/] that the d r a m a t i c change in p o l a r i z a 16t" tion at It ~ 0.4 (GeV/c) 2 can be aeeounted for within the context of the Reggeized a b s o r p t i o n model by using a moving P o m e r a n c h u k pole r a t h e r than a fixed J = 1 pole as we have done. F o r the n u m e r i c a l work we employed an exact p a r t i a l - w a v e s u m m a t i o n
396
P . A . COLLINS et al.
u s i n g 20 p a r t i a l w a v e s c a l c u l a t e d with a 4 8 - p o i n t G a u s s i a n q u a d r a t u r e . T h e s e c a l c u l a t i o n s w e r e r e p e a t e d u s i n g 30 p a r t i a l w a v e s a n d a 5 1 2 - p o i n t G a u s s i a n q u a d r a t u r e . T h e r e s u l t s w e r e f o u n d not to c h a n g e s i g n i f i c a n t l y . We w i s h to t h a n k P r o f e s s o r P. T. M a t t h e w s f o r e n c o u r a g e m e n t i n t h i s w o r k a n d f o r a c r i t i c a l r e a d i n g of the m a n u s c r i p t , D. D a l l m a n f o r c o m p i l i n g the d a t a , R. E h r l i c h f o r v e r y k i n d l y g i v i n g u s h i s d a t a in t a b u l a r f o r m , a n d R. C. B e c k w i t h f o r a s s i s t a n c e i n c o m p u t i n g . One of u s (B.J.H.) w i s h e s to t h a n k the U n i v e r s i t y of L o n d o n f o r t h e a w a r d of the G e o r g e W i l l i a m B r i t t ( J u n i o r ) S t u d e n t s h i p a n d two o t h e r s ( P . A . C . a n d R.W.M.) w i s h to t h a n k the Science Research Council for Research Studentships.
RE F E RE NCE S [1] B. J. Hartley, R.W. Moore and K. J. M. Moriarty, Phys. Rev. 187 (1969) 1921; D1 (1970) 954. [2] R.J. Eden, High-energy collisions of elementary particles (Cambridge University P r e s s , 1967). [3] M.Jacob and G.C.Wiek, Ann. of Phys. 7 (1959) 404. [4] R.Delbourgo, A.Salam and J.Strathdee, Phys. Rev. 179 (1968) 1487; 172 (1968) 1727; 186 (1969~ 1516. [5] Ph. Salin, Nucl. Phys. B3 (1967) 323. [6] D . D . R e e d e r and K . V . L . S a r m a , Phys. Rev. 172 (1968) 1566. [7] E. Bertolueei, I. Mannelli, G. Pierazzini, A. Seribano, F. Sergiampietri, M.L. Vincelli, C. Caverzasio, J . P . G u i l l a u d , L. Holloway and M.Yvert, Nuovo Cimento Letters 2 (1969) 149. [8] R . E h r l i c h , W.Selove and H.Yuta, Phys. Rev. 152 (1966) 1194. [9] D . J . C r e n n e l l , G.R.Kalbfleisch, K . W . L a i , J . M . S c a r r , T.G.Schumann, I.O. Skillicorn and M.S.Webster, Phys. Rev. Letters 18 (1967) 86. [10] W . L . Y e n , A . C . A m m a n n , D.D. Carmony, R . L . E i s n e r , A . F . G a r f i n k e l , L . J . Gutay, R.V. Lakshmi, D.H. Miller and G. W. Tautfest, Phys. Rev. Letters 22 (1969) 963. [11] W.A.Cooper, W.Manner, B.Musgrave and L.Voyvodic, Phys. Rev. Letters 20 (1968) 472. [12] S . M . P r u s s , C.W.Akerlof, D . I . M e y e r , S.P.Ying, J . L a l e s , R.A.Lundy, D.R. Rust, C . E . W . W a r d and D.D.Yovanovitch, Phys. Rev. Letters 23 (1969) 189. [13] R.R.Kofler, R.W.Hartung a n d D . D . R e e d e r , Phys. Rev. 163 (1967) 1479. [14] J.S. Loos, U . E . K r u s e a n d E . L . G o l d w a s s e r , Phys. Rev. 173 (1968) 1330. [15] D.Birnbaum, R . M . E d e l s t e i n , N.C.Hien, T.J.McMahon, J . F . M u c c i , J.S. Russ, E . W . A n d e r s o n , E . J . B l e s e r , H.R.Blieden, G . B . C o l l i n s , D.Garelick, J. Menes and F. Turkot, Experimental study of the reactions K-p -* ~-~+ and K-p - ' y-Y*+ at 8 and 16 GeV/c, submitted to the 14th Int. Conf. on high-energy physics, Vienna, 1968. [16] C.Lovelace, Nucl. Phys. B12 (1969) 253. [17] A.Krzywicki and J . T r a n Thanh Van, Phys. Letters 30B (1969) 185.
