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The ρ + R Regge-pole model in the reaction K− + p → K0 + n
-~
Nuclear Physics B3 (1967) 628-636. North-Holland Publ. Comp., Amsterdam
THE o + R REGGE-POLE MODEL I N T H E R E A C T I O N K" + p - - ~ K ° + n A. D E R E M * and G. SMADJA
D~partement de Physique des Particules El~menlaires, CEN-Saclay Received 2 October 1967
Abstract: It is shown that the high-energy data for the K-p charge-exchange s c a t t e r ing are in agreement with the p * R Regge-pole model of Phillips and Rarita. A new determination of the parameters of the model is given. The observed energy dependence of the differential cross sections is well accounted for over a wide range of energies. Evidence is found that the residue functions are all positive at t-:0.
P h i l l i p s and R a r i t a [1] s u g g e s t e d that two R e g g e p o l e s m i g h t d o m i n a t e the t - c h a n n e l of the K - p c h a r g e - e x c h a n g e s c a t t e r i n g at high e n e r g y : the p o l e s of the p (/(7 = 1 +, j P = 1-) and of the A 2 (/(7 = 1 - , j k - = 2+; a l s o c a l l e d R - p o l e ) . It i s the p u r p o s e of t h i s p a p e r to show that the p + R m o d e l [2] is c o n s i s t e n t with the n e w h i g h - e n e r g y CERN d a t a [4]. E x t r a p o l a t i o n of the m o d e l to l o w e r e n e r g i e s is m a d e . A new d e t e r m i n a t i o n of the p a r a m e t e r s of the m o d e l i s given. T h e s e p a r a m e t e r s a r e s t r o n g l y t i e d to t h o s e w h i c h a r e u s e d to d e s c r i b e the h i g h - e n e r g y r e a c t i o n s 7r- + p - , ~ o + n and 7r- + p ~ 7}o + n. As a c o n s e q u e n c e , we i n d e e d o b t a i n e d an i n d i c a t i o n of a g l o b a l a g r e e ment between e x p e r i m e n t a l differential c r o s s sections and Regge models f o r the t h r e e r e a c t i o n s ~- + p ~ o + n, ~ - + p __.wo + n and K- + p _ ~ o + n and for s q u a r e d m o m e n t u m t r a n s f e r 'tl -<.0.85 ( G e V / c ) 2. T h e f o r m a l i s m w i l l be b r i e f l y s k e t c h e d . T h e 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 ~N and KN s c a t t e r i n g i s w r i t t e n :
d~_ dt
1 ( 16n
) ~
1-
IA'I 4m 2
+ -.
4m 2
(s
) iB: 2 1-t/4m 2
(1)
in t e r m s of the two t - c h a n n e l a m p l i t u d e s A'(s, t) and B(s, t), as d e f i n e d by Singh [5]; s and - t a r e the s q u a r e s of c . m . e n e r g y and m o m e n t u m t r a n s f e r , m is the n u c l e o n m a s s , kla b and Wla b a r e the m o l . l e n t u m and e n e r g y in the l a b s y s t e m of the i n c i d e n t m e s o n . N a t u r a l u n i t s (/~ = c = 1) a r e u s e d . T h e c o n t r i b u t i o n s f r o m the p o l e s of the p and the A 2 to A ' a r e w r i t t e n :
* Chercheur agrdg IISN, Bruxelles.
REGGE-POLE MODEL
629
A' = - v~ A ; ,
(2)
A' : - A{~,
(3)
A. = 2 A K -
2A K,
(4)
f o r the r e a c t i o n s ~ - + p - ' n o + n , n - + p ~ ? o + n a n d K - + p - - * ~ ° + n , r e s p e c t i v e l y * . S i m i l a r r e l a t i o n s e x i s t f o r the B - a m p l i t u d e s . The f a c t o r i z a t i o n p r i n c i p l e [6] e n a b l e s to link c o n t r i b u t i o n s a r i s i n g f r o m m e s o n e x c h a n g e s in d i f f e r e n t r e a c t i o n s in the following f o r m :
Ap/ApK, :
= P(O ,
(5)
K AR/A R = B~R/B~ = Q(t) ,
(6)
w h e r e P(t) and Q(t) a r e r e a l f a c t o r i z a t i o n functions. F o r the a s y m p t o t i c f o r m of t h e s e a m p l i t u d e s , we have
A~p(S, l) = C~(t) (
A~(s, t) = C~(t)
Wlab )ap(t) 1 - e-i~aP (t) ~ sin .ap(t)
,
(7)
Wla b . aR(t) ( ~ ) 1 sin + e-i~aR(t),oR(t )
(8)
w h e r e ap(t) and aR(t ) a r e the t r a j e c t o r i e s of the two R e g g e p o l e s ; the s c a l ing f a c t o r Wo is c o n v e n i e n t l y c h o s e n equal to 1 GeV. E q u a t i o n s s i m i l a r to (7) and (8) r e l a t e B~(s, t) and B~R(S, t) to D~(t) and DE(t), except f o r the e x p o n e n t of Wlab/W o which is r e p l a c e d by ax(t ) - 1 (x s t a n d s f o r p o r R). The f u n c t i o n s C(t) and D(t) a r e r e a l f o r n e g a t i v e t and p r o p o r t i o n a l to the r e s i d u e s at the p o l e s of A' and B. R e s i d u e f u n c t i o n s can a l s o be defined f o r K- + p __.~o + n, u s i n g r e l a t i o n s (5) and (6):
cK(t) : P(t) C~)(t) ,
(9)
cK(t) = V(t) C~(t) ,
(10)
and s i m i l a r l y f o r D~(t) and DK(t). Now we t u r n to a d i s c u s s i o n about the v a r i o u s f u n c t i o n s i m p l i e d in the R e g g e - p o l e f o r m a l i s m . We a r e left with f o u r r e s i d u e f u n c t i o n s : C~(t), D~p(t), cK(t) and DRK(t), and two t r a j e c t o r i e s : ~ ( t ) and aR(t ). We f i r s t d e * For y- + p--~ r]° + n. elastic kinematics is then used; ~ - . mass difference effects are considered unimportant at high energy; this point is discussed in a recent paper by Phillips [17].
