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Sharp states in the spectrum of low-lying mesons
Volume 33B, n u m b e r 7
SHARP
STATES
PHYSICS
IN
THE
LETTERS
SPECTRUM
OF
7 D e c e m b e r 1970
LOW-LYING
MESONS
t
J . L. R O S N E R School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Received 23 October 1970
The existence of a family of narrow, low-lying mesons of ordinar3 I, Y is suggested. These would coexist with the wider s t a t e s usually assigned to q~: L = 1 in a quark model, and in p a r t i c u l a r would give r i s e to the A 2 splitting. The a s s i g n m e n t of these states to a q q ~ multiplet leads to t h e i r n a r r o w n e s s . Such s t a t e s have been predicted by applying duality to b a r y o n - a n t i b a r y o n scattering. Their p a r t n e r s of % x o t i c ' I and Y would lie higher in m a s s (at least one below 1.9 GeV, probably), and need not be so narrow.
L o w - l y i n g m e s o n s a r e o f t e n c l a s s i f i e d a s if c o m p o s e d of a q u a r k (q) a n d a n a n t i q u a r k (el) in a r o u g h l y h a r m o n i c p o t e n t i a l [1]. T h e o b s e r v a t i o n of c a n d i d a t e s f o r a l l q~; L = 0 a n d m o s t q~; L = 1 s t a t e s s u p p o r t s t h i s s c h e m e [2]. T h e s e r e s o n a n c e s a r e t y p i c a l l y b r o a d ( F > 50 MeV) a n d p r o minent. Their masses obey simple systematics
[1,21. Evidence that the above scheme is incomplete c o m e s f r o m t h e A 2 s p l i t t i n g [3]. W h i l e not e v e r y high-statistics experiment sees the splitting [4], a n e x t r a j P C = 2++ s t a t e d o e s s e e m r e q u i r e d . O n e i n t e r p r e t a t i o n of t h i s s t a t e i s t h a t it i s n a r r o w ( F ~ 12 MeV) a n d i n t e r f e r e s w i t h a w i d e r one with varying phase in different experiments [5]. W e s h a l l a d o p t t h i s p o i n t of v i e w , c a l l i n g t h e b r o a d s t a t e A 2 a n d t h e n a r r o w o n e A~. T h e p r e s e n t n o t e s u g g e s t s t h a t t h e A~, h a s c o m p a n i o n l o w - l y i n g s t a t e s w h i c h a r e all n a r r o w , a n d w h i c h m a y b e v i e w e d a s b e l o n g i n g to a qqq~ m u l t i p l e t TT. T h e c o e x i s t e n c e of s u c h a m u l t i p l e t w i t h t h e u s u a l q~; L s t a t e s i s p r e d i c t e d by d u a l i t y 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 ~ . In t h i s r e s p e c t o u r a p p r o a c h d i f f e r s f r o m o n e s [8~9] which describe the lowest positive-parity mesons by a q q q q m u l t i p l e t alone:~. T h e q q ~ a s s i g n m e n t i n v o l v e s s t a t e s of " e x o -
~"Work supported in p a r t by the U.S. Atomic Energy C o m m i s s i o n through Contract No. AT-(11-1)-1764. "~ The interplay of two such nearly degenerate s t a t e s as A 2 and A~ is a complex p r o b l e m [e.g. 6] and a detailed fit to A 2 s p e c t r a will t h e r e f o r e not be attempted h e r e . ~ ' ~ The importance of this coexistence was s t r e s s e d to us e a r l y in 1969 by K. Lassila and by C. Lovelace. :~ This was an e a r l y d e s c r i p t i o n of the p o s i t i v e - p a r i ty mesons [10, 11].
