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Regge poles and polarization in K−p elastic scattering
~
Nuclear Physics B9 (1969) 306-314. North-Holland Publ. Comp., Amsterdam
REGGE POLES A N D P O L A R I Z A T I O N IN K p ELASTIC SCATTERING G. P L A U T * Laboratoire de Physique Thborique el Hautes Energies Facult~ des Sciences. Orsay**
Heceived 30 September 1968
Abstract: A five Regge-pole model is presented and used to fit polarization data in K-p elastic scattering between 2 and 2.4 GeV/c. Following a phenomenological analysis, we obtain solutions consistent with high-energy data and we give sample predictions for polarization in other KN scattering experiments.
1. INTRODUCTION We shall study f r o m the Regge pole s t a n d p o i n t a K - p e l a s t i c s c a t t e r i n g e x p e r i m e n t in which the p r o t o n p o l a r i z a t i o n h a s b e e n m e a s u r e d . In sect. 2 we f i r s t r e c a l l the g e n e r a l f o r m a l i s m of KN e l a s t i c (or c h a r g e exchange) s c a t t e r i n g in the f r a m e w o r k of the Regge p o l e model. Then we give our p a r a m e t r i z a t i o n . In sect. 3, we a n a l y s e the K - p e x p e r i m e n t of Daum et al. [5], in o r d e r to d e t e r m i n e the c o n d i t i o n s that Regge t r a j e c t o r i e s and r e s i d u e s m u s t s a t i s f y in o r d e r to d e s c r i b e c o r r e c t l y the r e s u l t s of t h i s e x p e r i m e n t . In s e c t . 4 we p r e s e n t and d i s c u s s two e x p l i c i t s o l u t i o n s o b t a i n e d f o r the Regge p a r a m e t e r s ; in sect. 5 we c o m p u t e e x p e c t e d v a l u e s of p o l a r i z a t i o n in K - p , K+p and 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 with o u r model.
2. FORMALISM AND P A R A M E T R I Z A T I O N The f o r m a l i s m of K - p e l a s t i c s c a t t e r i n g i s given in ref. [1]. With the help of the two a m p l i t u d e s B and A ' : A' =A +
w+ t / 4 m
B,
1 -t/4m 2
( w h e r e ¢o is the t o t a l l a b e n e r g y of the K-), which a r e p r o p o r t i o n a l to t - c h a n nel h e l i c i t y flip and h e l i c l t y n o n - f l i p a m p l i t u d e s r e s p e c t i v e l y , we w r i t e the o b s e r v a b l e q u a n t i t i e s a s [1,2] * Postal address: Laboratoire de Physique Th~orique, b~tt. 211, Facult~ des Sciences, 91 Orsay. ** Laboratoire associ6 au CNRS.
K-p ELASTIC SCATTERING
2
307
t (s (m+¢o)2 ~1 2] +4m 2 - 1 - t / 4 m 2lIB[
,
~ t o t ( S ) = 1q Im A ' ( s , t = 0)
sin0 Im A'B* P = 16n~f~ da/dt
'
w h e r e q is the K- l a b o r a t o r y m o m e n t u m and k the c e n t e r - o f - m a s s m o m e n tum. k i and k f being the c e n t e r - o f - m a s s m o m e n t a of initial and final kaons, the p o l a r i z a t i o n is counted positive against ~ = k i x k f / I k i x kfl (Basel convention). The r e a d e r is r e f e r r e d to Philipps and R a r i t a [2] for the e x p r e s sions of the p h y s i c a l s - c h a n n e l a m p l i t u d e s in t e r m s of the five t-channel Regge t r a j e c t o r i e s (P, P ' , p, ¢o and A2). We shall now make explicit our p a r a m e t r i z a t i o n . When the complex ang u l a r m o m e n t u m a(t) of a Regge t r a j e c t o r y g o e s to z e r o , r e q u i r e d f a c t o r s p r o p o r t i o n a l to a may be a s s i g n e d to the r e s i d u e s in different ways [3]. We a s s u m e d that odd t r a j e c t o r i e s obey the " s e n s e - c h o o s i n g " m e c h a n i s m $. F o r even t r a j e c t o r i e s we t r i e d two m e c h a n i s m s : Chew m e c h a n i s m (solution 1), N o - c o m p e n s a t i o n m e c h a n i s m (solution 2). We g e n e r a l l y a p p r o x i m a t e d Regge t r a j e c t o r i e s by straight lines a(t) = a(o) + a ' ( o ) x t
but in some e a s e s we added a quadratic t e r m for the P ' (solution 2).
Even trajectories.
The amplitude a r i s i n g f r o m each t r a j e c t o r y is p a r a m e t -
r i z e d as follows:
A ~(t) = -C o exp (C l t ) ai(a i+ 1) el(t) ( A~(t)
: - Co exp(C 1 t)ai(a i2
Bi(t )
=
Here,
_DO e x p ( D l t ) ~i(t)
+1)2
a i2 (oti+ 1)
~i(t )
¢o+t/4rn )ai -~0 /
for the Chew m e c h a n i s m ,
t/4rn) ai k
b~
~i(t)( ¢o+t/4m ~0 ) ~i-1
is the well-known s i g n a t u r e f a c t o r :
for the n o - c o m p e n s a t i o n mechanism, for both m e c h a n i s m s .
~i(t)
= (e -i~c~ + 1)/sin (u~).
:~ As regards the p trajectory, there are several experimental facts
The polariza-
" ~"±p e l a s t i c s c a t t e r i n g a r e o p p o s i t e in s i g n and a p p r o x i m a t e l y e q u a l , b o t h t oi n s m c h a n g i n g s i g n a t t ~ - 0 . 6 GeV2 w h e r e t h e r e i s a dip in t h e c h a r g e - e x c h a n g e d i f f e r -
ential cross section. All these features are very well accounted for by a "sensechoosing" p-trajectory with a complex angular momentum which goes to zero at t ~ -0.6 GeV2.