630
A. DEREM and G. SMADJA
fine the following empirical parametrization, P h i l l i p s and R a r i t a * :
Cp(t) : C~ no(O) [2no(t) + 1]
a n a l o g o u s to t h e one u s e d by
Fp(t) ,
D~(t) : D~ ap(t) Ao(t) , for~-
+p~°+n,
(11)
and:
C~(t) = C~
aR(t ) [2a,R(t ) +1]
D~(t) : D~
aR(t)
eDl t
e C~t (12)
f o r u - + p ~ 77° + n. T h e n w e p r o c e e d to i m p o s e a c e r t a i n n u m b e r of r e s trictions: (a) The amplitudes C~(t) and D~(t) c a n b e known f r o m ~ - + p ~Tr ° + n at h i g h e n e r g y . H e r e we h a v e a d o p t e d p a r t i c u l a r s o l u t i o n s [8] c h a r a c t e r i z e d by a z e r o of Fp(t) at t ~ - 0 . 1 5 ( G e V / c ) 2 ( s e e fig. 1). T h i s t y p e of s o l u t i o n w i t h a z e r o of C~(t) r e c e i v e s s t r o n g s u p p o r t f r o m two h i g h - e n e r g y f e a t u r e s : (i) it o f f e r s a n a t u r a l e x p l a n a t i o n of t h e c r o s s o v e r e f f e c t o b s e r v e d f o r the ~ - p a n d ~+p e l a s t i c s c a t t e r i n g 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 ; (ii) the v a l u e at t = m ~ of t h e r a t i o of t h e m a g n e t i c to t h e e l e c t r i c i s o v e c t o r f o r m f a c t o r s s t r o n g l y f a v o u r s a z e r o of t h e n o n - s p i n - f l i p r e s i d u e n e a r t = - 0 . 2 ( C e V / c ) 2 [18]. It ~ u s t b e n o t e d t h a t Fp(t) c a n n o t b e u s e d b e y o n d t ~ -1 ( G e V / c ) 2. C~(t) a n d Dr~(t) are now d e t e r m i n e d t h r o u g h t h e f a c t o r i z a t i o n f u n c t i o n P(t). W i t h an e x a c t SU(3) s y m m e t r y , P(t) w o u l d b e c o n s t a n t a n d e q u a l to ½. E x p e r i m e n t a l d a t a c a n t e l l u s l i t t l e a b o u t t h e t - d e p e n d e n c e of P(t), f o r m u c h of t h e v a r i a t i o n of C~(t) and D~(t) s t e m s f r o m t h e f a c t o r s r p ( t ) (with a z e r o at t ~ - 0 . 1 5 ( G e V / c ) ) and ap(t) (with a z e r o at t ~ - 0 . 5 ( G e V / c ) 2 ) . A s f u r t h e r m o r e w e o b t a i n e d g o o d f i t s w i t h P(t) = c o n s t a n t = P o , t h i s w a s t a k e n a s a constraint. (b) /n an analogous fashion, CK(t) and D~(t) w e r e r e l a t e d to t h e c o r r e s p o n d i n g f u n c t i o n s in ~ - + p ~ 7/0 + n [cf. f o r m u l a (6)]. T h i s r e a c t i o n w a s a l r e a d y s u c c e s s f u l l y a n a l y s e d in t h e R e g g e - p o l e a p p r o a c h by P h i l l i p s and R a r i t a [14]. But h e r e C~(t) and D~(t) a r e not s o t i e d to t h e e x p e r i m e n t a l p o i n t s a s a x e C~p(t) and DE(t). S o l u t i o n s in e x p o n e n t i a l f o r m a r e f a r f r o m b e i n g unique. A c c o r d i n g l y we f i t t e d a l t o g e t h e r t h e h i g h - e n e r g y d a t a f o r K - + p ~ ~ o + n a n d ~ - + p ~ ~?o + n, t h u s r e d u c i n g t h e d e g r e e of a r b i t r a r i n e s s : t h e e x p o n e n t i a l s l o p e s of t h e r e s i d u e f u n c t i o n s a r e t i e d t o g e t h e r t h r o u g h t h e f a c t o r i z a t i o n t h e o r e m . E x a c t SU(3) w o u l d g i v e Q(t) = Q(O) = v r 3 / 2 ¢ 2 . In f a c t , v i o l a t i o n of SU(3) w a s p e r m i t t e d t h r o u g h t h e i n t r o d u c t i o n of two p a r a m e t e r s Qo and Q I : * This p a r a m e t r i z a t i o n is perhaps not the best one [1]], but it can be hoped that the global t-dependence of C(t) and D (t) is approximately c o r r e c t .
REGGE-POLE MODEL
631
Ap(t)
~R(t) - t (GeV/(;) 2 0.5
ap
-0.5'
rpct) -I.0.
Fig. 1. Otp(t) and OCR(t) trajectories; the functions Fp(t) and Ap(t) have been adjusted to ~'N charge-exchange scattering.
Q(t) : Qo eQ1 t
(13)
(c) The Regge formalism d e s c r i b e d above is such that A~ and BE have the s a m e phase and hence the p o l a r i z a t i o n of the nucleon is z e r o in the ~-N c h a r g e - e x c h a n g e s c a t t e r i n g . P r o b a b l y the p - p o l e model can be looked upon as a z e r o - o r d e r a p p r o x i m a t i o n [9]: only small background t e r m s a r e to be added to the Regge amplitudes to give a r e a s o n a b l e p o l a r i z a t i o n [7]. It is
632
A. DEREM and G. SMADJA Table 1 Particular solutions used in the fit. •- + p--~T/° + n (rcf. [8])
ap(t)
= 0.57 + 0.98 l - 0.52 t2 - 0.73 t 3
Fp(t) = 1 + 7.63t + 6.53t 2 - 0.02t 3 Ap(t) = 1 + t + 0.69t 2 + 0.16t 3 = 2.03 mb. GeV;
Dpo = 51.7
mb
7:- + p--~U ° + n (ref. [I01)
.