Nuclear Physics B20 (1970) 381-396. North-Holland Publishing Company
THE REGGEIZED U(6) ® U(6) ® 0 ( 3 ) ABSORPTIVE PERIPHERAL MODEL FOR o - ½ + --,- 0 - ~ + H Y P E R C H A R G E - E X C H A N G E REACTIONS P . A. C O L L I N S , B . J . H A R T L E Y , R . W . MOORE a n d K. J . M. M O R I A R T Y
Physics Department, Imperial College, London SW. 7 Received 11 F e b r u a r y 1970 Abstract: The O-½ + ~ O-½ + hypercharge-exchange reactions: y - p --* K°A(5-0), ?r-p ~ K°A, K-n --* ~-A, K-n -- ~ - ~ o , ~+p --. K+~+ and K-p -- ?r-Z+ a r e studied within the f r a m e w o r k of the Reggeized U(6) ® U(6) ® 0(3) absorptive p e r i p h e r a l model. The differential c r o s s - s e c t i o n data for 7r-p ~ K°h(~ °) is used to determine the p a r a m e t e r s for the K*(890) and the KN(1420 ) exchanges. With these p a r a m e t e r s absolute predictions a r e made for the differential c r o s s sections for the other r e actions. Agreement with experiment is reasonable. The polarization p a r a m e t e r P(t) is presented. Good agreement is obtained for Itl < 03
1. INTRODUCTION In two r e c e n t p a p e r s [1] t h e O-½ + ~ O-½ + c h a r g e - e x c h a n g e r e a c t i o n s ~ - p -~ ~On, ~ - p ~ TOn, K - p ~ KOn a n d K+n ~ KOp w e r e w e l l e x p l a i n e d u s i n g R e g g e i z e d U(6) ® U(6) ® 0(3) a n d a b s o r p t i v e c o r r e c t i o n c u t s in t h e e n e r g y r a n g e 5.0 < P~ab "-< 1~.2 G e V / c . T h e c a l c u l a t i o n s h a v e now b e e n e x t e n d e d to O-½ +-~ O - ~ + h y p e r c h a r g e - e x c h a n g e r e a c t i o n s . In r e f . [1] t h e only s i g n i f i c a n t c o n s t r a i n t i m p o s e d by t h e h i g h e r - s y m m e t r y s c h e m e w a s in f i x i n g t h e r a t i o of t h e h e l i c i t y f l i p to t h e h e l i c i t y n o n flip amplitude. The baryon-baryon vertex involved both D- and F-couplings, t h e r a t i o of w h i c h w a s not f i x e d by SU(3) but h a d to b e f i x e d by s o m e h i g h e r s y m m e t r y s c h e m e . S i n c e t h e r e w a s only one t y p e of b a r y o n - b a r y o n v e r t e x , n a m e l y pn, t h e D / F r a t i o c o u l d b e e f f e c t i v e l y a b s o r b e d into t h e r e s i d u e s . T h u s , t h e r e a c t i o n s w e r e s i m p l y r e l a t e d by SU(3). S i n c e t h e r e a r e two t y p e s of b a r y o n - b a r y o n v e r t i c e s , n a m e l y n u c l e o n - A a n d n u c l e o n - ~ , f o r t h e hypercharge-exchange reactions considered here, the D / F ratios cannot s i m p l y b e a b s o r b e d into t h e r e s i d u e s . T h e U(6) ® U(6) ® O(3) s y m m e t r y scheme thus provides a significant constraint. In s e c t . 2 we o u t l i n e t h e f o r m a l i s m f o r O - -~1 + - ~ O - -~1 + r e a c t i o n s in t e r m s of t h e t - c h a n n e l M - f u n c t i o n s a n d t h e s - c h a n n e l h e l i c i t y a m p l i t u d e s a n d p r e $ Science R e s e a r c h Council R e s e a r c h Fellow.
382
t~.A. COLLINS et al.