308
G. PLAUT
Odd trajectories. We p a r a m e t r i z e the amplitudes as follows: for the p t r a j e c t o r y : A~(t) = -(~ + 1) C0[(1 + C2) exp (C 1 t)
- C2]
~i(t)( ¢0+ t/4m )~ ¢o0
/
,
for the ¢o t r a j e c t o r y :
A~(t)
= -(~ + 1) C0(1 -
t/to)
exp (C 1 t)
(¢o+ t/4m (~ ~i(t) \ ~00 )
Both spin-flip amplitudes a r e p a r a m e t r i z e d by:
B (t) = -ai(oti+
1) DO exp ( n 1 t) ~z~t)(
w+ t//4m ) ~i-1 w0
The p a r a m e t r i z a t i o n chosen for the w allows one to fit the c r o s s - o v e r phenomenon in differential c r o s s sections for the K±p elastic s c a t t e r i n g and also to account for the a b s e n c e of a dip in the w contribution [4] to r e a c t i o n s uN -* pN. When t r a j e c t o r i e s and r e s i d u e s a r e a l r e a d y known f r o m the study of uN s c a t t e r i n g we make use of f a c t o r i z a t i o n to obtain a simple p a r a m e t r i z a t i o n
[2]: A 'z~KN) Bi(KN) A,z~7rE) - Bz(uN) - F 0 e x p ( F 1 t). 3. ANALYSIS OF DATA In fig. 1, we p r e s e n t m e a s u r e m e n t s f r o m Daum, Ern~, Lagnaux, Sens, Steuer and Udo [5] at t h r e e l a b o r a t o r y m o m e n t a above 2 GeV/c. It is noteworthy that at all m o m e n t a the p o l a r i z a t i o n is initially positive, goes to z e r o at about t = -0.9 GeV 2 and then b e c o m e s negative (a noticeable dependence of p o l a r i z a t i o n on m o m e n t u m a p p e a r s only at l a r g e r t). M o r e o v e r , t h e r e is a dip in the angular distribution at the s a m e m o m e n t u m t r a n s f e r as the z e r o in the p o l a r i z a t i o n (t = -0.9 GeV 2) (see for instance dc~/dt at 2.44 GeV/c given in fig. 2; see also ref. [5]). This striking stability of the s t r u c t u r e (already t r u e at lower e n e r g i e s [5]) s u g g e s t s that, despite the r a t h e r low value of m o m e n t u m , Daum et a l . ' s r e s u l t s may be r a t h e r well accounted for by c r o s s - c h a n n e l Regge t r a j e c t o r i e s . M o r e o v e r , Chin, Chu and Wang [3] have s u c c e s s f u l l y used Regge poles in ~N s c a t t e r i n g at the same energies. Now we wish to explain the dip in d(r/dt together with a change of sign in the polarization. The most natural explanation is to a s s u m e that the c o m plex angular m o m e n t u m of one of the Regge t r a j e c t o r i e s involved goes to z e r o and that one of the amplitudes due to this t r a j e c t o r y is p r o p o r t i o n a l to for ~ -~ 0. Taking into account the signature f a c t o r , the r e q u i r e d ~ f a c t o r a p p e a r s in the spin-flip amplitudes of odd t r a j e c t o r i e s , and also in the nonflip amplitudes of even t r a j e c t o r i e s which obey a n o - c o m p e n s a t i o n m e c h a nism. Note that we a r e not i n t e r e s t e d in the even t r a j e c t o r i e s contributions
K-p ELASTIC SCATTERING
309
I }. I-
Fig. 1. Polarization in K-p e l a s t i c s c a t tering. Comparison between e x p e r i m e n tal data [5] and our two solutions. The solid lines r e p r e s e n t solution 1 and the dashed ones solution 2.
, i$ OeVk:
Fig. 2. Comparison for d(y(K-p --* K-p)/dt. Solid lines r e p r e s e n t solution i and dashed lines solution 2.
to s p i n - f l i p a m p l i t u d e s w h i c h w e a s s u m e to b e m u c h w e a k e r t h a n t h a t of t h e odd trajectories * Now l e t u s c o n s i d e r t h e o d d t r a j e c t o r i e s . U s i n g t h e o p t i c a l t h e o r e m a n d a p p r o p r i a t e c o m b i n a t i o n s of t o t a l c r o s s s e c t i o n s , w e e s t i m a t e d f r o m e x p e r i m e n t a l d a t a t h e c o n t r i b u t i o n of e a c h t r a j e c t o r y to t h e a m p l i t u d e I m A '(t = 0). W e f o u n d t h a t t h e w c o n t r i b u t i o n w a s m u c h l a r g e r t h a n t h a t of t h e p; t h e A 2 c o n t r i b u t i o n w a s s t i l l s m a l l e r . On t h e o t h e r h a n d t h e SU(3) prediction:
s e e m s to b e w e l l v e r i f i e d a t t = 0 b y e x p e r i m e n t a l d a t a ( o u r a m p l i t u d e s a r e d e f i n e d a s in r e f . [2]). If t h e s p i n - f l i p a m p l i t u d e d u e to t h e p i s d o m i n a n t , i t s h o u l d p r o d u c e a s u b s t a n t i a l n e g a t i v e p o l a r i z a t i o n if o n e b e l i e v e s f a c t o r i z a t i o n (which i m p l i e s t h a t t h e r e l a t i v e s i g n of t h e a m p l i t u d e s A ' a n d B i s t h e s a m e in KN s c a t t e r ing a s in ~N s c a t t e r i n g . T h e r e f o r e it i s n a t u r a l to a s s u m e t h a t t h e ¢o s p i n f l i p a m p l i t u d e i s r e s p o n s i b l e f o r p o l a r i z a t i o n in K - p e l a s t i c s c a t t e r i n g n e a r the forward direction. This spin-flip amplitude should be much larger than t h a t a r i s i n g f r o m t h e p, a n d h a v e t h e o p p o s i t e s i g n . T h e c h a n g e of s i g n of p o l a r i z a t i o n a t t = - 0 . 9 GeV 2 w i l l t h e n b e e x p l a i n e d e i t h e r b y ~¢o(t) g o i n g to z e r o o r b y (~p,(t) g o i n g to z e r o ; in t h e l a t t e r c a s e t h e a m p l i t u d e A~,,(t) w h i c h i n t e r f e r e s w i t h Bw(t) c h a n g e s s i g n . * The fact that the polarizations in 7r±p e l a s t i c s c a t t e r i n g a r e approximately opposite in signs and equal is a strong argument for assuming that the P and P' contributions to spin-flip amplitudes a r e v e r y weak.