C~R(t) - -1
(I + 0.35) 2 l + 0.35 - 0.46/
hoped the ap(t) t r a j e c t o r y will not be s t r o n g l y m o d i f i e d when d e f i n i t e a c count will be t a k e n f o r the p o l a r i z a t i o n and we u s e d f o r it a p o l y n o m i a l f o r m [8], g i v e n in t a b l e 1. (d) F i n a l l y , we c h o s e f o r aR(t ) the P i g n o t t i f o r m 3b quoted in a p a p e r by P h i l l i p s and R a r i t a [10] (cf. t a b l e 1), w h i c h i s c l o s e s t to J = 2 when t = m2 2. T h i s t r a j e c t o r y is e x t r a c t e d d i r e c t l y f r o m the h i g h - e n e r g y 7r- + p ~ Uo + n d a t a . Fp(t), Ap(t), ap(t) and a~(t) a r e r e p r e s e n t e d on fig. 1. A g l a n c e at f o r m u l a e (10), (12) and (13) s h o w s that s e v e n r e a l p a r a m e t e r s a r e to be a d j u s t e d on the h i g h - e n e r g y d a t a for 7:- + p ~ U° + n and K-+p--i~°+ n. T h e s e a r e : Po; Qo, Q1; C~, C~; D~, D 1. D a t a f o r K - + p ~ g o + n a r e t a k e n f r o m r e f s . [3] and [4!~ f o u r e n e r g i e s axe a v a i l a b l e : 5, 7, 9.5 and 12.3 G e V / c . F o r lr- + p ~77 + n, r e s u l t s a r e f r o m r e f . [20]; t h e r e a r e two d i f f e r e n t s e t s of p o i n t s at 5.9, 13.3 and 18.2 G e V / c and one s e t at 9.8 G e V / c . F o r t h i s s e c o n d r e a c t i o n , the p o i n t s a r e to be c o n v e r t e d to c o m p l e t e p r o d u c t i o n : the b r a n c h i n g r a t i o (U ~ 2V)/(U ~ all) is n e e d e d and we took 0.314 a s g i v e n by R o s e r d e l d et al. [19]. C h o o s i n g - t = 0.85 ( G e V / c ) 2 a s a l i m i t f o r s q u a r e d m o m e n t u m t r a n s f e r , the t o t a l n u m b e r of e x p e r i m e n t a l p o i n t s f o r the two r e a c t i o n s is 76. T h e i d e a l ×2 is t h e n 76 - 7 = 69. Two t y p i c a l s o l u t i o n s a r e g i v e n in t a b l e 2, t o g e t h e r with the c o r r e s p o n d ing ×2. S o l u t i o n 2 is i l l u s t r a t e d on fig. 2, at 5, 7.1, 9.5 and 12.3 GeV/'c. S o l u t i o n 1 is not v e r y d i f f e r e n t : the s h a p e of the f o r w a r d p e a k i s s l i g h t l y c h a n g e d , but not the e n e r g y d e p e n d e n c e . T h i s i m p l i e s that f l u c t u a t i o n s a m o n g the p a r a m e t e r s a r e s t i l l p o s s i b l y l a r g e . A d d i t i o n a l i n f o r m a t i o n c a n be found for the p a r a m e t e r s at t = 0 (Po, Qo and C~) f r o m the t o t a l c r o s s s e c t i o n s . R e s u l t s o b t a i n e d in t h i s way have b e e n p u b l i s h e d by B a r g e r and O l s s o n [16] and axe i n d i c a t e d in t a b l e 2, t h i r d c o l u m n ( c o n v e r t e d to the p r e s e n t n o t a t i o n s ) . As t h e s e a u t h o r s found ap(O) = 0.57 ± 0.03 and aR(0 ) = 0.34 ± 0.03, the two s e t s of v a l u e s c a n be c o m p a r e d . With r e g a r d to the s i g n of the a m p l i t u d e DI~ (t), the p a r a m e t e r D~ w a s
REGGE-POLE MODEL
633
I i i
100 soo;
s 6°v/~
%
"t
.Q --L
lOO
•
t
lOO 50
lO 5
:
o.,/.
I
V
0.5 I
o.1 I o
1
I o.2
I
I 0.6
I
I ~.0
1
I ~.4
- t (GeV/c)
2
Fig. 2. /3 + R model fitted on the reaction K- + p---* R° ~ n at high energy. Measurements are from refs. [3], [4]. f o u n d to be p o s i t i v e . No s o l u t i o n could be g o t t e n with DK(t) n e g a t i v e , as w a s the c a s e f o r the fit r e a l i z e d by P h i l l i p s and R a r i t a [2]. T h i s fact is l i n k e d to the s h a p e of the f o r w a r d p e a k , w h i c h is s o m e w h a t c h a n g e d , and to i t s v a r i a t i o n with the e n e r g y . A s s u m i n g A~/B~p h a s the s a m e s i g n at t = 0 a s at the pole w h e r e t = m~ f i x e s D~ to b e p o s i t i v e . Now ~ b e i n g a l r e a d y p o s i t i v e , all the f o u r r e s i d u e f u n c t i o n s h a v e the s a m e p o s i t i v e s i g n at t= O. What i s p e r h a p s m o r e i m p o r t a n t i s the m a g n i t u d e of D ~ : h e r e a g a i n a n i n d i c a t i o n i s found that the p r o b a b i l i t y of s p i n - f l i p s c a t t e r i n g c o u l d r e m a i n i m p o r t a n t at high e n e r g y .
634
A. DEREM and G. SMADJA Table 2 Values of p a r a m e t e r s Solution 1 (X2=61)
Solution 2 (X2=63)
C~ mb. GeV
3.965
4.356
C1R GeV -2
0.645
0.811
D~ mb D 1 GeV -2
55.66 2.445
Ref. [ 16]
SU(3)
4.96 ±0.1
50.13 2.22
P0
0.602
0.583
0.56 ~0.06
0.5
Q0
0.526
0.554
0.655 + 0.092
0.612
Q1 GeV-2
0.003
0.106
0
If we a c c e p t DR ~ (t) to b e p o s i t i v e , t h e a g r e e m e n t with SU(3) i s good. It m u s t b e n o t i c e d t h a t h e r e t h e s y m m e t r y p l a y s a r o l e not only t h r o u g h t h e n o n - s p i n - f l i p a m p l i t u d e s A ' , but a l s o t h r o u g h t h e B - a m p l i t u d e s . F o r t h e f i r s t f a c t o r i z a t i o n f u n c t i o n P(t), w h i c h we a s s u m e d t - i n d e p e n d e n t , t h e SU(3) v i o l a t i o n i s w h o l l y c o n t a i n e d in t h e p a r a m e t e r P o . T a b l e 2 s h o w s t h a t in a g r e e m e n t w i t h B a r g e r and O l s s o n [16], we f i n d a v a l u e g r e a t e r t h a n t h e SU(3) p r e d i c t i o n of 0.5. On t h e c o n t r a r y , we f i n d f o r Q(t) an SU(3) b r e a k i n g t h a t g o e s in t h e o p p o s i t e d i r e c t i o n to r e f . [16]. In f a c t , t h e r e i s s t i l l a l a c k of e x p e r i m e n t a l a c c u r a c y to g i v e d e f i n i t e c o n c l u s i o n s . C o m p a r i s o n w i t h I
t
I
i
I
,
I
I
I
I
J
l
:b
6~x
40(
2O(
I
o
0.2
0.4
0 . 6 - I ( G ¢ ~ / c ) ;~
Fig. 3. Extrapolation of t h e p + R model down to 3 GeV/c (ref. [12]).
