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Quark model selection rules for Hadron couplings
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Nuclear Physics B13 f1969) 237-245. North-Holland Publ. Comp., Amsterdam
QUARK MODEL SELECTION RULES FOR HADRON COUPLINGS* P. G. O. FREUND and R. W A L T Z The Enrico ~'ermi Institute and the Department of Physics, The University of Chicago, Chicago, Illinois, 60637 and J . ROSNER School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, 55455
Received 7 August 1969
Abstract: Quark model selection rules are derived which apply to the couplings of hadrons composed of any number of quarks. These reduce to well-known and experimentally verified rules in the simplest cases, provide a necessary selection rule in the application of duality to baryon-antibaryon systems, and could explain the apparent difficulty of producing free quarks. Some experimental consequences of the rules are discussed.
1. ~ T R O D U C T I O N F i n i t e - e n e r g y s u m r u l e s (FESR) [1] allow the c o n s i s t e n t m a t c h i n g of d i r e c t - c h a n n e l and R e g g e - p o l e d e s c r i p t i o n s of h a d r o n i c r e a c t i o n s . The a p p r o x i m a t i o n of d i r e c t - c h a n n e l c o n t r i b u t i o n s by r e s o n a n c e s s e e m s to s a t u r a t e F E S R f o r t r a j e c t o r i e s o t h e r than the P o m e r a n c h u k (P) f a i r l y w e l l [2]. T h i s s u c c e s s h a s b e e n c a l l e d " d u a l i t y " [3] s i n c e it i m p l i e s that d i r e c t c h a n n e l r e s o n a n c e s and e x c h a n g e d Regge p o l e s a r e r e a l l y two alternative d e s c r i p t i o n s of the s a m e p r o c e s s , and should not be a d d e d to one a n o t h e r . D u a l i t y i s b e s t v i s u a l i z e d u s i n g q u a r k m o d e l d i a g r a m s [4]. Two d i a g r a m s
o b c Fig. 1. Quark model diagrams illustrating duality.
* Supported in part by the U.S. Atomic Energy Commission through contracts AT(11-1)-264 and AT(11-1)-1764. $ National Science Foundation predoctoral fellow.
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f o r m e s o n - m e s o n s c a t t e r i n g a r e i l l u s t r a t e d in fig. 1. F i g . l a h a s q~l i n t e r m e d i a t e s t a t e s in the s - c h a n n e l , s h o w n by t h e w a v y l i n e . T h e s e s t a t e s a r e e x p e c t e d to " b u i l d " a l a r g e a b s o r p t i v e c o n t r i b u t i o n a s s o c i a t e d w i t h t h e t c h a n n e l R e g g e p o l e t e r m . On the o t h e r hand fig. l b h a s no q~l i n t e r m e d i a t e s t a t e in the s - c h a n n e l , and h e n c e one e x p e c t s t h e a b s o r p t i v e c o n t r i b u t i o n s of the v a r i o u s t - c h a n n e l R e g g e p o l e t e r m s to c a n c e l one a n o t h e r . If t h i s i s the c a s e one d e r i v e s l a r g e s e t s of e x p e r i m e n t a l l y v e r i f i e d c o n s t r a i n t s , i n c l u d i n g d e g e n e r a c y of v a r i o u s t r a j e c t o r i e s c o u p l i n g r e l a t i o n s , and s p e c i f i c m e c h a n i s m s of SU(3) b r e a k i n g [ 2 , 4 - 7 ] . A s t r i c t a p p l i c a t i o n of d u a l i t y r e q u i r e s the e x i s t e n c e of c e r t a i n e x o t i c m e s o n s to " b u i l d " the e x p e c t e d n o n - P e x c h a n g e s in b a r y o n - a n t i b a r y o n s c a t t e r i n g [7]. T h i s s i t u a t i o n i s d e p i c t e d in fig. l c . T h e d o m i n a n t t - c h a n n e l e x c h a n g e s s e e m e x p e r i m e n t a l l y to b e c o n f i n e d to t h o s e i n v o l v i n g a s i n g l e q~l p a i r . The s - c h a n n e l s t a t e s t h a t " b u i l d " t h e s e e x c h a n g e s m u s t t h e n b e M4 = qq~*. If m e s o n s M4 e x i s t , t h e y m u s t not u p s e t o u r c o n c l u s i o n s in MM s c a t t e r ing. One p o s s i b i l i t y [7], w h i c h we s h a l l d i s c u s s h e r e , i s t h a t such m e s o n s do not couple to MM s y s t e m s **. M o r e g e n e r a l l y , i t h a s b e e n s h o w n [12] t h a t m e s o n s M = (q~t)v (v = 2, 3 , 4 , . . . ) a n d b a r y o n s B2v+3 = q V+3~lv(v = 1 , 2 , 3 , . . . ) a r e r e q u i r e d b y d u a l i t y . F o r t h e s e h i g h e r m e s o n s and b a r y o n s a s i m i l a r q u e s t i o n c a n b e a s k e d . How d o e s one a v o i d the c o u p l i n g of t h e s e h i g h e r h a d r o n s to l o w e r c h a n n e l s in w h i c h t h e i r p r e s e n c e s e e m s to b e e x c l u d e d b o t h by e x p e r i m e n t and by d u a l i t y ? We s h a l l a n s w e r t h e s e q u e s t i o n s in t e r m s of a s e l e c t i o n r u l e i n s p i r e d b y the q u a r k m o d e l .
2. T H E S E L E C T I O N R U L E L e t us f i r s t c o n s i d e r the c a s e s of M M M a n d B B M c o u p l i n g s [13]. T h e q u a r k m o d e l d i a g r a m s f o r t h e s e c o u p l i n g s a r e p r e s e n t e d in fig.2. T h e e x p e r i m e n t a l a b s e n c e of ~ -~ pu f o r b i d s d i a g r a m 2b. S i m i l a r l y gq~]~N = 0 f o r b i d s d i a g r a m 2d. T h e a l l o w e d d i a g r a m s 2a a n d 2c s h a r e the f o l l o w i n g i m portant properties: (A) E a c h of the t h r e e h a d r o n s m e e t i n g a t a v e r t e x e x c h a n g e s a t l e a s t one q u a r k l i n e w i t h e a c h of t h e r e m a i n i n g two h a d r o n s . (B) No q u a r k l i n e i s c o n n e c t e d w i t h a n a n t i q u a r k l i n e f r o m the s a m e hadron. In the c a s e of M M M a n d I~BM c o u p l i n g s e i t h e r p r o p e r t y (A) o r p r o p e r t y * Throughout this paper we shall label a hadron by its baryon number (M for m e s on, B for baryon) and an index equal to the total number of quarks and antiquarks contained in the hadron. We shall abbreviate M2 by M and B 3 by B. The p o s s i b i lity that states like M4, M 6, . . . , B 5, B 7, . , . do ffot correspond to resonances has been discussed elsewhere (see r e f s . [4, 7, 8]). We also a r e aware that duality may not be u n i v e r s a l l y applicable (see ref. [10]) when used to derive certain sets of exchange degeneracies (see ref. [11]). ** This is to be distinguished from the r e s u l t s of SU(6)W in which not all M4MM couplings a r e forbidden.