310
G. PLAUT
4. D E S C R I P T I O N O F OUR S O L U T I O N S L e t u s b e g i n with a f e w w o r d s a b o u t o u r c o m p u t a t i o n a l m e t h o d s . W e c o n s i d e r e d e x p e r i m e n t a l d a t a * on KN high e n e r g y t o t a l a n d 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 t o g e t h e r with d a t a on p o l a r i z a t i o n a n d 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 in K - p e l a s t i c s c a t t e r i n g b e t w e e n 2 a n d 3 G e V / c . I n d e e d it i s not s i g n i f i c a n t to f i t r a t h e r l o w e n e r g y e x p e r i m e n t a l d a t a in t h e R e g g e p o l e f r a m e w o r k if t h e p a r a m e t e r s o b t a i n e d do not a l s o d e s c r i b e high e n e r g y d a t a w e l l . O u r c o m p u t a t i o n s w e r e p e r f o r m e d on U n i v a c 1108 a t O r s a y with t h e C E R N m i n i m i z a t i o n p r o g r a m MINROS. W e f i r s t o b t a i n e d a f i t f o r ~N d a t a with two d i s t i n c t s o l u t i o n s : in t h e f i r s t one e v e n t r a j e c t o r i e s o b e y t h e C h e w m e c h a n i s m ; in t h e s e c o n d one t h e y o b e y t h e n o - c o m p e n s a t i o n m e c h a n i s m . T h e p, P a n d P ' t r a j e c t o r i e s a n d r e s i d u e s in nN s c a t t e r i n g w e r e d e t e r m i n e d f r o m t h i s fit. T h e r e s u l t s a r e r e p o r t e d in t a b l e 1. Table 1 Regge-pole p a r a m e t e r s in ~N scattering. Name
o~(0)
ol'(0)
o~"(0)
Dimension
-
GeV -2
GeV -4
CO
C1
C2
DO
D1
mb • GeV
GeV -2
-
mb.
GeV -2
-41.9
11.7
P ( s o l . 1)
1.0
0.4
-
10.3
1.86
-
P ( s o l . 2)
1.0
0.33
-
5.0
1.57
-
P ' ( s o l . 1)
0.57
1.89
-
15.7
-1.18
-
P ' ( s o l . 2)
0.61
1.24
0.23
15.2
-1.69
-
-
-
p ( s o l . 1)
0.57
1.0
-
1.59
1.79
1.74
29
0.08
p ( s o l . 2)
0.58
1.01
-
1.57
1.91
1.65
28
0.0
-126
5.7
In o r d e r to fit KN d a t a , we h a d to d e t e r m i n e , in a d d i t i o n to t h e w a n d A 2 trajectories and residues, the parameters relating, via factorization, the c o n t r i b u t i o n s of t h e p, P a n d P ' in ~N s c a t t e r i n g to t h o s e in KN s c a t t e r i n g . T h e s e p a r a m e t e r s a r e r e p o r t e d on t a b l e 2 a n d t h e ×2 v a l u e s on t a b l e 3. In s o l u t i o n 2 we i m p o s e d a z e r o s p i n - f l i p a m p l i t u d e s f o r P a n d P ' a s s u g g e s t e d to u s by R. J. N. P h i l i p p s ( p r i v a t e c o m m u n i c a t i o n ) in v i e w of t h e b e h a v i o u r of t h e p o l a r i z a t i o n s in n+p e l a s t i c s c a t t e r i n g ( s e e p r e c e d i n g n o t e s ) . S e v e r a l f e a t u r e s of o u r s o l u t i o n s s h o u l d b e p o i n t e d out. W i t h t h e r e c e n t nN t o t a l c r o s s s e c t i o n d a t a f r o m F o l e y et a l . [7] t h e p a r a m e t e r F 0 ( P ) t a k e s on v a l u e s of a b o u t 0.8 in both s o l u t i o n s ( t h i s i m p l i e s t h a t t h e a s y m p t o t i c t o t a l c r o s s s e c t i o n s h o u l d b e s m a l l e r in KN t h a n in ~N s c a t t e r i n g ) . T h e SU(3) p r e d i c t i o n f o r F0(P) i s r a t h e r w e l l v e r i f i e d . N o t e t h e w e a k n e s s of t h e a n d A 2 s l o p e s , e s p e c i a l l y with s o l u t i o n 2 w h e r e t h e c h a n g e of s i g n of t h e p o l a r i z a t i o n in K - p e l a s t i c s c a t t e r i n g i s due to t h e P ' a n d w h e r e t h e v a n i s h ing of a w ( t ) w o u l d p r o d u c e an u n w a n t e d s e c o n d z e r o in t h e p o l a r i z a t i o n . * See refs. [6], and also refs. [1,2] of our ref. [5].
K-p ELASTIC SCATTERING
311
Table 2 P, P' and p Regge-pole p a r a m e t e r s in KN scattering. Name
F0(P)
FI(P)
F0(P')
FI(P')
F0(P)
FI(P)
Sol. 1
0.82
-0.29
0.47
2.0
0.49
0.54
Sol. 2
0.81
-0.34
0.55
-0.26
0.57
1.4
A 2 and co Regge-pole parameters in K N scattering. Name
0/(0)
0/'(0)
C0
C1
t0(co)
DO
co(sol, i)
0.45
0.52
5.2
3.0
-0.15
-44
-0.15
-24.2
D 1 0.28
co(sol. 2)
0.42
0.28
5.5
3.6
A2(sol. 1)
0.34
0.37
4.2
1.24
-33
-0.38
0.64
A2(sol. 2)
0.37
0.30
6.7
-0.24
46
1.6
Table 3 Approximate X2 values for experimental points. Type
Number of points
Solution 1
Solution 2
d~ (K-p --, K-p) L.E. dt
25
83
51
d(~ (K-p --* K-p) H.E. dt
29
32
32
d ff (K+p -* K+p) H.E. dt
37
46
51
d(r (U-p --* K-'~n)H.E. d--/ P (K-p --* K-p) L.E.
36
32
33
25
22
19
(~tot H.E.
31
51
54
183
268
240
Total
W e o b t a i n e d a v e r y g o o d f i t to t h e p o l a r i z a t i o n d a t a of D a u m et al. f o r m o m e n t u m t r a n s f e r s up to - 1 . 2 GeV 2 (fig. 1). T h e p e a k in t h e p o l a r i z a t i o n a t I tl = 0.65 GeV 2 i s m o r e p r o n o u n c e d f o r o u r s o l u t i o n 2. A l t h o u g h )<2 v a l ues are rather large, the differential cross section for K-p elastic scatteri n g a r o u n d 2.5 G e V / ¢ i s r e a s o n a b l y w e l l d e s c r i b e d , i n c l u d i n g t h e d i p a t a b o u t t = - 0 . 9 GeV 2 ( s e e fig. 2). A t h i g h e n e r g y t h e a g r e e m e n t of o u r s o l u t i o n s w i t h e x p e r i m e n t a l d a t a i s a l m o s t t h e s a m e a s in t h e l i t e r a t u r e [2]. N o t e t h a t t h e p r e d i c t i o n s f o r d ~ / d t a n d ~tot a t h i g h e n e r g y a r e a l m o s t i d e n t i c a l f o r t h e two s o l u t i o n s . It i s n o t e w o r t h y t h a t t h e e x t r a p o l a t i o n of t h e t o t a l c r o s s s e c t i o n s to l o w e n e r g i e s (2 to 3 G e V / ¢ ) , c o m p u t e d f r o m o u r R e g g e - p o l e p a r a m e t e r s , f i t s e x p e r i m e n t a l d a t a v e r y w e l l a l t h o u g h t h e s e d a t a w e r e not t a k e n into a c c o u n t
312
G. PLAUT
_30-~. OTOT
\mb \
_25.