sent the R e g g e i z e d S u p e r m u l t i p l e t a m p l i t u d e s . Sect. 3 contains a b r i e f d i s c u s s i o n of the r e s u l t s of the model as applied to the r e a c t i o n s
(i)
n - p ~ KOA(GO) ,
(ii)
u - p ~ KOA , K - n -~ n - A
(iii) (iv) (v) (vi)
,
K - n -~ ~ - F~° ,
~+p -~ K+E+
K-p ~ ~-E +
2. FORMALISM F o r r e a c t i o n s of the type -I+__ o
-I+
O~
,
the M-function [2] is
M=A+~B
,
w h e r e A and B a r e the i n v a r i a n t a m p l i t u d e s and Q is half the s u m of the initial and final m e s o n f o u r - m o m e n t a . We introduce the i n v a r i a n t a m p l i tude A' defined by
A'=A+
Ela b + t/4m 1
1- t / 4 m 2
B,
w h e r e E l a b is the lab. e n e r g y of the incoming m e s o n , t is the f o u r - m o m e n t u m t r a n s f e r , and m 1 is the m a s s of the t a r g e t nucleon. The independent s - c h a n n e l helicity a m p l i t u d e s [3] a r e defined by
1 c o s ½ 0 { m l A + ( E l W - m21)B}, 4~++ - 4nw 1 sin½0{E1A+ml(W - E1)B } , ~ +- = 4nw w h e r e 0 is the s - c h a n n e l c.m. s c a t t e r i n g angle, w is the c.m. e n e r g y and E1 is the c . m . e n e r g y of the t a r g e t nucleon. In the U(6) ® U(6) ® O(3) c l a s s i f i c a t i o n s c h e m e [4] the i n v a r i a n t a m p l i tudes A' and B a r e
A'= (1+ 4-~)gF
I f l _ h F r ( 1 - a_)(1- e - i n a - ) ( s - m
2-2mp/~2+½t) c~-
1
+ $+hDF(I_ 0%)(1+e-i~'o~+) (s-m ~2- i.t2+½t)a+
HYPERCHARGE-EXCHANGE REACTIONS
B= p-1 (1+ ~)(1-
× (s-
4-~)
383
gD+~FI~-hF F(1- a_)(1-e -i~a-)
m 2 - tl 2 + ½ t ) a - - 1 2m~+ fi+hDr(1 _ a+)(1 + e - ~ + )
( s - m2--2mp~2+½t)a+-l]'
w h e r e the g and h a r e b a r y o n - b a r y o n - m e s o n and m e s o n - m e s o n - m e s o n couplings, m and p a r e the m a s s e s a s s o c i a t e d with the (56, 1: 0) and (6, 8;0) m u l t i p l e t s and the ± r e f e r to the even and odd s i g n a t u r e t r a j e c t o r i e s , r e s p e c t i v e l y . The G e l l - M a n n ghost-killing m e c h a n i s m is employed. P a i r w i s e e q u a l - m a s s k i n e m a t i c s a r e a s s u m e d in this derivation. The helicity a m p l i t u d e s qS++ and qS+_ a r e expanded in p a r t i a l - w a v e s e r i e s , the a b s o r p t i v e c o r r e c t i o n s applied using the technique outlined in r e f . [ 1], and the p a r t i a l - w a v e s e r i e s then r e s u m m e d to give the modified helicity Y a m p l i t u d e s ¢ ++ and q~+_. We take a G a u s s i a n f o r m for the non-flip s c a t t e r i n g amplitude ( a s s u m e d the s a m e in the initial and final states) S J = 1- c e - J ( J + l ) / u 2 p 2 , w h e r e J is the angular m o m e n t u m , p is the c.m. t h r e e - m o m e n t u m of the initiai p a r t i c l e s , and u is the e l a s t i c r a d i u s of i n t e r a c t i o n of the p a r t i c l e s in the initial state. In t e r m s of the modified helicity a m p l i t u d e s , the differential c r o s s s e c tion and the p o l a r i z a t i o n a r e I ~gg d~ ~ , 2 I m [qS++~+_]
a~ = ~ [1~++12+ 1~--]2] ,
P(t) : 1¢):~+]2+ I~LI2 •
3. DISCUSSION AND RESULTS 0 - -~1 + ~ 0 - - 1~+ h y p e r c h a r g e - e x c h a n g e r e a c t i o n s have been t r e a t e d in the conventional (no-cut contribution) Regge a p p r o a c h by Salin [5] and by R e e d e r and S a r m a [6] with r e a s o n a b l e s u c c e s s . As d i s c u s s e d in ref. [6] the only Regge poles which can be exchanged in the t - c h a n n e l a r e a s s o c i a t e d with the K*(890) ( J P = 1-) and KN(1420 ) ( J P = = 2 +) m e s o n s . The d i f f e r e n t i a l c r o s s - s e c t i o n data with the l a r g e s t v a r i a tions in s and t is for the r e a c t i o n ~ - p ~ K°A(~. °) (ref. [7]). We t h e r e f o r e fix the residue/3(t) and the t r a j e c t o r y function a(t) for both of t h e s e t r a j e c t o r i e s f r o m a X2 fit of the a b s o r b e d differential c r o s s section to this e x p e r imental data, and then p r e d i c t the 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 and p o l a r i z a t i o n s for the other r e a c t i o n s . Following ref. [ 1] we took the r e s i d u e s to be c o n s t a n t s and p a r a m e t r i z e d the t r a j e c t o r y functions for negative t by a(t) = a o+c~ l e ~ 2 t .
384
P . A . C O L L I N S et al.