t i c " I, Y ( o n e s w i t h I, Y not c h a r a c t e r i s t i c of a n SU(3) 1 o r 8 ) . T h e e v i d e n c e a g a i n s t s u c h s t a t e s a t low m a s s i s s u b s t a n t i a l , t h o u g h not a i r t i g h t [12]. F o r e x a m p l e , r e c e n t d a t a o n ~ - p -~ ~+p ( M M ) - - s h o w no e x o t i c s t a t e in t h e m i s s i n g m a s s (MM) s p e c t r u m up to 1450 M e V [13]. F o r t h i s r e a s o n we s h a l l a s s u m e t h a t t h e A~ i s not a m e m b e r of a n SU(3) 27 [8], a n d t h a t it d o e s not h a v e I = 2 [9]. B o t h t h e A 2 a n d t h e A ~ will b e t a k e n a s m e m b e r s of SU(3) 8 's. T h e 10 ' s , ~ ' s , a n d 2 7 ' s of t h e qqcl(t m u l t i p l e t a r e a s s u m e d to l i e h i g h e r , p e r h a p s e v e n c l o s e to 2 GeV. E v e n a t this mass, such states will help provide the n e e d e d s a t u r a t i o n of b a r y o n - a n t i b a r y o n f i n i t e e n e r g y s u m r u l e s ( F E S R ) in e x o t i c c h a n n e l s [7]. T h e o r b i t a l s t r u c t u r e of t h e l o w e s t qqctq m u l t i p l e t i s a s s u m e d to i n v o l v e a r e l a t i v e S w a v e b e t w e e n a n y p a i r of q u a r k s , a n d w i l l b e d e n o t e d ( q q ~ ; a l l L -- 0). A n i m m e d i a t e c o r o l l a r y of t h i s a s s i g n m e n t i s t h e p r e d i c t i o n of a n u m b e r of p o s i t i v e - p a r i t y s t a t e s of J = 0, 1 a n d 2, a l l of w h i c h a r e " s u p e r f l u o u s " f r o m t h e s t a n d p o i n t of t h e q~; L p i c t u r e w i t h r o u g h l y h a r m o n i c f o r c e s . T h e l o w e s t qqqq s t a t e s s h o u l d b e narrow, a s a r e s u l t of t w o e m p i r i c a l r u l e s p r o p o s e d e a r l i e r to i n s u r e t h a t v e r t e x f u n c t i o n s s a t i s f y c o n s t r a i n t s i m p o s e d b y d u a l i t y [14]. T h e r u l e s a r e s t a t e m e n t s a b o u t t r i l i n e a r c o u p l i n g s , e x p r e s s e d in t e r m s of t h e q u a r k m o d e l ( t h o u g h n o t r e q u i r i n g quarks to exist): (I) E a c h p a i r of h a d r o n s a t a v e r t e x i s c o n n e c t e d by >i 1 q u a r k l i n e . (II) No q u a r k l i n e b e g i n s a n d e n d s in t h e s a m e hadron. The few couplings violating these rules, as est i m a t e d f r o m ~ ~ pv o r Z~ ~ K + p (if t h e Z~ e x i s t s ) , s e e m a t l e a s t a n o r d e r of m a g n i t u d e weaker than allowed couplings. 493
Volume 33B, n u m b e r 7
PHYSICS
I n d e p e n d e n t m o t i v a t i o n f o r r u l e (II) in the d e c a y s of the (qqctq; a l l L = 0) m e s o n s c o m e s f r o m the odd r e l a t i v e p a r i t y of e v e r y q~ p a i r in such m e s o n s . Only when a qCl p a i r had j P C = 0++ c o u l d it " v a n i s h into the v a c u u m " . Such a p a i r w o u l d h a v e to be a 3P 0 s t a t e , c o n t r a r y to o u r a s sumption. R u l e s (I) and (II) c o r r e s p o n d to SU(6)W s t a t e m e n t s . F o r e x a m p l e , t h e d e c a y s into a p a i r of SU(6) 35 m e s o n s of a 405 m u l t i p l e t ~ a r e d e s c r i b e d by t h r e e SU(6) w a m p l i t u d e s A405, A35 and A 1 [11]. Fig. 1 s h o w s how t h e s e a r e r e l a t e d to v a r i o u s q u a r k s t r u c t u r e s . R u l e (I) f o r b i d s the c o u p l i n g s in f i g s . (la) and ( l c ) , i m p l y i n g A405 = 0
(1)
and A 4 0 5 - 7 A 3 5 / ( 3 ) 1/2 - 8 A 1 / ( 1 5 ) 1/2 = 0 , (2) w h i l e r u l e (II) f o r b i d s the c o u p l i n g s in figs. (lb) and ( i c ) , i m p l y i n g A405 - (3)I/2A35 = 0
(3)
as well as eq. (2). Together, the rules forbid all decays of qqqcl states (called "M4" [14]) into p a i r s of qq ("M2") states. Allowed decays, illustrated in figs. (Id) and (le), include M4 --~B3B3 Such a multiplet is the c o r r e c t one for the qq(lq r e s o nances required by duality in baryon-antibaryon s c a t t e r i n g [7].
