- Email: [email protected]
ϱ + R Regge-pole model for isobar production
~
Nuclear Physics B13 (1969) 651-661. North-Holland Publ. Comp., A m s t e r d a m
0 + R REGGE-POLE M O D E L FOR ISOBAR P R O D U C T I O N M. K R A M M E R Deutsches Elektronen-Synchrotron DESY, Hamburg and U. M A O R * Tel-Aviv University, Tel-Aviv Received 28 March 1969 Abstract: The reactions ?r+p---*yoA++, ?r+p---*?TA++ and K+p -'~ K oA++ a r e analyzed in a p +R Regge-pole model. Kinematmal s m g u l a r i t m s and threshold (pseudothreshold) conditions a r e incorporated in the p a r a m e t r i z a t l o n . The polarization data a r e examined and their i m p h c a h o n s with r e s p e c t to the dynamical s t r u c t u r e of the heliclty amphtudes a r e inferred. SU(3) s y m m e t r y and exchange degeneracy a r e apphed m o r d e r to minimize the number of free p a r a m e t e r s . A s a h s f a c t o r y d e scription of these i s o b a r production p r o c e s s e s is obtained.
1. I N T R O D U C T I O N ** In t h i s r e p o r t w e s u m m a r i z e t h e r e s u l t s of a d e t a i l e d R e g g e - p o l e a n a l y s i s w h i c h h a s b e e n c a r r i e d out f o r t h e r e a c t i o n s : ~+p ~ ~o~++(1238) ,
(1)
7r+p ~ ~?A++(1238) ,
(2)
K+p ~ K°A++(1238) .
(3)
An o b v i o u s s i m p l i f i c a t i o n of t h e a n a l y s i s of t h e s e i n e l a s t i c p r o c e s s e s c o n s i s t s in t h e l i m i t e d n u m b e r of R e g g e p o l e s e x c h a n g e d in t h e t - c h a n n e l . R e a c t i o n s (1) a n d (2) c a n p r o c e e d o n l y t h r o u g h a s i n g l e R e g g e - p o l e e x c h a n g e w i t h t h e q u a n t u m n u m b e r s of t h e p a n d A 2 (the R t r a j e c t o r y ) , r e s p e c t i v e l y . R e a c t i o n (3) p r o c e e d s t h r o u g h t h e e x c h a n g e of b o t h p a n d A2 t r a j e c t o r i e s . Complications are introduced, however, by the unequal mass kinematics a n d h i g h s p i n s of t h e e x t e r n a l p a r t i c l e s . T h r o u g h o u t o u r i n v e s t i g a t i o n w e * R e s e a r c h sponsored by the A i r F o r c e Offme of Sclenhfm Research, Offme of A e r o s p a c e R e s e a r c h , Umted States A i r F o r c e , under AFOSR grant number EOOAR-68-0010, through the European Office of A e r o s p a c e R e s e a r c h . ** A p r e l i m i n a r y version of this paper was submitted to the 14th Int. Conf. on highenergy physics, Vmnna, 1968.
652
M. KRAMMER and U. MAOR
a s s u m e the e x i s t e n c e of daughter t r a j e c t o r i e s which a s s u r e the p r o p e r Regge a s y m p t o t i c b e h a v i o r in the s - c h a n n e l f o r w a r d direction. T h e p r e s e n t w o r k e n l a r g e s on our p r e v i o u s p a p e r s [1-3] in that both the k i n e m a t i c a l s i n g u l a r i t i e s and t h r e s h o l d (pseudothreshold) conditions of the helicity a m p l i t u d e s a r e fully i n c o r p o r a t e d in the p a r a m e t r i z a t i o n . Also, the s i g n i f i c a n c e of s i m u l t a n e o u s l y fitting the different r e a c t i o n s is s t r e s s e d and i m p l i c a t i o n s of SU(3) s y m m e t r y and exchange d e g e n e r a c y a r e explored. In sect. 2 we s u m m a r i z e the R e g g e - p o l e p a r a m e t r i z a t i o n r e l e v a n t to our p r o b l e m and d i s c u s s s o m e qualitative f e a t u r e s of it. The data u s e d and the fits obtained a r e given in sect. 3. Our final c o n c l u s i o n s a r e contained in sect. 4.
2. P A R A M E T R I Z A T I O N
We c o n s i d e r the t - c h a n n e l p r o c e s s ~1+~2 ~ i S + A , w h e r e n l , ~2 stand f o r ~r, K and ~7 a p p r o p r i a t e l y . We denote the t-channel helicity a m p l i t u d e s by f¢i, w h e r e i and j r e f e r to the h e l i c i t i e s of i0 and A, and p stands for the Reg~e t r a j e c t o r y . T h e n o r m a l i z a t i o n is, a s b e f o r e [1-3],
d~_
dt
~
~ 12
sp2 n ½ i,j ~ [~f'" p U
"
(4)
Following the routine p r o c e d u r e [4, 5] one obtains the following a m p l i t u d e s f r e e f r o m k i n e m a t i c a l s i n g u l a r i t i e s , suitable for R e g g e i z a t i o n
p
(t)(sinO~li-JlKij(t) ~qq'
cos0
aP(t)-Ii-Jl
w h e r e q, q' and 0 t a r e the incoming m o m e n t a , outgoing m o m e n t a and s c a t t e r i n g a~gle in the t-channel c e n t r e - o f - m a s s s y s t e m ; So is a scaling constant, y~j(t) a r e the s i n g u l a r i t y f r e e r e s i d u e s , and
bP(t) bo(t)P r[ap(t)1 + a]
1 ± exp [- i~ap(t)] sin~rap(t) '
with
bPc(t) = ~2T h e Wang f a c t o r s [5]
Kij(t)
K½½(t)= { I t -
F[ap(t)+~] ~- 1
C6)
a r e given by
(m A - mp}2][t- (m A + rap)2]½ F 1 ,
(Ta)
p+R REGGE-POLE MODEL
653
~(t) = K½{(t) = l[t- (m~l- m~r2)2][t(mh- (m~lJ
-_-
rnp)2.
K½_~(t) = l [ t -
,
(7b)
(mlr 1 - rnTT2)2][t - (mlr 1 + mlT2)2][t - (mA+ mp)2] ½ .