The conventional linear parametrization e r e d in t h e p e r i p h e r a l r e g i o n
of a R e g g e t r a j e c t o r y i s r e c o v -
a(t) ~ (ao+ ~1) + ( c q a 2 ) t • U s i n g t h i s f o r m t h e R e g g e t r a j e c t o r i e s w e r e e x t r a p o l a t e d s o a s to p a s s through their respective poles, thus providing additional constraints. Alt h o u g h t h i s e o n s t r a i n t w a s not i m p o s e d f o r t h e P a n d A 2 t r a j e c t o r i e s in r e f . [1], t h e t r a j e c t o r i e s w e r e f o u n d to p a s s s u r p r i s i n g l y c l o s e t o t h e i r r e s p e c t i v e p o l e s . We t h e r e f o r e f e e l t h a t t h i s i s not a n u n n a t u r a l e o n s t r a i n t . H e n c e we h a v e t h r e e f r e e p a r a m e t e r s f o r e a c h p o l e , n a m e l y a c o n s t a n t r e s i d u e a n d t w o t r a j e c t o r y p a r a m e t e r s , g i v i n g a t o t a l of s i x f r e e p a r a m e ters. These were determined using MINUITS (CERN Program Library, n u m b e r D506). T h e D - a n d F - c o u p l i n g s f o r t h e p r o e e s s e s c o n s i d e r e d a r e s h o w n in t a b l e 1, a n d t h e e l a s t i c s c a t t e r i n g c o e f f i c i e n t s a r e s h o w n in t a b l e 2 ( s i n c e no d a t a i s a v a i l a b l e f o r t C n d i f f e r e n t i a l c r o s s s e c t i o n s we h a v e u s e d t h e K - p c o e f f i c i e n t s ) . T h e r e s u l t s of t h i s m i n i m i z a t i o n p r o c e d u r e a r e s h o w n in t a b l e 3. Table 1 D- and F - c o u p l i n g s . Baryon vertex Reaction
K*(890) and KN(1420 ) D + ~F
-p -~ KO A
?r-p -* K°~ K-n
°
--' rr-A
F
Meson vertex KN(1420 )
K*(890)
D-type
F-type
- V5
-45
~
-C5
1/3
-1
~/2
-'/-2
- ,/3
-if3
~/2
~/2
K-n ~ rr-E °
-1/3
1
~/2
~/2
#+p -* K+E +
ff-2/3
-V~
~-
-V~
K - p ~ 7r-E +
ff-2/3
- ~/2
~/2
~/2
A C h e w - F r a u t s c h i p l o t i s p r e s e n t e d in fig. 1. We n o t e t h a t t h e K*(tt90) a n d KN(1420 ) t r a j e c t o r i e s a r e a l m o s t p a r a l l e l a n d n e a r l y e x c h a n g e d e g e n e r a t e , a l t h o u g h t h i s w a s not i m p o s e d a s a c o n s t r a i n t in t h e d a t a f i t t i n g . W e now d i s c u s s e a c h r e a c t i o n s e p a r a t e l y : (i)
~ - p -~ K ° A ( E °) •
T h e m a s s r e s o l u t i o n of t h e r e c o i l h y p e r o n in t h i s e x p e r i m e n t w a s s u c h t h a t t h e A a n d E ° c o u l d not b e d i s t i n g u i s h e d . S i n c e t h e s e a r e in p r i n c i p l e distinguishable reactions d~ d(r dcr dr- (Tr-p --, K ° h ( E ° ) ) = d r - ( r r - p --' K°A) + ~ - ( r r - p --' K ° E °) .
385
HYPERCHARGE-EXCHANGE REACTIONS Table 2 Absorption coefficients. Channel
Plab
v-I(GeV-1)
c
7T+p
3.00 3.25 4.00 5.05 5.40 7.00
0.27 0.27 0.27 0.27 0.27 0.27
0.89 0.87 0.84 0.82 0.81 0.79
7r-p
6.00 7.91 8.00 10.00 11.20
0.26 0.26 0.26 0.26 0.26
0.79 0.76 0.76 0.74 0.73
K-p
4.07 4.25 5.47
0.26 0.26 0.26
0.78 0.77 0.73
Table 3 Regge p a r a m e t e r s . Trajectory
K* (890)
KN(1420)
oto
-0.829
-0.984
~1
1.098
1.177
or2 (GeV/c) -2
0.840
0.761
fl(GeV/c)-I
-7.162
4.712
No. of data points X2
76 65
A s d i s c u s s e d a b o v e t h i s r e a c t i o n w a s u s e d to d e t e r m i n e the Regge p a rameters. F i g . 2 s h o w s the e n e r g y d e p e n d e n c e of the 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 . T h e s - a n d t - d e p e n d e n c e of the d a t a a r e e x t r e m e l y well r e p r e s e n t e d . H a v i n g d e t e r m i n e d the R e g g e p a r a m e t e r s we c a l c u l a t e d t h e f o l l o w i n g r e a c t i o n s a n d c o m p a r e d the p r e d i c t i o n s with e x p e r i m e n t . (ii) ~ - p - * K°A. D a t a on t h i s e x p e r i m e n t e x i s t s at 7.91 G e V / c [8]. T h e 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 is s h o w n i n fig. 3. We s e e that the d a t a is w e l l r e p r e s e n t e d b y the t h e o r y f o r a l l v a l u e s of t. T h e r e is p o l a r i z a t i o n d a t a at 6.0 G e V / c [9]. A s s h o w n in fig. 4, f o r I tl < 0.3 ( G e V / c ) 2 t h e o r y a g r e e s with e x p e r i m e n t , but f o r > 0.3 the t h e o r y f a i l s t o fit the d a t a , i n t h a t we o b t a i n + 0 . 1 f o r t = - 0 . 3 5 ( G e V / c ) 2 w h e r e a s e x p e r i m e n t i s c o n s i s t e n t with p o l a r i z a t i o n b e t w e e n 0. a n d - 1.0.