\,, I..~.
K-p
_20.
~p ~l
IG,
I~ qLA8~Vk
Fig. 3. Comparison between experimental data f o r total cross sections of kaons on
protons and our solution 2 (the two solutions are undistinguishable). The black points are experimental points at low energy, not taken into account f o r the X2 minimization. f o r the ×2 minimization. This unexpected a g r e e m e n t is v e r y gratifying (see fig. 3).
5. PREDICTIONS If one believes that Regge poles r e p r e s e n t c o r r e c t l y the actual amplitude, a s is suggested by the a g r e e m e n t with experimental data of Daum et al., it would be i n t e r e s t i n g to test the p r e d i c t i o n s of our solutions for p o l a r i z a t i o n in high e n e r g y K p elastic s c a t t e r i n g and in other KN s c a t t e r i n g experiment. Note that at the p r e s e n t time the only available p o l a r i z a t i o n data in KN s c a t t e r i n g a r e the K-p p o l a r i z a t i o n data of Daum et al. Fig. 4 displays the p r e d i c t i o n s of each of our solutions for the p o l a r i z a tion in K+p elastic s c a t t e r i n g and 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 t h r e e l a b o r a t o r y momenta: 4, 10 and 50 GeV/c. In K-p elastic scattering, the p o l a r i z a t i o n is much l a r g e r with solution 2; at high energy it r e m a i n s n e g a tive beyond t = -1 GeV 2. In K+p elastic s c a t t e r i n g the s a m e solution p r e dicts a l m o s t no p o l a r i z a t i o n between 0 and -1 GeV 2 while solution 1 (Chew m e c h a n i s m ) p r e d i c t s a negative and m o r e substantial polarization. In c h a r g e - e x c h a n g e s c a t t e r i n g both solutions p r e d i c t a l a r g e polarization, negative at f i r s t , and then positive as It I i n c r e a s e s .
6. CONCLUSION With a five Regge pole model for KN s c a t t e r i n g , we were able to account r e a s o n a b l y well for the p o l a r i z a t i o n and differential c r o s s sections in K-p elastic s c a t t e r i n g above 2 G e V / c , t o g e t h e r with all high energy KN data. The fact that our solutions also d e s c r i b e v e r y well the total c r o s s sections between 2 and 3 G e V / c s e e m s to justify a p o s t e r i o r i the use of Regge p o l e s in this r a n g e of l a b o r a t o r y momenta, at l e a s t for c r o s s sections. A t high e n e r g i e s , out two s o l u t i o n s cannot be d i s t i n g u i s h e d by c o m p a r i s o n with e x p e r i m e n t a l data now available: t h e i r p r e d i c t i o n s a r e a l m o s t the same to
K-p ELASTIC SCATTERING PoL . .50
so
IPol ,/
/"
"~
1E_ ~, K'p--~g-p
•5
~P~
313 //~/'~
.50
~.
/.~h"J
1-~
K÷P
Y
I.
FIG 4
Fig. 4. Predictions from our two solutions for polarization in KN scattering. P r e dictions from solution 1 a r e shown in solid lines and from solution 2 in dashed lines.
the p r e dictions f o r the polarization strongly depend on the solution considered.
t h e p o i n t of b e i n g i n d i s t i n g u i s h a b l e in, e.g. fig. 3; on t h e c o n t r a r y ,
A n o t h e r r e s i d u e p a r a m e t r i z a t i o n - in t h e f r a m e w o r k of t h e s a m e m o d e l a n d t h e s a m e m e c h a n i s m s - w o u l d p e r h a p s g i v e o t h e r p r e d i c t i o n s b u t it c a n b e h o p e d t h a t t h e g e n e r a l b e h a v i o u r of t h e p o l a r i z a t i o n w i l l r e p r o d u c e a p p r o x i m a t e l y t h e g e n e r a l f e a t u r e s of o u r two s o l u t i o n s . In c o n c l u s i o n w e c a n n o t s t a t e t h a t o n e of o u r s o l u t i o n s i s f a v o u r e d a t p r e s e n t . M o r e o v e r , c a r e m u s t b e t a k e n in c o m p a r i n g o u r p r e d i c t i o n s w i t h future experimental data: while the qualitative behaviour predicted may p r o v i d e a r e a l t e s t of t h e g e n e r a l R e g g e p o l e m o d e l , p r e c i s e n u m e r i c a l p r e dictions can only test a particular solution. W e w i s h to t h a n k P r o f e s s o r J. C. S e n s a n d a l l t h e p h y s i c i s t s of h i s g r o u p , D r . M. K. G a i l l a r d a n d P r o f e s s o r A. P . C o n t o g o u r i s f o r t h e i r i n t e r e s t in t h i s w o r k a n d f o r i n v a l u a b l e d i s c u s s i o n s . W e a r e a l s o h a p p y to a c k n o w l e d g e fruitful discussions with Dr. Tran Thanh Van, Dr. Le Bellac, Professor R. J. N. P h i l i p p s , D r . Ph. S a l i n , P r o f e s s o r R. O m n ~ s a n d P r o f e s s o r C. B. Chiu.
REFERENCES [1] [2] [3] [4]
V. Singh, Phys. Rev. 129 (1963) 1889. R . J . N . Philipps and W. R a r i t a , Phys. Rev. 139B (1965} 1336. C.B. Chiu, S. Chu and L. Wang, Phys. Rev. 161 (1967) 1563. A. P. Contogouris, H. Lubatti and J. Tran Thanh Van, Phys. Rev. L e t t e r s 19 (1967) 1352. [5] C. Daum, F. C. Ern~, J . P . Lagnaux, J. C. Sens, M. Steuer, F. Udo, Nucl. Phys. Phys. B6 (1968) 273.
314
G. PLAUT
[6] W. Galbraith et al., Phys. Hev, 138B {1965} 913; Aderholz, CERN/D Ph. II, Physics 67-9; K. J. Foley et al., Phys. Rev. Letters 11 {1965) 503; Atsbury et al., Phys. Letters 16 {1965} 328 and CEHN 1966; R. Crittenden et al., Phys. Rev° Letters 12 {1964} 430; Mott et al., Phys. Letters 23 (1966) 171. [7] K.J. Foley et al., Phys. Rev. Letters 19 (1967} 330.