(o)
{b)
(d)
(c)
(e)~/
LETTERS
7 D e c e m b e r 1970
w h e r e B 3 r e f e r s to any t h r e e - q u a r k b a r y o n ~J~, and M 4 ~ M2M 4. We now d i s c u s s the e x o t i c - I , Y M 4 s t a t e s . T h e p r e s e n c e of n o n - P o m e r a n c h u k i m a g i n a r y p a r t s in c e r t a i n e x o t i c b a r y o n - a n t i b a r y o n c h a n n e l s f o l l o w s f r o m r e a s o n a b l e a s s u m p t i o n s [7,15,16] ~ l W h e t h e r t h e s e i m a g i n a r y p a r t s a r e p r o v i d e d by r e s o n a n c e s , a s h a s b e e n s u g g e s t e d to h o l d in g e n e r a l [18], is quite a n o t h e r s t o r y , q u e s t i o n e d in t h e o r y [19] but u n c h e c k e d in p r a c t i c e . To f a c i l i t a t e the o b s e r v a n c e of such e x o t i c m e s o n s (if t h e y exist) one n e e d s p r o b a b l e m a s s e s , p r o d u c t i o n m e c h a n i s m s , and d e c a y p r o p e r t i e s . (a) P r o b a b l e m a s s e s o f M 4 m e s o n s w i t h e x o t i c I and Y
1. T h e f o l l o w i n g e m p i r i c a l r u l e h o l d s f o r the o r d i n a r y h a d r o n s , and it is a m u s i n g to s p e c u l a t e that it m i g h t h o l d f o r M 4 s t a t e s a s w e l l : R u l e o f t h u m b : If the s e l e c t i o n r u l e s (I) and (II) a l l o w two h a d r o n s to c o u p l e to any r e s o n a n c e , t h e y f o r m at l e a s t one such s t a t e b e t w e e n t h r e s h old and the p o i n t w h e r e k, the m a g n i t u d e of the CM 3 - m o m e n t u m , r e a c h e s 0.5 G e V / c . A s e x a m p l e s , a n+ and n+ a r e not a l l o w e d by r u l e (I) to f o r m r e s o n a n c e s , w h e r e a s s i n c e a n+ and a n - c a n do so, t h e y f o r m at l e a s t one s u c h s t a t e (the p and t h e or) b e l o w 1 GeV. T h e r u l e of t h u m b s e e m s to apply s e p a r a t e l y to e a c h i s o s p i n channel. Exceptions for conventional hadrons s e e m to r e f l e c t p u r e l y e x p e r i m e n t a l d i f f i c u l t i e s . F o r e x a m p l e , both SU(3) and the q u a r k m o d e l r e q u i r e an I = 0 p~ s t a t e in the v i c i n i t y of the B m e s o n (or l o w e r ) , and the t e m p o r a r y l a c k of such a s t a t e ( p r e d i c t e d by o u r " r u l e of t h u m b " ) is not disturbing. A c c o r d i n g to s e l e c t i o n r u l e s (I) and (II), the s y s t e m s A++fi, A-p, and t h e i r c h a r g e - c o n j u g a t e s ( t h e s e a r e the l o w e s t e x o t i c b a r y o n - a n t i b a r y o n c h a n n e l s ) a r e a l l o w e d to c o u p l e to M 4. By the " r u l e of t h u m b " , at l e a s t one such s t a t e s h o u l d l i e b e t w e e n the b o u n d s 2.2 G e V < M ( I =
2, Y = 0 ) < 2 . 5 G e V .
(4)
S i m i l a r l y , the A~-+~ and Y~+p s y s t e m s i m p l y the bounds 2.35 G e V < M ( I = Fig. 1. (a)-(c). Tensor structure of couplings to pairs of 35 mesons of a 405 meson; quark lines represent tensor indices: (a) Coupling proportional to A405; pure S wave; (b) Coupling proportional to A405 - (3) -1/2 A35; S and D waves; (c) Coupling ~A405 - 7 A35/(3)I/2 - 8 AI/(15)I/2; S, D, and G waves. (d) Coupling of M4 to B3B3. (e) Coupling of Ni4 to M2M4.
494
3/2,
IY 1 = 1)<2.6 G e V .
A p p l y i n g t h e s a m e r u l e to the ~ - A~. s y s t e m , which is a l l o w e d to f o r m an I = 2 M 4 s t a t e s i n c e t h e A~ is a s s u m e d to be an M4, one would e x p e c t
$$ The subscript denotes the number of quarks plus antiquarks. M is a meson, B a baryon. "~$~"See, however, ref. [17].
Volume 33B, number 7
PHYSICS
an I = 2, Y = 0 state:~ b e t w e e n the b o u n d s 1.45 GeV < M(I= 2, Y = 0 ) < I . 9 G e V . 2. Dependence o f m a s s on SU(3) r e p r e s e n t a tion. T h e s i m p l e s t s u c h d e p e n d e n c e w h i c h c a n a p p e a r in an SU(6) m a s s f o r m u l a c o m e s in t h r o u g h the C a s i m i r o p e r a t o r C~3) [20], which is 0 f o r 1, 6 f o r 8, 12 f o r 10 and ~0, and 16 f o r 27. T h e s i m p l e s t SU(3) b r e a k i n g t e r m is of the f o r m ~y2 - I(I+l). F o r s t a t e s of J = 2 we c a n t h e n g u e s s at r e a s o n a b l e p a r a m e t e r s in the s a m p l e m a s s formula M 2 = M 2 + aC(23) + b[~Y 2 - I(I+ 1)]
Y = 0) ~ 1.7 G e V .
(6)
H o w e v e r , the d e p e n d e n c e on C~3) n e e d not be l i n e a r [11], o t h e r SU(6) b r e a k i n g c o u l d be p r e s ent, and t h i s e x o t i c s t a t e c o u l d e a s i l y l i e s o m e what h i g h e r . O u r only p o i n t is that with the s i m p l e s t p o s s i b l e a s s u m p t i o n s , it is not d i f f i c u l t to p u s h the e x o t i c m e s o n s q u i t e high in m a s s , w h i l e c o n t i n u i n g to "count s t a t e s " by t h e b a d l y b r o k e n s y m m e t r y SU(6). If t h e J = 2 SU(3) s i n g l e t w e r e i n d e e d to l i e a s low a s 1 GeV, of c o u r s e , nonet s y m m e t r y w o u l d b e p o o r f o r the 1 and 8 J = 2 M 4 s t a t e s . T h e A 2 s p l i t t i n g , then, n e e d not i m p l y that of a l l its n o n e t p a r t n e r s . T h e A 2 - A~. d e g e n e r a c y , if not accidental, would then follow from considerat i o n s o t h e r than t h e p r e s e n t e l e m e n t a r y " t a x o nomic" ones. $ If this state belonged to ( q q ~ ; all L = 0) its JP would be 2+; a 0 + state cannot couple by J, P conservation, and the 1 + state has G = -.