I~EGGE-I~OLE MODEL
635
SU(3) s h o u l d b e m a d e at t h e p o l e , i n s t e a d of at t = 0. But t h i s k e e p s s e n s e if t h e t - d e p e n d e n c e of P(t) and Q(t) c a n b e c o n s i d e r e d a s a s l o w one. T h i s c o n d i t i o n s e e m s a c t u a l l y to b e s a t i s f i e d , s i n c e P(t) = P(0) a n d Q1 i s s m a l l . T h e p r e v i o u s f i t b y P h i l l i p s a n d R a r i t a , n o r m a l i z e d on t h e 9.5 G e V / c e x p e r i m e n t , w a s not p e r f e c t when e x t e n d e d to o t h e r e n e r g i e s . E f f e c t i v e l y t h e f o r w a r d p e a k is l o w e r in t h e new fit, r e d u c i n g t h e t o t a l c r o s s s e c t i o n at 9.5 G e V / c f r o m 70 p b to a r o u n d 60 tzb. N e v e r t h e l e s s , t h e t h e o r e t i c a l d i f f e r e n t i a l c r o s s s e c t i o n i s s t i l l c o m p a t i b l e w i t h the e x p e r i m e n t a l e r r o r s . W e h a v e m a d e e x t r a p o l a t i o n s of the m o d e l to l o w e r e n e r g i e s : fig. 3 and 4 s h o w t h e r e s u l t f o r two e x p e r i m e n t s at 3 and 2.24 G e V / c , f r o m r e f s . [12] and [13]. It i s s u r p r i s i n g l y good, though t h e f o r w a r d p e a k i s not e x p e r i m e n t a l l y w e l l e s t a b l i s h e d . S l o p e and n o r m a l i z a t i o n a r e c o r r e c t l y g i v e n , w h i c h i s r e m a r k a b l e when n o t i c i n g t h a t the c r o s s s e c t i o n i n c r e a s e s f r o m 45 ± 5 p b at 12.3 G e V / c to 782 ± 72 t~b at 2.24 G e V / c . The following conclusions will be drawn. (a) T h e r e is a g l o b a l a g r e e m e n t b e t w e e n t h e R e g g e f o r m a l i s m and e x p e r i m e n t f o r t h e t h r e e r e a c t i o n s ~ - + p - - . ~ ° + n, n - + p ~ 77° + n and K- + p ~ ~ o + n, at high e n e r g y a n d f o r it! £ 1 (GeV//c) 2. A m o r e d e t a i l e d c h e c k i s o n l y l i m i t e d by t h e e x p e r i m e n t a l e r r o r s . U s e of t o t a l c r o s s s e c t i o n s g i v e a d d i t i o n a l i n f o r m a t i o n on t h e p a r a m e t e r s i n v o l v e d in the m o d e l ; no d i s i
1000
- t (GeV/c)2
0
0:~
0:4
o:G
~8
Fig. 4. Extrapolation of the tO + R model down to 2.24 GeV/c (ref. [13]).
636
A. DEREM and G. SMADJA
a g r e e m e n t c a n be d e t e c t e d w i t h i n the p r e s e n t a c c u r a c y . (b) As w a s e m p h a s i z e d by B a r g e r and O l s s o n [16], the R e g g e - p o l e t h e o r y p r o v i d e s a u s e f u l d y n a m i c s to m a k e c r i t i c a l t e s t s of SU(3). H e r e s p i n flip a m p l i t u d e s as well a s n o n - s p i n - f l i p o n e s a r e r e l a t e d t h r o u g h the f a c t o r i z a t i o n f u n c t i o n s P(t) and Q(t). Though the v i o l a t i o n s h o u l d not be l a r g e r t h a n 10% at t = 0, it i s d i f f i c u l t to d e t e r m i n e it a c c u r a t e l y at p r e s e n t . (c) F o r K - + p ~ K ° + n, a l t h o u g h we a r e not r e a l l y in the a s y m p t o t i c r e g i o n , the R e g g e h y p o t h e s i s of the d o m i n a n c e of two p o l e s in the c r o s s e d c h a n n e l p r o v i d e s a s a t i s f a c t o r y t h e o r e t i c a l a p p r o x i m a t i o n o v e r a wide r a n g e of e n e r g y . A b e t t e r e x p e r i m e n t a l d e f i n i t i o n of the f o r w a r d p e a k would r e s u l t i n a b e t t e r k n o w l e d g e of the r e s i d u e f u n c t i o n s at s m a l l [t [ - v a l u e s . K + + n K ° + p e x p e r i m e n t s will help to t e s t the v a l i d i t y of the p + R m o d e l . (d) T h e r e s t i l l r e m a i n s g r e a t a r b i t r a r i n e s s c o n c e r n i n g the f o r m of the r e s i d u e f u n c t i o n s . C e r t a i n l y the p + R m o d e l can be f i t t e d at l a r g e ]t' L !-valu e s . P e r h a p s c o u l d one a s s o c i a t e a c h a n g e of s l o p e n e a r t = -1 ( G e V / c ) 2 with a z e r o of eR(t ) a s s u g g e s t e d by F r a u t s c h i [15]. But it f i r s t r e q u i r e s a b e t t e r k n o w l e d g e of the i n v o l v e d r e s i d u e f u n c t i o n s . One of u s (A. D.) i s g r a t e f u l to P r o f e s s o r A. B e r t h e l o t , f o r the h o s p i t a l ity e x t e n d e d to h i m at the D P h P E (Saclay). We a / s o want to t h a n k P r o f e s s o r J. M e y e r for his s y m p a t h e t i c s u p p o r t , and P r o f e s s o r R. B a r l o u t a u d f o r m a n y helpful d i s c u s s i o n s .
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [ 11] [12] [13] [14] [15] [16] [17] [18] [ 19] [20]
R . J . N . Phillips a n d W . R a r i t a , Phys. Rev. 138 (1965) B723. R . J . N . Phillips a n d W . R a r i t a , Phys. Rev. 139 (1965) B1336. P.Astbury et al.. Phys. Letters 16 (1965} 328. P.Asthury et al., Phys. Letters 23 (1966) 396. V.Singh, Phys. Rev. 129 (1963) 1889. M.Gcll-Mann. Phys. Rev. Letters 8 (1962) 263 V.N.Gribov, l. Ya. Pomeranchuk. Phys. Rev. Letters 8 (1962) 343. R . J . N . Phillips, Nuovo Cimento 45 (1966) A245; R.K. Logan, L.Sertorio, Phys. Rev. Letters 17 (1966) 834. A. Derem. Echange de charge 7r-nucl4on et p61e du p, Journal de Physique, to be published. G.HShler, J.Baacke, G . E i s c n b e i s s , Phys. Letters 22 (1966) 203. R . J . N . Phillips, W.Rarita. Phys. Letters 19 (1965} 598. R . J . N . Phillips, private communication. J . B a d i c r et al.. Rapport CEA - R 3037 (1966). G.W.l,ondon et al., Phys. Rev. 143 (1966) 1034. R . J . N . Phillips and W.Rarita, Phys. Rev. Letters 15 (1965) 807. S . F r a u t s c h i , Phys. Hcv. Letters 17 {1966) 722. V.Barger and M.Olsson, Phys. Rev. Letters 18 (1967) 294. R . J . N . P h i l l i p s , Nucl. Phys. B1 {1967) 572. MengTa-Chung. Nucl. Phys. B1 (1967) 219. A. H. Roscnfcld et al., Revs. Mod. Phys. 39 (1967) 1. O.Guisan et al., Phys. Letters 18 (1965) 200.