Nuclear Physics B9 (1969) 306-314. North-Holland Publ. Comp., Amsterdam
REGGE POLES A N D P O L A R I Z A T I O N IN K p ELASTIC SCATTERING G. P L A U T * Laboratoire de Physique Thborique el Hautes Energies Facult~ des Sciences. Orsay**
Heceived 30 September 1968
Abstract: A five Regge-pole model is presented and used to fit polarization data in K-p elastic scattering between 2 and 2.4 GeV/c. Following a phenomenological analysis, we obtain solutions consistent with high-energy data and we give sample predictions for polarization in other KN scattering experiments.
1. INTRODUCTION We shall study f r o m the Regge pole s t a n d p o i n t a K - p e l a s t i c s c a t t e r i n g e x p e r i m e n t in which the p r o t o n p o l a r i z a t i o n h a s b e e n m e a s u r e d . In sect. 2 we f i r s t r e c a l l the g e n e r a l f o r m a l i s m of KN e l a s t i c (or c h a r g e exchange) s c a t t e r i n g in the f r a m e w o r k of the Regge p o l e model. Then we give our p a r a m e t r i z a t i o n . In sect. 3, we a n a l y s e the K - p e x p e r i m e n t of Daum et al. [5], in o r d e r to d e t e r m i n e the c o n d i t i o n s that Regge t r a j e c t o r i e s and r e s i d u e s m u s t s a t i s f y in o r d e r to d e s c r i b e c o r r e c t l y the r e s u l t s of t h i s e x p e r i m e n t . In s e c t . 4 we p r e s e n t and d i s c u s s two e x p l i c i t s o l u t i o n s o b t a i n e d f o r the Regge p a r a m e t e r s ; in sect. 5 we c o m p u t e e x p e c t e d v a l u e s of p o l a r i z a t i o n in K - p , K+p and 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 with o u r model.
2. FORMALISM AND P A R A M E T R I Z A T I O N The f o r m a l i s m of K - p e l a s t i c s c a t t e r i n g i s given in ref. [1]. With the help of the two a m p l i t u d e s B and A ' : A' =A +
w+ t / 4 m
B,
1 -t/4m 2
( w h e r e ¢o is the t o t a l l a b e n e r g y of the K-), which a r e p r o p o r t i o n a l to t - c h a n nel h e l i c i t y flip and h e l i c l t y n o n - f l i p a m p l i t u d e s r e s p e c t i v e l y , we w r i t e the o b s e r v a b l e q u a n t i t i e s a s [1,2] * Postal address: Laboratoire de Physique Th~orique, b~tt. 211, Facult~ des Sciences, 91 Orsay. ** Laboratoire associ6 au CNRS.
K-p ELASTIC SCATTERING
2
307
t (s (m+¢o)2 ~1 2] +4m 2 - 1 - t / 4 m 2lIB[
,
~ t o t ( S ) = 1q Im A ' ( s , t = 0)
sin0 Im A'B* P = 16n~f~ da/dt
'
w h e r e q is the K- l a b o r a t o r y m o m e n t u m and k the c e n t e r - o f - m a s s m o m e n tum. k i and k f being the c e n t e r - o f - m a s s m o m e n t a of initial and final kaons, the p o l a r i z a t i o n is counted positive against ~ = k i x k f / I k i x kfl (Basel convention). The r e a d e r is r e f e r r e d to Philipps and R a r i t a [2] for the e x p r e s sions of the p h y s i c a l s - c h a n n e l a m p l i t u d e s in t e r m s of the five t-channel Regge t r a j e c t o r i e s (P, P ' , p, ¢o and A2). We shall now make explicit our p a r a m e t r i z a t i o n . When the complex ang u l a r m o m e n t u m a(t) of a Regge t r a j e c t o r y g o e s to z e r o , r e q u i r e d f a c t o r s p r o p o r t i o n a l to a may be a s s i g n e d to the r e s i d u e s in different ways [3]. We a s s u m e d that odd t r a j e c t o r i e s obey the " s e n s e - c h o o s i n g " m e c h a n i s m $. F o r even t r a j e c t o r i e s we t r i e d two m e c h a n i s m s : Chew m e c h a n i s m (solution 1), N o - c o m p e n s a t i o n m e c h a n i s m (solution 2). We g e n e r a l l y a p p r o x i m a t e d Regge t r a j e c t o r i e s by straight lines a(t) = a(o) + a ' ( o ) x t
but in some e a s e s we added a quadratic t e r m for the P ' (solution 2).
Even trajectories.
The amplitude a r i s i n g f r o m each t r a j e c t o r y is p a r a m e t -
r i z e d as follows:
A ~(t) = -C o exp (C l t ) ai(a i+ 1) el(t) ( A~(t)
: - Co exp(C 1 t)ai(a i2
Bi(t )
=
Here,
_DO e x p ( D l t ) ~i(t)
+1)2
a i2 (oti+ 1)
~i(t )
¢o+t/4rn )ai -~0 /
for the Chew m e c h a n i s m ,
t/4rn) ai k
b~
~i(t)( ¢o+t/4m ~0 ) ~i-1
is the well-known s i g n a t u r e f a c t o r :
for the n o - c o m p e n s a t i o n mechanism, for both m e c h a n i s m s .
~i(t)
= (e -i~c~ + 1)/sin (u~).
:~ As regards the p trajectory, there are several experimental facts
The polariza-
" ~"±p e l a s t i c s c a t t e r i n g a r e o p p o s i t e in s i g n and a p p r o x i m a t e l y e q u a l , b o t h t oi n s m c h a n g i n g s i g n a t t ~ - 0 . 6 GeV2 w h e r e t h e r e i s a dip in t h e c h a r g e - e x c h a n g e d i f f e r -
ential cross section. All these features are very well accounted for by a "sensechoosing" p-trajectory with a complex angular momentum which goes to zero at t ~ -0.6 GeV2.