7 December 1970
(b) Production m e c h a n i s m s T h e b e s t way to p r o d u c e M 4 s t a t e s s h o u l d be v i a " a l l o w e d " c o u p l i n g s [14]. T h e B3B3 c o u p l i n g s of such s t a t e s m a y be i n v e s t i g a t e d in b a r y o n - e x c h a n g e r e a c t i o n s , which i n c l u d e ,pp ~ n n i h i l a t i o n s [12] and b a c k w a r d m e s o n - b a r y o n : s c a t t e r i n g [21,14]. A f u r t h e r m e c h a n i s m , c o n s i s t e n t with the r u l e s of r e f . [14] and m e n t i o n e d p r e v i o u s l y by A r n o l d [22], is the u t i l i z a t i o n of t h e a l l o w e d (M2M4M4) and (M4B3B3) c o u p l i n g s t h r o u g h p r o c e s s e s of the f o r m M2 B3 ~ M4 (forward) B3 ,
(5)
u s i n g the f o l l o w i n g inputs: (i) T h e A~ b e l o n g s to an o c t e t , with M(A~) 1.3 GeV. (ii) T h e v a l u e of b is a p p r o x i m a t e l y the s a m e a s that which l e a d s to the K** - A2, K* - p, o r K - 7r m a s s d i f f e r e n c e . One w o u l d be s u r p r i s e d if SU(3) w e r e b e t t e r f o r qqq~ m a s s e s t h a n f o r qcl m a s s e s . (In the c a s e of n u c l e i , i s o s p i n is m o r e b a d l y b r o k e n in the m a s s e s of m o r e c o m p l e x systems.) (iii) T h e m a s s of the J = 2 u n i t a r y s i n g l e t e x c e e d s 1 GeV o r so. A l t h o u g h f o r b i d d e n by r u l e s (I) and (II) to c o u p l e to uTr, such a s t a t e b e l o w 1 GeV w o u l d s e e m h a r d to m i s s u n l e s s the s e l e c t i o n r u l e s w e r e c o n s i d e r a b l y b e t t e r t h a n we b e l i e v e t h e m to be. M o r e o v e r , a J = 2 s t a t e b e l o w 1 GeV w o u l d h a v e to lie on a R e g g e t r a j e c t o r y of a b n o r m a l l y g r e a t s l o p e , a p o s s i b l e if u n l i k e l y circumstance. G i v e n t h e s e t h r e e i n p u t s , one then p r e d i c t s t h a t the J= 2, I= 2, Y= O p a r t n e r of t h e A ~ i n s 405 m u l t i p l e t w o u l d l i e b e l o w M ( J = 2, I = 2 ,
LETTERS
(7)
i n v o l v i n g M 4 e x c h a n g e . An SU(3) 1 M 4 t r a j e c t o r y c o u l d l i e quite high without h a v i n g b e e n i d e n t i f i e d p r e v i o u s l y ~ . F o r e x a m p l e , the l a r g e C2(3) d e p e n d e n c e of the s h o r t d i s c u s s e d a b o v e c o u l d l e a d to an i n t e r c e p t ~(0) of n e a r l y 1. On the o t h e r hand, the h i g h e s t - l y i n g I : 1, Y = 0 M 4 t r a j e c t o r y w o u l d be that of the A2, with i n t e r c e p t p e r h a p s h a l f a unit l o w e r . In s h o r t , p r o c e s s e s i n v o l v i n g exchange o f no quantum n u m b e r s c o u l d be a g o o d way to p r o d u c e q q ~ s t a t e s at high e n e r g y . T h e d o m i n a n c e of I = Y = 0 e x c h a n g e in r e a c t i o n (7) w o u l d h a v e several effects: 1. A 0 v e r s u s A ~ production. In c o m p a r i n g , for example, v
g+ n ~ A 0 p
(8)
with ~-p ~A~p
,
(9)
one m i g h t e x p e c t the c r o s s s e c t i o n f o r r e a c t i o n (9) to l e v e l off at high e n e r g y if c o n s i d e r a b l e A~. w e r e b e i n g p r o d u c e d . An e x i s t i n g c o m p i l a t i o n [23] s h o w s a d i f f e r e n c e b e t w e e n the two p r o c e s s e s t e n d i n g in t h i s d i r e c t i o n , but with l a r g e e r r o r s . 2. P r o m i n e n c e o f G = - states in ~ -p ~ M~p at high energy. T h i s p r e d i c t i o n f o l l o w s f r o m the G = + n a t u r e of the d o m i n a n t M 4 e x c h a n g e . If the s t a t e s s e e n in h i g h - e n e r g y m i s s i n g - m a s s e x p e r i m e n t s of the f o r m [24] n - p ~ X - p a r e i n d e e d M 4 ' s , a s h a s b e e n s u g g e s t e d [9,14], and if t h e y c o n t i n u e to a l t e r n a t e in G - p a r i t y a s in the s e q u e n c e p - A 2 - g, the d o m i n a n t p e a k s at high e n e r g y should be the S(1930), the U(2382) and the X-(2800), with the T(2195) and X-(2620) s u p pressed. 3. E x o t i c s t a t e s . T h e p r o c e s s (7), f o l l o w e d by the d e c a y M 4-~ M~ (exotic I, Y) M 2 ,