(7c)
T h e functions g~/~(t) depend on the coupling and the ghost-killing m e c h a n i s m c h o s e n by the tr~tjectory at ap(t) = O. Optional g~(t) f o r m s c o r r e s p o n d i n g to different m e c h a n i s m s a r e given in table 1. The ~xperimental information on pion-nucleon c h a r g e - e x c h a n g e s c a t t e r i n g suggests a s e n s e - c h o o s i n g coupling f o r the p t r a j e c t o r y [6]. Unless a v e r y flat A 2 t r a j e c t o r y is t o l e r a t e d , only the Gell-Mann m e c h a n i s m is compatible [2] with the e x p e r i m e n t a l data on r e a c t i o n s (2) and (3). Table 1 Optional couplings and ghost-killing mechanisms for reactions (1-3). t-channel hehcity amplitude
Sense choosing mechanism
Gell-Mann mechanism
Chew mechanism
No compensation mechanism
f~½
1
Olp(ap +1)
a'p
q~(ap +1) 2
f;~
Otp ap(ap-
.f~_~
O~p 1)
ap(ap-
1)
qp
a 2p
q2(ap - 1)
q~(ap-1)
We have to c o n s i d e r also the k i n e m a t i c a l c o n s t r a i n t s imposed on the helicity amplitudes at the P - A t h r e s h o l d and pseudothreshold. In p a r t i c u l a r , ignoring the t h r e s h o l d conditions at t+ = (mA+mp) = 4.72 GeV 2 and conside r i n g only the conditions at the p s e u d o t h r e s h o l d t_ = (m A -rap) 2 =0.09 GeV2, one obtains [7, 8] 1 1 + (.f½_~-- d-3 f½2) 11 i(f½~ - v~ f½_~) = O(t - t-)
1 3 (t- t_)[i(4~ f½~ +-f~__~_)(d-~ f½_: 2 2"
(t- t-)[i(~
f½~ + f_,_;). ~ ~- + ( ¢ 5
+f½½)] = O(t- t_)
f½_~+f½½)] = o ( t 1
3
,
t_) ,
(8)
w h e r e fij stands f o r the full t-channel helicity amplitude. A lengthy discussion on the implications of t h e s e c o n s t r a i n t s has been given in ref. [7]. We
M. KRAMMER and U. MAOR
654
onl Y want to r e c a l l that s i m p l e p a r a m e t r i z a t i o n s of the fii, s a t i s f y i n g eqs. (8), lead to s p i n - d e n s i t y m a t r i x e l e m e n t s which a r e v e r y different f r o m t h o s e s e e n in A ++ decay in r e a c t i o n s (1) and (3). In o r d e r to r e m o v e this difficulty a l e s s d i v e r g e n t f o r m f o r the t - c h a n n e l helicity a m p l i t u d e s was s u g g e s t e d [7], yielding a new p a r a m e t r i z a t i o n in which
yPx = [- ~ X4(t - mp)2
+Xl(t_
22
/,
, =[
-~--~
t_)]aP(t_) , __ 22
+ x (t- t_)],,,,___(t_),
x4 (t-
2
(t- t_)]a
(t_) ,
= [Z4(t- % 2)
Xl+(
X2_X3)(t_t_)]aP - - -
aPij (t-)
=(t_) ½ li-J
7_1~ : [ - X 4 ( t -
m )+
2
22
2
,
(9)
2
where
(- 1)i-J •. l (So) li-J l gPij(t_)
(10)
The functions X1, X~., X 3, X4 a r e a r b i t r a r y , and we will a s s u m e that they a r e constants. S o m e qualitative c o n c l u s i o n s can be drawn even at this e a r l y s t a g e of our investigation. The p o l a r i z a t i o n of the i s o b a r in r e a c t i o n s (1)-(3) is known to be in r e a s o n a b l e good a g r e e m e n t with an M1 t r a n s i t i o n , i.e.: f ~ ! =fl_~_ - 0 , 22
2
2
f la = f ' 3 f l _ _ l • 22
2
2
It is evident, t h e r e f o r e , that any fit to the e x p e r i m e n t a l data on r e a c t i o n s (1)-(3) is bound to f o r c e X 4 to be v e r y s m a l l . F o r the 0 t r a j e c t o r y exchange we conclude , t h e r e f o r e , that the k i n e m a t i c a l f a c t o r s at t = ( m . A - m/)) ~ 2 de-. g e n e r a t e to the B o r n p - e x c h a n g e f o r m ( d y n a m i c a l exception m J a c k s o n - H l t e d e s c r i p t i o n [7]). No such s i m p l e c o r r e s p o n d e n c e can be m a d e f o r the A 2 t r a j e c t o r y exchange for which the A2(2 +-) is the l o w e s t B o r n pole. D e s p i t e that, we o b s e r v e e m p i r i c a l l y that r e s i d u e s of the A 2 t r a j e c t o r y choose a l e s s d i v e r g e n t f o r m tha~0the one ~ptioned by the Wang method [5]. The conclusion that f ~ and f ~ have the s a m e d y n a m i c a l s t r u c t u r e is f u r t h e r s u p p o r t e d by the~observation that the P t r a j e c t o r y c h o o s e s a s e n s e m e c h a n i s m at ap = 0, w h e r e a s the A 2 t r a j e c t o r y c h o o s e s the G e l l - M a n n m e c h a n i s m at aA2 = 0 (see table 1). Consequently, all the helicity a m p l i tudes but f½½ have the s a m e a f a c t o r s f o r P and A 2 exchange. As we shall s e e in sect. 3, the r e l a t i v e contribution of f½½ is r a t h e r s m a l l and i n s e n s i t i v e to the detailed a dependence. We thus find s u p p o r t f o r m o d e l s a s s u m i n g the s a m e i n t e r a c t i o n f o r 0 and A 2 t r a j e c t o r y e x c h a n g e s [9]. It is p a r t i c u l a r ly i n t e r e s t i n g to c h e c k an e x c h a n g e - d e g e n e r a c y m o d e l [10] which a s s u m e s the identity of the P and A 2 r e s i d u e s [11].
p+R REGGE-POLE MODEL
655
3. DATA C O L L E C T I O N AND FITS P E R F O R M E D We used 78 data points f o r r e a c t i o n (1) c o v e r i n g the t - d e p e n d e n c e of both the c r o s s section and 4 ++ p o l a r i z a t i o n a r o u n d 3.5 G e V / c [12], at 4.0 G e V / c [13] and at 8.0 G e V / c [14]. A total of 14 data points f o r the differential c r o s s section of r e a c t i o n (2) a r o u n d 3.5 G e V / c [15] * and at 8.0 G e V / c [14] w e r e taken. Finally, we combined to 88 data points the values of da/dt and Prnrn' for r e a c t i o n (3) obtained by the C E R N - B r u x e l l e s c o l l a b o r a t i o n at 3.0, 3.5, 5.0 and 8.25 G e V / c and quoted in a r e c e n t compilation [16]. Additional i n f o r m a t i o n on r e a c t i o n (3) at 10.0 G e V / c [17] and 13.0 G e V / c [18] has been obtained r e c e n t l y . T h e s e points were not included in the actual fit but we shall c o m m e n t on the c o m p a r i s o n of our calculations with t h e s e experimental results. A s i m u l t a n e o u s fit to r e a c t i o n s (1)-(3) is i m p o r t a n t as, with the r e q u i r e ment that t r a j e c t o r i e s with the s a m e quantum n u m b e r s contributing to diff e r e n t p r o c e s s e s a r e identical, we can i m p o s e additional c o n s t r a i n t s such a s f a c t o r i z a t i o n of the r e s i d u e s and SU(3) s y m m e t r y . Throughout this calculation we have a s s u m e d the SU(3) r e l a t i o n s :
~y+TIA2 ~ 2 )'K+~OA2 7n+~op
= ~
VK+EOp •
(11)
Deviations f r o m exact s y m m e t r y w e r e investigated by introducing different s c a l i n g c o n s t a n t s So for different r e a c t i o n s . Five p o s s i b l e solutions w e r e u s e d in the fitting p r o c e d u r e : Solution 1. We a s s u m e exact SU(3) s y m m e t r y and use the t r a j e c t o r i e s d e t e r m i n e d [19] f r o m p i o n - n u c l e o n c h a r g e exchange and compatible ** with 7/production.
