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Polarization and unnatural parity exchange in K− p → (ω, ϕ) Λ and π− p → K∗0 Λ
Volume 39B, number 3
PHYSICS
POLARIZATION
LETTERS
AND UNNATURAL I N K - p - ' (~0 , g o ) A A N D
1 May 1972
PARITY EXCHANGE ~ - p -" K * o h $
R. D. F I E L D J r . Brookhaven National Laboratory, Upton, New York 11973, uSA Received 24 March 1972
We use newly obtained joint density matrix elements for K-p --* (¢o, go)A and ~-p ~ K*°A to isolate three interference t e r m s contributing to the polarization. The data show large interference between natural parity exchanges (K**,K*) and substantial interference between unnatural parity exchanges (K-KA). We use SU(3) to compare ~0,go and K*O production.
In t h i s l e t t e r we p r e s e n t r e s u l t s of a R e g g e p o l e - e x c h a n g e a n a l y s i s of t h e r e a c t i o n s [1]:
f:+;lo(S,t)
= (go(s,t)) 1/2 (IV1 + U1) -~ N1 + U1
(4)
K - p ~ wA
(1)
ft_;lo(S,t)
= cos 2 ½0t(~2+~2)
(5)
K - p ~ q~A
(2)
f t + ; 1 0 ( s ,t) = sin 2 ½0t(TV 2 - US)
(6)
7r-p ~ K * ° A .
(3)
ft_;lo(S,t)
(7)
We show that i n t e r f e r e n c e s b e t w e e n e x c h a n g e d Regge trajectories can be isolated by observing t h e j o i n t d e n s i t y m a t r i x e l e m e n t s of the a b o v e r e a c t i o n s in a d d i t i o n to the A p o l a r i z a t i o n ~ . T h e data show large interferences between natural p a r i t y e x c h a n g e s and s u b s t a n t i a l i n t e r f e r e n c e s b e t w e e n u n n a t u r a l p a r i t y e x c h a n g e s . By c o m p a r ing r e a c t i o n s (1) and (2) we c o n c l u d e that t h e unnatural parity terms arise partly from the i n t e r f e r e n c e b e t w e e n t h e K and K A t r a j e c t o r i e s T T . W e point out that an e f f e c t i v e R e g g e p o l e m o d e l with SU(3) g i v e s a good fit to a l l the o b s e r v a b l e s f o r r e a c t i o n s (1)~(3) i n c l u d i n g the j o i n t d e n s i t y matrix elements. ~T S i n c e we a r e d e a l i n g with R e g g e t r a j e c t o r i e s e x c h a n g e d in the t - c h a n n e l we i n t r o d u c e t h e s i x t - c h a n n e l h e i i c i t y a m p l i t u d e s f o r r e a c t i o n s (1)-(3). T h e s e a m p l i t u d e s can b e e x p r e s s e d in t e r m s of a m p l i t u d e s of d e f i n i t e i n t r i n s i c p a r i t y a s f o l l o w s : Work performed under the auspices of the U. S. Atomic Energy Commission. ~" The idea of using the joint density matrix elements to isolate various interferences is not a new one [e.g.2]. Fox [2] discusses the use of a polarized proton target to detect possible ~-A 1 interference in ~-p--~pOn. "~ The K A is the Y = 1, I = ~ member of the J PC = 1++ nonet to which the A 1 is a member. ~']~" A more complete treatment of the analysis, including many details not mentioned here, will be published elsewhere.
= (~s,t))l/2
(N1 - U1)-- N1 - U 1
ft++;oo(S,t) = u3
(8)
ft+_;oo(S,t) = (q~(s,t)) 1/2 U 4 =- V 4 ,
(9)
w h e r e go(s,t) i s the K i b b l e f u n c t i o n and 0t i s t h e t - c h a n n e l s c a t t e r i n g a n g l e [3]. A w a y f r o m the s - c h a n n e l f o r w a r d d i r e c t i o n (0 s = 0 °) c o s 0 t r a p i d l y b e c o m e s l a r g e so that c o s 2 ½0t ~ - sin 2 ½0t .
(10)
E x p r e s s i o n s (5) and (6) c a n t h e n b e w r i t t e n a s f t + _ ; 1 0 ~ N2+ V2
(5')
f t + ; 1 0 ~ - g 2 + U2 ,
(6')
w h e r e N 2 ~ ½ cos0tTV 2 and U2 - ½ c o s 0 t U 2. T h e n a t u r a l p a r i t y a m p l i t u d e s N i and the u n n a t u r a l p a r i t y a m p l i t u d e s U i can b e d e c o m p o s e d into a m p l i t u d e s of d e f i n i t e s i g n a t u r e r a s f o l l o w s :
w h e r e the + s i g n c o r r e s p o n d s to r = ±1. T h e R e g g e t r a j e c t o r y c o n t r i b u t i n g to e a c h of t h e s e a m p l i t u d e s a l o n g with the C - p a r i t y of the honer 389
PHYSICS
Volume 39B, n u m b e r 3
LETTERS
1 May 1972
t o w h i c h it b e l o n g s i s s h o w n in t a b l e 1. E a c h of t h e a m p l i t u d e s AT_.(+) and UJ~:) h a s o n e t r a j e c t o r y " ~ e x c e p t -u~ ~ + ) a n d -U,~ -I.+) f o r w h m . h c o n t r i •b u t i "n g tO it no t r a j e c t o r y w i t h t h e p r o p e r q u a n t u m n u m b e r s (TC = - 1, T = + I ) e x i s t s . The polariziation and double density matrix e l e m e n t s pnmnm,' f o r r e a c t i o n s ( 1 ) - ( 3 ) c a n b e e x p r e s s e d in t e r m s of t h e a m p l i t u d e s N i a n d Ui. In p a r t i c u l a r t h e p o l a r i z a t i o n of t h e A i s g i v e n
w h e r e Pun, a r e m e a s u r e d in t h e J a c k s o n G o t t f r i e d f r a m e [4]. In o r d e r to a v o i d l a r g e e r r o r s we h a v e r e - e x p r e s s e d t h e a n g u l a r d i s t r i b u t i o n f u n c t i o n s i n t e r m s of I1,I2, a n d 13 a n d d e t e r m i n e d t h e m d i r e c t l y f r o m t h e data.~ T h e r e s u i t s a r e s h o w n i n fig. 1. W e a l s o e x h i b i t t h e polarization, where
by
P =I 1 -X 2 -r 3 .