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_JL_
v
o
b
JL c
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u d
Fig. 2. Quark model diagrams for MMM (a, and b) and BBM (c and d) couplings. Diagrams a and c are allowed, diagrams b and d forbidden. (b) alone suffices to exclude the couplings in figs. 2b and 2d. This is not the c a s e in g e n e r a l , however. We shall adopt (A) and (B) as g e n e r a l p r i n c i p l e s defining an allowed vertex. We can then establish the following selection rule as a c o r o l l a r y to (A) and (B): (S) Let HI, H2, and H 3 be any hadrons p e r m i t t e d to couple at a v e r t e x by b a r y o n n u m b e r , h y p e r c h a r g e , and isospin conservation. Let vi and Pi (i = 1, 2, 3) be the n u m b e r of quarks and antiquarks in H i. Define n i = v i + vi" Then each of the t h r e e inequalities n i ~nj
+n k - 2(i,j,k
p e r m . of 1 , 2 , 3 )
(1)
m u s t hold or the HIH2H 3 coupling is forbidden *. By (A), the total n u m b e r of lines N i running f r o m the I.iii.ik s y s t e m to H i m u s t be l e s s than o r equal to n.~ + n k - 2, since at l e a s t on6 line of Hj is conJ nected to one of Hk. However, by (B) we m u s t have N i = n i. Hence the s e lection rule is proved. An i m m e d i a t e application of (S) is that M4 ~ MM is forbidden but M 4 B ~ allowed. G e n e r a l l y the coupling Ba ~ B b + Me, M a --. M b + M c or M a -* B b + B c is allowed only ff la-b]
(2)
O b s e r v e that for given a and b (a and c) there a r e only a f i n i t e n u m b e r of c(b) values f o r which the inequalities (2) can be obeyed. The p r e s e n t r u l e s may be g e n e r a l i z e d to include N - p o i n t functions, N >~ 4, by demanding that they be topologically equivalent to t r e e d i a g r a m s c o m p o s e d of internal lines and t h r e e - p o i n t functions, as is seen to be the c a s e in the d i a g r a m s of fig. 1. In p a r t i c u l a r we forbid s c a t t e r i n g d i a g r a m s in which all lines just "go through". Such d i a g r a m s could provide an a l t e r native to the m e c h a n i s m of P o m e r a n c h u k s c a t t e r i n g p r o p o s e d in ref. [12]. T h e r e is a f u r t h e r i n t e r e s t i n g application of our selection rule. A s s u m e (we s t r e s s here that this is an additional assumption) that the selection rule (S) e n c o m p a s s e s h a d r o n s with non-vanishing t r i a l i t y (in p a r t i c u l a r , quarks). It follows that the m e s o n - q u a r k - a n t i q u a r k coupling is forbidden. This is p r e c i s e l y what one would expect in a h a r m o n i c o s c i l l a t o r model and m a y be * These three inequalities are not all independent. The one with greatest n i on the left hand side implies the other two.
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r e l a t e d to the difficulty of producing f r e e quarks. By the same a r g u m e n t higher m e s o n s (M4, M 6 , . . . ) or b a r y o n s (B, B5, B 7 , . . . ) cannot couple to q u a r k s either. Rule (A) implies that at least one additional qcl p a i r m u s t be produced in any decay p r o c e s s , as i l l u s t r a t e d for M4 ~ B ~ in fig. 3a. Rule (B) then ins u r e s that M4, for example, will not mix with M 2 under the action of the f o r c e that s e e m s to produce and annihilate q~l p a i r s so easily.
M4
:
M4 a o
M2
:
"
" b
Fig. 3. Quark model diagrams illustrating the allowed M4 ---*B]~ coupling (a), and the forbidden M4M2 mixing (b). It has been suggested that p r o d u c t i o n and annihilation of q~i p a i r s takes place in the isosinglet 3P o state, which has the quantum n u m b e r of the vacuum [14]. An appealing candidate f o r M4 obeying (B) would then be one in which no qq p a i r was in a 3P o state. The non-exotic m e m b e r s of the M 4 multiplets would then be prevented f r o m mixing with M2. A r e m a r k about the expected a c c u r a c y of our selection rule is in o r d e r . If the decay A ~ B + C and the r e a c t i o n B + C ~ D + E a r e allowed, then final state i n t e r a c t i o n s will induce the decay A ~ D + E whether o r not a d i r e c t A ~ D + E coupling is allowed. Thus, f o r example, the d e c a y s f ' ~ ~ or M4 ~ MM, forbidden by the selection rule, can still p r o c e e d through the chains f' ~ KI~ ~ ~Tr and M4 ~ BB ~ MM, as i l l u s t r a t e d in fig. 4. E x p e r i mentally f' ~ ~ is r a r e , indicating that the net effects of such r e s c a t t e r ing c o r r e c t i o n s a r e expected to be small. We could u n d e r s t a n d this on the b a s i s of our r u l e s (A) and (B) if we w e r e to impose these r u l e s (at least approximately) no m a t t e r w h a t o t h e r c l o s e d loops e n t e r the d i a g r a m s . One then s e e s , f o r example, that figs. 4a and 4b violate the so extended condition (B), while fig. 4c violates the extended condition A. Through p a r t i a l l y u n i t a r i z e d v e r t i c e s of the type 4 a - c one would induce s m a l l r e s o n a n t c o n t r i butions to channels which in the z e r o width a p p r o x i m a t i o n would be empty of r e s o n a n c e s . F o r example the d i a g r a m 4d induces a B 5 pole in MB s c a t t e r ing. T h e r e would then be s m a l l r e s o n a n t contributions to exotic channels such as K+p ~ K+p, K°p - , K°p, etc. If the selection rule we have p r e s e n t e d is as well obeyed in all c a s e s as it s e e m s to be f o r the MMM and BBM couplings, we expect the "forbidden" decay modes of hadrons (such as M4 ~ MM) to be c h a r a c t e r i z e d by p a r t i a l widths about an o r d e r of magnitude below the c o r r e s p o n d i n g p a r t i a l widths of allowed modes. The selection rule against the p r o d u c t i o n of f r e e q u a r k s is not affected by unitarity c o r r e c t i o n s .