Up(t) = 0.57 + 0 . 9 6 t aA2(t) = 0.4 + 0 . 9 t
(12)
No r e l a t i o n between the p and A2 r e s i d u e s is imposed. Solution 2. We a s s u m e exact SU(3) and ' a p p r o x i m a t e ' exchange d e g e n e r acy. The t r a j e c t o r i e s a r e given by eq. (12). The Xi p a r a m e t e r s of eq. (9) a r e a s s u m e d to be identical f o r p and A2 exchange. The a f a c t o r s for f~_~ and f A ~ a r e kept different. 22 22
* Since no absolute values are given, we used as normalization the total cross section at 3.5 GeV/c obtained by interpolating data quoted in: D. R. O. Morrison, invited paper presented at the Conf. on high-energy two-body reactions, Stony Brook, CERN/TC/PHYSICS 66-20. The A++ polarization m this experiment was found to be similar to the one observed m reactions (1) and (3) (G.Gidal, private communication). ** The reactmn ?r-p--~7;n can be fitted well w~th a A = 0.4+0.9t. P.Mannhe~m, private communication; see also footnote (10) m re~. [2].
656
M. KRAMMER and U. MAOR
S o l u t i o n 3. S a m e a s 2, b u t w i t h o u t e x a c t SU(3) - i . e . w e u s e d i f f e r e n t scaling constants for reactions (1)-(3). S o l u t i o n 4. W e a s s u m e e x a c t SU(3) a n d e x c h a n g e d e g e n e r a c y i n w h i c h ap(t) = a A 2 ( t ) = 0 . 5 + 0 . 9 t . All the p and A2 residues multiplies fP~).
(13)
a r e i d e n t i c a l ( i . e . , a n a d d i t i o n a l e p ( e p + 1) f a c t o r
22
S o l u t i o n 5. S a m e a s 4, b u t w i t h o u t e x a c t SU(3). Table 2 P a r a m e t e r s for the best fits obtained Dimension
XCGeV
Solution 1
Solution 2
Solution 3
Solution 4
Solution 5
0.348
0.290
0.319
0.270
0.262
X2p
GeV
0.427
0.450
0.427
0.410
0.433
X3p
GeV
1.528
1.444
1.460
1.333
1.336
X4p
GeV
0.095
0.084
0.088
0.084
0.078
X~I2 GeV
-0.268
0.290
0.319
0.270
0.262
X~22 GeV
0.965
0.450
0.427
0.410
0.433
X~32 GeV
1.129
1.444
1.460
1.333
1.336
X~42 GeV
-0.042
0.084
0.088
0.084
0.078
S 7r GeV 2
1.630
1.561
1.600
0.785
0.642
S T} GeV 2
1.630
1.561
1.314
0.785
1.594
S K GeV 2
1.630
1.561
1.471
0.785
0.799
O
O
O
X2
288.5
311.0
308.0
391.3
346.5
d e g r e e s of freedom
171
175
173
175
173
Ozp(t) = 0 . 5 7 + 0 . 9 6 t CCA2(t) -- 0.40 +0.90 t P a r a m e t e r s for r e a c t i o n (1): X p ~/2. P a r a m e t e r s for r e a c t i o n (2): X A2 P a r a m e t e r s for reaction (3): X/p, X A2.
'Otp(t) = °tA2(t)7 = 0.50 +0.90t
657
p +R R E G G E - P O L E MODEL
The p a r a m e t e r s a d j u s t e d to l o w e s t X2 f o r 180 data points a r e s u m m a r i z e d in t a b l e 2, and the c u r v e s c o r r e s p o n d i n g to solution 1 a r e shown in figs. 1 to 5. The v a l u e s of X2 p e r d e g r e e of f r e e d o m a r e s o m e w h a t l e s s s p e c t a c u l a r than t h o s e obtained in the e a r l i e r fits [1, 2]. T h i s should not be a t t r i b u t e d to the new p a r a m e t r i z a t i o n . The fit fails, w h e r e we expect the ' n o n s e n s e ' dip due to the v a n i s h i n g of ap(t), s i n c e the dip s t r u c t u r e is m u c h l e s s p r o n o u n c e d (see fig. 1) than r e q u i r e d by the b u i l t - i n a - d e p e n d e n c e . An a p p r o p r i a t e e s t i m a t e of the ~p b a c k g r o u n d would be d e s i r a b l e in o r d e r to c l a r i f y w h e t h e r r e a c t i o n (1) does not show a p r o n o u n c e d dip, or w h e t h e r it is filled up by a b a c k g r o u n d with f l a t t e r t - d e p e n d e n c e than genuine A production. Also, this s i m p l e R e g g e - p o l e m o d e l p r e d i c t s e n e r g y independent spin density m a t r i x e l e m e n t s . The e x p e r i m e n t a l o b s e r v a t i o n s (see fig. 2) c o n f i r m this b e h a v i o u r only qualitatively. The o b s e r v e d d i f f e r e n c e s m a y n e c e s s i t a t e s o m e r e f i n e m e n t s of the p r e s e n t s i m p l e model. Another p r o b l e m a t i c a s p e c t of the fit is the e n e r g y dependence of the total c r o s s s e c t i o n f o r r e a c t i o n (3). Our input contains only data points with incoming m o m e n t a f r o m 3.0 to 8.25 G e V / c . The new p r e l i m i n a r y data at 10.0 and 13.0 G e V / c [17, 18] indicate a d i f f e r e n t i a l d i s t r i b u t i o n and spin d e n s i t y m a t r i x e l e m e n t s c o n s i s t e n t with our calculations. The i n t e g r a t e d c r o s s section, h o w e v e r , s e e m s to d e c r e a s e f a s t e r than anticipated. T h i s is p a r t i c u l a r l y t r u e f o r the 10.0 G e V / c data which a r e r e l a t i v e l y s m a l l . If t h e s e c r o s s s e c t i o n s a r e c o n f i r m e d , a value of aA2(0) is needed which is m u c h l o w e r than the v a l u e s u s e d in the p r e s e n t calculations.
10-
~+ p ~
r~° A +÷ 1t* p ~
~
I~° A ++
3 5 GeV/c 0 GeV/c
E o
0,I
Ol 02 03 O~ 050G 07 OB 0910 II 12 13 I/, 15 - t [{GeV/c)2 ]
£2. £,
£6
1:0 ,'1 ,z 13 ,:, 15
-t ((6eV/c) 2]
F i g . 1. 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 f o r r e a c t i o n (1). The s o l i d h n e c o r r e s p o n d s to s o l u t i o n 1.
658
M. K R A M M E R and U. MAOR
O3
]i,.)" o~ J/L
II II
o 2
• 3,5GeV/c I o ~.o6,w~ P~
r
O0
a
-01
[
_ _
_
-02 -03
01 O1 02 03 OZ, 0 5 0 5 07 08 09 10 11 - t [[GeV/c)2 ]
Fig. 2. Spin denszty m a t r i x e l e m e n t s f o r r e a c t i o n (1). The s o h d l i n e c o r r e s p o n d s to s o l u t i o n 1.
011 f
~
T~+P~
io,:
L~''+
L 60 GeV/c
0.01.
\
o:~ o:~ 03 o:, o:5 ~6 ~7 o:a ~9/o '
-t [( GeV/c)z)
Fzg. 3. Dzfferentzal c r o s s s e c t z o n s f o r r e a c t i o n (2). The s o h d line c o r r e s p o n d s to solutxon 1.
p + R R E G G E - P O L E MODEL
lO
'
< ~%00eV/c
659
K+p--KO& ++ 10.