(13)
P ~ ( 8 I m N 1 N 2 * - 8 I m U1U2* - 4 I m U 3 U4 . ) / F .
,
where = 41Nll 2 + 41N212 +
41Vx[ 2 + 41 u212 + 21 u3l 2 +
2]U412 $. There are three interference terms contributing to the polarization: one arising from natural p a r i t y o b j e c t s ( I m N 1 N 2 * ) a n d two a r i s i n g f r o m u n n a t u r a l p a r i t y o b j e c t s ( I m U1 U2 . a n d I m U3 U4*). F r o m t a b l e 1 we s e e t h a t I m N1N 2 * a r i s e s f r o m K * * - K * i n t e r f e r e n c e , w h e r e a s I m U1U 2 . a n d I m U 3 U4 . a r i s e s f r o m K B - K A o r K - K A i n t e r f e r e n c e . If t h e r e i s n o K A t r a j e c t o r y t h e n we w o u l d e x p e c t I m U 1 U2 * = I m U 3 U4 . = 0 a n d t h e polarization would be given entirely by 8 I m g l N 2 . / Z ~ ~$$, By f o r m i n g l i n e a r c o m b i n a t i o n s of d o u b l e density matrix elements and the polarization one c a n i s o l a t e t h e t h r e e t e r m s in (13). N a m e l y , 8 I m N 1 N 2 * ~ 11 = ( P + 2 I m p
0 _ 2im0
+
- 3Im 01+ =1 + 3im01 - 1 ) / 3 N 8 I m U 1 U 2 * ~ I2 = - ( P +
2 I m p -00 +_
+ 3Imp+l_ -1 - 3 I m p 1 _ + - 1 ) / 3 ~ ,
(14) 2Im01+l_ +
rnm'
8Imp 00-
8Imol+l)/6~
-
-
(16)
(17)
T h e d a t a s h o w l a r g e v a l u e s of 11 a n d n o n - z e r o v a l u e s of 12 + 13 f o r a l l t h r e e r e a c t i o n s . R e a c t i o n s (2) a n d (3) a r e r e m a r k a b l y s i m i l a r w i t h 12 ~ 0 a n d 1 3 ~ 1 1 for to-t <0.3 resulting ina s m a l l p o l a r i z a t i o n . F o r l a r g e r v a l u e s of t o - t I 1 b e g i n s to d o m i n a t e o v e r 13 a n d t h e p o l a r i z a t i o n t h u s b e c o m e s l a r g e a n d n e g a t i v e . In r e a c t i o n (1) t h e l a r g e p o l a r i z a t i o n f o r t o - t > 0.2 i s due to I 1 , (-12), a n d (-•3) a l l c o n t r i b u t i n g p o s i t i v e l y to P . T h e o b s e r v a t i o n t h a t 12 ¢ 0 o r 13 ¢ 0 f o r r e a c t i o n s ( 1 ) - ( 3 ) i n d i c a t e s t h e p r e s e n c e of t h e K A t r a j e c t o r y ' ~ ' ~ . By c o m p a r i n g r e a c t i o n s (1) a n d (2) a n d u s i n g SU(3) we c a n d e t e r m i n e w h e t h e r t h i s observed unnatural parity interference term is due to K B - K A o r K - K A i n t e r f e r e n c e . T h e m e s o n meson-Regge pole vertices for reactions (1)-(3) a r e r e l a t e d b y SU(3) a s f o l l o w s : (K*O}R~ - > = (~b IRK-> = C ( 2 ) 1/2 ( w } R K - > ,
(18)
where R is the Regge trajectory exchanged and C i s t h e C - p a r i t y of t h e n o n e t to w h i c h R b e l o n g s ( s e e t a b l e 1).Jt"'~ T h u s SU(3) p r e d i c t s t h a t a l l t h e o b s e r v a b l e s f o r r e a c t i o n s (2) a n d (3) b e t h e s a m e . Fig. 1 s h o w s t h a t t h e o b s e r v a b l e s I1,I2,I3, a n d P a r e in e x c e l l e n t a g r e e m e n t w i t h t h i s p r e d i c t i o n S . A l s o we s e e t h a t I 1 c h a n g e s s i g n u n d e r
(15)
The ~ sign means that the e x p r e s s i o n holds as long as (1) is valid. F r o m our effective pole fits we find that the c o r r e c t i o n t e r m s a r e indeed s m a l l except p r e c i s e l y at 0 s = 0 °. ~:~ The polarization for r e a c t i o n s (2) and (3) looks s i m i l a r to the polarization in Ir-p--~ KOA, while the polarization for r e a c t i o n (1) looks s i m i l a r to that for K-p ---~ff°A. The reactions K - p ~ ffOA and ~'-p-+ K°A have no unnatural parity t r a j e c t o r i e s contributing (only the f i r s t t e r m in eq. (13) is present). Thus before confronting the data one would suppose that Im UIU2* and Im U3U4* a r e zero in r e a c t i o n s (1)-(3), which would imply no K A contributions. ~ This conclusion is not altered by introducing a b sorptive effects. The P o m e r o n - R e g g e pole cut (P.*H) has the s a m e signature and C - p a r i t y as the Hegge pole H. T h e r e f o r e an object with ~'C = - 1 (KA) cannot be produced by absorbing the other TC= + 1 Regge poles in table 1.