HADRON COUPLINGS
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a
b
C |
B M
v
-
[,
l
M
?_
B
$I
~
_
L
D
M4
M4
d Fig. 4. Quark model d i a g r a m s for inducing forbidden couplings through final state interactions: a) (p --. p~; b) and c) M4 --'MM; d) MB --~B 5 --. MB. F o r the case d) t h e r e a r e also d i a g r a m s with c r o s s e d lines like the d i a g r a m c) but we do not d r a w them all h e r e as our purpose is m e r e l y i l l u s t r a t i v e .
3. E X P E R I M E N T A L C O N S E Q U E N C E S A s t h e p r e s e n t r u l e s r e d u c e to p r e v i o u s l y s u g g e s t e d o n e s f o r t h e c o u p l i n g s of t h e u s u a l m e s o n s (M2) a n d b a r y o n s (B3) , t h e i r u s e f u l n e s s d e p e n d s on t h e e x i s t e n c e of " e x o t i c " s t a t e s ( o n e s o t h e r t h a n M 2 o r B3). A l l of t h e k n o w n p a r t i c l e s a r e n o n - e x o t i c , a s a r e t h e d o m i n a n t e x c h a n g e s . In b o t h f o r m a t i o n (AB ~ E) a n d p r o d u c t i o n (AB --. E C ) t h e r e l e v a n t c o u p l i n g of t h e e x o t i c p a r t i c l e E i s t h e r e f o r e (XYE), w h e r e n e i t h e r X n o r Y i s e x o t i c . A c c o r d i n g to t h e p r e s e n t r u l e s , t h e o n l y s u c h s i t u a t i o n o c c u r s in t h e (BBM4) c o u p l i n g . H e n c e M4 s h o u l d b e t h e e a s i e s t e x o t i c s t a t e t o s e e , a n d s h o u l d b e p r o d u c e d p r i m a r i l y v i a c o u p l i n g to b a r y o n s *. S e v e r a l e x p e r i m e n t s in a d d i t i o n t o t h o s e a l r e a d y s u g g e s t e d [7] w o u l d b e h e l p f u l i n c o n f i r m ing this prediction. * This c o n t r a s t s with the suggestion of ref. [15] that an exotic p a r t i c l e t~ near the A 2 m a s s m a y be produced by mesons via meson exchange.
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3.1. M i s s i n g m a s s s p e c t r o m e t e r It i s p o s s i b l e t h a t s o m e of the l o w e s t - l y i n g M 4 m e s o n s l i e b e l o w the baryon-antibaryon threshold. Therefore, althoufh they could be produced b y b a r y o n s v i a b a r y o n e x c h a n g e , t h e y w o u l d h a v e to d e c a y into m a n y d i f f e r e n t t y p e s of m u l t i m e s o n s t a t e s . In e a c h i n d i v i d u a l s t a t e the d e t e c t i o n of an e x o t i c r e s o n a n c e m i g h t b e q u i t e d i f f i c u l t . A w a y of c i r c u m v e n t i n g t h i s p r o b l e m w o u l d b e in b a c k w a r d m i s s i n g - m a s s e x p e r i m e n t s s u c h a s y+p -~ n(MM) ++ .
(3)
T h e n e u t r o n w o u l d b e f a s t and m o r e o r l e s s f o r w a r d , s o d e t e c t i o n b y c o u n t e r w o u l d b e l e s s of a p r o b l e m t h a n one m i g h t s u s p e c t *. In g e n e r a l f o r any r e a c t i o n MB ~ BE i t i s p o s s i b l e f o r E to b e e x o t i c w h e n p r o d u c e d n e a r t h e b a c k w a r d d i r e c t i o n . E x a m p l e s of s u c h p r o c e s s e s w h i c h m i g h t b e s e e n m o r e e a s i l y in b u b b l e c h a m b e r e x p e r i m e n t s a r e pn--~ 7r+(MM) - - ;
(4)
~Op ~ ~_(MM)++ , K - n ~ Z+(MM) - - ;
(5)
l ~ ° p - - , ~ - ( M M ) ++ , K-n-~ ~°:(MM)- ;
(6)
~r+p-~ A(MM) ++ , ~-n ~ A(MM)- .
(7)
In e a c h c a s e s e l e c t i o n of f o r w a r d b a r y o n s w o u l d p r o b a b l y e n h a n c e t h e e f fect. Another missing-mass experiment would use a double-arm spectrometer, a s in p p ~ 7r+~+(MM) - -
(8)
7r-p ~ p n + ( M M ) - - .
(9)
or
One s h o u l d l o o k f o r b a c k w a r d ~ + ' s in r e a c t i o n (8), and f o r a f o r w a r d p a n d b a c k w a r d ~+ in r e a c t i o n (9) **. The n a r r o w n e s s of o b s e r v e d p e a k s in m i s s i n g - m a s s e x p e r i m e n t s [16] m a y b e a p o i n t in f a v o r of o u r s e l e c t i o n r u l e [15]. If s u c h p e a k s i n d i c a t e d M4 o b j e c t s t h e y w o u l d b e m o r e p r o n o u n c e d f o r f o r w a r d f i n a l b a r y o n s .
* One of us (J. R.) thanks Prof. D . R e e d e r for this observation. ** We thank Prof. B. Maglic for informing us that the l a t t e r reaction is in fact being studied at present.