~
2"
5 0 GeVlc
~tO
w
01
Ol
01 112 03 l& 05 0.6 ~7 08 ~9 10 -t llGeV/c)2]
- t [(GeV/c) 2 ]
F i g . 4. 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 f o r r e a c t i o n (3). T h e s o l i d l i n e c o r r e s p o n d s to s o l u t i o n 1. T h e d a t a at 3.0 G e V / c h a v e r a t h e r l a r g e s t a t i s t i c a l f l u c t u a t i o n s , as can be s e e n by c o m p a r i n g t h e full p o i n t s and c r o s s e s which dzffer by the chozce of t - i n t e r v a l s only. (The fit i s b a s e d on the full p o i n t s . )
OG 03 02 O1 ,
,
,
+ ' T 8 25Ge'J/c
02 i . . . . Ol O0 -01 -02 -03 Ot 03 02 Ol
t
Ol 02 03 0/, 05 06 07 08 Og -t [(GeV;c)2]
F i g . 5. Spzn d e n s i t y m a t r i x e l e m e n t s f o r r e a c t z o n (3). The s o l i d line c o r r e s p o n d s to s o l u t i o n 1.
660
M. KRAMMER and U. MAOR
4. FINAL REMARKS The c h a r a c t e r i s t i c s of our p r e s e n t solutions a r e c o n s i s t e n t with the e a r l i e r fits [1, 2]. Our p r e v i o u s t r e a t m e n t was incomplete insofar as the kinem a t i c a l c o n s t r a i n t s at the p - A p s e u d o t h r e s h o l d (t_ = 0.09 GeV 2) w e r e not i n c o r p o r a t e d explicitly in the formulation. The old calculations r e q u i r e d a s t r o n g t - d e p e n d e n c e of the fitted r e s i d u e s ; this t - d e p e n d e n c e is in qualitative a g r e e m e n t * with the p r e s e n t p a r a m e t r i z a t i o n . Extrapolating the M1 type r e s i d u e s to t = m 2, we obtain (G~+pA++)/47r 30. Usin~ the v e c t o r m e s o n dominance model, with a~/7 2 = 0 . 3 6 5 . 1 0 -2, the value GZopA+/4~ ~ 20 is c o n s i s t e n t with the photoexcitation of the A+(1238)" r Our a n a l y s i s of the e x p e r i m e n t a l data shows that the t - c h a n n e l helicity amplitudes choose a l e s s divergent f o r m than the one optioned by the c r o s s ing m a t r i x [5]. This f e a t u r e is found for both the p and the A2 t r a j e c t o r y exchanges. The s i m i l a r i t y in f o r m and magnitude of the p and A 2 r e s i d u e s is consistent with an exchange d e g e n e r a c y hypothesis. When combined with SU(3) relations it is supported by the data a n a l y s i s only in the ' a p p r o x i m a t e ' f o r m s (solutions 2, 3). The hypothesis of exact SU(3) and exchange d e g e n e r a c y (solution 4) gives, within the model p r e s e n t e d here, r e s u l t s which a r e not quite s a t i s f a c t o r y . A somewhat b e t t e r fit is obtained if we a s s u m e exchange d e g e n e r a c y without exact SU(3) (solution 5). We note that ' a p p r o x i m a t e ' exchange d e g e n e r a c y does not support s y m m e t r y b r e a k i n g effects, as can be seen by c o m p a r i n g the X2 values for solutions 2 and 3. The conclusion of our a n a l y s i s is that a simple R e g g e - p o l e model, c o m bined with SU(3) r e l a t i o n s between the a p p r o p r i a t e r e s i d u e s , gives a r e a sonably a c c u r a t e d e s c r i p t i o n of the inelastic r e a c t i o n s d i s c u s s e d . In our opinion s o m e of the f e a t u r e s indicated by the data and lying outside the r a n g e of this model need f u r t h e r e x p e r i m e n t a l investigation b e f o r e m o r e ambitious evaluations, like inclusion of a b s o r p t i o n effects [20] or investigation of fixed poles connected with the t h i r d double s p e c t r a l function [21], will b e c o m e significant. After the Vienna Conference we r e c e i v e d a p r e p r i n t by G. H. Renninger and K. V. L. S a r m a , in which they p r e s e n t fits for the i s o b a r production p r o c e s s e s within the f r a m e w o r k of the invariant amplitudes.
RE FE RENCES [1] [2] [3] [4]
M.Krammer and U.Maor, Nuovo Clmento 50A (1967) 963. M.Krammer and U.Maor, Nuovo Clmento 52A (1967) 308. M.Krammer, Nuovo Clmento 52A {1967) 931. M.Gell-Mann, M.Goldberger, F. Low, E. Marx and F. Zacharmsen, Phys. Rev. 133 (1964) B145.
* Observing that the old parametrlzatlon 7(t) ~ (1 - t / t o ) gave good results for 6 ~< ..< 1 / t o ~< 15 GeV-2, 1.e. for t o ~ (rnA - rnN)2, it is compatible with the present one.
p + R REGGE-POLE MODEL
661
[5] L . L . W a n g , Phys. Rev. 142 (1966) 1187. [6] See, e.g., L.Bertocchl, in: Proc. of the Heidelberg Int. Conf. on elementary particles (North-Holland Publ. Comp., Amsterdam, 1968). [7] J . D . J a c k s o n and G . E . H i t e , Phys. Rev. 169 (1968) 1248. [8] E.Gotsman and U.Maor, Phys. Rev. 171 (1968) 1495. [9] R.Dashen and S . F r a u t s c h i , Phys. Rev. 152 (1966) 1450. [10] R . C . A r n o l d , Phys. Rev. Letters 14 (1965) 657; Phys. Rev. 153 (1967) 1506. [11] A.Ahmadzadeh, Phys. Rev. Letters 16 (1966) 952. [12] G.Gidal, G . B o r r e a n l , D.Brown, F . L o t t , S.Y. Fung, W.Jackson and R . P u , UCRL-18351. [13] G e r m a n - B r i t i s h Collaboration, Phys. Letters 10 (1964) 229 and private communication, N. Schmitz, Proc. of the 1965 E a s t e r School for Physicists, CERN 65-24/I. [14] A a c h e n - B e r h n - C E R N Collaboration, Nucl. Phys. B8 (1968) 45; D. R. O. Morrlson and P. F. Dalplaz, private communicahon. [15] D. Brown, G.Gidal, R . W . B l r g e , R.Bacastow, S.Y. Fung, W, Jackson and R . P u , Phys. Rev. Letters 19 (1967) 664. [16] CERN-Bruxelles Collaboration, Vmnna Int. Conf. on high-energy physics (paper nr. 445) ; Y. Goldschmldt-Clermont and F. Muller, private communication. ~ I T ~ i r m i n g h a m - G l a s g o w - O x f o r d Collaboration, Proc. of the Topical Conf. on highenergy colhslons of hadrons, Vol. II, p. 121, CERN 68-7. [18] J . C . B e r h n g h i e r i , M . S . F a r b e r , T . F e r b e l , B . E . F o r m a n , A . C . M e l i s s m o s , B. F. Slattery, T. Yamanouchi and H. Yuta, Proc. of the Topical Conf. on highenergy collisions of hadrons, Vol. II, p. 172, CERN 68-7. [19] G.H(ihler, J.Baacke and G.Eisenbe, ss, Phys. Letters 22 (1966) 203. [20] F.Schrempp, Nucl. Phys. B6 (1968) 487. [21] S.Mandelstam and L . L . W a n g , Phys. Rev. 160 (1967) 1490.