390
4ImU3U4* =I 3 = (-2P+
"~ The angular distributions e x p~9~l~ r e s s e d in t e r m s of • the double density m a t r i c e s Pnn, a r e p r e s e n t e d in e. g. ref. [5]. ~J The quantities l 1 , P and hence 12 + 13 a r e i n v a r i a n t under rotation about the y - a x i s . The o b s e r v a t i o n that 12 + 13 ¢ 0 in the J a c k s o n - G o t t f r i e d f r a m e means that it is also n o n - z e r o in the s-channel helicity f r a m e . Hence, the conclusion that a KA contribution is needed is independent of the f r a m e one chooses to use in d e t e r m i n i n g the quantities in eqs. (14)-(16). Since we are dealing with t - c h a n n e l effective poles we use the J a c k s o n - G o t t f r i e d f r a m e . ~-'~'~In deriving (18) we have a s s u m e d ideal mixing of the 0~ and ~b and have used the e x p e r i m e n t a l obs e r v a t i o n that r ( ~ -'~ plr) is small. Also, in o r d e r for (18) to hold for the KH we must a s s u m e B - / ~ , e. g. [6]. The s i m i l a r i t y of the differential c r o s s section, polarization, and single density m a t r i x elements for K-p--- ~A and ~-p --- K*oA has been pointed out by A g u i l a r - B e n i t e z [7].
Volume 39B, n u m b e r 3
PHYSICS
Table 1 Regge t r a j e c t o r i e s R contributing to the amplitudes defined in the text, where v is the s i g n a t u r e , ~7 the i n t r i n s i c p a r i t y , and C is the C - p a r i t y of the nonet to which R is a m e m b e r Amplitude
T
TC
C
trajectory
N1
41
+1
+1
K**
N1
-1
+1
-1
K*
N2
+1
+1
+1
K**
N;
-1
+1
-1
K*
U +1
+1
+1
+1
K
1 May 1972
LETTERS K-p--'," ~A '
I
'
I
K-p--"@A '
'
I
'
I
"n'-p"-'*Kw°A '
'
I
'
I
'
'
I
'
~
'
I
0.5 11
0.0
-0.5 i
12 o.o,
~
,
.+, 0.5 13 o.o
-0.5 t ~I
'
l~
,
%
0.5~ ' ~ U1
-i
U +2
+1
U2
-i
-i
+i
KA
U +3
+i
+1
+i
K
U3
-1
+1
-i
KB
+
+1
U4
-1
u4
+i
-i
KB none
+i
-0.5 -
0.0
none
-i
P 0.0
KA
(1) ~ (2), w h i c h it m u s t do if it i s a r e s u l t of K * * - K * i n t e r f e r e n c e a n d (18) h o l d s . T h e q u a n t i t i e s 12 a n d 13 n e e d n o t h a v e a n y d e f i n i t e p r o p e r t i e s u n d e r (1) ~ (2). T h e y a r e e a c h c o m p o s e d of a term arising from KB-K A interference which s h o u l d c h a n g e s i g n u n d e r ( 1 ) 4 (2) a n d a t e r m arising from K-KA interference which should not. T h e d a t a s h o w a s u b s t a n t i a l c o n t r i b u t i o n to 13 t h a t d o e s n o t c h a n g e s i g n u n d e r (1) ~ (2), w h i c h we i d e n t i f y a s K - K A i n t e r f e r e n c e . $ T h e c u r v e s d i s p l a y e d i n fig. 1 a r e f r o m a R e g g e p o l e a n a l y s i s of r e a c t i o n s (1) a n d (2).$$ W e r e move the kinematic singularities from the Regge p o l e a m p l i t u d e s in t a b l e 1 a n d a s s u m e t h e r e An object with the quantum n u m b e r s of the KB(TC = = + 1) could be g e n e r a t e d by a b s o r b i n g other l~egge t r a j e c t o r i e s in table 1. We can not say at this point whether any o b s e r v e d KI~ -K A i n t e r f e r e n c e a r i s e s f r o m an actual KB t r a j e c t o r y o r f r o m i n t e r f e r e n c e between the KA and cuts. $$ T h e s e poles a r e to be c o n s i d e r e d as effective poles into which the effects of Regge cuts have been a b sorbed.