Nuclear Physics B3 (1967) 628-636. North-Holland Publ. Comp., Amsterdam
THE o + R REGGE-POLE MODEL I N T H E R E A C T I O N K" + p - - ~ K ° + n A. D E R E M * and G. SMADJA
D~partement de Physique des Particules El~menlaires, CEN-Saclay Received 2 October 1967
Abstract: It is shown that the high-energy data for the K-p charge-exchange s c a t t e r ing are in agreement with the p * R Regge-pole model of Phillips and Rarita. A new determination of the parameters of the model is given. The observed energy dependence of the differential cross sections is well accounted for over a wide range of energies. Evidence is found that the residue functions are all positive at t-:0.
P h i l l i p s and R a r i t a [1] s u g g e s t e d that two R e g g e p o l e s m i g h t d o m i n a t e the t - c h a n n e l of the K - p c h a r g e - e x c h a n g e s c a t t e r i n g at high e n e r g y : the p o l e s of the p (/(7 = 1 +, j P = 1-) and of the A 2 (/(7 = 1 - , j k - = 2+; a l s o c a l l e d R - p o l e ) . It i s the p u r p o s e of t h i s p a p e r to show that the p + R m o d e l [2] is c o n s i s t e n t with the n e w h i g h - e n e r g y CERN d a t a [4]. E x t r a p o l a t i o n of the m o d e l to l o w e r e n e r g i e s is m a d e . A new d e t e r m i n a t i o n of the p a r a m e t e r s of the m o d e l i s given. T h e s e p a r a m e t e r s a r e s t r o n g l y t i e d to t h o s e w h i c h a r e u s e d to d e s c r i b e the h i g h - e n e r g y r e a c t i o n s 7r- + p - , ~ o + n and 7r- + p ~ 7}o + n. As a c o n s e q u e n c e , we i n d e e d o b t a i n e d an i n d i c a t i o n of a g l o b a l a g r e e ment between e x p e r i m e n t a l differential c r o s s sections and Regge models f o r the t h r e e r e a c t i o n s ~- + p ~ o + n, ~ - + p __.wo + n and K- + p _ ~ o + n and for s q u a r e d m o m e n t u m t r a n s f e r 'tl -<.0.85 ( G e V / c ) 2. T h e f o r m a l i s m w i l l be b r i e f l y s k e t c h e d . T h e 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 ~N and KN s c a t t e r i n g i s w r i t t e n :
d~_ dt
1 ( 16n
) ~
1-
IA'I 4m 2
+ -.
4m 2
(s
) iB: 2 1-t/4m 2
(1)
in t e r m s of the two t - c h a n n e l a m p l i t u d e s A'(s, t) and B(s, t), as d e f i n e d by Singh [5]; s and - t a r e the s q u a r e s of c . m . e n e r g y and m o m e n t u m t r a n s f e r , m is the n u c l e o n m a s s , kla b and Wla b a r e the m o l . l e n t u m and e n e r g y in the l a b s y s t e m of the i n c i d e n t m e s o n . N a t u r a l u n i t s (/~ = c = 1) a r e u s e d . T h e c o n t r i b u t i o n s f r o m the p o l e s of the p and the A 2 to A ' a r e w r i t t e n :
* Chercheur agrdg IISN, Bruxelles.
REGGE-POLE MODEL
629
A' = - v~ A ; ,
(2)
A' : - A{~,
(3)
A. = 2 A K -
2A K,
(4)
f o r the r e a c t i o n s ~ - + p - ' n o + n , n - + p ~ ? o + n a n d K - + p - - * ~ ° + n , r e s p e c t i v e l y * . S i m i l a r r e l a t i o n s e x i s t f o r the B - a m p l i t u d e s . The f a c t o r i z a t i o n p r i n c i p l e [6] e n a b l e s to link c o n t r i b u t i o n s a r i s i n g f r o m m e s o n e x c h a n g e s in d i f f e r e n t r e a c t i o n s in the following f o r m :
Ap/ApK, :
= P(O ,
(5)
K AR/A R = B~R/B~ = Q(t) ,
(6)
w h e r e P(t) and Q(t) a r e r e a l f a c t o r i z a t i o n functions. F o r the a s y m p t o t i c f o r m of t h e s e a m p l i t u d e s , we have
A~p(S, l) = C~(t) (
A~(s, t) = C~(t)
Wlab )ap(t) 1 - e-i~aP (t) ~ sin .ap(t)
,
(7)
Wla b . aR(t) ( ~ ) 1 sin + e-i~aR(t),oR(t )
(8)
w h e r e ap(t) and aR(t ) a r e the t r a j e c t o r i e s of the two R e g g e p o l e s ; the s c a l ing f a c t o r Wo is c o n v e n i e n t l y c h o s e n equal to 1 GeV. E q u a t i o n s s i m i l a r to (7) and (8) r e l a t e B~(s, t) and B~R(S, t) to D~(t) and DE(t), except f o r the e x p o n e n t of Wlab/W o which is r e p l a c e d by ax(t ) - 1 (x s t a n d s f o r p o r R). The f u n c t i o n s C(t) and D(t) a r e r e a l f o r n e g a t i v e t and p r o p o r t i o n a l to the r e s i d u e s at the p o l e s of A' and B. R e s i d u e f u n c t i o n s can a l s o be defined f o r K- + p __.~o + n, u s i n g r e l a t i o n s (5) and (6):
cK(t) : P(t) C~)(t) ,
(9)
cK(t) = V(t) C~(t) ,
(10)
and s i m i l a r l y f o r D~(t) and DK(t). Now we t u r n to a d i s c u s s i o n about the v a r i o u s f u n c t i o n s i m p l i e d in the R e g g e - p o l e f o r m a l i s m . We a r e left with f o u r r e s i d u e f u n c t i o n s : C~(t), D~p(t), cK(t) and DRK(t), and two t r a j e c t o r i e s : ~ ( t ) and aR(t ). We f i r s t d e * For y- + p--~ r]° + n. elastic kinematics is then used; ~ - . mass difference effects are considered unimportant at high energy; this point is discussed in a recent paper by Phillips [17].