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P r e s e n t data on b a c k w a r d boson production [17] do not s e e m to show an o v e r w h e l m i n g enhancement, but t h e i r s t a t i s t i c s a r e as yet limited c o m p a r e d with those of the CERN e x p e r i m e n t [16]. C e r t a i n p e a k s do s e e m to a p p e a r only in the b a c k w a r d d i r e c t i o n (e.g. M = 2260 ± 18, F --< 25). These would be e s p e c i a l l y s t r o n g candidates f o r M4 m e s o n s . One should s t r e s s that s o m e M4 m e s o n s will have conventional Y and I: for example, the 2_7 contains I = 0 and I = 1 multiplets of Y = 0 and I = ½ multiplets.of Y = +1. That s o m e of the o b s e r v e d m i s s i n g - m a s s peaks a r e c o r r e l a t e d with NI~ r e s o n a n c e s , and hence c o r r e s p o n d to I = 1 (ref. [16]), should t h e r e f o r e not be taken as evidence against their M4 a s s i g n m e n t . 3.2. M a s s distributions in known f i n a l s t a t e s The r e l a t i v e a b s e n c e of BB r e s o n a n c e s in I~B ~ BB + m e s o n s involves competition f r o m d o u b l e - i s o b a r production. This m e s o n - e x c h a n g e p r o c e s s tends to o v e r w h e l m the w e a k e r b a r y o n - e x c h a n g e n e c e s s a r y for BB r e s o nance production. It is t h e r e f o r e not s u r p r i s i n g that the f i r s t r e p o r t of such a r e s o n a n c e in pp ~ pp~+~-~o came t h r e e y e a r s after the exposure on which it was b a s e d [18]. I n f o r m a t i o n on couplings to BB of h i g h e r - m a s s r e s o n a n c e s with c o n v e n tional Y and I (M2 or M4) would be of g r e a t help in s u g g e s t i n g e x p e r i m e n t s where the exotic r e s o n a n c e s might be seen. As e x a m p l e s , c o m p a r e pn ~ C[~ + or K + (backward)] ,
(10a)
pp -~ D[n- or K ° (backward)] ,
(lOb)
with
w h e r e C and D contain a BB p a i r and p o s s i b l y some m e s o n s . In the r e a c tions (10a) C will be exotic, and thus M4, while in (10b) D will have conventional quantum n u m b e r s and m a y c o r r e s p o n d to M2 or M 4. The exchanges will be the s a m e in both c a s e s . The c o m p a r i s o n will then give g(A---= p C - - ) / g ( A ~4 pD+), g(Y= p C - - ) / g ~ "~ pDO), etc. V e r y s m a l l values f o r these r a t i o s would indicate that D is indeed M2, and that B3 B3 M2 couplings p r e d o m i n a t e over B3 B3 M4 couplings even for M2 or M 4 above BB threshold. Resonant s a t u r a t i o n of FESR in BB s c a t t e r i n g would indicate the r e v e r s e to be true u n l e s s t h e r e a r e many m o r e M4 m e s o n s in this e n e r g y r e g i o n than t h e r e a r e M2. One type of p r o c e s s in which s o m e groups claim to see b a r y o n - a n t i b a r y o n r e s o n a n c e s is K+N ~ A N ' + . . . [19]. One channel of this s o r t can actually yield an exotic I~B state: it is K+p --" [~p~+]n
(11)
f o r which the world data above 5 G e V / c include some 235 events. (The exotic state is shown in b r a c k e t s . ) To our knowledge no compilation of world data on p r o c e s s (11) (or r e lated ones) has been made. This would c e r t a i n l y be of value.
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3.3. Role of observed exotic exchanges E x o t i c e x c h a n g e s h a v e b e e n r e p o r t e d in r e a c t i o n s of t h e t y p e [20]
MB - " M ' D
,
(12)
w h e r e in (12) B = ½+ o c t e t , D = 3 + d e c i m e t . A r e m a r k a b l e f e a t u r e of t h e differential cross sections for the above processes is their sharp forward p e a k i n g , r e m i n i s c e n t of the s a m e e f f e c t in 7 p --- ~+n a n d pn - - np. T h e s u c c e s s i v e e x c h a n g e of two m e s o n s one of w h i c h i s a p i o n (the s y s t e m of two m e s o n s h a v i n g e x o t i c q u a n t u m n u m b e r s ) c a n l e a d to s u c h a n a r r o w p e a k in (12) in a n a t u r a l w a y *. T h u s a l t h o u g h M M ' M 4 w o u l d b e f o r b i d d e n b y o u r r u l e s , it i s q u i t e p o s s i b l e t h a t (12) i n v o l v e s m u l t i p l e e x c h a n g e s , e a c h of w h i c h o b e y s the r u l e s . If p r o c e s s (12) i n d e e d r e p r e s e n t s a s m a l l M M ' M ~ c o u p l i n g w e w o u l d e x p e c t m u c h l a r g e r e f f e c t s if M ' , w e r e r e p l a c e d b y M 4 o r if M - - . B 3 , M ' B 5. In e i t h e r c a s e t h e c o u p l i n g w o u l d t h e n b e one a l l o w e d b y o u r r u l e s . R e a c t i o n s of i n t e r e s t w o u l d t h e n i n c l u d e ~ - p - . ( M 4 ) - - A + + a n d p n - . Y * - (B5)++. Such p r o c e s s e s w o u l d p r o b a b l y b e s t b e s t u d i e d in b u b b l e - c h a m ber missing-mass analyses. One of u s ( J . R . ) w o u l d l i k e to t h a n k P r o f . G. Z w e i g , who h a s c o n s i d e r e d s i m i l a r p r o b l e m s , f o r a v e r y p r o d u c t i v e c o n v e r s a t i o n . Two of u s ( P . F . a n d J . R . ) w o u l d l i k e t o t h a n k D r s . R. C. A r n o l d a n d K. C. W a l i f o r t h e h o s p i t a l i t y of the H i g h E n e r g y P h y s i c s D i v i s i o n of t h e A r g o n n e N a t i o n a l L a b o r a t o y , w h e r e p a r t of t h i s w o r k w a s p e r f o r m e d . * This has also been r e a l i z e d by R. Arnold (private communication).
REFERENCES [1] K. Igi, Phys. Rev. L e t t e r s 9 (1962) 76; R.Dolen, D . H o r n a n d C.Schmid, Phys. Rev. 166 (1968) 1768. [2] P . G . O . F r e u n d , Phys. Rev. L e t t e r s 20 (1968) 235; H . H a r a r i , Phys. Rev. L e t t e r s 20 (1968) 1395. [3] G . F . Chew, Comments on Nuclear and P a r t i c l e P h y s i c s 2 (1968) 74. [4] H. H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 562; J . R o s n e r , Phys. Rev. L e t t e r s 22 (1969) 689. [5] C.Schmid, Phys. Rev. L e t t e r s 20 (1968) 628. [6] C.B. Chiu and J. Finkelstein, Phys. L e t t e r s 27B (1968) 510. [7] J . R o s n e r , Phys. Rev. L e t t e r s 21 (1968) 950. [8] H . J . Lipkin, Nucl. Phys. B9 (1969) 349. [9] D. Horn, H . J . Lipkin and S.Meshkov, Phys. Rev. L e t t e r s 17 (1966) 1200. [10] J. Mandula, J. Weyers and G. Zweig, Phys. Rev. L e f t e r s , to be published. [11] J. Mandula, C. Rebbi, R. Slansky, J. Weyers and G. Zweig, Phys. Rev. L e t t e r s 22 (1969) 1147. [12] P. G. O. Freund and R. J. R i v e r s , Phys. L e t t e r s , to be published. [13] G. Zweig, CERN r e p o r t CERN-Th-402, 1964, unpublished. [14] J . R o s n e r , ref.[4]. [15] R. C. Arnold and J. L. Uretsky, to be published.
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[16] B. Maglic, Report to the Lund International Conference on Elementary P a r t i c l e s , June, 1969. [17] E . W . A n d e r s o n et al., Phys. Rev. Letters 22 (1969~ 1390. [18] K. Myklebost, quoted by B. F r e n c h in Proceedings of the 14th International Conference on High Energy Physics, Vienna, 1968, p. 130. [19] B. French, ref. [18]. [20] M.A.Abolins et al., to be published.