0.4
0.8
0.0
0.4
to-t
0.0 0.8 (GeV/c) 2
o.4
o.8
1.2
Fig. 1. The i n t e r f e r e n c e t e r m s 11,12,I 3 and the A p o l a r i z a t i o n P, where P = 11 - 12 - 13, for K-p ~ (co,q~)A at 4.6 G e V / c andTr-p ~ K*oA at 4.5 G e V / c as a function of t o - L (GeV/c) 2. The c u r v e s a r e f r o m an effect i v e - R e g g e - p o l e model with SU(3). maining reduced residues are constant and obey t h e SU(3) r e l a t i o n s h i p s (18). W e v a r i e d t h e r e duced residues and the trajectory functions (ass u m e d l i n e a r ) in a n a t t e m p t to f i t r e a c t i o n s (1) a n d (2). T h e o b s e r v a b l e s f o r r e a c t i o n (3) w e r e p r e d i c t e d f r o m t h i s fit. T h i s e f f e c t i v e p o l e m o d e l w i t h SU(3) g i v e s a r e m a r k a b l y good f i t to t h e o b s e r v a b l e s f o r r e a c t i o n s (1), (2), a n d (3) i n c l u d ing the differential cross sections and other dens i t y m a t r i x e l e m e n t s n o t s h o w n h e r e [4]. A s e x pected from our discussion above the solution h a s a s i z a b l e KA R e g g e p o l e r e s i d u e . T h e s o l u t i o n a l s o e x h i b i t s a s i z a b l e K B - K A i n t e r f e r e n c e i n 12. :~ In a d d i t i o n , t h e e f f e c t i v e K * * a n d K * p o l e , i n o r d e r to p r o d u c e t h e e x p e r i m e n t a l l y o b s e r v e d l a r g e v a l u e s of I 1 , e x h i b i t l a r g e e x c h a n g e d e g e n e r a c y b r e a k i n g . T h i s i s s i m i l a r to w h a t i s f o u n d if o n e a t t e m p t s to fit t h e K - n - - - ~ - A p o l a r i z a t i o n w i t h a n e f f e c t i v e K * * a n d K* p o l e . t T o s u m m a r i z e , we s h o w t h a t t h e d a t a f o r K - p ~ (co ,~0 )A a n d ~ - p ~ K * ° A i n d i c a t e t h e p r e s "~ It is possible in K-p--~TrOAand 7r-p--~K°A to a s s u m e EXD K** and K* poles and to fit the o b s e r v e d large polarization by introducing P o m e r o n - R e g g e cuts [e.g.8]. We a r e investigating this possibility for r e actions (1)-(3). 391
Volume 39B, number 3
PHYSICS LETTERS
ence of the KA R eg g e t r a j e c t o r y . We b e l i e v e this to be the f i r s t e x p e r i m e n t a l e v i d e n c e f o r the KA t r a j e c t o r y in the n e g a t i v e t region. We p r e s e n t r e s u l t s of a Regge pole a n a l y s i s c o n s i s t i n g of a K**, K*, K, KB, and KA Regge t r a j e c t o r y . T h e s e Regge poles a r e a s s u m e d to obey SU(3) r e l a t i o n s and give a good fit to the o b s e r v a b l e s (including joint density m a t r i x e l e m e n t s ) for K-p --. (w, ~o) A and ~ -p --* K*OA.
I would like to thank M. A g u i l a r - B e n i t e z , S. U. Chung, R. L. E i s n e r , and N. P. Samios f o r making a v a i l a b l e to m e the data for K-p-~ ( w , ¢ ) A . I also thank D. J. C r e n n e l l , H.A. Gordon, J. Louie, K.-W. Lai and J . M. S c a r f f o r allowing me a c c e s s to t h e i r ~ - p ~ K*°A data b e f o r e publication. I acknowledge many s t i m u l a t i n g d i s c u s s i o n s with Drs. C. Quigg and J. M. S c a r r . F i n a l l y , s p e c i a l thanks goes to Dr. R. E i s n e r whose help with the e x p e r i m e n t a l a s p e c t s of this p a p e r has been i n valuable.
392
1 May 1972
~ferences [1] The K-p--~ (u),q~)A data are from exposures of the BNL 80-in hydrogen filled bubble chamber at 3.9 and 4.6 GeV/c by M. Aguilar-Benitez, S.U. Chung, R. L. Eisner and N. P. Samios, Phys. Rev. D, to be published. The u-p---, K*OA data at 4.5 GeV/c are from an experiment done on the SLAC 82-in chamber by D. J. Crennell, H. A. Gordon, J. Louie, K.-W. Lai and J. M. Scarr, to be published. [2] G. Fox, Secon0 Intern. Conf. on Polarization and polarized targets, Berkeley, 1971. [3] T. Kibble, Phys. Rev. 117 (1959} 1159. [4] K. Gottfried and J. D. Jackson, Nuovo Cimento 33 (1964) 309. [5] H. Pilkuhn, The interaetiong of hadrons (NorthHolland, Amsterdam) chapt. 9. [6] H.J. Lipkin, Nuel.Phys. B7 (1968) 321. [7] M. Aguilar-Benit~z, S. U. Chung, R.L. Eisner and N. P. Samios, BNL 16393 (1971), Phys. Rev. Letters, to be published. [8] A. C. Irving, A.D.Martin and C. Michael, Nucl. Phys. B32 (1970) 1 or R. D. Field Jr. , Phys. Rev. D5 (1972) 86.