630
A. DEREM and G. SMADJA
fine the following empirical parametrization, P h i l l i p s and R a r i t a * :
Cp(t) : C~ no(O) [2no(t) + 1]
a n a l o g o u s to t h e one u s e d by
Fp(t) ,
D~(t) : D~ ap(t) Ao(t) , for~-
+p~°+n,
(11)
and:
C~(t) = C~
aR(t ) [2a,R(t ) +1]
D~(t) : D~
aR(t)
eDl t
e C~t (12)
f o r u - + p ~ 77° + n. T h e n w e p r o c e e d to i m p o s e a c e r t a i n n u m b e r of r e s trictions: (a) The amplitudes C~(t) and D~(t) c a n b e known f r o m ~ - + p ~Tr ° + n at h i g h e n e r g y . H e r e we h a v e a d o p t e d p a r t i c u l a r s o l u t i o n s [8] c h a r a c t e r i z e d by a z e r o of Fp(t) at t ~ - 0 . 1 5 ( G e V / c ) 2 ( s e e fig. 1). T h i s t y p e of s o l u t i o n w i t h a z e r o of C~(t) r e c e i v e s s t r o n g s u p p o r t f r o m two h i g h - e n e r g y f e a t u r e s : (i) it o f f e r s a n a t u r a l e x p l a n a t i o n of t h e c r o s s o v e r e f f e c t o b s e r v e d f o r the ~ - p a n d ~+p e l a s t i c s c a t t e r i n g 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 ; (ii) the v a l u e at t = m ~ of t h e r a t i o of t h e m a g n e t i c to t h e e l e c t r i c i s o v e c t o r f o r m f a c t o r s s t r o n g l y f a v o u r s a z e r o of t h e n o n - s p i n - f l i p r e s i d u e n e a r t = - 0 . 2 ( C e V / c ) 2 [18]. It ~ u s t b e n o t e d t h a t Fp(t) c a n n o t b e u s e d b e y o n d t ~ -1 ( G e V / c ) 2. C~(t) a n d Dr~(t) are now d e t e r m i n e d t h r o u g h t h e f a c t o r i z a t i o n f u n c t i o n P(t). W i t h an e x a c t SU(3) s y m m e t r y , P(t) w o u l d b e c o n s t a n t a n d e q u a l to ½. E x p e r i m e n t a l d a t a c a n t e l l u s l i t t l e a b o u t t h e t - d e p e n d e n c e of P(t), f o r m u c h of t h e v a r i a t i o n of C~(t) and D~(t) s t e m s f r o m t h e f a c t o r s r p ( t ) (with a z e r o at t ~ - 0 . 1 5 ( G e V / c ) ) and ap(t) (with a z e r o at t ~ - 0 . 5 ( G e V / c ) 2 ) . A s f u r t h e r m o r e w e o b t a i n e d g o o d f i t s w i t h P(t) = c o n s t a n t = P o , t h i s w a s t a k e n a s a constraint. (b) /n an analogous fashion, CK(t) and D~(t) w e r e r e l a t e d to t h e c o r r e s p o n d i n g f u n c t i o n s in ~ - + p ~ 7/0 + n [cf. f o r m u l a (6)]. T h i s r e a c t i o n w a s a l r e a d y s u c c e s s f u l l y a n a l y s e d in t h e R e g g e - p o l e a p p r o a c h by P h i l l i p s and R a r i t a [14]. But h e r e C~(t) and D~(t) a r e not s o t i e d to t h e e x p e r i m e n t a l p o i n t s a s a x e C~p(t) and DE(t). S o l u t i o n s in e x p o n e n t i a l f o r m a r e f a r f r o m b e i n g unique. A c c o r d i n g l y we f i t t e d a l t o g e t h e r t h e h i g h - e n e r g y d a t a f o r K - + p ~ ~ o + n a n d ~ - + p ~ ~?o + n, t h u s r e d u c i n g t h e d e g r e e of a r b i t r a r i n e s s : t h e e x p o n e n t i a l s l o p e s of t h e r e s i d u e f u n c t i o n s a r e t i e d t o g e t h e r t h r o u g h t h e f a c t o r i z a t i o n t h e o r e m . E x a c t SU(3) w o u l d g i v e Q(t) = Q(O) = v r 3 / 2 ¢ 2 . In f a c t , v i o l a t i o n of SU(3) w a s p e r m i t t e d t h r o u g h t h e i n t r o d u c t i o n of two p a r a m e t e r s Qo and Q I : * This p a r a m e t r i z a t i o n is perhaps not the best one [1]], but it can be hoped that the global t-dependence of C(t) and D (t) is approximately c o r r e c t .
REGGE-POLE MODEL
631
Ap(t)
~R(t) - t (GeV/(;) 2 0.5
ap
-0.5'
rpct) -I.0.
Fig. 1. Otp(t) and OCR(t) trajectories; the functions Fp(t) and Ap(t) have been adjusted to ~'N charge-exchange scattering.
Q(t) : Qo eQ1 t
(13)
(c) The Regge formalism d e s c r i b e d above is such that A~ and BE have the s a m e phase and hence the p o l a r i z a t i o n of the nucleon is z e r o in the ~-N c h a r g e - e x c h a n g e s c a t t e r i n g . P r o b a b l y the p - p o l e model can be looked upon as a z e r o - o r d e r a p p r o x i m a t i o n [9]: only small background t e r m s a r e to be added to the Regge amplitudes to give a r e a s o n a b l e p o l a r i z a t i o n [7]. It is
632
A. DEREM and G. SMADJA Table 1 Particular solutions used in the fit. •- + p--~T/° + n (rcf. [8])
ap(t)
= 0.57 + 0.98 l - 0.52 t2 - 0.73 t 3
Fp(t) = 1 + 7.63t + 6.53t 2 - 0.02t 3 Ap(t) = 1 + t + 0.69t 2 + 0.16t 3 = 2.03 mb. GeV;
Dpo = 51.7
mb
7:- + p--~U ° + n (ref. [I01)
.