308
G. PLAUT
Odd trajectories. We p a r a m e t r i z e the amplitudes as follows: for the p t r a j e c t o r y : A~(t) = -(~ + 1) C0[(1 + C2) exp (C 1 t)
- C2]
~i(t)( ¢0+ t/4m )~ ¢o0
/
,
for the ¢o t r a j e c t o r y :
A~(t)
= -(~ + 1) C0(1 -
t/to)
exp (C 1 t)
(¢o+ t/4m (~ ~i(t) \ ~00 )
Both spin-flip amplitudes a r e p a r a m e t r i z e d by:
B (t) = -ai(oti+
1) DO exp ( n 1 t) ~z~t)(
w+ t//4m ) ~i-1 w0
The p a r a m e t r i z a t i o n chosen for the w allows one to fit the c r o s s - o v e r phenomenon in differential c r o s s sections for the K±p elastic s c a t t e r i n g and also to account for the a b s e n c e of a dip in the w contribution [4] to r e a c t i o n s uN -* pN. When t r a j e c t o r i e s and r e s i d u e s a r e a l r e a d y known f r o m the study of uN s c a t t e r i n g we make use of f a c t o r i z a t i o n to obtain a simple p a r a m e t r i z a t i o n
[2]: A 'z~KN) Bi(KN) A,z~7rE) - Bz(uN) - F 0 e x p ( F 1 t). 3. ANALYSIS OF DATA In fig. 1, we p r e s e n t m e a s u r e m e n t s f r o m Daum, Ern~, Lagnaux, Sens, Steuer and Udo [5] at t h r e e l a b o r a t o r y m o m e n t a above 2 GeV/c. It is noteworthy that at all m o m e n t a the p o l a r i z a t i o n is initially positive, goes to z e r o at about t = -0.9 GeV 2 and then b e c o m e s negative (a noticeable dependence of p o l a r i z a t i o n on m o m e n t u m a p p e a r s only at l a r g e r t). M o r e o v e r , t h e r e is a dip in the angular distribution at the s a m e m o m e n t u m t r a n s f e r as the z e r o in the p o l a r i z a t i o n (t = -0.9 GeV 2) (see for instance dc~/dt at 2.44 GeV/c given in fig. 2; see also ref. [5]). This striking stability of the s t r u c t u r e (already t r u e at lower e n e r g i e s [5]) s u g g e s t s that, despite the r a t h e r low value of m o m e n t u m , Daum et a l . ' s r e s u l t s may be r a t h e r well accounted for by c r o s s - c h a n n e l Regge t r a j e c t o r i e s . M o r e o v e r , Chin, Chu and Wang [3] have s u c c e s s f u l l y used Regge poles in ~N s c a t t e r i n g at the same energies. Now we wish to explain the dip in d(r/dt together with a change of sign in the polarization. The most natural explanation is to a s s u m e that the c o m plex angular m o m e n t u m of one of the Regge t r a j e c t o r i e s involved goes to z e r o and that one of the amplitudes due to this t r a j e c t o r y is p r o p o r t i o n a l to for ~ -~ 0. Taking into account the signature f a c t o r , the r e q u i r e d ~ f a c t o r a p p e a r s in the spin-flip amplitudes of odd t r a j e c t o r i e s , and also in the nonflip amplitudes of even t r a j e c t o r i e s which obey a n o - c o m p e n s a t i o n m e c h a nism. Note that we a r e not i n t e r e s t e d in the even t r a j e c t o r i e s contributions
K-p ELASTIC SCATTERING
309
I }. I-
Fig. 1. Polarization in K-p e l a s t i c s c a t tering. Comparison between e x p e r i m e n tal data [5] and our two solutions. The solid lines r e p r e s e n t solution 1 and the dashed ones solution 2.
, i$ OeVk:
Fig. 2. Comparison for d(y(K-p --* K-p)/dt. Solid lines r e p r e s e n t solution i and dashed lines solution 2.
to s p i n - f l i p a m p l i t u d e s w h i c h w e a s s u m e to b e m u c h w e a k e r t h a n t h a t of t h e odd trajectories * Now l e t u s c o n s i d e r t h e o d d t r a j e c t o r i e s . U s i n g t h e o p t i c a l t h e o r e m a n d a p p r o p r i a t e c o m b i n a t i o n s of t o t a l c r o s s s e c t i o n s , w e e s t i m a t e d f r o m e x p e r i m e n t a l d a t a t h e c o n t r i b u t i o n of e a c h t r a j e c t o r y to t h e a m p l i t u d e I m A '(t = 0). W e f o u n d t h a t t h e w c o n t r i b u t i o n w a s m u c h l a r g e r t h a n t h a t of t h e p; t h e A 2 c o n t r i b u t i o n w a s s t i l l s m a l l e r . On t h e o t h e r h a n d t h e SU(3) prediction:
s e e m s to b e w e l l v e r i f i e d a t t = 0 b y e x p e r i m e n t a l d a t a ( o u r a m p l i t u d e s a r e d e f i n e d a s in r e f . [2]). If t h e s p i n - f l i p a m p l i t u d e d u e to t h e p i s d o m i n a n t , i t s h o u l d p r o d u c e a s u b s t a n t i a l n e g a t i v e p o l a r i z a t i o n if o n e b e l i e v e s f a c t o r i z a t i o n (which i m p l i e s t h a t t h e r e l a t i v e s i g n of t h e a m p l i t u d e s A ' a n d B i s t h e s a m e in KN s c a t t e r ing a s in ~N s c a t t e r i n g . T h e r e f o r e it i s n a t u r a l to a s s u m e t h a t t h e ¢o s p i n f l i p a m p l i t u d e i s r e s p o n s i b l e f o r p o l a r i z a t i o n in K - p e l a s t i c s c a t t e r i n g n e a r the forward direction. This spin-flip amplitude should be much larger than t h a t a r i s i n g f r o m t h e p, a n d h a v e t h e o p p o s i t e s i g n . T h e c h a n g e of s i g n of p o l a r i z a t i o n a t t = - 0 . 9 GeV 2 w i l l t h e n b e e x p l a i n e d e i t h e r b y ~¢o(t) g o i n g to z e r o o r b y (~p,(t) g o i n g to z e r o ; in t h e l a t t e r c a s e t h e a m p l i t u d e A~,,(t) w h i c h i n t e r f e r e s w i t h Bw(t) c h a n g e s s i g n . * The fact that the polarizations in 7r±p e l a s t i c s c a t t e r i n g a r e approximately opposite in signs and equal is a strong argument for assuming that the P and P' contributions to spin-flip amplitudes a r e v e r y weak.