Nuclear Physics B13 (1969) 651-661. North-Holland Publ. Comp., A m s t e r d a m
0 + R REGGE-POLE M O D E L FOR ISOBAR P R O D U C T I O N M. K R A M M E R Deutsches Elektronen-Synchrotron DESY, Hamburg and U. M A O R * Tel-Aviv University, Tel-Aviv Received 28 March 1969 Abstract: The reactions ?r+p---*yoA++, ?r+p---*?TA++ and K+p -'~ K oA++ a r e analyzed in a p +R Regge-pole model. Kinematmal s m g u l a r i t m s and threshold (pseudothreshold) conditions a r e incorporated in the p a r a m e t r i z a t l o n . The polarization data a r e examined and their i m p h c a h o n s with r e s p e c t to the dynamical s t r u c t u r e of the heliclty amphtudes a r e inferred. SU(3) s y m m e t r y and exchange degeneracy a r e apphed m o r d e r to minimize the number of free p a r a m e t e r s . A s a h s f a c t o r y d e scription of these i s o b a r production p r o c e s s e s is obtained.
1. I N T R O D U C T I O N ** In t h i s r e p o r t w e s u m m a r i z e t h e r e s u l t s of a d e t a i l e d R e g g e - p o l e a n a l y s i s w h i c h h a s b e e n c a r r i e d out f o r t h e r e a c t i o n s : ~+p ~ ~o~++(1238) ,
(1)
7r+p ~ ~?A++(1238) ,
(2)
K+p ~ K°A++(1238) .
(3)
An o b v i o u s s i m p l i f i c a t i o n of t h e a n a l y s i s of t h e s e i n e l a s t i c p r o c e s s e s c o n s i s t s in t h e l i m i t e d n u m b e r of R e g g e p o l e s e x c h a n g e d in t h e t - c h a n n e l . R e a c t i o n s (1) a n d (2) c a n p r o c e e d o n l y t h r o u g h a s i n g l e R e g g e - p o l e e x c h a n g e w i t h t h e q u a n t u m n u m b e r s of t h e p a n d A 2 (the R t r a j e c t o r y ) , r e s p e c t i v e l y . R e a c t i o n (3) p r o c e e d s t h r o u g h t h e e x c h a n g e of b o t h p a n d A2 t r a j e c t o r i e s . Complications are introduced, however, by the unequal mass kinematics a n d h i g h s p i n s of t h e e x t e r n a l p a r t i c l e s . T h r o u g h o u t o u r i n v e s t i g a t i o n w e * R e s e a r c h sponsored by the A i r F o r c e Offme of Sclenhfm Research, Offme of A e r o s p a c e R e s e a r c h , Umted States A i r F o r c e , under AFOSR grant number EOOAR-68-0010, through the European Office of A e r o s p a c e R e s e a r c h . ** A p r e l i m i n a r y version of this paper was submitted to the 14th Int. Conf. on highenergy physics, Vmnna, 1968.
652
M. KRAMMER and U. MAOR
a s s u m e the e x i s t e n c e of daughter t r a j e c t o r i e s which a s s u r e the p r o p e r Regge a s y m p t o t i c b e h a v i o r in the s - c h a n n e l f o r w a r d direction. T h e p r e s e n t w o r k e n l a r g e s on our p r e v i o u s p a p e r s [1-3] in that both the k i n e m a t i c a l s i n g u l a r i t i e s and t h r e s h o l d (pseudothreshold) conditions of the helicity a m p l i t u d e s a r e fully i n c o r p o r a t e d in the p a r a m e t r i z a t i o n . Also, the s i g n i f i c a n c e of s i m u l t a n e o u s l y fitting the different r e a c t i o n s is s t r e s s e d and i m p l i c a t i o n s of SU(3) s y m m e t r y and exchange d e g e n e r a c y a r e explored. In sect. 2 we s u m m a r i z e the R e g g e - p o l e p a r a m e t r i z a t i o n r e l e v a n t to our p r o b l e m and d i s c u s s s o m e qualitative f e a t u r e s of it. The data u s e d and the fits obtained a r e given in sect. 3. Our final c o n c l u s i o n s a r e contained in sect. 4.
2. P A R A M E T R I Z A T I O N
We c o n s i d e r the t - c h a n n e l p r o c e s s ~1+~2 ~ i S + A , w h e r e n l , ~2 stand f o r ~r, K and ~7 a p p r o p r i a t e l y . We denote the t-channel helicity a m p l i t u d e s by f¢i, w h e r e i and j r e f e r to the h e l i c i t i e s of i0 and A, and p stands for the Reg~e t r a j e c t o r y . T h e n o r m a l i z a t i o n is, a s b e f o r e [1-3],
d~_
dt
~
~ 12
sp2 n ½ i,j ~ [~f'" p U
"
(4)
Following the routine p r o c e d u r e [4, 5] one obtains the following a m p l i t u d e s f r e e f r o m k i n e m a t i c a l s i n g u l a r i t i e s , suitable for R e g g e i z a t i o n
p
(t)(sinO~li-JlKij(t) ~qq'
cos0
aP(t)-Ii-Jl
w h e r e q, q' and 0 t a r e the incoming m o m e n t a , outgoing m o m e n t a and s c a t t e r i n g a~gle in the t-channel c e n t r e - o f - m a s s s y s t e m ; So is a scaling constant, y~j(t) a r e the s i n g u l a r i t y f r e e r e s i d u e s , and
bP(t) bo(t)P r[ap(t)1 + a]
1 ± exp [- i~ap(t)] sin~rap(t) '
with
bPc(t) = ~2T h e Wang f a c t o r s [5]
Kij(t)
K½½(t)= { I t -
F[ap(t)+~] ~- 1
C6)
a r e given by
(m A - mp}2][t- (m A + rap)2]½ F 1 ,
(Ta)
p+R REGGE-POLE MODEL
653
~(t) = K½{(t) = l[t- (m~l- m~r2)2][t(mh- (m~lJ
-_-
rnp)2.
K½_~(t) = l [ t -
,
(7b)
(mlr 1 - rnTT2)2][t - (mlr 1 + mlT2)2][t - (mA+ mp)2] ½ .