C~R(t) - -1
(I + 0.35) 2 l + 0.35 - 0.46/
hoped the ap(t) t r a j e c t o r y will not be s t r o n g l y m o d i f i e d when d e f i n i t e a c count will be t a k e n f o r the p o l a r i z a t i o n and we u s e d f o r it a p o l y n o m i a l f o r m [8], g i v e n in t a b l e 1. (d) F i n a l l y , we c h o s e f o r aR(t ) the P i g n o t t i f o r m 3b quoted in a p a p e r by P h i l l i p s and R a r i t a [10] (cf. t a b l e 1), w h i c h i s c l o s e s t to J = 2 when t = m2 2. T h i s t r a j e c t o r y is e x t r a c t e d d i r e c t l y f r o m the h i g h - e n e r g y 7r- + p ~ Uo + n d a t a . Fp(t), Ap(t), ap(t) and a~(t) a r e r e p r e s e n t e d on fig. 1. A g l a n c e at f o r m u l a e (10), (12) and (13) s h o w s that s e v e n r e a l p a r a m e t e r s a r e to be a d j u s t e d on the h i g h - e n e r g y d a t a for 7:- + p ~ U° + n and K-+p--i~°+ n. T h e s e a r e : Po; Qo, Q1; C~, C~; D~, D 1. D a t a f o r K - + p ~ g o + n a r e t a k e n f r o m r e f s . [3] and [4!~ f o u r e n e r g i e s axe a v a i l a b l e : 5, 7, 9.5 and 12.3 G e V / c . F o r lr- + p ~77 + n, r e s u l t s a r e f r o m r e f . [20]; t h e r e a r e two d i f f e r e n t s e t s of p o i n t s at 5.9, 13.3 and 18.2 G e V / c and one s e t at 9.8 G e V / c . F o r t h i s s e c o n d r e a c t i o n , the p o i n t s a r e to be c o n v e r t e d to c o m p l e t e p r o d u c t i o n : the b r a n c h i n g r a t i o (U ~ 2V)/(U ~ all) is n e e d e d and we took 0.314 a s g i v e n by R o s e r d e l d et al. [19]. C h o o s i n g - t = 0.85 ( G e V / c ) 2 a s a l i m i t f o r s q u a r e d m o m e n t u m t r a n s f e r , the t o t a l n u m b e r of e x p e r i m e n t a l p o i n t s f o r the two r e a c t i o n s is 76. T h e i d e a l ×2 is t h e n 76 - 7 = 69. Two t y p i c a l s o l u t i o n s a r e g i v e n in t a b l e 2, t o g e t h e r with the c o r r e s p o n d ing ×2. S o l u t i o n 2 is i l l u s t r a t e d on fig. 2, at 5, 7.1, 9.5 and 12.3 GeV/'c. S o l u t i o n 1 is not v e r y d i f f e r e n t : the s h a p e of the f o r w a r d p e a k i s s l i g h t l y c h a n g e d , but not the e n e r g y d e p e n d e n c e . T h i s i m p l i e s that f l u c t u a t i o n s a m o n g the p a r a m e t e r s a r e s t i l l p o s s i b l y l a r g e . A d d i t i o n a l i n f o r m a t i o n c a n be found for the p a r a m e t e r s at t = 0 (Po, Qo and C~) f r o m the t o t a l c r o s s s e c t i o n s . R e s u l t s o b t a i n e d in t h i s way have b e e n p u b l i s h e d by B a r g e r and O l s s o n [16] and axe i n d i c a t e d in t a b l e 2, t h i r d c o l u m n ( c o n v e r t e d to the p r e s e n t n o t a t i o n s ) . As t h e s e a u t h o r s found ap(O) = 0.57 ± 0.03 and aR(0 ) = 0.34 ± 0.03, the two s e t s of v a l u e s c a n be c o m p a r e d . With r e g a r d to the s i g n of the a m p l i t u d e DI~ (t), the p a r a m e t e r D~ w a s
REGGE-POLE MODEL
633
I i i
100 soo;
s 6°v/~
%
"t
.Q --L
lOO
•
t
lOO 50
lO 5
:
o.,/.
I
V
0.5 I
o.1 I o
1
I o.2
I
I 0.6
I
I ~.0
1
I ~.4
- t (GeV/c)
2
Fig. 2. /3 + R model fitted on the reaction K- + p---* R° ~ n at high energy. Measurements are from refs. [3], [4]. f o u n d to be p o s i t i v e . No s o l u t i o n could be g o t t e n with DK(t) n e g a t i v e , as w a s the c a s e f o r the fit r e a l i z e d by P h i l l i p s and R a r i t a [2]. T h i s fact is l i n k e d to the s h a p e of the f o r w a r d p e a k , w h i c h is s o m e w h a t c h a n g e d , and to i t s v a r i a t i o n with the e n e r g y . A s s u m i n g A~/B~p h a s the s a m e s i g n at t = 0 a s at the pole w h e r e t = m~ f i x e s D~ to b e p o s i t i v e . Now ~ b e i n g a l r e a d y p o s i t i v e , all the f o u r r e s i d u e f u n c t i o n s h a v e the s a m e p o s i t i v e s i g n at t= O. What i s p e r h a p s m o r e i m p o r t a n t i s the m a g n i t u d e of D ~ : h e r e a g a i n a n i n d i c a t i o n i s found that the p r o b a b i l i t y of s p i n - f l i p s c a t t e r i n g c o u l d r e m a i n i m p o r t a n t at high e n e r g y .
634
A. DEREM and G. SMADJA Table 2 Values of p a r a m e t e r s Solution 1 (X2=61)
Solution 2 (X2=63)
C~ mb. GeV
3.965
4.356
C1R GeV -2
0.645
0.811
D~ mb D 1 GeV -2
55.66 2.445
Ref. [ 16]
SU(3)
4.96 ±0.1
50.13 2.22
P0
0.602
0.583
0.56 ~0.06
0.5
Q0
0.526
0.554
0.655 + 0.092
0.612
Q1 GeV-2
0.003
0.106
0
If we a c c e p t DR ~ (t) to b e p o s i t i v e , t h e a g r e e m e n t with SU(3) i s good. It m u s t b e n o t i c e d t h a t h e r e t h e s y m m e t r y p l a y s a r o l e not only t h r o u g h t h e n o n - s p i n - f l i p a m p l i t u d e s A ' , but a l s o t h r o u g h t h e B - a m p l i t u d e s . F o r t h e f i r s t f a c t o r i z a t i o n f u n c t i o n P(t), w h i c h we a s s u m e d t - i n d e p e n d e n t , t h e SU(3) v i o l a t i o n i s w h o l l y c o n t a i n e d in t h e p a r a m e t e r P o . T a b l e 2 s h o w s t h a t in a g r e e m e n t w i t h B a r g e r and O l s s o n [16], we f i n d a v a l u e g r e a t e r t h a n t h e SU(3) p r e d i c t i o n of 0.5. On t h e c o n t r a r y , we f i n d f o r Q(t) an SU(3) b r e a k i n g t h a t g o e s in t h e o p p o s i t e d i r e c t i o n to r e f . [16]. In f a c t , t h e r e i s s t i l l a l a c k of e x p e r i m e n t a l a c c u r a c y to g i v e d e f i n i t e c o n c l u s i o n s . C o m p a r i s o n w i t h I
t
I
i
I
,
I
I
I
I
J
l
:b
6~x
40(
2O(
I
o
0.2
0.4
0 . 6 - I ( G ¢ ~ / c ) ;~
Fig. 3. Extrapolation of t h e p + R model down to 3 GeV/c (ref. [12]).
