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Pion-ΛΣ coupling from forward scattering dispersion relations?
Volume 29B, number 8
PHYSICS L E T T E R S
21 July 1969
PION-A~ COUPLING FROM FORWARD S C A T T E R I N G DISPERSION RELATIONS ? C. WEIL Department o f P h y s i c s , Technion - I s r a e l Institute o f Technology, Haifa, I s r a e l
Received 4 June 1969
It has been proposed that the availability of the recent effective range multichannel K-matrix parametrization of low energy KN scattering data enables us to determine the ~ and y ~ couplings. We suggest that this method of determination is incapable of yielding reliable numerical results at present.
Chan and M e i e r e suggested [1] a f o r w a r d s c a t t e r i n g d i s p e r s i o n r e l a t i o n (FSDR) evaluated at the ~N t h r e s h o l d to d e t e r m i n e the ~A~ and ~ couplings. The data for f o r w a r d ~A and ~ e l a s t i c s c a t t e r i n g w e r e obtained as follows : F o r low e n e r g i e s they u s e d K i m ' s m u l t i c h a n n e l effective r a n g e K - m a t r i x p a r a m e t r i z a t i o n for EN s c a t t e r i n g [2]. F o r the i n t e r m e d i a t e r e s o n a n c e r e g i o n they u s e d the n a r r o w width approximation. The c o n t r i b u t i o n f r o m e n e r g i e s beyond the r e s o n a n c e r e g i o n was found to be negligible. T h e i r r e s u l t for the ~A~ coupling is
g2 = 21.5~7
(I)
with the s a m e convention that yields g 2 = 14.5 for the TrNN coupling. Next, they c o m p a r e this r e s u l t with the p r e d i c t i o n of SU(3) with F / D ~ ~. This p r e d i c t i o n i s g 2 ~ 7, and the d i s a g r e e m e n t with eq. (I) i s obvious. Chan and M e i e r e u s e d the s a m e FSDR method to d e t e r m i n e a v a l ue for the 7r~; coupling, with t h e i r given 7rA~ coupling. It is the p u r p o s e of this l e t t e r to suggest that one should r e g a r d eq. (I) with caution. F i r s t , one should note that s e r i o u s questions about the validity and m e r i t s of K i m ' s p a r a m e t r i zation [2] have b e e n r a i s e d r e c e n t l y [3]. A diff e r e n t p a r a m e t r i z a t i o n [4] of the low e n e r g y data has also been suggested, and s e e m s to have s a m e advantages over K i m ' s , in p a r t i c u l a r as r e g a r d s c o n s i s t e n c y test [3]. The r e s u l t of Chan and M e i e r e for g2 [our eq. (1)] is p r o p o r t i o n a l to the s u m of t h r e e t e r m s : (i) A c o n t r i b u t i o n of -10.79 fm 2 f r o m the v a l ue of the amplitude at the KN threshold. (ii) A c o n t r i b u t i o n of 33.32 fm 2 f r o m the i n t e g r a l over the low e n e r g y region.
(iii) A c o n t r i b u t i o n of 5.16 fm 2 f r o m the r e s o n a n c e region. The e r r o r of ± 7 for g 2 in eq. (1) a r i s e s solely f r o m c o n t r i b u t i o n s (i) and (ii) and r e p r e s e n t s a typical v a r i a t i o n r e s u l t i n g f r o m the quoted e r r o r for K i m ' s p a r a m e t e r s . A quick c a l c u l a t i o n shows that this e r r o r constitute 40% of the s u m (i) + (ii). In view of the r e m a r k s in the p r e c e d ing p a r a g r a p h , even this e r r o r s e e m s o p t i m i s t i c . Second Chan and M e i e r e play down the c o n t r i bution (iii) f r o m the r e s o n a n c e region. However, i s t u r n s out t h a t 90~ ('.) of (iii) c o m e s f r o m a ~,(1690), JP = (2)-, ~ F = 120 MeV with a branching ratio of 100~ into the 7rA channel, which they use. Hence the contributionfrom the resonance region is extremely sensitive to which resonances, quantum numbers, widths and branching ratios are used in the 1600-1800 MeV region. We have used recent data [5, 6] to reevaluate (iii). The state closest to the ~(1690) of Chan and Meiere is the ~(1660), JP = (~)-, F = 50 MeV, with a 29% branching ratio into the AIr channel. One sees that its contributionis reduced by a factor of 8 ' Other states contribute at least as much, sometimes more, the contributions being of both signs. If one plays the game in various way, one gets 16.5
(2)