(7c)
T h e functions g~/~(t) depend on the coupling and the ghost-killing m e c h a n i s m c h o s e n by the tr~tjectory at ap(t) = O. Optional g~(t) f o r m s c o r r e s p o n d i n g to different m e c h a n i s m s a r e given in table 1. The ~xperimental information on pion-nucleon c h a r g e - e x c h a n g e s c a t t e r i n g suggests a s e n s e - c h o o s i n g coupling f o r the p t r a j e c t o r y [6]. Unless a v e r y flat A 2 t r a j e c t o r y is t o l e r a t e d , only the Gell-Mann m e c h a n i s m is compatible [2] with the e x p e r i m e n t a l data on r e a c t i o n s (2) and (3). Table 1 Optional couplings and ghost-killing mechanisms for reactions (1-3). t-channel hehcity amplitude
Sense choosing mechanism
Gell-Mann mechanism
Chew mechanism
No compensation mechanism
f~½
1
Olp(ap +1)
a'p
q~(ap +1) 2
f;~
Otp ap(ap-
.f~_~
O~p 1)
ap(ap-
1)
qp
a 2p
q2(ap - 1)
q~(ap-1)
We have to c o n s i d e r also the k i n e m a t i c a l c o n s t r a i n t s imposed on the helicity amplitudes at the P - A t h r e s h o l d and pseudothreshold. In p a r t i c u l a r , ignoring the t h r e s h o l d conditions at t+ = (mA+mp) = 4.72 GeV 2 and conside r i n g only the conditions at the p s e u d o t h r e s h o l d t_ = (m A -rap) 2 =0.09 GeV2, one obtains [7, 8] 1 1 + (.f½_~-- d-3 f½2) 11 i(f½~ - v~ f½_~) = O(t - t-)
1 3 (t- t_)[i(4~ f½~ +-f~__~_)(d-~ f½_: 2 2"
(t- t-)[i(~
f½~ + f_,_;). ~ ~- + ( ¢ 5
+f½½)] = O(t- t_)
f½_~+f½½)] = o ( t 1
3
,
t_) ,
(8)
w h e r e fij stands f o r the full t-channel helicity amplitude. A lengthy discussion on the implications of t h e s e c o n s t r a i n t s has been given in ref. [7]. We
M. KRAMMER and U. MAOR
654
onl Y want to r e c a l l that s i m p l e p a r a m e t r i z a t i o n s of the fii, s a t i s f y i n g eqs. (8), lead to s p i n - d e n s i t y m a t r i x e l e m e n t s which a r e v e r y different f r o m t h o s e s e e n in A ++ decay in r e a c t i o n s (1) and (3). In o r d e r to r e m o v e this difficulty a l e s s d i v e r g e n t f o r m f o r the t - c h a n n e l helicity a m p l i t u d e s was s u g g e s t e d [7], yielding a new p a r a m e t r i z a t i o n in which
yPx = [- ~ X4(t - mp)2
+Xl(t_
22
/,
, =[
-~--~
t_)]aP(t_) , __ 22
+ x (t- t_)],,,,___(t_),
x4 (t-
2
(t- t_)]a
(t_) ,
= [Z4(t- % 2)
Xl+(
X2_X3)(t_t_)]aP - - -
aPij (t-)
=(t_) ½ li-J
7_1~ : [ - X 4 ( t -
m )+
2
22
2
,
(9)
2
where
(- 1)i-J •. l (So) li-J l gPij(t_)
(10)
The functions X1, X~., X 3, X4 a r e a r b i t r a r y , and we will a s s u m e that they a r e constants. S o m e qualitative c o n c l u s i o n s can be drawn even at this e a r l y s t a g e of our investigation. The p o l a r i z a t i o n of the i s o b a r in r e a c t i o n s (1)-(3) is known to be in r e a s o n a b l e good a g r e e m e n t with an M1 t r a n s i t i o n , i.e.: f ~ ! =fl_~_ - 0 , 22
2
2
f la = f ' 3 f l _ _ l • 22
2
2
It is evident, t h e r e f o r e , that any fit to the e x p e r i m e n t a l data on r e a c t i o n s (1)-(3) is bound to f o r c e X 4 to be v e r y s m a l l . F o r the 0 t r a j e c t o r y exchange we conclude , t h e r e f o r e , that the k i n e m a t i c a l f a c t o r s at t = ( m . A - m/)) ~ 2 de-. g e n e r a t e to the B o r n p - e x c h a n g e f o r m ( d y n a m i c a l exception m J a c k s o n - H l t e d e s c r i p t i o n [7]). No such s i m p l e c o r r e s p o n d e n c e can be m a d e f o r the A 2 t r a j e c t o r y exchange for which the A2(2 +-) is the l o w e s t B o r n pole. D e s p i t e that, we o b s e r v e e m p i r i c a l l y that r e s i d u e s of the A 2 t r a j e c t o r y choose a l e s s d i v e r g e n t f o r m tha~0the one ~ptioned by the Wang method [5]. The conclusion that f ~ and f ~ have the s a m e d y n a m i c a l s t r u c t u r e is f u r t h e r s u p p o r t e d by the~observation that the P t r a j e c t o r y c h o o s e s a s e n s e m e c h a n i s m at ap = 0, w h e r e a s the A 2 t r a j e c t o r y c h o o s e s the G e l l - M a n n m e c h a n i s m at aA2 = 0 (see table 1). Consequently, all the helicity a m p l i tudes but f½½ have the s a m e a f a c t o r s f o r P and A 2 exchange. As we shall s e e in sect. 3, the r e l a t i v e contribution of f½½ is r a t h e r s m a l l and i n s e n s i t i v e to the detailed a dependence. We thus find s u p p o r t f o r m o d e l s a s s u m i n g the s a m e i n t e r a c t i o n f o r 0 and A 2 t r a j e c t o r y e x c h a n g e s [9]. It is p a r t i c u l a r ly i n t e r e s t i n g to c h e c k an e x c h a n g e - d e g e n e r a c y m o d e l [10] which a s s u m e s the identity of the P and A 2 r e s i d u e s [11].
p+R REGGE-POLE MODEL
655
3. DATA C O L L E C T I O N AND FITS P E R F O R M E D We used 78 data points f o r r e a c t i o n (1) c o v e r i n g the t - d e p e n d e n c e of both the c r o s s section and 4 ++ p o l a r i z a t i o n a r o u n d 3.5 G e V / c [12], at 4.0 G e V / c [13] and at 8.0 G e V / c [14]. A total of 14 data points f o r the differential c r o s s section of r e a c t i o n (2) a r o u n d 3.5 G e V / c [15] * and at 8.0 G e V / c [14] w e r e taken. Finally, we combined to 88 data points the values of da/dt and Prnrn' for r e a c t i o n (3) obtained by the C E R N - B r u x e l l e s c o l l a b o r a t i o n at 3.0, 3.5, 5.0 and 8.25 G e V / c and quoted in a r e c e n t compilation [16]. Additional i n f o r m a t i o n on r e a c t i o n (3) at 10.0 G e V / c [17] and 13.0 G e V / c [18] has been obtained r e c e n t l y . T h e s e points were not included in the actual fit but we shall c o m m e n t on the c o m p a r i s o n of our calculations with t h e s e experimental results. A s i m u l t a n e o u s fit to r e a c t i o n s (1)-(3) is i m p o r t a n t as, with the r e q u i r e ment that t r a j e c t o r i e s with the s a m e quantum n u m b e r s contributing to diff e r e n t p r o c e s s e s a r e identical, we can i m p o s e additional c o n s t r a i n t s such a s f a c t o r i z a t i o n of the r e s i d u e s and SU(3) s y m m e t r y . Throughout this calculation we have a s s u m e d the SU(3) r e l a t i o n s :
~y+TIA2 ~ 2 )'K+~OA2 7n+~op
= ~
VK+EOp •
(11)
Deviations f r o m exact s y m m e t r y w e r e investigated by introducing different s c a l i n g c o n s t a n t s So for different r e a c t i o n s . Five p o s s i b l e solutions w e r e u s e d in the fitting p r o c e d u r e : Solution 1. We a s s u m e exact SU(3) s y m m e t r y and use the t r a j e c t o r i e s d e t e r m i n e d [19] f r o m p i o n - n u c l e o n c h a r g e exchange and compatible ** with 7/production.