PHYSICS
POLARIZATION
LETTERS
AND UNNATURAL I N K - p - ' (~0 , g o ) A A N D
1 May 1972
PARITY EXCHANGE ~ - p -" K * o h $
R. D. F I E L D J r . Brookhaven National Laboratory, Upton, New York 11973, uSA Received 24 March 1972
We use newly obtained joint density matrix elements for K-p --* (¢o, go)A and ~-p ~ K*°A to isolate three interference t e r m s contributing to the polarization. The data show large interference between natural parity exchanges (K**,K*) and substantial interference between unnatural parity exchanges (K-KA). We use SU(3) to compare ~0,go and K*O production.
In t h i s l e t t e r we p r e s e n t r e s u l t s of a R e g g e p o l e - e x c h a n g e a n a l y s i s of t h e r e a c t i o n s [1]:
f:+;lo(S,t)
= (go(s,t)) 1/2 (IV1 + U1) -~ N1 + U1
(4)
K - p ~ wA
(1)
ft_;lo(S,t)
= cos 2 ½0t(~2+~2)
(5)
K - p ~ q~A
(2)
f t + ; 1 0 ( s ,t) = sin 2 ½0t(TV 2 - US)
(6)
7r-p ~ K * ° A .
(3)
ft_;lo(S,t)
(7)
We show that i n t e r f e r e n c e s b e t w e e n e x c h a n g e d Regge trajectories can be isolated by observing t h e j o i n t d e n s i t y m a t r i x e l e m e n t s of the a b o v e r e a c t i o n s in a d d i t i o n to the A p o l a r i z a t i o n ~ . T h e data show large interferences between natural p a r i t y e x c h a n g e s and s u b s t a n t i a l i n t e r f e r e n c e s b e t w e e n u n n a t u r a l p a r i t y e x c h a n g e s . By c o m p a r ing r e a c t i o n s (1) and (2) we c o n c l u d e that t h e unnatural parity terms arise partly from the i n t e r f e r e n c e b e t w e e n t h e K and K A t r a j e c t o r i e s T T . W e point out that an e f f e c t i v e R e g g e p o l e m o d e l with SU(3) g i v e s a good fit to a l l the o b s e r v a b l e s f o r r e a c t i o n s (1)~(3) i n c l u d i n g the j o i n t d e n s i t y matrix elements. ~T S i n c e we a r e d e a l i n g with R e g g e t r a j e c t o r i e s e x c h a n g e d in the t - c h a n n e l we i n t r o d u c e t h e s i x t - c h a n n e l h e i i c i t y a m p l i t u d e s f o r r e a c t i o n s (1)-(3). T h e s e a m p l i t u d e s can b e e x p r e s s e d in t e r m s of a m p l i t u d e s of d e f i n i t e i n t r i n s i c p a r i t y a s f o l l o w s : Work performed under the auspices of the U. S. Atomic Energy Commission. ~" The idea of using the joint density matrix elements to isolate various interferences is not a new one [e.g.2]. Fox [2] discusses the use of a polarized proton target to detect possible ~-A 1 interference in ~-p--~pOn. "~ The K A is the Y = 1, I = ~ member of the J PC = 1++ nonet to which the A 1 is a member. ~']~" A more complete treatment of the analysis, including many details not mentioned here, will be published elsewhere.
= (~s,t))l/2
(N1 - U1)-- N1 - U 1
ft++;oo(S,t) = u3
(8)
ft+_;oo(S,t) = (q~(s,t)) 1/2 U 4 =- V 4 ,
(9)
w h e r e go(s,t) i s the K i b b l e f u n c t i o n and 0t i s t h e t - c h a n n e l s c a t t e r i n g a n g l e [3]. A w a y f r o m the s - c h a n n e l f o r w a r d d i r e c t i o n (0 s = 0 °) c o s 0 t r a p i d l y b e c o m e s l a r g e so that c o s 2 ½0t ~ - sin 2 ½0t .
(10)
E x p r e s s i o n s (5) and (6) c a n t h e n b e w r i t t e n a s f t + _ ; 1 0 ~ N2+ V2
(5')
f t + ; 1 0 ~ - g 2 + U2 ,
(6')
w h e r e N 2 ~ ½ cos0tTV 2 and U2 - ½ c o s 0 t U 2. T h e n a t u r a l p a r i t y a m p l i t u d e s N i and the u n n a t u r a l p a r i t y a m p l i t u d e s U i can b e d e c o m p o s e d into a m p l i t u d e s of d e f i n i t e s i g n a t u r e r a s f o l l o w s :
w h e r e the + s i g n c o r r e s p o n d s to r = ±1. T h e R e g g e t r a j e c t o r y c o n t r i b u t i n g to e a c h of t h e s e a m p l i t u d e s a l o n g with the C - p a r i t y of the honer 389
PHYSICS
Volume 39B, n u m b e r 3
LETTERS
1 May 1972
t o w h i c h it b e l o n g s i s s h o w n in t a b l e 1. E a c h of t h e a m p l i t u d e s AT_.(+) and UJ~:) h a s o n e t r a j e c t o r y " ~ e x c e p t -u~ ~ + ) a n d -U,~ -I.+) f o r w h m . h c o n t r i •b u t i "n g tO it no t r a j e c t o r y w i t h t h e p r o p e r q u a n t u m n u m b e r s (TC = - 1, T = + I ) e x i s t s . The polariziation and double density matrix e l e m e n t s pnmnm,' f o r r e a c t i o n s ( 1 ) - ( 3 ) c a n b e e x p r e s s e d in t e r m s of t h e a m p l i t u d e s N i a n d Ui. In p a r t i c u l a r t h e p o l a r i z a t i o n of t h e A i s g i v e n
w h e r e Pun, a r e m e a s u r e d in t h e J a c k s o n G o t t f r i e d f r a m e [4]. In o r d e r to a v o i d l a r g e e r r o r s we h a v e r e - e x p r e s s e d t h e a n g u l a r d i s t r i b u t i o n f u n c t i o n s i n t e r m s of I1,I2, a n d 13 a n d d e t e r m i n e d t h e m d i r e c t l y f r o m t h e data.~ T h e r e s u i t s a r e s h o w n i n fig. 1. W e a l s o e x h i b i t t h e polarization, where
by
P =I 1 -X 2 -r 3 .