Nuclear Physics B13 f1969) 237-245. North-Holland Publ. Comp., Amsterdam
QUARK MODEL SELECTION RULES FOR HADRON COUPLINGS* P. G. O. FREUND and R. W A L T Z The Enrico ~'ermi Institute and the Department of Physics, The University of Chicago, Chicago, Illinois, 60637 and J . ROSNER School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, 55455
Received 7 August 1969
Abstract: Quark model selection rules are derived which apply to the couplings of hadrons composed of any number of quarks. These reduce to well-known and experimentally verified rules in the simplest cases, provide a necessary selection rule in the application of duality to baryon-antibaryon systems, and could explain the apparent difficulty of producing free quarks. Some experimental consequences of the rules are discussed.
1. ~ T R O D U C T I O N F i n i t e - e n e r g y s u m r u l e s (FESR) [1] allow the c o n s i s t e n t m a t c h i n g of d i r e c t - c h a n n e l and R e g g e - p o l e d e s c r i p t i o n s of h a d r o n i c r e a c t i o n s . The a p p r o x i m a t i o n of d i r e c t - c h a n n e l c o n t r i b u t i o n s by r e s o n a n c e s s e e m s to s a t u r a t e F E S R f o r t r a j e c t o r i e s o t h e r than the P o m e r a n c h u k (P) f a i r l y w e l l [2]. T h i s s u c c e s s h a s b e e n c a l l e d " d u a l i t y " [3] s i n c e it i m p l i e s that d i r e c t c h a n n e l r e s o n a n c e s and e x c h a n g e d Regge p o l e s a r e r e a l l y two alternative d e s c r i p t i o n s of the s a m e p r o c e s s , and should not be a d d e d to one a n o t h e r . D u a l i t y i s b e s t v i s u a l i z e d u s i n g q u a r k m o d e l d i a g r a m s [4]. Two d i a g r a m s
o b c Fig. 1. Quark model diagrams illustrating duality.
* Supported in part by the U.S. Atomic Energy Commission through contracts AT(11-1)-264 and AT(11-1)-1764. $ National Science Foundation predoctoral fellow.
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f o r m e s o n - m e s o n s c a t t e r i n g a r e i l l u s t r a t e d in fig. 1. F i g . l a h a s q~l i n t e r m e d i a t e s t a t e s in the s - c h a n n e l , s h o w n by t h e w a v y l i n e . T h e s e s t a t e s a r e e x p e c t e d to " b u i l d " a l a r g e a b s o r p t i v e c o n t r i b u t i o n a s s o c i a t e d w i t h t h e t c h a n n e l R e g g e p o l e t e r m . On the o t h e r hand fig. l b h a s no q~l i n t e r m e d i a t e s t a t e in the s - c h a n n e l , and h e n c e one e x p e c t s t h e a b s o r p t i v e c o n t r i b u t i o n s of the v a r i o u s t - c h a n n e l R e g g e p o l e t e r m s to c a n c e l one a n o t h e r . If t h i s i s the c a s e one d e r i v e s l a r g e s e t s of e x p e r i m e n t a l l y v e r i f i e d c o n s t r a i n t s , i n c l u d i n g d e g e n e r a c y of v a r i o u s t r a j e c t o r i e s c o u p l i n g r e l a t i o n s , and s p e c i f i c m e c h a n i s m s of SU(3) b r e a k i n g [ 2 , 4 - 7 ] . A s t r i c t a p p l i c a t i o n of d u a l i t y r e q u i r e s the e x i s t e n c e of c e r t a i n e x o t i c m e s o n s to " b u i l d " the e x p e c t e d n o n - P e x c h a n g e s in b a r y o n - a n t i b a r y o n s c a t t e r i n g [7]. T h i s s i t u a t i o n i s d e p i c t e d in fig. l c . T h e d o m i n a n t t - c h a n n e l e x c h a n g e s s e e m e x p e r i m e n t a l l y to b e c o n f i n e d to t h o s e i n v o l v i n g a s i n g l e q~l p a i r . The s - c h a n n e l s t a t e s t h a t " b u i l d " t h e s e e x c h a n g e s m u s t t h e n b e M4 = qq~*. If m e s o n s M4 e x i s t , t h e y m u s t not u p s e t o u r c o n c l u s i o n s in MM s c a t t e r ing. One p o s s i b i l i t y [7], w h i c h we s h a l l d i s c u s s h e r e , i s t h a t such m e s o n s do not couple to MM s y s t e m s **. M o r e g e n e r a l l y , i t h a s b e e n s h o w n [12] t h a t m e s o n s M = (q~t)v (v = 2, 3 , 4 , . . . ) a n d b a r y o n s B2v+3 = q V+3~lv(v = 1 , 2 , 3 , . . . ) a r e r e q u i r e d b y d u a l i t y . F o r t h e s e h i g h e r m e s o n s and b a r y o n s a s i m i l a r q u e s t i o n c a n b e a s k e d . How d o e s one a v o i d the c o u p l i n g of t h e s e h i g h e r h a d r o n s to l o w e r c h a n n e l s in w h i c h t h e i r p r e s e n c e s e e m s to b e e x c l u d e d b o t h by e x p e r i m e n t and by d u a l i t y ? We s h a l l a n s w e r t h e s e q u e s t i o n s in t e r m s of a s e l e c t i o n r u l e i n s p i r e d b y the q u a r k m o d e l .
2. T H E S E L E C T I O N R U L E L e t us f i r s t c o n s i d e r the c a s e s of M M M a n d B B M c o u p l i n g s [13]. T h e q u a r k m o d e l d i a g r a m s f o r t h e s e c o u p l i n g s a r e p r e s e n t e d in fig.2. T h e e x p e r i m e n t a l a b s e n c e of ~ -~ pu f o r b i d s d i a g r a m 2b. S i m i l a r l y gq~]~N = 0 f o r b i d s d i a g r a m 2d. T h e a l l o w e d d i a g r a m s 2a a n d 2c s h a r e the f o l l o w i n g i m portant properties: (A) E a c h of the t h r e e h a d r o n s m e e t i n g a t a v e r t e x e x c h a n g e s a t l e a s t one q u a r k l i n e w i t h e a c h of t h e r e m a i n i n g two h a d r o n s . (B) No q u a r k l i n e i s c o n n e c t e d w i t h a n a n t i q u a r k l i n e f r o m the s a m e hadron. In the c a s e of M M M a n d I~BM c o u p l i n g s e i t h e r p r o p e r t y (A) o r p r o p e r t y * Throughout this paper we shall label a hadron by its baryon number (M for m e s on, B for baryon) and an index equal to the total number of quarks and antiquarks contained in the hadron. We shall abbreviate M2 by M and B 3 by B. The p o s s i b i lity that states like M4, M 6, . . . , B 5, B 7, . , . do ffot correspond to resonances has been discussed elsewhere (see r e f s . [4, 7, 8]). We also a r e aware that duality may not be u n i v e r s a l l y applicable (see ref. [10]) when used to derive certain sets of exchange degeneracies (see ref. [11]). ** This is to be distinguished from the r e s u l t s of SU(6)W in which not all M4MM couplings a r e forbidden.