not taking into account nay errors from (i) and (ii). The e x p e r i m e n t a l situation in this r e s o n a n c e e n e r g y r e g i o n is s t i l l v e r y ambiguous, due to its weak coupling to the K~N channel [5]. F u r t h e r m o r e , on the b a s i s of SU(3) s y m m e t r y one m a y expect [7] at l e a s t nine (.') additional ~ . - r e s o n a n c e s in the 1800-2300 MeV e n e r g y region. The point is thus that CM's value of (iii) is s i m p l y a c 501
Volume 29B, n u m b e r 8
PHYSICS
cidental. If w e t a k e i n t o a c c o u n t t h e q u e s t i o n s r a i s e d a b o v e , w e o b s e r v e t h a t w i ~ a l i t t l e b i t of l u c k one could get a value for g~ which would be comp a t i b l e w i t h SU(3). It i s o u r o p i n i o n , t h e r e f o r e , t h a t w e h a v e to w a i t f o r b e t t e r e x p e r i m e n t a l d a t a f o r b o t h low e n e r g y a n d r e s o n a n c e r e g i o n s i n order to be able to obtain a meaningful result for t h e ~A ~, c o u p l i n g w i t h t h i s F S D R m e t h o d . The value for the ~ coupling is similarly o b t a i n e d if w e m a k e u s e of a g i v e n v a l u e of t h e 7rA~ c o u p l i n g . T h e a b o v e r e m a r k s c e r t a i n l y a p p l y h e r e , too.
502
LETTERS
21 July 1969
Reference s 1. C.H. Chan and F. T. Meiere, Phys. L e t t e r s 28B (1968) 125. 2. J . K . Kim, Phys. Rev. L e t t e r s 19 (1967) 1074. 3. A. D. Martin, N . M . Q u e e n and G. Violini, Nuel. Phys. 10B (1969) 481. 4. B. R. M a r t i n and M. Sakitt, Phys. Rev., to be published. 5. R. D. Tripp, r a p p o r t e u e r talk in Proe. 14th Intern. Conf. on High energy physics, Vienna, 1968. 6. P a r t i c l e Data Group, Rev. Mod. Phys. 41 (1969) 109. 7. H. Harari, r a p p o r t e u r talk in Proc. 14th Intern. Conf. on High energy physics, Vienna, 1968.
PHYSICS L E T T E R S
21 July 1969
PION-A~ COUPLING FROM FORWARD S C A T T E R I N G DISPERSION RELATIONS ? C. WEIL Department o f P h y s i c s , Technion - I s r a e l Institute o f Technology, Haifa, I s r a e l
Received 4 June 1969
It has been proposed that the availability of the recent effective range multichannel K-matrix parametrization of low energy KN scattering data enables us to determine the ~ and y ~ couplings. We suggest that this method of determination is incapable of yielding reliable numerical results at present.
Chan and M e i e r e suggested [1] a f o r w a r d s c a t t e r i n g d i s p e r s i o n r e l a t i o n (FSDR) evaluated at the ~N t h r e s h o l d to d e t e r m i n e the ~A~ and ~ couplings. The data for f o r w a r d ~A and ~ e l a s t i c s c a t t e r i n g w e r e obtained as follows : F o r low e n e r g i e s they u s e d K i m ' s m u l t i c h a n n e l effective r a n g e K - m a t r i x p a r a m e t r i z a t i o n for EN s c a t t e r i n g [2]. F o r the i n t e r m e d i a t e r e s o n a n c e r e g i o n they u s e d the n a r r o w width approximation. The c o n t r i b u t i o n f r o m e n e r g i e s beyond the r e s o n a n c e r e g i o n was found to be negligible. T h e i r r e s u l t for the ~A~ coupling is
g2 = 21.5~7
(I)
with the s a m e convention that yields g 2 = 14.5 for the TrNN coupling. Next, they c o m p a r e this r e s u l t with the p r e d i c t i o n of SU(3) with F / D ~ ~. This p r e d i c t i o n i s g 2 ~ 7, and the d i s a g r e e m e n t with eq. (I) i s obvious. Chan and M e i e r e u s e d the s a m e FSDR method to d e t e r m i n e a v a l ue for the 7r~; coupling, with t h e i r given 7rA~ coupling. It is the p u r p o s e of this l e t t e r to suggest that one should r e g a r d eq. (I) with caution. F i r s t , one should note that s e r i o u s questions about the validity and m e r i t s of K i m ' s p a r a m e t r i zation [2] have b e e n r a i s e d r e c e n t l y [3]. A diff e r e n t p a r a m e t r i z a t i o n [4] of the low e n e r g y data has also been suggested, and s e e m s to have s a m e advantages over K i m ' s , in p a r t i c u l a r as r e g a r d s c o n s i s t e n c y test [3]. The r e s u l t of Chan and M e i e r e for g2 [our eq. (1)] is p r o p o r t i o n a l to the s u m of t h r e e t e r m s : (i) A c o n t r i b u t i o n of -10.79 fm 2 f r o m the v a l ue of the amplitude at the KN threshold. (ii) A c o n t r i b u t i o n of 33.32 fm 2 f r o m the i n t e g r a l over the low e n e r g y region.