Itl
386
P.A.COLLINS et al. 7.0 6.0
5.0 4.0 3.0 R e a l o=(t)
2.0 1.0
t"
0.0
-1.0 -2":3.o
'
-;.o
'
-,:o
'
o.o
'
I.'o
'
21o
'
3.0
t r(c.vlc~ 3
Fig. 1. Plot of ~(t) against t for the K*(890) trajectory ( ) and the KN(1420) tra. ' jectory ( . . . . ). Parameters from table 3. 100
. . . . . . . . . . . . . . 't["p~ KOA{,r o )
101
~
10'
rl
162 ~
11.2 GeV~:
1." . . . . . . 04)0
'
'
'
.50
t
. . . .
.
~
I.(30 . . . .
1.50
- t EfGWIc )z'l
Fig. 2. Differential cross section for ~-p ~ K°A(~°). Data from ref. [7].
HYPERCHARGE-EXCHANGE REACTIONS
1
387
ffp--K°A at 7.91 GeVIc
f T\
K~3/ . . . . . . . . . . . . . . 0.00 .50 1.00 -t r'( GeVic)2"l
1.50
Fig. 3. Differential cross section for ?r-p ~ K ° A . Data from r ef. [8].
~'p--,-K°^ at 6.0 C,eV/c
0.50
Z 0
N_ n,
-1.00 0.50 tOO -t C(GeVIc )2:]
Fig. 4. Polarization for 7r-p -~ K°A. Data from ref. [9]. (iii) K - n ~ v - A . T h e t h e o r e t i c a l p r e d i c t i o n f o r the d i f f e r e n t i a l c r o s s s e c t i o n for t h i s r e a c t i o n is p l o t t e d a g a i n s t the e x p e r i m e n t a l d a t a at 4.25 G e V / c [10] i n fig. 5. T h e n o r m a l i z a t i o n in the f o r w a r d d i r e c t i o n i s r e p r o d u c e d , b u t the t h e o r y p r e d i c t s too m u c h s c a t t e r i n g f o r l a r g e t. In fig. 6 the t h e o r e t i c a l a n d e x p e r i m e n t a l p o l a r i z a t i o n s [ 10] a r e c o m p a r e d . T h e t h e o r y d o e s not r e p r e s e n t the data.
388
P.A.COLLINS '
I
.
.
.
.
et al. i
,
,
•
,
[
K-n~"E-A at 4.25 GeV/c
101 I"I
.o E
u
1152
.
i
,
F i g . 5. D i f f e r e n t i a l c r o s s
I
,
,
,
i
~0
I - t C(C_~//c)2 D
i
I i
.
I
1.50
,
s e c t i o n f o r K - n - * 7r-A. D a t a f r o m r e f . [ i 0 ] ~n..,,~
ot &.25 C~k:
Z O
N
0D0 F i g . 6. P o l a r i z a t i o n
0.50 1.00 1.50 -t E(6eWct 3 f o r K - n -~ 7r-A. D a t a f r o m r e f . [10].
HYPERCHARGE-EXCHANGE REACTIONS
10(
,
,
,
i
I
.
.
.
.
i
.
.
K-rt~
.
.
389
i
tr'~ ° Qt 4.25GeV/c
u
E U
10
16: . . . . . . . . 0.00 0.50
~ j 1.00
, , 1.5O
-t E (GeV/c)2~
F i g . 7. D i f f e r e n t i a l c r o s s s e c t i o n f o r K - n - * 7r-S ° . Data f r o m r e f . [10].
tOO
.
.
.
.
i
,
,
,
050
•
i
,
•
,
,
!
K-n - - tY'L"° at 4.25GeVIc
~oo N
-050
--1.0(
,
0.00
.
,
,
i
,
0.50
,
,
,
I
.
-t
E( GeV/c)2 ~
1,00
,
.
.
i
,
1.50
F i g . 8. P o l a r i z a t i o n f o r K - n --* 7r-S ° .
•
390
P . A . COLLINS et al.