(10)
~"Such a trajectory, called X by Arnold, has been postulated in order to explain certain features of interactions of conventional hadrons [22]. 495
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LETTERS
7 December 1970
Table 1 "Better" and "worse" meson exchange reactions for M4 production at high energy. "Better" (No quantum numbers exchanged)
"Worse" (Charge or strangeness exchanged) 17r-p
K ~ p ~ p X -~
K~p
- A~
X--
- - n X ~ ~+
[~ Xprr ~
~Tr-X +~ ~ A pn ~ pp ' p X ~
pp ~ A Z * * ~
~,K- Z*~ ~ 7r-p
~ pX-
~-p
~
[~ pp'n's h o u l d be a good way to p r o d u c e e x o t i c m e s o n s . T h e a b s e n c e of such s t a t e s b e l o w 1450 MeV in n - p ~ n+p ( M M ) - - [13] s h o u l d t h e n be c o n s i d e r e d good evidence a g a i n s t such s t a t e s in t h i s m a s s r e g i o n , p a r t i c u l a r l y if the n+ ( M M ) - - m a s s in t h i s r e a c t i o n e m b r a c e s the r a n g e up to 1.9 GeV or soS. 4. " B e t t e r " and " w o r s e " e x p e r i m e n t s . If I = Y : 0 M 4 e x c h a n g e i n d e e d d o m i n a t e s r e a c t i o n (7), some meson exchange reactions are "better" for M 4 production, while others are "worse". Some c o m p a r i s o n s a r e m a d e in t a b l e 1. 5. P o m e r a n c h u k o n exchange. It is p o s s i b l e f o r the P o m e r a n c h u k o n (whose q u a r k s t r u c t u r e is u n c e r t a i n ) to p l a y a r o l e in r e a c t i o n (7). H o w e v e r , (i) it would not be s p e c i f i c to M 4 p r o d u c t i o n (one c o u l d t e s t f o r it in p r i n c i p l e by looking f o r e v i d e n c e of M6, M8, e t c . ) , and (ii) it w o u l d be e x p e c t e d to h a v e a s h a r p l y p e a k e d t d i s t r i b u t i o n , at l e a s t f r o m o u r p a s t e x p e r i e n c e . If the t d i s t r i b u t i o n a s s o c i a t e d with M 4 e x c h a n g e is any f l a t t e r t h a n that a s s o c i a t e d with P o m e r a n c h u k o n e x c h a n g e , the p r o d u c t i o n of M 4 r e s o n a n c e s at m o d e r a t e t (as in the c o n f i g u r a t i o n of r e f s . [13] and [24]) m a y win out o v e r d i f f r a c t i v e b a c k g r o u n d . (c) Decay P r o p e r t i e s If the l o w e s t e x o t i c m e s o n s lie b e l o w b a r y o n antibaryon threshold, as seems quite reasonable f r o m the a r g u m e n t in the s e c o n d p a r t of s e c t i o n (a), the M2M 4 d e c a y s of t h e s e m e s o n s w i l l s t i l l be a l l o w e d . In p r i n c i p l e , t h e y c o u l d a c q u i r e c o n s i d e r a b l e width t h r o u g h t h e s e c e d a y s , u n l e s s all c o u p l i n g s i n v o l v i n g M 4 ' s t u r n out to be w e a k $$. ~" This is because the "rule of thumb" would predict an M~ resonance in the M~-~"+ channel below 1.9 GeV if an M~- lay below 1.45 GeV. ~$ Ref, [25] predicts a normal strength for these couplings. 496
nX ° [~pl~
I n d e e d , the only M 4 ' s a s n a r r o w a s the A½ m a y be the l o w e s t o n e s , which we h a v e a s s u m e d to be n o n - e x o t i c . T h e s e s t a t e s w o u l d be n a r r o w s i n c e t h e y w o u l d h a v e to d e c a y by a v i o l a t i o n of the s e l e c t i o n r u l e s of r e f . [14]. A c r u c i a l t e s t of the s e l e c t i o n r u l e s would be to f o l l o w the s u b s e q u e n t d e c a y s of any c a n d i d a t e s f o r M4 s t a t e s , s u c h a s t h o s e p r o d u c e d in m i s s i n g m a s s e x p e r i m e n t s ( w h e t h e r of e x o t i c [13] o r o r d i n a r y [24] I and Y). T h e s e d e c a y s s h o u l d o c c u r v i a the c a s c a d e M4~
~4M2
--~M~' (narrow) M_~ •
(11)