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_JL_
v
o
b
JL c
239
u d
Fig. 2. Quark model diagrams for MMM (a, and b) and BBM (c and d) couplings. Diagrams a and c are allowed, diagrams b and d forbidden. (b) alone suffices to exclude the couplings in figs. 2b and 2d. This is not the c a s e in g e n e r a l , however. We shall adopt (A) and (B) as g e n e r a l p r i n c i p l e s defining an allowed vertex. We can then establish the following selection rule as a c o r o l l a r y to (A) and (B): (S) Let HI, H2, and H 3 be any hadrons p e r m i t t e d to couple at a v e r t e x by b a r y o n n u m b e r , h y p e r c h a r g e , and isospin conservation. Let vi and Pi (i = 1, 2, 3) be the n u m b e r of quarks and antiquarks in H i. Define n i = v i + vi" Then each of the t h r e e inequalities n i ~nj
+n k - 2(i,j,k
p e r m . of 1 , 2 , 3 )
(1)
m u s t hold or the HIH2H 3 coupling is forbidden *. By (A), the total n u m b e r of lines N i running f r o m the I.iii.ik s y s t e m to H i m u s t be l e s s than o r equal to n.~ + n k - 2, since at l e a s t on6 line of Hj is conJ nected to one of Hk. However, by (B) we m u s t have N i = n i. Hence the s e lection rule is proved. An i m m e d i a t e application of (S) is that M4 ~ MM is forbidden but M 4 B ~ allowed. G e n e r a l l y the coupling Ba ~ B b + Me, M a --. M b + M c or M a -* B b + B c is allowed only ff la-b]
(2)
O b s e r v e that for given a and b (a and c) there a r e only a f i n i t e n u m b e r of c(b) values f o r which the inequalities (2) can be obeyed. The p r e s e n t r u l e s may be g e n e r a l i z e d to include N - p o i n t functions, N >~ 4, by demanding that they be topologically equivalent to t r e e d i a g r a m s c o m p o s e d of internal lines and t h r e e - p o i n t functions, as is seen to be the c a s e in the d i a g r a m s of fig. 1. In p a r t i c u l a r we forbid s c a t t e r i n g d i a g r a m s in which all lines just "go through". Such d i a g r a m s could provide an a l t e r native to the m e c h a n i s m of P o m e r a n c h u k s c a t t e r i n g p r o p o s e d in ref. [12]. T h e r e is a f u r t h e r i n t e r e s t i n g application of our selection rule. A s s u m e (we s t r e s s here that this is an additional assumption) that the selection rule (S) e n c o m p a s s e s h a d r o n s with non-vanishing t r i a l i t y (in p a r t i c u l a r , quarks). It follows that the m e s o n - q u a r k - a n t i q u a r k coupling is forbidden. This is p r e c i s e l y what one would expect in a h a r m o n i c o s c i l l a t o r model and m a y be * These three inequalities are not all independent. The one with greatest n i on the left hand side implies the other two.
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r e l a t e d to the difficulty of producing f r e e quarks. By the same a r g u m e n t higher m e s o n s (M4, M 6 , . . . ) or b a r y o n s (B, B5, B 7 , . . . ) cannot couple to q u a r k s either. Rule (A) implies that at least one additional qcl p a i r m u s t be produced in any decay p r o c e s s , as i l l u s t r a t e d for M4 ~ B ~ in fig. 3a. Rule (B) then ins u r e s that M4, for example, will not mix with M 2 under the action of the f o r c e that s e e m s to produce and annihilate q~l p a i r s so easily.
M4
:
M4 a o
M2
:
"
" b
Fig. 3. Quark model diagrams illustrating the allowed M4 ---*B]~ coupling (a), and the forbidden M4M2 mixing (b). It has been suggested that p r o d u c t i o n and annihilation of q~i p a i r s takes place in the isosinglet 3P o state, which has the quantum n u m b e r of the vacuum [14]. An appealing candidate f o r M4 obeying (B) would then be one in which no qq p a i r was in a 3P o state. The non-exotic m e m b e r s of the M 4 multiplets would then be prevented f r o m mixing with M2. A r e m a r k about the expected a c c u r a c y of our selection rule is in o r d e r . If the decay A ~ B + C and the r e a c t i o n B + C ~ D + E a r e allowed, then final state i n t e r a c t i o n s will induce the decay A ~ D + E whether o r not a d i r e c t A ~ D + E coupling is allowed. Thus, f o r example, the d e c a y s f ' ~ ~ or M4 ~ MM, forbidden by the selection rule, can still p r o c e e d through the chains f' ~ KI~ ~ ~Tr and M4 ~ BB ~ MM, as i l l u s t r a t e d in fig. 4. E x p e r i mentally f' ~ ~ is r a r e , indicating that the net effects of such r e s c a t t e r ing c o r r e c t i o n s a r e expected to be small. We could u n d e r s t a n d this on the b a s i s of our r u l e s (A) and (B) if we w e r e to impose these r u l e s (at least approximately) no m a t t e r w h a t o t h e r c l o s e d loops e n t e r the d i a g r a m s . One then s e e s , f o r example, that figs. 4a and 4b violate the so extended condition (B), while fig. 4c violates the extended condition A. Through p a r t i a l l y u n i t a r i z e d v e r t i c e s of the type 4 a - c one would induce s m a l l r e s o n a n t c o n t r i butions to channels which in the z e r o width a p p r o x i m a t i o n would be empty of r e s o n a n c e s . F o r example the d i a g r a m 4d induces a B 5 pole in MB s c a t t e r ing. T h e r e would then be s m a l l r e s o n a n t contributions to exotic channels such as K+p ~ K+p, K°p - , K°p, etc. If the selection rule we have p r e s e n t e d is as well obeyed in all c a s e s as it s e e m s to be f o r the MMM and BBM couplings, we expect the "forbidden" decay modes of hadrons (such as M4 ~ MM) to be c h a r a c t e r i z e d by p a r t i a l widths about an o r d e r of magnitude below the c o r r e s p o n d i n g p a r t i a l widths of allowed modes. The selection rule against the p r o d u c t i o n of f r e e q u a r k s is not affected by unitarity c o r r e c t i o n s .