(iv) K - n ~ ~ - ~ o . Fig. 7 s h o w s a c o m p a r i s o n of the r e c e n t d a t a [10] on the d i f f e r e n t i a l c r o s s s e c t i o n f o r t h i s r e a c t i o n at 4.25 G e V / c with the p r e d i c t i o n of this m o d e l . T h e a g r e e m e n t is m o s t e n c o u r a g i n g . Both the f o r w a r d n o r m a l i z a tion and the t - d e p e n d e n c e a r e well r e p r o d u c e d . T h e p o l a r i z a t i o n d i s t r i b u t i o n f o r which no d a t a e x i s t s at p r e s e n t is shown in fig. 8. (v) ~+p ~ K+~ +. T h e h i g h - e n e r g y e x p e r i m e n t a l d a t a on dg/dt f o r t h i s r e a c t i o n h a s b e e n m e a s u r e d by two e x p e r i m e n t a l g r o u p s [11, 12]. In fig. 9 the r e s u l t s of o u r c a l c u l a t i o n s a r e c o m p a r e d with the d a t a of ref. [ 11]. Both the n o r m a l i z a t i o n and t - d e p e n d e n c e of the m o d e l a r e c o n s i s t e n t with the data. A plot of the 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 and the c o r r e s p o n d i n g e x p e r i m e n t a l d a t a of ref. [12] is shown in fig. 10. We o b s e r v e that in this c a s e we do not obt a i n the c o r r e c t n o r m a l i z a t i o n . T h e r i g h t t - d e p e n d e n c e is o b t a i n e d out to about t = - 0 . 4 ( G e V / c ) 2, but b e y o n d this t h e r e a p p e a r s to be s o m e s t r u c t u r e which is not r e p r e d u c e d by our m o d e l . Since m u c h of this d a t a is at low ene r g y , and t h i s s t r u c t u r e is not p r e s e n t at the h i g h e s t a v a i l a b l e e n e r g y , we do not take this d i s a g r e e m e n t in t - d e p e n d e n c e too s e r i o u s l y . In figs. 11-13 the p o l a r i z a t i o n p r e d i c t i o n s a r e c o m p a r e d with the a v a i l able d a t a [11-13]. ~Figs. 11 and 13 s h o w that the p o l a r i z a t i o n is s m a l l and n e g a t i v e f o r it I < 0.3. H o w e v e r , b e t w e e n t = - 0 . 3 and t = - 0 . 5 the data
11"*p ~ K" l;'at 5.4 GeVIc
tt n
E
16" 0.0o
'
'
J
'
i~0'
0
J
'
J
I
J
I.O0 - t t-(GeVlc) 2 3
i
I
I
I
I
I
1.50
F i g . 9. D i f f e r e n t i a l c r o s s s e c t i o n f o r ~+p ~ K+L~+. Data f r o m r e f . [11].
HYPERCHARGE-EXCHANGE REACTIONS
391
zr'p - - K"£"
3.0 Ge~:
OOO~
O
O
3.25 C-~//c
0
o
O
w
4.0 GW/c
I-J
0
0
0
0
o
o
o
5.05
0
0
GeVIc
0
9
7.0 GeV/k:
16'I
o.oo
.~o
. . . . ,:oo . . . . -t r'(GeVlc) z D
,.so
Fig. I0. Differential cross section for ~+p -~ K+~+. Data from ref. [12]. shows a d r a m a t i c change to a l a r g e positive value of about 0.7 at all e n e r gies, and this p e r s i s t s to l a r g e r values of t. Our model c o r r e c t l y r e p r o duces the p o l a r i z a t i o n for I tt < 0.3, which c o r r e s p o n d s to the f o r w a r d peak region in the differential c r o s s section, but beyond this the model f a i l s (vi) K-p -~ ~-~+. Our prediction of the differential cross section for this reaction is compared with the available experimental data [14, 15] in fig. 14. As in the previous reaction (v), we fail to obtain the correct normalization, although the t-dependence is reasonable. Since reactions (v) and (vi) are related by charge conjugation at the meson vertex, the K*(890) contribution simply changes sign from one reaction to the other. However, the K*(890) residue is approximately twice the KN(1420) residue and therefore this change in
lr’p-
-1.001
Fig.
11.
3
8
Polarization
1.50 -
1.00 -
a50 -
p
’
*
K’ C*at
3.23
1 ’ ’ 0 ’ ’ ’ 0.50 1.OQ -t C(G@.‘/C)~~
for 7~+p---) K%+.
*
G&c
’
’
1.50
Data from
ref.
[13].
+-I U’P-K’Z’at
5.4 GeVl
ifi EOOOT 5’ 8
-0.50 -
-1.00 ’
1 0.50 -t C(GcVk
Fig.
12.
Polarization
I 1.00 ,‘3
for ?‘r+p + K%+.
Data from
ref.
[ll].
HYPERCHARGE-EXCHANGE REACTIONS 1.2
,
'
,
. . . . . .
393 i
. . . . .
1.0 .8
.L .i "0 |
[ T T
1
C
N -1.0
1.o~ .....
I
'~ .........
'f: ,,!!! t 1 I =1
_1;i
i
0
A
i
i
I
0.50
L
i
i
z
i
,
1.00
,
,
,
-1.0
i
. . . .
1.50
' . . . . 0.50
I , 1.00
~
~
•
I , 1.50
,
:t0 .... ti .......... 12 ~4
1.