( v i o l a t i o n of s e l e c t i o n r u l e s ) T h e n a r r o w M 4 s t a t e s w o u l d be the ground slale o f exotic m a t t e r , with o r d i n a r y I, Y which w o u l d not b e t r a y t h e i r q q q q n a t u r e . At any p o i n t in (11), if an M 4 s t i l l lay a b o v e B3B 3 t h r e s h o l d , t h e " c h a i n " c o u l d be " b r o k e n " by M 4 ~ B3B 3. Both m u l t i m e s o n and BB f i n a l s t a t e s w i l l be v i s i b l e in f o r t h c o m i n g e x p e r i m e n t s at S e r p u k h o v [26]. P r o c e s s e s such a s M 2 B 3 ~ M4 ~5
L B~
(12) (exotic) M 2
a r e a l l o w e d by t h e s e l e c t i o n r u l e s , and m a y be w o r t h looking f o r . Some further remarks: (d) T h e L - e x c i t e d s t a t e s of q q ~ s y s t e m s m a y p o s s e s s c o n s i d e r a b l e d e g e n e r a c y , g i v i n g r i s e to b e h a v i o r s i m i l a r to that s u g g e s t e d in r e f . [27]. (e) In the a b s e n c e of e x o t i c p a r t n e r s of the A~.,
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implied by the present discussion, one would be t e m p t e d to a d o p t a m o d e l s u c h a s t h a t of r e f . [28], w h i c h i n t r o d u c e s e x t r a d e g r e e s of f r e e d o m i n t o t h e q(l; L p i c t u r e w i t h o u t r e q u i r i n g s t a t e s of i n c r e a s i n g I a n d Y. (f) A s c o m p a n i o n s of t h e A½ a t low m a s s e s w o u l d b e n a r r o w b u t not n e c e s s a r i l y d e g e n e r a t e w i t h b r o a d e r s t a t e s , t h e y m i g h t b e v e r y h a r d to s e e . H o w e v e r , c l a i m s f o r s o m e of t h e s e l o w l y i n g " s h a r p s t a t e s " a l r e a d y do e x i s t [29,30]. A c o m b i n a t i o n of h i g h s t a t i s t i c s a n d h i g h r e s o l u t i o n w i l l b e n e e d e d to c o n f i r m s u c h s t a t e s . T h e y must be distinguished from statistical fluctuations, which are also typically narrow~. Facilities such as the CERN Omega spectrometer s h o u l d b e h e l p f u l in t h i s r e s p e c t . T o s u m m a r i z e , we h a v e s u g g e s t e d t h a t t h e e x p e r i m e n t a l p r o b l e m of t h e A 2 s p l i t t i n g a n d t h e t h e o r e t i c a l p r o b l e m of d u a l i t y in b a r y o n - a n t i b a r y o n s y s t e m s m a y b o t h b e s o l v e d in a q u a l i t a t i v e w a y by t h e e x i s t e n c e of qqqq s t a t e s s u b j e c t to c e r t a i n s e l e c t i o n r u l e s I $ . T h i s w o u l d b e e n c o u r a g i n g f o r m o d e l s of n - p o i n t f u n c t i o n s i n c o r p o r a t i n g d u a l i t y g r a p h s [32], w h i c h r e q u i r e such states for BB scattering$'~I. One should be cautious, however, about assuming that they s o l v e t h e B B s a t u r a t i o n p r o b l e m exactly. E v e n f o r t h e b e s t - k n o w n of p r o c e s s e s , t h e quantitative c o n c e p t of " d u a l i t y " ( a c c u r a t e r e s o n a n c e s a t u r a t i o n of n o n - P o m e r a n c h u k F E S R ) s e e m s to h a v e i t s l i m i t a t i o n s [35]. T h i s w o r k w a s d o n e in p a r t a t C a l t e c h , a t t h e U n i v e r s i t y of C a l i f o r n i a i n B e r k e l e y , a n d a t C E R N . I a m g r a t e f u l to M. G e l l - M a n n , G . F . C h e w a n d S. M a n d e l s t a m , a n d J. P r e n t k i f o r t h e h o s p i t a l i t y of t h e i r r e s p e c t i v e i n s t i t u t i o n s , a n d to R. A r n o l d , M. B a n d e r , G. F. C h e w , t h e C h u n g k i n g R e s t a u r a n t , P. G. O. F r e u n d , M. G e l l - M a n n , G. G o l d h a b e r , D. H o r n , M. J a c o b , W. K i e n z l e , J. M a n d u l a , S. M e s h k o v , N. P . S a m i o s , J. W e y e r s a n d G. Z w e i g f o r h e l p f u l a d v i c e . "~The author thanks D. R. O. M o r r i s o n and N. P. Samios for reminding him of this i m p o r t a n t point. "~'~'Similar (but more limited) r u l e s following f r o m SU(6)W were noted much e a r l i e r [31]. They forbid, e.g., the state in eq. (6) to decay to 7r+~+. "~'~"~The Utower n of q~ excitations does not close ]33,34] {see, e.g., eq. (12)).