I~EGGE-I~OLE MODEL
635
SU(3) s h o u l d b e m a d e at t h e p o l e , i n s t e a d of at t = 0. But t h i s k e e p s s e n s e if t h e t - d e p e n d e n c e of P(t) and Q(t) c a n b e c o n s i d e r e d a s a s l o w one. T h i s c o n d i t i o n s e e m s a c t u a l l y to b e s a t i s f i e d , s i n c e P(t) = P(0) a n d Q1 i s s m a l l . T h e p r e v i o u s f i t b y P h i l l i p s a n d R a r i t a , n o r m a l i z e d on t h e 9.5 G e V / c e x p e r i m e n t , w a s not p e r f e c t when e x t e n d e d to o t h e r e n e r g i e s . E f f e c t i v e l y t h e f o r w a r d p e a k is l o w e r in t h e new fit, r e d u c i n g t h e t o t a l c r o s s s e c t i o n at 9.5 G e V / c f r o m 70 p b to a r o u n d 60 tzb. N e v e r t h e l e s s , t h e t h e o r e t i c a l d i f f e r e n t i a l c r o s s s e c t i o n i s s t i l l c o m p a t i b l e w i t h the e x p e r i m e n t a l e r r o r s . W e h a v e m a d e e x t r a p o l a t i o n s of the m o d e l to l o w e r e n e r g i e s : fig. 3 and 4 s h o w t h e r e s u l t f o r two e x p e r i m e n t s at 3 and 2.24 G e V / c , f r o m r e f s . [12] and [13]. It i s s u r p r i s i n g l y good, though t h e f o r w a r d p e a k i s not e x p e r i m e n t a l l y w e l l e s t a b l i s h e d . S l o p e and n o r m a l i z a t i o n a r e c o r r e c t l y g i v e n , w h i c h i s r e m a r k a b l e when n o t i c i n g t h a t the c r o s s s e c t i o n i n c r e a s e s f r o m 45 ± 5 p b at 12.3 G e V / c to 782 ± 72 t~b at 2.24 G e V / c . The following conclusions will be drawn. (a) T h e r e is a g l o b a l a g r e e m e n t b e t w e e n t h e R e g g e f o r m a l i s m and e x p e r i m e n t f o r t h e t h r e e r e a c t i o n s ~ - + p - - . ~ ° + n, n - + p ~ 77° + n and K- + p ~ ~ o + n, at high e n e r g y a n d f o r it! £ 1 (GeV//c) 2. A m o r e d e t a i l e d c h e c k i s o n l y l i m i t e d by t h e e x p e r i m e n t a l e r r o r s . U s e of t o t a l c r o s s s e c t i o n s g i v e a d d i t i o n a l i n f o r m a t i o n on t h e p a r a m e t e r s i n v o l v e d in the m o d e l ; no d i s i
1000
- t (GeV/c)2
0
0:~
0:4
o:G
~8
Fig. 4. Extrapolation of the tO + R model down to 2.24 GeV/c (ref. [13]).
636
A. DEREM and G. SMADJA
a g r e e m e n t c a n be d e t e c t e d w i t h i n the p r e s e n t a c c u r a c y . (b) As w a s e m p h a s i z e d by B a r g e r and O l s s o n [16], the R e g g e - p o l e t h e o r y p r o v i d e s a u s e f u l d y n a m i c s to m a k e c r i t i c a l t e s t s of SU(3). H e r e s p i n flip a m p l i t u d e s as well a s n o n - s p i n - f l i p o n e s a r e r e l a t e d t h r o u g h the f a c t o r i z a t i o n f u n c t i o n s P(t) and Q(t). Though the v i o l a t i o n s h o u l d not be l a r g e r t h a n 10% at t = 0, it i s d i f f i c u l t to d e t e r m i n e it a c c u r a t e l y at p r e s e n t . (c) F o r K - + p ~ K ° + n, a l t h o u g h we a r e not r e a l l y in the a s y m p t o t i c r e g i o n , the R e g g e h y p o t h e s i s of the d o m i n a n c e of two p o l e s in the c r o s s e d c h a n n e l p r o v i d e s a s a t i s f a c t o r y t h e o r e t i c a l a p p r o x i m a t i o n o v e r a wide r a n g e of e n e r g y . A b e t t e r e x p e r i m e n t a l d e f i n i t i o n of the f o r w a r d p e a k would r e s u l t i n a b e t t e r k n o w l e d g e of the r e s i d u e f u n c t i o n s at s m a l l [t [ - v a l u e s . K + + n K ° + p e x p e r i m e n t s will help to t e s t the v a l i d i t y of the p + R m o d e l . (d) T h e r e s t i l l r e m a i n s g r e a t a r b i t r a r i n e s s c o n c e r n i n g the f o r m of the r e s i d u e f u n c t i o n s . C e r t a i n l y the p + R m o d e l can be f i t t e d at l a r g e ]t' L !-valu e s . P e r h a p s c o u l d one a s s o c i a t e a c h a n g e of s l o p e n e a r t = -1 ( G e V / c ) 2 with a z e r o of eR(t ) a s s u g g e s t e d by F r a u t s c h i [15]. But it f i r s t r e q u i r e s a b e t t e r k n o w l e d g e of the i n v o l v e d r e s i d u e f u n c t i o n s . One of u s (A. D.) i s g r a t e f u l to P r o f e s s o r A. B e r t h e l o t , f o r the h o s p i t a l ity e x t e n d e d to h i m at the D P h P E (Saclay). We a / s o want to t h a n k P r o f e s s o r J. M e y e r for his s y m p a t h e t i c s u p p o r t , and P r o f e s s o r R. B a r l o u t a u d f o r m a n y helpful d i s c u s s i o n s .
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R . J . N . Phillips a n d W . R a r i t a , Phys. Rev. 138 (1965) B723. R . J . N . Phillips a n d W . R a r i t a , Phys. Rev. 139 (1965) B1336. P.Astbury et al.. Phys. Letters 16 (1965} 328. P.Asthury et al., Phys. Letters 23 (1966) 396. V.Singh, Phys. Rev. 129 (1963) 1889. M.Gcll-Mann. Phys. Rev. Letters 8 (1962) 263 V.N.Gribov, l. Ya. Pomeranchuk. Phys. Rev. Letters 8 (1962) 343. R . J . N . Phillips, Nuovo Cimento 45 (1966) A245; R.K. Logan, L.Sertorio, Phys. Rev. Letters 17 (1966) 834. A. Derem. Echange de charge 7r-nucl4on et p61e du p, Journal de Physique, to be published. G.HShler, J.Baacke, G . E i s c n b e i s s , Phys. Letters 22 (1966) 203. R . J . N . Phillips, W.Rarita. Phys. Letters 19 (1965} 598. R . J . N . Phillips, private communication. J . B a d i c r et al.. Rapport CEA - R 3037 (1966). G.W.l,ondon et al., Phys. Rev. 143 (1966) 1034. R . J . N . Phillips and W.Rarita, Phys. Rev. Letters 15 (1965) 807. S . F r a u t s c h i , Phys. Hcv. Letters 17 {1966) 722. V.Barger and M.Olsson, Phys. Rev. Letters 18 (1967) 294. R . J . N . P h i l l i p s , Nucl. Phys. B1 {1967) 572. MengTa-Chung. Nucl. Phys. B1 (1967) 219. A. H. Roscnfcld et al., Revs. Mod. Phys. 39 (1967) 1. O.Guisan et al., Phys. Letters 18 (1965) 200.