.0O
0.50 -t
1.00 E(GeVIv ) 2-j
1.50
Fig. 13. Polarization for ~'+p -~ K+~,+. Data from ref. [12]. sign p r o d u c e s little difference in the f o r w a r d n o r m a l i z a t i o n s of t h e s e two reactions. Fig. 15 shows the p r e d i c t i o n for the p o l a r i z a t i o n for r e a c t i o n (vi). No h i g h - e n e r g y data is available at p r e s e n t . We note that in the c h a r g e - e x c h a n g e r e a c t i o n s of ref. [ 1] the data indic a t e s the helicity flip amplitude dominates in the f o r w a r d direction giving a t u r n o v e r , w h e r e a s in the h y p e r c h a r g e c a s e the helicity non-flip amplitude dominates giving a f o r w a r d peak. This f e a t u r e is c o r r e c t l y r e p r e s e n t e d by the U(6) ® U(6) ® O(3) s y m m e t r y scheme. Within the confines of our model, we feel that the a g r e e m e n t between theory and e x p e r i m e n t is s a t i s f a c t o r y . The conventional Regge a p p r o a c h of R e e d e r and S a r m a [6] contained many m o r e p a r a m e t e r s than the p r e s e n t work. In p a r t i c u l a r , they had r e s i d u e s c o r r e s p o n d i n g to v e c t o r and t e n s o r exchange, helicity flip and non-flip and A and ~, r e a c t i o n s , a total of eight
394
P. A. COLLINS et al. 100
.
.
.
.
i
.
.
.
.
i
.
.
.
.
K-p-- ~1:"
td
~
.07 G~/c
5.47 GeVk:
t-t
E U
8.0 GeV/c
16.0 GeVlc
(3.00
.50
1.00
1.50
-t E(GeV/c)23 Fig. 14. Differential c r o s s section for K-p --o ?r-E +. Data from refs. [14, 15]. residue parameters. The 'so-called' scale parameters for vector and tensor exchange and helicity flip and non-flip were also included as parameters, adding four more. A 'turnover' parameter was included to obtain the d r a m a t i c c h a n g e in t h e ?r-p -o K+E+ p o l a r i z a t i o n at ! = - 0 . 4 ( G e V / c ) 2. H o w e v e r t h e i r t r a j e c t o r i e s w e r e not f r e e in t h a t t h e i r K*(890) t r a j e c t o r y w a s o b t a i n e d f r o m t h e i r p - t r a j e c t o r y a n d t h e i r KN(1420 ) t r a j e c t o r y f r o m t h e i r A 2 t r a j e c t o r y b y a d i s p l a c e m e n t e q u a l to t h e d i f f e r e n c e of t h e s q u a r e s of t h e r e s p e c t i v e m a s s e s . T h e F/D r a t i o f o r t h e v e c t o r a n d t e n s o r e x c h a n g e s and helicity flip and non-flip were effectively parameters, adding another f o u r f r e e p a r a m e t e r s . T h i s w a s a t o t a l of s i x t e e n p a r a m e t e r s a n d g a v e a g o o d r e p r e s e n t a t i o n of t h e d a t a . C o n s i d e r i n g t h e f a c t t h a t we h a v e o n l y s i x f r e e p a r a m e t e r s w e c o n s i d e r o u r f i t to b e a n a c c e p t a b l e r e p r e s e n t a t i o n of the data. Discrepancies between our model and the data could possibly be improved by different final- state absorption coefficients. However, these c a n n o t b e d e t e r m i n e d , a n d we f e e l t h a t l i t t l e i n s i g h t into t h e p h y s i c s i s to
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be gained by making these p a r a m e t e r s . Another possibility is that exact SU(3) s y m m e t r y does not hold for the couplings. We o b s e r v e that in the work of Salin [5] and R e e d e r and S a r m a [6] the D/F r a t i o obtained is significantly different f r o m the SU(6) prediction of 2. This may be another s o u r c e of trouble. P r e l i m i n a r y application of this model to the r e a c t i o n yp ~ ~-A++ has shown that exact k i n e m a t i c s is important. P a i r - w i s e equal m a s s - k i n e m a t ics with the U(6) ® U(6) ® O(3) p r e d i c t e d r a t i o of helicity amplitudes leads to a pion f o r w a r d peak r a t h e r than the t u r n o v e r exhibited by the data. However, exact k i n e m a t i c s r e p r o d u c e s this f e a t u r e of the data. We t h e r e f o r e feel that the effect of exact k i n e m a t i c s may be significant for these h y p e r charge reactions. It m u s t also be b o r n e in mind that t h e r e a r e n o r m a l i s a t i o n e r r o r s in the e x p e r i m e n t a l data, typically of the o r d e r of 20-30%, which could account for some of our n o r m a l i z a t i o n difficulties. However, we see no point in r e adjusting the data within the stated n o r m a l i z a t i o n e r r o r s in o r d e r to improve the a p p e a r a n c e of our fits. We a r e generally able to account for the p o l a r i z a t i o n data for It l < 0.3 (GeV/c) 2, but beyond this the e x p e r i m e n t a l data undergoes a d r a m a t i c change which we cannot explain. Since p o l a r i z a t i o n is a subtle effect this m a y be explainable in t e r m s of the h i g h e r - o r d e r c o r r e c t i o n s to the amplitudes [ It has been suggested [1'/] that the d r a m a t i c change in p o l a r i z a 16t" tion at It ~ 0.4 (GeV/c) 2 can be aeeounted for within the context of the Reggeized a b s o r p t i o n model by using a moving P o m e r a n c h u k pole r a t h e r than a fixed J = 1 pole as we have done. F o r the n u m e r i c a l work we employed an exact p a r t i a l - w a v e s u m m a t i o n
396
P . A . COLLINS et al.