Up(t) = 0.57 + 0 . 9 6 t aA2(t) = 0.4 + 0 . 9 t
(12)
No r e l a t i o n between the p and A2 r e s i d u e s is imposed. Solution 2. We a s s u m e exact SU(3) and ' a p p r o x i m a t e ' exchange d e g e n e r acy. The t r a j e c t o r i e s a r e given by eq. (12). The Xi p a r a m e t e r s of eq. (9) a r e a s s u m e d to be identical f o r p and A2 exchange. The a f a c t o r s for f~_~ and f A ~ a r e kept different. 22 22
* Since no absolute values are given, we used as normalization the total cross section at 3.5 GeV/c obtained by interpolating data quoted in: D. R. O. Morrison, invited paper presented at the Conf. on high-energy two-body reactions, Stony Brook, CERN/TC/PHYSICS 66-20. The A++ polarization m this experiment was found to be similar to the one observed m reactions (1) and (3) (G.Gidal, private communication). ** The reactmn ?r-p--~7;n can be fitted well w~th a A = 0.4+0.9t. P.Mannhe~m, private communication; see also footnote (10) m re~. [2].
656
M. KRAMMER and U. MAOR
S o l u t i o n 3. S a m e a s 2, b u t w i t h o u t e x a c t SU(3) - i . e . w e u s e d i f f e r e n t scaling constants for reactions (1)-(3). S o l u t i o n 4. W e a s s u m e e x a c t SU(3) a n d e x c h a n g e d e g e n e r a c y i n w h i c h ap(t) = a A 2 ( t ) = 0 . 5 + 0 . 9 t . All the p and A2 residues multiplies fP~).
(13)
a r e i d e n t i c a l ( i . e . , a n a d d i t i o n a l e p ( e p + 1) f a c t o r
22
S o l u t i o n 5. S a m e a s 4, b u t w i t h o u t e x a c t SU(3). Table 2 P a r a m e t e r s for the best fits obtained Dimension
XCGeV
Solution 1
Solution 2
Solution 3
Solution 4
Solution 5
0.348
0.290
0.319
0.270
0.262
X2p
GeV
0.427
0.450
0.427
0.410
0.433
X3p
GeV
1.528
1.444
1.460
1.333
1.336
X4p
GeV
0.095
0.084
0.088
0.084
0.078
X~I2 GeV
-0.268
0.290
0.319
0.270
0.262
X~22 GeV
0.965
0.450
0.427
0.410
0.433
X~32 GeV
1.129
1.444
1.460
1.333
1.336
X~42 GeV
-0.042
0.084
0.088
0.084
0.078
S 7r GeV 2
1.630
1.561
1.600
0.785
0.642
S T} GeV 2
1.630
1.561
1.314
0.785
1.594
S K GeV 2
1.630
1.561
1.471
0.785
0.799
O
O
O
X2
288.5
311.0
308.0
391.3
346.5
d e g r e e s of freedom
171
175
173
175
173
Ozp(t) = 0 . 5 7 + 0 . 9 6 t CCA2(t) -- 0.40 +0.90 t P a r a m e t e r s for r e a c t i o n (1): X p ~/2. P a r a m e t e r s for r e a c t i o n (2): X A2 P a r a m e t e r s for reaction (3): X/p, X A2.
'Otp(t) = °tA2(t)7 = 0.50 +0.90t
657
p +R R E G G E - P O L E MODEL
The p a r a m e t e r s a d j u s t e d to l o w e s t X2 f o r 180 data points a r e s u m m a r i z e d in t a b l e 2, and the c u r v e s c o r r e s p o n d i n g to solution 1 a r e shown in figs. 1 to 5. The v a l u e s of X2 p e r d e g r e e of f r e e d o m a r e s o m e w h a t l e s s s p e c t a c u l a r than t h o s e obtained in the e a r l i e r fits [1, 2]. T h i s should not be a t t r i b u t e d to the new p a r a m e t r i z a t i o n . The fit fails, w h e r e we expect the ' n o n s e n s e ' dip due to the v a n i s h i n g of ap(t), s i n c e the dip s t r u c t u r e is m u c h l e s s p r o n o u n c e d (see fig. 1) than r e q u i r e d by the b u i l t - i n a - d e p e n d e n c e . An a p p r o p r i a t e e s t i m a t e of the ~p b a c k g r o u n d would be d e s i r a b l e in o r d e r to c l a r i f y w h e t h e r r e a c t i o n (1) does not show a p r o n o u n c e d dip, or w h e t h e r it is filled up by a b a c k g r o u n d with f l a t t e r t - d e p e n d e n c e than genuine A production. Also, this s i m p l e R e g g e - p o l e m o d e l p r e d i c t s e n e r g y independent spin density m a t r i x e l e m e n t s . The e x p e r i m e n t a l o b s e r v a t i o n s (see fig. 2) c o n f i r m this b e h a v i o u r only qualitatively. The o b s e r v e d d i f f e r e n c e s m a y n e c e s s i t a t e s o m e r e f i n e m e n t s of the p r e s e n t s i m p l e model. Another p r o b l e m a t i c a s p e c t of the fit is the e n e r g y dependence of the total c r o s s s e c t i o n f o r r e a c t i o n (3). Our input contains only data points with incoming m o m e n t a f r o m 3.0 to 8.25 G e V / c . The new p r e l i m i n a r y data at 10.0 and 13.0 G e V / c [17, 18] indicate a d i f f e r e n t i a l d i s t r i b u t i o n and spin d e n s i t y m a t r i x e l e m e n t s c o n s i s t e n t with our calculations. The i n t e g r a t e d c r o s s section, h o w e v e r , s e e m s to d e c r e a s e f a s t e r than anticipated. T h i s is p a r t i c u l a r l y t r u e f o r the 10.0 G e V / c data which a r e r e l a t i v e l y s m a l l . If t h e s e c r o s s s e c t i o n s a r e c o n f i r m e d , a value of aA2(0) is needed which is m u c h l o w e r than the v a l u e s u s e d in the p r e s e n t calculations.
10-
~+ p ~
r~° A +÷ 1t* p ~
~
I~° A ++
3 5 GeV/c 0 GeV/c
E o
0,I
Ol 02 03 O~ 050G 07 OB 0910 II 12 13 I/, 15 - t [{GeV/c)2 ]
£2. £,
£6
1:0 ,'1 ,z 13 ,:, 15
-t ((6eV/c) 2]
F i g . 1. 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 f o r r e a c t i o n (1). The s o l i d h n e c o r r e s p o n d s to s o l u t i o n 1.
658
M. K R A M M E R and U. MAOR
O3
]i,.)" o~ J/L
II II
o 2
• 3,5GeV/c I o ~.o6,w~ P~
r
O0
a
-01
[
_ _
_
-02 -03
01 O1 02 03 OZ, 0 5 0 5 07 08 09 10 11 - t [[GeV/c)2 ]
Fig. 2. Spin denszty m a t r i x e l e m e n t s f o r r e a c t i o n (1). The s o h d l i n e c o r r e s p o n d s to s o l u t i o n 1.
011 f
~
T~+P~
io,:
L~''+
L 60 GeV/c
0.01.
\
o:~ o:~ 03 o:, o:5 ~6 ~7 o:a ~9/o '
-t [( GeV/c)z)
Fzg. 3. Dzfferentzal c r o s s s e c t z o n s f o r r e a c t i o n (2). The s o h d line c o r r e s p o n d s to solutxon 1.
p + R R E G G E - P O L E MODEL
lO
'
< ~%00eV/c
659
K+p--KO& ++ 10.