(13)
P ~ ( 8 I m N 1 N 2 * - 8 I m U1U2* - 4 I m U 3 U4 . ) / F .
,
where = 41Nll 2 + 41N212 +
41Vx[ 2 + 41 u212 + 21 u3l 2 +
2]U412 $. There are three interference terms contributing to the polarization: one arising from natural p a r i t y o b j e c t s ( I m N 1 N 2 * ) a n d two a r i s i n g f r o m u n n a t u r a l p a r i t y o b j e c t s ( I m U1 U2 . a n d I m U3 U4*). F r o m t a b l e 1 we s e e t h a t I m N1N 2 * a r i s e s f r o m K * * - K * i n t e r f e r e n c e , w h e r e a s I m U1U 2 . a n d I m U 3 U4 . a r i s e s f r o m K B - K A o r K - K A i n t e r f e r e n c e . If t h e r e i s n o K A t r a j e c t o r y t h e n we w o u l d e x p e c t I m U 1 U2 * = I m U 3 U4 . = 0 a n d t h e polarization would be given entirely by 8 I m g l N 2 . / Z ~ ~$$, By f o r m i n g l i n e a r c o m b i n a t i o n s of d o u b l e density matrix elements and the polarization one c a n i s o l a t e t h e t h r e e t e r m s in (13). N a m e l y , 8 I m N 1 N 2 * ~ 11 = ( P + 2 I m p
0 _ 2im0
+
- 3Im 01+ =1 + 3im01 - 1 ) / 3 N 8 I m U 1 U 2 * ~ I2 = - ( P +
2 I m p -00 +_
+ 3Imp+l_ -1 - 3 I m p 1 _ + - 1 ) / 3 ~ ,
(14) 2Im01+l_ +
rnm'
8Imp 00-
8Imol+l)/6~
-
-
(16)
(17)
T h e d a t a s h o w l a r g e v a l u e s of 11 a n d n o n - z e r o v a l u e s of 12 + 13 f o r a l l t h r e e r e a c t i o n s . R e a c t i o n s (2) a n d (3) a r e r e m a r k a b l y s i m i l a r w i t h 12 ~ 0 a n d 1 3 ~ 1 1 for to-t <0.3 resulting ina s m a l l p o l a r i z a t i o n . F o r l a r g e r v a l u e s of t o - t I 1 b e g i n s to d o m i n a t e o v e r 13 a n d t h e p o l a r i z a t i o n t h u s b e c o m e s l a r g e a n d n e g a t i v e . In r e a c t i o n (1) t h e l a r g e p o l a r i z a t i o n f o r t o - t > 0.2 i s due to I 1 , (-12), a n d (-•3) a l l c o n t r i b u t i n g p o s i t i v e l y to P . T h e o b s e r v a t i o n t h a t 12 ¢ 0 o r 13 ¢ 0 f o r r e a c t i o n s ( 1 ) - ( 3 ) i n d i c a t e s t h e p r e s e n c e of t h e K A t r a j e c t o r y ' ~ ' ~ . By c o m p a r i n g r e a c t i o n s (1) a n d (2) a n d u s i n g SU(3) we c a n d e t e r m i n e w h e t h e r t h i s observed unnatural parity interference term is due to K B - K A o r K - K A i n t e r f e r e n c e . T h e m e s o n meson-Regge pole vertices for reactions (1)-(3) a r e r e l a t e d b y SU(3) a s f o l l o w s : (K*O}R~ - > = (~b IRK-> = C ( 2 ) 1/2 ( w } R K - > ,
(18)
where R is the Regge trajectory exchanged and C i s t h e C - p a r i t y of t h e n o n e t to w h i c h R b e l o n g s ( s e e t a b l e 1).Jt"'~ T h u s SU(3) p r e d i c t s t h a t a l l t h e o b s e r v a b l e s f o r r e a c t i o n s (2) a n d (3) b e t h e s a m e . Fig. 1 s h o w s t h a t t h e o b s e r v a b l e s I1,I2,I3, a n d P a r e in e x c e l l e n t a g r e e m e n t w i t h t h i s p r e d i c t i o n S . A l s o we s e e t h a t I 1 c h a n g e s s i g n u n d e r
(15)
The ~ sign means that the e x p r e s s i o n holds as long as (1) is valid. F r o m our effective pole fits we find that the c o r r e c t i o n t e r m s a r e indeed s m a l l except p r e c i s e l y at 0 s = 0 °. ~:~ The polarization for r e a c t i o n s (2) and (3) looks s i m i l a r to the polarization in Ir-p--~ KOA, while the polarization for r e a c t i o n (1) looks s i m i l a r to that for K-p ---~ff°A. The reactions K - p ~ ffOA and ~'-p-+ K°A have no unnatural parity t r a j e c t o r i e s contributing (only the f i r s t t e r m in eq. (13) is present). Thus before confronting the data one would suppose that Im UIU2* and Im U3U4* a r e zero in r e a c t i o n s (1)-(3), which would imply no K A contributions. ~ This conclusion is not altered by introducing a b sorptive effects. The P o m e r o n - R e g g e pole cut (P.*H) has the s a m e signature and C - p a r i t y as the Hegge pole H. T h e r e f o r e an object with ~'C = - 1 (KA) cannot be produced by absorbing the other TC= + 1 Regge poles in table 1.