310
G. PLAUT
4. D E S C R I P T I O N O F OUR S O L U T I O N S L e t u s b e g i n with a f e w w o r d s a b o u t o u r c o m p u t a t i o n a l m e t h o d s . W e c o n s i d e r e d e x p e r i m e n t a l d a t a * on KN high e n e r g y t o t a l a n d 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 t o g e t h e r with d a t a on p o l a r i z a t i o n a n d 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 in K - p e l a s t i c s c a t t e r i n g b e t w e e n 2 a n d 3 G e V / c . I n d e e d it i s not s i g n i f i c a n t to f i t r a t h e r l o w e n e r g y e x p e r i m e n t a l d a t a in t h e R e g g e p o l e f r a m e w o r k if t h e p a r a m e t e r s o b t a i n e d do not a l s o d e s c r i b e high e n e r g y d a t a w e l l . O u r c o m p u t a t i o n s w e r e p e r f o r m e d on U n i v a c 1108 a t O r s a y with t h e C E R N m i n i m i z a t i o n p r o g r a m MINROS. W e f i r s t o b t a i n e d a f i t f o r ~N d a t a with two d i s t i n c t s o l u t i o n s : in t h e f i r s t one e v e n t r a j e c t o r i e s o b e y t h e C h e w m e c h a n i s m ; in t h e s e c o n d one t h e y o b e y t h e n o - c o m p e n s a t i o n m e c h a n i s m . T h e p, P a n d P ' t r a j e c t o r i e s a n d r e s i d u e s in nN s c a t t e r i n g w e r e d e t e r m i n e d f r o m t h i s fit. T h e r e s u l t s a r e r e p o r t e d in t a b l e 1. Table 1 Regge-pole p a r a m e t e r s in ~N scattering. Name
o~(0)
ol'(0)
o~"(0)
Dimension
-
GeV -2
GeV -4
CO
C1
C2
DO
D1
mb • GeV
GeV -2
-
mb.
GeV -2
-41.9
11.7
P ( s o l . 1)
1.0
0.4
-
10.3
1.86
-
P ( s o l . 2)
1.0
0.33
-
5.0
1.57
-
P ' ( s o l . 1)
0.57
1.89
-
15.7
-1.18
-
P ' ( s o l . 2)
0.61
1.24
0.23
15.2
-1.69
-
-
-
p ( s o l . 1)
0.57
1.0
-
1.59
1.79
1.74
29
0.08
p ( s o l . 2)
0.58
1.01
-
1.57
1.91
1.65
28
0.0
-126
5.7
In o r d e r to fit KN d a t a , we h a d to d e t e r m i n e , in a d d i t i o n to t h e w a n d A 2 trajectories and residues, the parameters relating, via factorization, the c o n t r i b u t i o n s of t h e p, P a n d P ' in ~N s c a t t e r i n g to t h o s e in KN s c a t t e r i n g . T h e s e p a r a m e t e r s a r e r e p o r t e d on t a b l e 2 a n d t h e ×2 v a l u e s on t a b l e 3. In s o l u t i o n 2 we i m p o s e d a z e r o s p i n - f l i p a m p l i t u d e s f o r P a n d P ' a s s u g g e s t e d to u s by R. J. N. P h i l i p p s ( p r i v a t e c o m m u n i c a t i o n ) in v i e w of t h e b e h a v i o u r of t h e p o l a r i z a t i o n s in n+p e l a s t i c s c a t t e r i n g ( s e e p r e c e d i n g n o t e s ) . S e v e r a l f e a t u r e s of o u r s o l u t i o n s s h o u l d b e p o i n t e d out. W i t h t h e r e c e n t nN t o t a l c r o s s s e c t i o n d a t a f r o m F o l e y et a l . [7] t h e p a r a m e t e r F 0 ( P ) t a k e s on v a l u e s of a b o u t 0.8 in both s o l u t i o n s ( t h i s i m p l i e s t h a t t h e a s y m p t o t i c t o t a l c r o s s s e c t i o n s h o u l d b e s m a l l e r in KN t h a n in ~N s c a t t e r i n g ) . T h e SU(3) p r e d i c t i o n f o r F0(P) i s r a t h e r w e l l v e r i f i e d . N o t e t h e w e a k n e s s of t h e a n d A 2 s l o p e s , e s p e c i a l l y with s o l u t i o n 2 w h e r e t h e c h a n g e of s i g n of t h e p o l a r i z a t i o n in K - p e l a s t i c s c a t t e r i n g i s due to t h e P ' a n d w h e r e t h e v a n i s h ing of a w ( t ) w o u l d p r o d u c e an u n w a n t e d s e c o n d z e r o in t h e p o l a r i z a t i o n . * See refs. [6], and also refs. [1,2] of our ref. [5].
K-p ELASTIC SCATTERING
311
Table 2 P, P' and p Regge-pole p a r a m e t e r s in KN scattering. Name
F0(P)
FI(P)
F0(P')
FI(P')
F0(P)
FI(P)
Sol. 1
0.82
-0.29
0.47
2.0
0.49
0.54
Sol. 2
0.81
-0.34
0.55
-0.26
0.57
1.4
A 2 and co Regge-pole parameters in K N scattering. Name
0/(0)
0/'(0)
C0
C1
t0(co)
DO
co(sol, i)
0.45
0.52
5.2
3.0
-0.15
-44
-0.15
-24.2
D 1 0.28
co(sol. 2)
0.42
0.28
5.5
3.6
A2(sol. 1)
0.34
0.37
4.2
1.24
-33
-0.38
0.64
A2(sol. 2)
0.37
0.30
6.7
-0.24
46
1.6
Table 3 Approximate X2 values for experimental points. Type
Number of points
Solution 1
Solution 2
d~ (K-p --, K-p) L.E. dt
25
83
51
d(~ (K-p --* K-p) H.E. dt
29
32
32
d ff (K+p -* K+p) H.E. dt
37
46
51
d(r (U-p --* K-'~n)H.E. d--/ P (K-p --* K-p) L.E.
36
32
33
25
22
19
(~tot H.E.
31
51
54
183
268
240
Total
W e o b t a i n e d a v e r y g o o d f i t to t h e p o l a r i z a t i o n d a t a of D a u m et al. f o r m o m e n t u m t r a n s f e r s up to - 1 . 2 GeV 2 (fig. 1). T h e p e a k in t h e p o l a r i z a t i o n a t I tl = 0.65 GeV 2 i s m o r e p r o n o u n c e d f o r o u r s o l u t i o n 2. A l t h o u g h )<2 v a l ues are rather large, the differential cross section for K-p elastic scatteri n g a r o u n d 2.5 G e V / ¢ i s r e a s o n a b l y w e l l d e s c r i b e d , i n c l u d i n g t h e d i p a t a b o u t t = - 0 . 9 GeV 2 ( s e e fig. 2). A t h i g h e n e r g y t h e a g r e e m e n t of o u r s o l u t i o n s w i t h e x p e r i m e n t a l d a t a i s a l m o s t t h e s a m e a s in t h e l i t e r a t u r e [2]. N o t e t h a t t h e p r e d i c t i o n s f o r d ~ / d t a n d ~tot a t h i g h e n e r g y a r e a l m o s t i d e n t i c a l f o r t h e two s o l u t i o n s . It i s n o t e w o r t h y t h a t t h e e x t r a p o l a t i o n of t h e t o t a l c r o s s s e c t i o n s to l o w e n e r g i e s (2 to 3 G e V / ¢ ) , c o m p u t e d f r o m o u r R e g g e - p o l e p a r a m e t e r s , f i t s e x p e r i m e n t a l d a t a v e r y w e l l a l t h o u g h t h e s e d a t a w e r e not t a k e n into a c c o u n t
312
G. PLAUT
_30-~. OTOT
\mb \
_25.