References [1] M. Gell-Mann, Phys. L e t t e r s 8 (1964) 2]4; G. Zweig, CERN r e p o r t s TH-401 and TH-412 (1964), unpublished.
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[2] G. Goldhaber, in Proe. 13th Intern. Conf. on Highe n e r g ) , p h y s i c s , Berkeley, 1966, ed. M.AlstonGarnjost (Berkeley, University of California P r e s s , 1967) p. 103: R . H . D a l i t z , Ibid., p. 215. G. Zweig, in P r o c . 1968 Philadelphia Meson Conf., eds. C . B a l t a y and A. Rosenfeld (New York, Benjamin, 1968) p. 485. [3] B. L e v r a t et al., Phys. L e t t e r s 22 (1966) 714: M:N. F o e a c c i e t a l . , Phys. Rev. Letters 17 (1966i 890. [4] A. B a r b a r o - G a l t i e r i , in P r e c . 1970 Philadelphia Meson Conf., eds. C . B a l t a y and A. Rosenfeld (New York, Columbia University P r e s s , 1.970)~ [5] G. Goldhaber, in Proc. 15th Intern. Conf. on Highenergy physics, Kiev, USSR, 1970, to be published. [6] Y.Dothan and D. Horn, Phys. Rev. D1 (19701 916. [7] J. Rosner, Phys. Rev. Letters 21 (1968~ 950, 1422(E). [8] K. L a s s i l a and P. Ruuskanen, Phys. Rev. [.etters 19 (1967) 762; S. Meshkov, in P r o e . 1970 Philadelphia Meson Conf., op. cit. [9] R. Arnold and J. Uretsky, Phys. Rev. Letters 23 (1969) 444: S.F. Tuan, Phys. Rev. L e t t e r s 23 (1969) 1198. [10] Y. Dothan, M. Gell-Mann and ¥. Nereman, Phys. L e t t e r s 17 (1965) 148. [11] D.Horn, J. Coyne, S. Meshkov and J. C a r t e r , Phys. Rev. 147 (1966) 980. [12] J. Rosner, in P r o c . 1970 Philadelphia Meson Conf., op. cit. [13] C. Lechanoine, r e p o r t on the CERN-Serpukhov m i s s i n g - m a s s s p e c t r o m e t e r collaboration to the 1970 Kiev Conference, op. cir. [14] P. G. O. Freund, R. Waltz and J. Rosner, Nucl. Phys. B13 (1969) 237. [15] H.J. Lipkin, Phys. Rev. L e t t e r s 16 (1966) 1015: R.H. Capps, Phys. Rev. 168 (1968)173]. [16] J. Rosner, C . R e b b i a n d R. Slansky, Phys. Rev. 188 (1969) 2367. [17] M.Kugler, Phys. L e t t e r s 32B (1970) 107. [18] P . G . O . Freund, Phys. Rev. L e t t e r s 20 (1969) 235: H . H a r a r i , Phys. Rev. L e t t e r s 20 (1969) 1395. [19] H.J. Lipkin, Nucl. Phys. B9 (1969) 349: D . P . R o y and M.Suzuki, Phys. L e t t e r s 28B (1969) 558; M. Kugler, Phys. Rev. 180 (1969) 1538: J. Mandula, J. Weyers and G. Zweig, Phys. Rev. L e t t e r s 23 (1969) 266: R . H . C a p p s , Phys. Rev. 185 (1969)2008: H.J. Lipkin, Phys. L e t t e r s 32B (1970) 301. [20] M . A . B . B ~ g a n d V. Singh, Phys. Rev. L e t t e r s 13 (1964) 418. [21] M. J a c o b a n d J. Weyers, C E R N p r e p r i n t TH-1169, June, 1970, to be published; J. Rosner, in Proc. 8th E a s t e r n Theoretical P h y s i c s Conf., Syracuse, New York, October, 1969, ed. F . R o h r l i c h (Syracuse, P h y s i c s D e p a r t ment of Syracuse University, 1969) p. 231; J. Rosner, invited talk at the Washington Meeting of the A m e r i c a n Physical Society, April, 1970, Caltech r e p o r t CALT-68-254, unpublished. [22] R. Arnold, private communication, Argonne p r e print, Aug. 1970, to be published. [23] J. T. C a r r o l l et al., Production of A 2 mesons, W i s c o n s i n - T o r o n t o preprint, to be published. 497
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[24] M. Martin, in P r o c . 1970 Philadelphia Meson Conf., op. cit. [25] P. H. Frampton and P. G. O. Freund, Enrico F e r m i Institute p r e p r i n t EFI-70-44, August 1970, to be published. [26] W. Kienzle, private communication. [27] L. -H. Chan, R. Slansky and D. Sutherland, Phys. Rev. Letters 25 (1970) 482. [28] D.Horn, Nuovo Cimento 62A (1969) 581. [29] P a r t i c l e Data Group, Phys. L e t t e r s 33B {1970) 1. [30] A. Astier, in P r o c . 1970 Kiev Conference, op. cit. [31] D.Horn, H . J . Lipkin and S. Meshkov, Phys. Rev. L e t t e r s 17 (1966) 1200.