390
4ImU3U4* =I 3 = (-2P+
"~ The angular distributions e x p~9~l~ r e s s e d in t e r m s of • the double density m a t r i c e s Pnn, a r e p r e s e n t e d in e. g. ref. [5]. ~J The quantities l 1 , P and hence 12 + 13 a r e i n v a r i a n t under rotation about the y - a x i s . The o b s e r v a t i o n that 12 + 13 ¢ 0 in the J a c k s o n - G o t t f r i e d f r a m e means that it is also n o n - z e r o in the s-channel helicity f r a m e . Hence, the conclusion that a KA contribution is needed is independent of the f r a m e one chooses to use in d e t e r m i n i n g the quantities in eqs. (14)-(16). Since we are dealing with t - c h a n n e l effective poles we use the J a c k s o n - G o t t f r i e d f r a m e . ~-'~'~In deriving (18) we have a s s u m e d ideal mixing of the 0~ and ~b and have used the e x p e r i m e n t a l obs e r v a t i o n that r ( ~ -'~ plr) is small. Also, in o r d e r for (18) to hold for the KH we must a s s u m e B - / ~ , e. g. [6]. The s i m i l a r i t y of the differential c r o s s section, polarization, and single density m a t r i x elements for K-p--- ~A and ~-p --- K*oA has been pointed out by A g u i l a r - B e n i t e z [7].
Volume 39B, n u m b e r 3
PHYSICS
Table 1 Regge t r a j e c t o r i e s R contributing to the amplitudes defined in the text, where v is the s i g n a t u r e , ~7 the i n t r i n s i c p a r i t y , and C is the C - p a r i t y of the nonet to which R is a m e m b e r Amplitude
T
TC
C
trajectory
N1
41
+1
+1
K**
N1
-1
+1
-1
K*
N2
+1
+1
+1
K**
N;
-1
+1
-1
K*
U +1
+1
+1
+1
K
1 May 1972
LETTERS K-p--'," ~A '
I
'
I
K-p--"@A '
'
I
'
I
"n'-p"-'*Kw°A '
'
I
'
I
'
'
I
'
~
'
I
0.5 11
0.0
-0.5 i
12 o.o,
~
,
.+, 0.5 13 o.o
-0.5 t ~I
'
l~
,
%
0.5~ ' ~ U1
-i
U +2
+1
U2
-i
-i
+i
KA
U +3
+i
+1
+i
K
U3
-1
+1
-i
KB
+
+1
U4
-1
u4
+i
-i
KB none
+i
-0.5 -
0.0
none
-i
P 0.0
KA
(1) ~ (2), w h i c h it m u s t do if it i s a r e s u l t of K * * - K * i n t e r f e r e n c e a n d (18) h o l d s . T h e q u a n t i t i e s 12 a n d 13 n e e d n o t h a v e a n y d e f i n i t e p r o p e r t i e s u n d e r (1) ~ (2). T h e y a r e e a c h c o m p o s e d of a term arising from KB-K A interference which s h o u l d c h a n g e s i g n u n d e r ( 1 ) 4 (2) a n d a t e r m arising from K-KA interference which should not. T h e d a t a s h o w a s u b s t a n t i a l c o n t r i b u t i o n to 13 t h a t d o e s n o t c h a n g e s i g n u n d e r (1) ~ (2), w h i c h we i d e n t i f y a s K - K A i n t e r f e r e n c e . $ T h e c u r v e s d i s p l a y e d i n fig. 1 a r e f r o m a R e g g e p o l e a n a l y s i s of r e a c t i o n s (1) a n d (2).$$ W e r e move the kinematic singularities from the Regge p o l e a m p l i t u d e s in t a b l e 1 a n d a s s u m e t h e r e An object with the quantum n u m b e r s of the KB(TC = = + 1) could be g e n e r a t e d by a b s o r b i n g other l~egge t r a j e c t o r i e s in table 1. We can not say at this point whether any o b s e r v e d KI~ -K A i n t e r f e r e n c e a r i s e s f r o m an actual KB t r a j e c t o r y o r f r o m i n t e r f e r e n c e between the KA and cuts. $$ T h e s e poles a r e to be c o n s i d e r e d as effective poles into which the effects of Regge cuts have been a b sorbed.