\,, I..~.
K-p
_20.
~p ~l
IG,
I~ qLA8~Vk
Fig. 3. Comparison between experimental data f o r total cross sections of kaons on
protons and our solution 2 (the two solutions are undistinguishable). The black points are experimental points at low energy, not taken into account f o r the X2 minimization. f o r the ×2 minimization. This unexpected a g r e e m e n t is v e r y gratifying (see fig. 3).
5. PREDICTIONS If one believes that Regge poles r e p r e s e n t c o r r e c t l y the actual amplitude, a s is suggested by the a g r e e m e n t with experimental data of Daum et al., it would be i n t e r e s t i n g to test the p r e d i c t i o n s of our solutions for p o l a r i z a t i o n in high e n e r g y K p elastic s c a t t e r i n g and in other KN s c a t t e r i n g experiment. Note that at the p r e s e n t time the only available p o l a r i z a t i o n data in KN s c a t t e r i n g a r e the K-p p o l a r i z a t i o n data of Daum et al. Fig. 4 displays the p r e d i c t i o n s of each of our solutions for the p o l a r i z a tion in K+p elastic s c a t t e r i n g and 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 t h r e e l a b o r a t o r y momenta: 4, 10 and 50 GeV/c. In K-p elastic scattering, the p o l a r i z a t i o n is much l a r g e r with solution 2; at high energy it r e m a i n s n e g a tive beyond t = -1 GeV 2. In K+p elastic s c a t t e r i n g the s a m e solution p r e dicts a l m o s t no p o l a r i z a t i o n between 0 and -1 GeV 2 while solution 1 (Chew m e c h a n i s m ) p r e d i c t s a negative and m o r e substantial polarization. In c h a r g e - e x c h a n g e s c a t t e r i n g both solutions p r e d i c t a l a r g e polarization, negative at f i r s t , and then positive as It I i n c r e a s e s .
6. CONCLUSION With a five Regge pole model for KN s c a t t e r i n g , we were able to account r e a s o n a b l y well for the p o l a r i z a t i o n and differential c r o s s sections in K-p elastic s c a t t e r i n g above 2 G e V / c , t o g e t h e r with all high energy KN data. The fact that our solutions also d e s c r i b e v e r y well the total c r o s s sections between 2 and 3 G e V / c s e e m s to justify a p o s t e r i o r i the use of Regge p o l e s in this r a n g e of l a b o r a t o r y momenta, at l e a s t for c r o s s sections. A t high e n e r g i e s , out two s o l u t i o n s cannot be d i s t i n g u i s h e d by c o m p a r i s o n with e x p e r i m e n t a l data now available: t h e i r p r e d i c t i o n s a r e a l m o s t the same to
K-p ELASTIC SCATTERING PoL . .50
so
IPol ,/
/"
"~
1E_ ~, K'p--~g-p
•5
~P~
313 //~/'~
.50
~.
/.~h"J
1-~
K÷P
Y
I.
FIG 4
Fig. 4. Predictions from our two solutions for polarization in KN scattering. P r e dictions from solution 1 a r e shown in solid lines and from solution 2 in dashed lines.
the p r e dictions f o r the polarization strongly depend on the solution considered.
t h e p o i n t of b e i n g i n d i s t i n g u i s h a b l e in, e.g. fig. 3; on t h e c o n t r a r y ,
A n o t h e r r e s i d u e p a r a m e t r i z a t i o n - in t h e f r a m e w o r k of t h e s a m e m o d e l a n d t h e s a m e m e c h a n i s m s - w o u l d p e r h a p s g i v e o t h e r p r e d i c t i o n s b u t it c a n b e h o p e d t h a t t h e g e n e r a l b e h a v i o u r of t h e p o l a r i z a t i o n w i l l r e p r o d u c e a p p r o x i m a t e l y t h e g e n e r a l f e a t u r e s of o u r two s o l u t i o n s . In c o n c l u s i o n w e c a n n o t s t a t e t h a t o n e of o u r s o l u t i o n s i s f a v o u r e d a t p r e s e n t . M o r e o v e r , c a r e m u s t b e t a k e n in c o m p a r i n g o u r p r e d i c t i o n s w i t h future experimental data: while the qualitative behaviour predicted may p r o v i d e a r e a l t e s t of t h e g e n e r a l R e g g e p o l e m o d e l , p r e c i s e n u m e r i c a l p r e dictions can only test a particular solution. W e w i s h to t h a n k P r o f e s s o r J. C. S e n s a n d a l l t h e p h y s i c i s t s of h i s g r o u p , D r . M. K. G a i l l a r d a n d P r o f e s s o r A. P . C o n t o g o u r i s f o r t h e i r i n t e r e s t in t h i s w o r k a n d f o r i n v a l u a b l e d i s c u s s i o n s . W e a r e a l s o h a p p y to a c k n o w l e d g e fruitful discussions with Dr. Tran Thanh Van, Dr. Le Bellac, Professor R. J. N. P h i l i p p s , D r . Ph. S a l i n , P r o f e s s o r R. O m n ~ s a n d P r o f e s s o r C. B. Chiu.
REFERENCES [1] [2] [3] [4]
V. Singh, Phys. Rev. 129 (1963) 1889. R . J . N . Philipps and W. R a r i t a , Phys. Rev. 139B (1965} 1336. C.B. Chiu, S. Chu and L. Wang, Phys. Rev. 161 (1967) 1563. A. P. Contogouris, H. Lubatti and J. Tran Thanh Van, Phys. Rev. L e t t e r s 19 (1967) 1352. [5] C. Daum, F. C. Ern~, J . P . Lagnaux, J. C. Sens, M. Steuer, F. Udo, Nucl. Phys. Phys. B6 (1968) 273.
314
G. PLAUT
[6] W. Galbraith et al., Phys. Hev, 138B {1965} 913; Aderholz, CERN/D Ph. II, Physics 67-9; K. J. Foley et al., Phys. Rev. Letters 11 {1965) 503; Atsbury et al., Phys. Letters 16 {1965} 328 and CEHN 1966; R. Crittenden et al., Phys. Rev° Letters 12 {1964} 430; Mott et al., Phys. Letters 23 (1966) 171. [7] K.J. Foley et al., Phys. Rev. Letters 19 (1967} 330.