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a
b
C |
B M
v
-
[,
l
M
?_
B
$I
~
_
L
D
M4
M4
d Fig. 4. Quark model d i a g r a m s for inducing forbidden couplings through final state interactions: a) (p --. p~; b) and c) M4 --'MM; d) MB --~B 5 --. MB. F o r the case d) t h e r e a r e also d i a g r a m s with c r o s s e d lines like the d i a g r a m c) but we do not d r a w them all h e r e as our purpose is m e r e l y i l l u s t r a t i v e .
3. E X P E R I M E N T A L C O N S E Q U E N C E S A s t h e p r e s e n t r u l e s r e d u c e to p r e v i o u s l y s u g g e s t e d o n e s f o r t h e c o u p l i n g s of t h e u s u a l m e s o n s (M2) a n d b a r y o n s (B3) , t h e i r u s e f u l n e s s d e p e n d s on t h e e x i s t e n c e of " e x o t i c " s t a t e s ( o n e s o t h e r t h a n M 2 o r B3). A l l of t h e k n o w n p a r t i c l e s a r e n o n - e x o t i c , a s a r e t h e d o m i n a n t e x c h a n g e s . In b o t h f o r m a t i o n (AB ~ E) a n d p r o d u c t i o n (AB --. E C ) t h e r e l e v a n t c o u p l i n g of t h e e x o t i c p a r t i c l e E i s t h e r e f o r e (XYE), w h e r e n e i t h e r X n o r Y i s e x o t i c . A c c o r d i n g to t h e p r e s e n t r u l e s , t h e o n l y s u c h s i t u a t i o n o c c u r s in t h e (BBM4) c o u p l i n g . H e n c e M4 s h o u l d b e t h e e a s i e s t e x o t i c s t a t e t o s e e , a n d s h o u l d b e p r o d u c e d p r i m a r i l y v i a c o u p l i n g to b a r y o n s *. S e v e r a l e x p e r i m e n t s in a d d i t i o n t o t h o s e a l r e a d y s u g g e s t e d [7] w o u l d b e h e l p f u l i n c o n f i r m ing this prediction. * This c o n t r a s t s with the suggestion of ref. [15] that an exotic p a r t i c l e t~ near the A 2 m a s s m a y be produced by mesons via meson exchange.
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3.1. M i s s i n g m a s s s p e c t r o m e t e r It i s p o s s i b l e t h a t s o m e of the l o w e s t - l y i n g M 4 m e s o n s l i e b e l o w the baryon-antibaryon threshold. Therefore, althoufh they could be produced b y b a r y o n s v i a b a r y o n e x c h a n g e , t h e y w o u l d h a v e to d e c a y into m a n y d i f f e r e n t t y p e s of m u l t i m e s o n s t a t e s . In e a c h i n d i v i d u a l s t a t e the d e t e c t i o n of an e x o t i c r e s o n a n c e m i g h t b e q u i t e d i f f i c u l t . A w a y of c i r c u m v e n t i n g t h i s p r o b l e m w o u l d b e in b a c k w a r d m i s s i n g - m a s s e x p e r i m e n t s s u c h a s y+p -~ n(MM) ++ .
(3)
T h e n e u t r o n w o u l d b e f a s t and m o r e o r l e s s f o r w a r d , s o d e t e c t i o n b y c o u n t e r w o u l d b e l e s s of a p r o b l e m t h a n one m i g h t s u s p e c t *. In g e n e r a l f o r any r e a c t i o n MB ~ BE i t i s p o s s i b l e f o r E to b e e x o t i c w h e n p r o d u c e d n e a r t h e b a c k w a r d d i r e c t i o n . E x a m p l e s of s u c h p r o c e s s e s w h i c h m i g h t b e s e e n m o r e e a s i l y in b u b b l e c h a m b e r e x p e r i m e n t s a r e pn--~ 7r+(MM) - - ;
(4)
~Op ~ ~_(MM)++ , K - n ~ Z+(MM) - - ;
(5)
l ~ ° p - - , ~ - ( M M ) ++ , K-n-~ ~°:(MM)- ;
(6)
~r+p-~ A(MM) ++ , ~-n ~ A(MM)- .
(7)
In e a c h c a s e s e l e c t i o n of f o r w a r d b a r y o n s w o u l d p r o b a b l y e n h a n c e t h e e f fect. Another missing-mass experiment would use a double-arm spectrometer, a s in p p ~ 7r+~+(MM) - -
(8)
7r-p ~ p n + ( M M ) - - .
(9)
or
One s h o u l d l o o k f o r b a c k w a r d ~ + ' s in r e a c t i o n (8), and f o r a f o r w a r d p a n d b a c k w a r d ~+ in r e a c t i o n (9) **. The n a r r o w n e s s of o b s e r v e d p e a k s in m i s s i n g - m a s s e x p e r i m e n t s [16] m a y b e a p o i n t in f a v o r of o u r s e l e c t i o n r u l e [15]. If s u c h p e a k s i n d i c a t e d M4 o b j e c t s t h e y w o u l d b e m o r e p r o n o u n c e d f o r f o r w a r d f i n a l b a r y o n s .
* One of us (J. R.) thanks Prof. D . R e e d e r for this observation. ** We thank Prof. B. Maglic for informing us that the l a t t e r reaction is in fact being studied at present.