u s i n g 20 p a r t i a l w a v e s c a l c u l a t e d with a 4 8 - p o i n t G a u s s i a n q u a d r a t u r e . T h e s e c a l c u l a t i o n s w e r e r e p e a t e d u s i n g 30 p a r t i a l w a v e s a n d a 5 1 2 - p o i n t G a u s s i a n q u a d r a t u r e . T h e r e s u l t s w e r e f o u n d not to c h a n g e s i g n i f i c a n t l y . We w i s h to t h a n k P r o f e s s o r P. T. M a t t h e w s f o r e n c o u r a g e m e n t i n t h i s w o r k a n d f o r a c r i t i c a l r e a d i n g of the m a n u s c r i p t , D. D a l l m a n f o r c o m p i l i n g the d a t a , R. E h r l i c h f o r v e r y k i n d l y g i v i n g u s h i s d a t a in t a b u l a r f o r m , a n d R. C. B e c k w i t h f o r a s s i s t a n c e i n c o m p u t i n g . One of u s (B.J.H.) w i s h e s to t h a n k the U n i v e r s i t y of L o n d o n f o r t h e a w a r d of the G e o r g e W i l l i a m B r i t t ( J u n i o r ) S t u d e n t s h i p a n d two o t h e r s ( P . A . C . a n d R.W.M.) w i s h to t h a n k the Science Research Council for Research Studentships.
RE F E RE NCE S [1] B. J. Hartley, R.W. Moore and K. J. M. Moriarty, Phys. Rev. 187 (1969) 1921; D1 (1970) 954. [2] R.J. Eden, High-energy collisions of elementary particles (Cambridge University P r e s s , 1967). [3] M.Jacob and G.C.Wiek, Ann. of Phys. 7 (1959) 404. [4] R.Delbourgo, A.Salam and J.Strathdee, Phys. Rev. 179 (1968) 1487; 172 (1968) 1727; 186 (1969~ 1516. [5] Ph. Salin, Nucl. Phys. B3 (1967) 323. [6] D . D . R e e d e r and K . V . L . S a r m a , Phys. Rev. 172 (1968) 1566. [7] E. Bertolueei, I. Mannelli, G. Pierazzini, A. Seribano, F. Sergiampietri, M.L. Vincelli, C. Caverzasio, J . P . G u i l l a u d , L. Holloway and M.Yvert, Nuovo Cimento Letters 2 (1969) 149. [8] R . E h r l i c h , W.Selove and H.Yuta, Phys. Rev. 152 (1966) 1194. [9] D . J . C r e n n e l l , G.R.Kalbfleisch, K . W . L a i , J . M . S c a r r , T.G.Schumann, I.O. Skillicorn and M.S.Webster, Phys. Rev. Letters 18 (1967) 86. [10] W . L . Y e n , A . C . A m m a n n , D.D. Carmony, R . L . E i s n e r , A . F . G a r f i n k e l , L . J . Gutay, R.V. Lakshmi, D.H. Miller and G. W. Tautfest, Phys. Rev. Letters 22 (1969) 963. [11] W.A.Cooper, W.Manner, B.Musgrave and L.Voyvodic, Phys. Rev. Letters 20 (1968) 472. [12] S . M . P r u s s , C.W.Akerlof, D . I . M e y e r , S.P.Ying, J . L a l e s , R.A.Lundy, D.R. Rust, C . E . W . W a r d and D.D.Yovanovitch, Phys. Rev. Letters 23 (1969) 189. [13] R.R.Kofler, R.W.Hartung a n d D . D . R e e d e r , Phys. Rev. 163 (1967) 1479. [14] J.S. Loos, U . E . K r u s e a n d E . L . G o l d w a s s e r , Phys. Rev. 173 (1968) 1330. [15] D.Birnbaum, R . M . E d e l s t e i n , N.C.Hien, T.J.McMahon, J . F . M u c c i , J.S. Russ, E . W . A n d e r s o n , E . J . B l e s e r , H.R.Blieden, G . B . C o l l i n s , D.Garelick, J. Menes and F. Turkot, Experimental study of the reactions K-p -* ~-~+ and K-p - ' y-Y*+ at 8 and 16 GeV/c, submitted to the 14th Int. Conf. on high-energy physics, Vienna, 1968. [16] C.Lovelace, Nucl. Phys. B12 (1969) 253. [17] A.Krzywicki and J . T r a n Thanh Van, Phys. Letters 30B (1969) 185.