0.4
0.8
0.0
0.4
to-t
0.0 0.8 (GeV/c) 2
o.4
o.8
1.2
Fig. 1. The i n t e r f e r e n c e t e r m s 11,12,I 3 and the A p o l a r i z a t i o n P, where P = 11 - 12 - 13, for K-p ~ (co,q~)A at 4.6 G e V / c andTr-p ~ K*oA at 4.5 G e V / c as a function of t o - L (GeV/c) 2. The c u r v e s a r e f r o m an effect i v e - R e g g e - p o l e model with SU(3). maining reduced residues are constant and obey t h e SU(3) r e l a t i o n s h i p s (18). W e v a r i e d t h e r e duced residues and the trajectory functions (ass u m e d l i n e a r ) in a n a t t e m p t to f i t r e a c t i o n s (1) a n d (2). T h e o b s e r v a b l e s f o r r e a c t i o n (3) w e r e p r e d i c t e d f r o m t h i s fit. T h i s e f f e c t i v e p o l e m o d e l w i t h SU(3) g i v e s a r e m a r k a b l y good f i t to t h e o b s e r v a b l e s f o r r e a c t i o n s (1), (2), a n d (3) i n c l u d ing the differential cross sections and other dens i t y m a t r i x e l e m e n t s n o t s h o w n h e r e [4]. A s e x pected from our discussion above the solution h a s a s i z a b l e KA R e g g e p o l e r e s i d u e . T h e s o l u t i o n a l s o e x h i b i t s a s i z a b l e K B - K A i n t e r f e r e n c e i n 12. :~ In a d d i t i o n , t h e e f f e c t i v e K * * a n d K * p o l e , i n o r d e r to p r o d u c e t h e e x p e r i m e n t a l l y o b s e r v e d l a r g e v a l u e s of I 1 , e x h i b i t l a r g e e x c h a n g e d e g e n e r a c y b r e a k i n g . T h i s i s s i m i l a r to w h a t i s f o u n d if o n e a t t e m p t s to fit t h e K - n - - - ~ - A p o l a r i z a t i o n w i t h a n e f f e c t i v e K * * a n d K* p o l e . t T o s u m m a r i z e , we s h o w t h a t t h e d a t a f o r K - p ~ (co ,~0 )A a n d ~ - p ~ K * ° A i n d i c a t e t h e p r e s "~ It is possible in K-p--~TrOAand 7r-p--~K°A to a s s u m e EXD K** and K* poles and to fit the o b s e r v e d large polarization by introducing P o m e r o n - R e g g e cuts [e.g.8]. We a r e investigating this possibility for r e actions (1)-(3). 391
Volume 39B, number 3
PHYSICS LETTERS
ence of the KA R eg g e t r a j e c t o r y . We b e l i e v e this to be the f i r s t e x p e r i m e n t a l e v i d e n c e f o r the KA t r a j e c t o r y in the n e g a t i v e t region. We p r e s e n t r e s u l t s of a Regge pole a n a l y s i s c o n s i s t i n g of a K**, K*, K, KB, and KA Regge t r a j e c t o r y . T h e s e Regge poles a r e a s s u m e d to obey SU(3) r e l a t i o n s and give a good fit to the o b s e r v a b l e s (including joint density m a t r i x e l e m e n t s ) for K-p --. (w, ~o) A and ~ -p --* K*OA.
I would like to thank M. A g u i l a r - B e n i t e z , S. U. Chung, R. L. E i s n e r , and N. P. Samios f o r making a v a i l a b l e to m e the data for K-p-~ ( w , ¢ ) A . I also thank D. J. C r e n n e l l , H.A. Gordon, J. Louie, K.-W. Lai and J . M. S c a r f f o r allowing me a c c e s s to t h e i r ~ - p ~ K*°A data b e f o r e publication. I acknowledge many s t i m u l a t i n g d i s c u s s i o n s with Drs. C. Quigg and J. M. S c a r r . F i n a l l y , s p e c i a l thanks goes to Dr. R. E i s n e r whose help with the e x p e r i m e n t a l a s p e c t s of this p a p e r has been i n valuable.
392
1 May 1972
~ferences [1] The K-p--~ (u),q~)A data are from exposures of the BNL 80-in hydrogen filled bubble chamber at 3.9 and 4.6 GeV/c by M. Aguilar-Benitez, S.U. Chung, R. L. Eisner and N. P. Samios, Phys. Rev. D, to be published. The u-p---, K*OA data at 4.5 GeV/c are from an experiment done on the SLAC 82-in chamber by D. J. Crennell, H. A. Gordon, J. Louie, K.-W. Lai and J. M. Scarr, to be published. [2] G. Fox, Secon0 Intern. Conf. on Polarization and polarized targets, Berkeley, 1971. [3] T. Kibble, Phys. Rev. 117 (1959} 1159. [4] K. Gottfried and J. D. Jackson, Nuovo Cimento 33 (1964) 309. [5] H. Pilkuhn, The interaetiong of hadrons (NorthHolland, Amsterdam) chapt. 9. [6] H.J. Lipkin, Nuel.Phys. B7 (1968) 321. [7] M. Aguilar-Benit~z, S. U. Chung, R.L. Eisner and N. P. Samios, BNL 16393 (1971), Phys. Rev. Letters, to be published. [8] A. C. Irving, A.D.Martin and C. Michael, Nucl. Phys. B32 (1970) 1 or R. D. Field Jr. , Phys. Rev. D5 (1972) 86.