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P r e s e n t data on b a c k w a r d boson production [17] do not s e e m to show an o v e r w h e l m i n g enhancement, but t h e i r s t a t i s t i c s a r e as yet limited c o m p a r e d with those of the CERN e x p e r i m e n t [16]. C e r t a i n p e a k s do s e e m to a p p e a r only in the b a c k w a r d d i r e c t i o n (e.g. M = 2260 ± 18, F --< 25). These would be e s p e c i a l l y s t r o n g candidates f o r M4 m e s o n s . One should s t r e s s that s o m e M4 m e s o n s will have conventional Y and I: for example, the 2_7 contains I = 0 and I = 1 multiplets of Y = 0 and I = ½ multiplets.of Y = +1. That s o m e of the o b s e r v e d m i s s i n g - m a s s peaks a r e c o r r e l a t e d with NI~ r e s o n a n c e s , and hence c o r r e s p o n d to I = 1 (ref. [16]), should t h e r e f o r e not be taken as evidence against their M4 a s s i g n m e n t . 3.2. M a s s distributions in known f i n a l s t a t e s The r e l a t i v e a b s e n c e of BB r e s o n a n c e s in I~B ~ BB + m e s o n s involves competition f r o m d o u b l e - i s o b a r production. This m e s o n - e x c h a n g e p r o c e s s tends to o v e r w h e l m the w e a k e r b a r y o n - e x c h a n g e n e c e s s a r y for BB r e s o nance production. It is t h e r e f o r e not s u r p r i s i n g that the f i r s t r e p o r t of such a r e s o n a n c e in pp ~ pp~+~-~o came t h r e e y e a r s after the exposure on which it was b a s e d [18]. I n f o r m a t i o n on couplings to BB of h i g h e r - m a s s r e s o n a n c e s with c o n v e n tional Y and I (M2 or M4) would be of g r e a t help in s u g g e s t i n g e x p e r i m e n t s where the exotic r e s o n a n c e s might be seen. As e x a m p l e s , c o m p a r e pn ~ C[~ + or K + (backward)] ,
(10a)
pp -~ D[n- or K ° (backward)] ,
(lOb)
with
w h e r e C and D contain a BB p a i r and p o s s i b l y some m e s o n s . In the r e a c tions (10a) C will be exotic, and thus M4, while in (10b) D will have conventional quantum n u m b e r s and m a y c o r r e s p o n d to M2 or M 4. The exchanges will be the s a m e in both c a s e s . The c o m p a r i s o n will then give g(A---= p C - - ) / g ( A ~4 pD+), g(Y= p C - - ) / g ~ "~ pDO), etc. V e r y s m a l l values f o r these r a t i o s would indicate that D is indeed M2, and that B3 B3 M2 couplings p r e d o m i n a t e over B3 B3 M4 couplings even for M2 or M 4 above BB threshold. Resonant s a t u r a t i o n of FESR in BB s c a t t e r i n g would indicate the r e v e r s e to be true u n l e s s t h e r e a r e many m o r e M4 m e s o n s in this e n e r g y r e g i o n than t h e r e a r e M2. One type of p r o c e s s in which s o m e groups claim to see b a r y o n - a n t i b a r y o n r e s o n a n c e s is K+N ~ A N ' + . . . [19]. One channel of this s o r t can actually yield an exotic I~B state: it is K+p --" [~p~+]n
(11)
f o r which the world data above 5 G e V / c include some 235 events. (The exotic state is shown in b r a c k e t s . ) To our knowledge no compilation of world data on p r o c e s s (11) (or r e lated ones) has been made. This would c e r t a i n l y be of value.
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P . G . O . FREUND et al.
3.3. Role of observed exotic exchanges E x o t i c e x c h a n g e s h a v e b e e n r e p o r t e d in r e a c t i o n s of t h e t y p e [20]
MB - " M ' D
,
(12)
w h e r e in (12) B = ½+ o c t e t , D = 3 + d e c i m e t . A r e m a r k a b l e f e a t u r e of t h e differential cross sections for the above processes is their sharp forward p e a k i n g , r e m i n i s c e n t of the s a m e e f f e c t in 7 p --- ~+n a n d pn - - np. T h e s u c c e s s i v e e x c h a n g e of two m e s o n s one of w h i c h i s a p i o n (the s y s t e m of two m e s o n s h a v i n g e x o t i c q u a n t u m n u m b e r s ) c a n l e a d to s u c h a n a r r o w p e a k in (12) in a n a t u r a l w a y *. T h u s a l t h o u g h M M ' M 4 w o u l d b e f o r b i d d e n b y o u r r u l e s , it i s q u i t e p o s s i b l e t h a t (12) i n v o l v e s m u l t i p l e e x c h a n g e s , e a c h of w h i c h o b e y s the r u l e s . If p r o c e s s (12) i n d e e d r e p r e s e n t s a s m a l l M M ' M ~ c o u p l i n g w e w o u l d e x p e c t m u c h l a r g e r e f f e c t s if M ' , w e r e r e p l a c e d b y M 4 o r if M - - . B 3 , M ' B 5. In e i t h e r c a s e t h e c o u p l i n g w o u l d t h e n b e one a l l o w e d b y o u r r u l e s . R e a c t i o n s of i n t e r e s t w o u l d t h e n i n c l u d e ~ - p - . ( M 4 ) - - A + + a n d p n - . Y * - (B5)++. Such p r o c e s s e s w o u l d p r o b a b l y b e s t b e s t u d i e d in b u b b l e - c h a m ber missing-mass analyses. One of u s ( J . R . ) w o u l d l i k e to t h a n k P r o f . G. Z w e i g , who h a s c o n s i d e r e d s i m i l a r p r o b l e m s , f o r a v e r y p r o d u c t i v e c o n v e r s a t i o n . Two of u s ( P . F . a n d J . R . ) w o u l d l i k e t o t h a n k D r s . R. C. A r n o l d a n d K. C. W a l i f o r t h e h o s p i t a l i t y of the H i g h E n e r g y P h y s i c s D i v i s i o n of t h e A r g o n n e N a t i o n a l L a b o r a t o y , w h e r e p a r t of t h i s w o r k w a s p e r f o r m e d . * This has also been r e a l i z e d by R. Arnold (private communication).
REFERENCES [1] K. Igi, Phys. Rev. L e t t e r s 9 (1962) 76; R.Dolen, D . H o r n a n d C.Schmid, Phys. Rev. 166 (1968) 1768. [2] P . G . O . F r e u n d , Phys. Rev. L e t t e r s 20 (1968) 235; H . H a r a r i , Phys. Rev. L e t t e r s 20 (1968) 1395. [3] G . F . Chew, Comments on Nuclear and P a r t i c l e P h y s i c s 2 (1968) 74. [4] H. H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 562; J . R o s n e r , Phys. Rev. L e t t e r s 22 (1969) 689. [5] C.Schmid, Phys. Rev. L e t t e r s 20 (1968) 628. [6] C.B. Chiu and J. Finkelstein, Phys. L e t t e r s 27B (1968) 510. [7] J . R o s n e r , Phys. Rev. L e t t e r s 21 (1968) 950. [8] H . J . Lipkin, Nucl. Phys. B9 (1969) 349. [9] D. Horn, H . J . Lipkin and S.Meshkov, Phys. Rev. L e t t e r s 17 (1966) 1200. [10] J. Mandula, J. Weyers and G. Zweig, Phys. Rev. L e f t e r s , to be published. [11] J. Mandula, C. Rebbi, R. Slansky, J. Weyers and G. Zweig, Phys. Rev. L e t t e r s 22 (1969) 1147. [12] P. G. O. Freund and R. J. R i v e r s , Phys. L e t t e r s , to be published. [13] G. Zweig, CERN r e p o r t CERN-Th-402, 1964, unpublished. [14] J . R o s n e r , ref.[4]. [15] R. C. Arnold and J. L. Uretsky, to be published.
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[16] B. Maglic, Report to the Lund International Conference on Elementary P a r t i c l e s , June, 1969. [17] E . W . A n d e r s o n et al., Phys. Rev. Letters 22 (1969~ 1390. [18] K. Myklebost, quoted by B. F r e n c h in Proceedings of the 14th International Conference on High Energy Physics, Vienna, 1968, p. 130. [19] B. French, ref. [18]. [20] M.A.Abolins et al., to be published.