(iii) A c o n t r i b u t i o n of 5.16 fm 2 f r o m the r e s o n a n c e region. The e r r o r of ± 7 for g 2 in eq. (1) a r i s e s solely f r o m c o n t r i b u t i o n s (i) and (ii) and r e p r e s e n t s a typical v a r i a t i o n r e s u l t i n g f r o m the quoted e r r o r for K i m ' s p a r a m e t e r s . A quick c a l c u l a t i o n shows that this e r r o r constitute 40% of the s u m (i) + (ii). In view of the r e m a r k s in the p r e c e d ing p a r a g r a p h , even this e r r o r s e e m s o p t i m i s t i c . Second Chan and M e i e r e play down the c o n t r i bution (iii) f r o m the r e s o n a n c e region. However, i s t u r n s out t h a t 90~ ('.) of (iii) c o m e s f r o m a ~,(1690), JP = (2)-, ~ F = 120 MeV with a branching ratio of 100~ into the 7rA channel, which they use. Hence the contributionfrom the resonance region is extremely sensitive to which resonances, quantum numbers, widths and branching ratios are used in the 1600-1800 MeV region. We have used recent data [5, 6] to reevaluate (iii). The state closest to the ~(1690) of Chan and Meiere is the ~(1660), JP = (~)-, F = 50 MeV, with a 29% branching ratio into the AIr channel. One sees that its contributionis reduced by a factor of 8 ' Other states contribute at least as much, sometimes more, the contributions being of both signs. If one plays the game in various way, one gets 16.5
(2)
not taking into account nay errors from (i) and (ii). The e x p e r i m e n t a l situation in this r e s o n a n c e e n e r g y r e g i o n is s t i l l v e r y ambiguous, due to its weak coupling to the K~N channel [5]. F u r t h e r m o r e , on the b a s i s of SU(3) s y m m e t r y one m a y expect [7] at l e a s t nine (.') additional ~ . - r e s o n a n c e s in the 1800-2300 MeV e n e r g y region. The point is thus that CM's value of (iii) is s i m p l y a c 501
Volume 29B, n u m b e r 8
PHYSICS
cidental. If w e t a k e i n t o a c c o u n t t h e q u e s t i o n s r a i s e d a b o v e , w e o b s e r v e t h a t w i ~ a l i t t l e b i t of l u c k one could get a value for g~ which would be comp a t i b l e w i t h SU(3). It i s o u r o p i n i o n , t h e r e f o r e , t h a t w e h a v e to w a i t f o r b e t t e r e x p e r i m e n t a l d a t a f o r b o t h low e n e r g y a n d r e s o n a n c e r e g i o n s i n order to be able to obtain a meaningful result for t h e ~A ~, c o u p l i n g w i t h t h i s F S D R m e t h o d . The value for the ~ coupling is similarly o b t a i n e d if w e m a k e u s e of a g i v e n v a l u e of t h e 7rA~ c o u p l i n g . T h e a b o v e r e m a r k s c e r t a i n l y a p p l y h e r e , too.
502
LETTERS
21 July 1969
Reference s 1. C.H. Chan and F. T. Meiere, Phys. L e t t e r s 28B (1968) 125. 2. J . K . Kim, Phys. Rev. L e t t e r s 19 (1967) 1074. 3. A. D. Martin, N . M . Q u e e n and G. Violini, Nuel. Phys. 10B (1969) 481. 4. B. R. M a r t i n and M. Sakitt, Phys. Rev., to be published. 5. R. D. Tripp, r a p p o r t e u e r talk in Proe. 14th Intern. Conf. on High energy physics, Vienna, 1968. 6. P a r t i c l e Data Group, Rev. Mod. Phys. 41 (1969) 109. 7. H. Harari, r a p p o r t e u r talk in Proc. 14th Intern. Conf. on High energy physics, Vienna, 1968.