~
2"
5 0 GeVlc
~tO
w
01
Ol
01 112 03 l& 05 0.6 ~7 08 ~9 10 -t llGeV/c)2]
- t [(GeV/c) 2 ]
F i g . 4. 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 f o r r e a c t i o n (3). T h e s o l i d l i n e c o r r e s p o n d s to s o l u t i o n 1. T h e d a t a at 3.0 G e V / c h a v e r a t h e r l a r g e s t a t i s t i c a l f l u c t u a t i o n s , as can be s e e n by c o m p a r i n g t h e full p o i n t s and c r o s s e s which dzffer by the chozce of t - i n t e r v a l s only. (The fit i s b a s e d on the full p o i n t s . )
OG 03 02 O1 ,
,
,
+ ' T 8 25Ge'J/c
02 i . . . . Ol O0 -01 -02 -03 Ot 03 02 Ol
t
Ol 02 03 0/, 05 06 07 08 Og -t [(GeV;c)2]
F i g . 5. Spzn d e n s i t y m a t r i x e l e m e n t s f o r r e a c t z o n (3). The s o l i d line c o r r e s p o n d s to s o l u t i o n 1.
660
M. KRAMMER and U. MAOR
4. FINAL REMARKS The c h a r a c t e r i s t i c s of our p r e s e n t solutions a r e c o n s i s t e n t with the e a r l i e r fits [1, 2]. Our p r e v i o u s t r e a t m e n t was incomplete insofar as the kinem a t i c a l c o n s t r a i n t s at the p - A p s e u d o t h r e s h o l d (t_ = 0.09 GeV 2) w e r e not i n c o r p o r a t e d explicitly in the formulation. The old calculations r e q u i r e d a s t r o n g t - d e p e n d e n c e of the fitted r e s i d u e s ; this t - d e p e n d e n c e is in qualitative a g r e e m e n t * with the p r e s e n t p a r a m e t r i z a t i o n . Extrapolating the M1 type r e s i d u e s to t = m 2, we obtain (G~+pA++)/47r 30. Usin~ the v e c t o r m e s o n dominance model, with a~/7 2 = 0 . 3 6 5 . 1 0 -2, the value GZopA+/4~ ~ 20 is c o n s i s t e n t with the photoexcitation of the A+(1238)" r Our a n a l y s i s of the e x p e r i m e n t a l data shows that the t - c h a n n e l helicity amplitudes choose a l e s s divergent f o r m than the one optioned by the c r o s s ing m a t r i x [5]. This f e a t u r e is found for both the p and the A2 t r a j e c t o r y exchanges. The s i m i l a r i t y in f o r m and magnitude of the p and A 2 r e s i d u e s is consistent with an exchange d e g e n e r a c y hypothesis. When combined with SU(3) relations it is supported by the data a n a l y s i s only in the ' a p p r o x i m a t e ' f o r m s (solutions 2, 3). The hypothesis of exact SU(3) and exchange d e g e n e r a c y (solution 4) gives, within the model p r e s e n t e d here, r e s u l t s which a r e not quite s a t i s f a c t o r y . A somewhat b e t t e r fit is obtained if we a s s u m e exchange d e g e n e r a c y without exact SU(3) (solution 5). We note that ' a p p r o x i m a t e ' exchange d e g e n e r a c y does not support s y m m e t r y b r e a k i n g effects, as can be seen by c o m p a r i n g the X2 values for solutions 2 and 3. The conclusion of our a n a l y s i s is that a simple R e g g e - p o l e model, c o m bined with SU(3) r e l a t i o n s between the a p p r o p r i a t e r e s i d u e s , gives a r e a sonably a c c u r a t e d e s c r i p t i o n of the inelastic r e a c t i o n s d i s c u s s e d . In our opinion s o m e of the f e a t u r e s indicated by the data and lying outside the r a n g e of this model need f u r t h e r e x p e r i m e n t a l investigation b e f o r e m o r e ambitious evaluations, like inclusion of a b s o r p t i o n effects [20] or investigation of fixed poles connected with the t h i r d double s p e c t r a l function [21], will b e c o m e significant. After the Vienna Conference we r e c e i v e d a p r e p r i n t by G. H. Renninger and K. V. L. S a r m a , in which they p r e s e n t fits for the i s o b a r production p r o c e s s e s within the f r a m e w o r k of the invariant amplitudes.
RE FE RENCES [1] [2] [3] [4]
M.Krammer and U.Maor, Nuovo Clmento 50A (1967) 963. M.Krammer and U.Maor, Nuovo Clmento 52A (1967) 308. M.Krammer, Nuovo Clmento 52A {1967) 931. M.Gell-Mann, M.Goldberger, F. Low, E. Marx and F. Zacharmsen, Phys. Rev. 133 (1964) B145.
* Observing that the old parametrlzatlon 7(t) ~ (1 - t / t o ) gave good results for 6 ~< ..< 1 / t o ~< 15 GeV-2, 1.e. for t o ~ (rnA - rnN)2, it is compatible with the present one.
p + R REGGE-POLE MODEL
661
[5] L . L . W a n g , Phys. Rev. 142 (1966) 1187. [6] See, e.g., L.Bertocchl, in: Proc. of the Heidelberg Int. Conf. on elementary particles (North-Holland Publ. Comp., Amsterdam, 1968). [7] J . D . J a c k s o n and G . E . H i t e , Phys. Rev. 169 (1968) 1248. [8] E.Gotsman and U.Maor, Phys. Rev. 171 (1968) 1495. [9] R.Dashen and S . F r a u t s c h i , Phys. Rev. 152 (1966) 1450. [10] R . C . A r n o l d , Phys. Rev. Letters 14 (1965) 657; Phys. Rev. 153 (1967) 1506. [11] A.Ahmadzadeh, Phys. Rev. Letters 16 (1966) 952. [12] G.Gidal, G . B o r r e a n l , D.Brown, F . L o t t , S.Y. Fung, W.Jackson and R . P u , UCRL-18351. [13] G e r m a n - B r i t i s h Collaboration, Phys. Letters 10 (1964) 229 and private communication, N. Schmitz, Proc. of the 1965 E a s t e r School for Physicists, CERN 65-24/I. [14] A a c h e n - B e r h n - C E R N Collaboration, Nucl. Phys. B8 (1968) 45; D. R. O. Morrlson and P. F. Dalplaz, private communicahon. [15] D. Brown, G.Gidal, R . W . B l r g e , R.Bacastow, S.Y. Fung, W, Jackson and R . P u , Phys. Rev. Letters 19 (1967) 664. [16] CERN-Bruxelles Collaboration, Vmnna Int. Conf. on high-energy physics (paper nr. 445) ; Y. Goldschmldt-Clermont and F. Muller, private communication. ~ I T ~ i r m i n g h a m - G l a s g o w - O x f o r d Collaboration, Proc. of the Topical Conf. on highenergy colhslons of hadrons, Vol. II, p. 121, CERN 68-7. [18] J . C . B e r h n g h i e r i , M . S . F a r b e r , T . F e r b e l , B . E . F o r m a n , A . C . M e l i s s m o s , B. F. Slattery, T. Yamanouchi and H. Yuta, Proc. of the Topical Conf. on highenergy collisions of hadrons, Vol. II, p. 172, CERN 68-7. [19] G.H(ihler, J.Baacke and G.Eisenbe, ss, Phys. Letters 22 (1966) 203. [20] F.Schrempp, Nucl. Phys. B6 (1968) 487. [21] S.Mandelstam and L . L . W a n g , Phys. Rev. 160 (1967) 1490.