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[32] M. Imachi et al.~ P r o g . Theor. Phys. (Kyoto) 40 (1968) 353; H . H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 562; J. Rosner, Phys. Rev. L e t t e r s 22 (1969) 689; D. E. Neville, Phys. Rev. L e t t e r s 22 (1969) 494; Chan Hong-Mo and J . E . P a t o n , Nucl. Phys. B10 (1969) 516; Z. Koba and H. B. Nielsen, Nucl. Phys. B10 (1969) 633. [33] D. P. Roy and M. Suzuki, Phys. L e t t e r s 28B {1969) 508. [34] S. Mandelstam, Phys. Rev. D1 (1970) 1720. [35] R.Dolen, D.Horn and C.Schmid, Phys. Rev. 166 (1968) 1768.
SHARP
STATES
PHYSICS
IN
THE
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LOW-LYING
MESONS
t
J . L. R O S N E R School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Received 23 October 1970
The existence of a family of narrow, low-lying mesons of ordinar3 I, Y is suggested. These would coexist with the wider s t a t e s usually assigned to q~: L = 1 in a quark model, and in p a r t i c u l a r would give r i s e to the A 2 splitting. The a s s i g n m e n t of these states to a q q ~ multiplet leads to t h e i r n a r r o w n e s s . Such s t a t e s have been predicted by applying duality to b a r y o n - a n t i b a r y o n scattering. Their p a r t n e r s of % x o t i c ' I and Y would lie higher in m a s s (at least one below 1.9 GeV, probably), and need not be so narrow.
L o w - l y i n g m e s o n s a r e o f t e n c l a s s i f i e d a s if c o m p o s e d of a q u a r k (q) a n d a n a n t i q u a r k (el) in a r o u g h l y h a r m o n i c p o t e n t i a l [1]. T h e o b s e r v a t i o n of c a n d i d a t e s f o r a l l q~; L = 0 a n d m o s t q~; L = 1 s t a t e s s u p p o r t s t h i s s c h e m e [2]. T h e s e r e s o n a n c e s a r e t y p i c a l l y b r o a d ( F > 50 MeV) a n d p r o minent. Their masses obey simple systematics
[1,21. Evidence that the above scheme is incomplete c o m e s f r o m t h e A 2 s p l i t t i n g [3]. W h i l e not e v e r y high-statistics experiment sees the splitting [4], a n e x t r a j P C = 2++ s t a t e d o e s s e e m r e q u i r e d . O n e i n t e r p r e t a t i o n of t h i s s t a t e i s t h a t it i s n a r r o w ( F ~ 12 MeV) a n d i n t e r f e r e s w i t h a w i d e r one with varying phase in different experiments [5]. W e s h a l l a d o p t t h i s p o i n t of v i e w , c a l l i n g t h e b r o a d s t a t e A 2 a n d t h e n a r r o w o n e A~. T h e p r e s e n t n o t e s u g g e s t s t h a t t h e A~, h a s c o m p a n i o n l o w - l y i n g s t a t e s w h i c h a r e all n a r r o w , a n d w h i c h m a y b e v i e w e d a s b e l o n g i n g to a qqq~ m u l t i p l e t TT. T h e c o e x i s t e n c e of s u c h a m u l t i p l e t w i t h t h e u s u a l q~; L s t a t e s i s p r e d i c t e d by d u a l i t y 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 ~ . In t h i s r e s p e c t o u r a p p r o a c h d i f f e r s f r o m o n e s [8~9] which describe the lowest positive-parity mesons by a q q q q m u l t i p l e t alone:~. T h e q q ~ a s s i g n m e n t i n v o l v e s s t a t e s of " e x o -
~"Work supported in p a r t by the U.S. Atomic Energy C o m m i s s i o n through Contract No. AT-(11-1)-1764. "~ The interplay of two such nearly degenerate s t a t e s as A 2 and A~ is a complex p r o b l e m [e.g. 6] and a detailed fit to A 2 s p e c t r a will t h e r e f o r e not be attempted h e r e . ~ ' ~ The importance of this coexistence was s t r e s s e d to us e a r l y in 1969 by K. Lassila and by C. Lovelace. :~ This was an e a r l y d e s c r i p t i o n of the p o s i t i v e - p a r i ty mesons [10, 11].
t i c " I, Y ( o n e s w i t h I, Y not c h a r a c t e r i s t i c of a n SU(3) 1 o r 8 ) . T h e e v i d e n c e a g a i n s t s u c h s t a t e s a t low m a s s i s s u b s t a n t i a l , t h o u g h not a i r t i g h t [12]. F o r e x a m p l e , r e c e n t d a t a o n ~ - p -~ ~+p ( M M ) - - s h o w no e x o t i c s t a t e in t h e m i s s i n g m a s s (MM) s p e c t r u m up to 1450 M e V [13]. F o r t h i s r e a s o n we s h a l l a s s u m e t h a t t h e A~ i s not a m e m b e r of a n SU(3) 27 [8], a n d t h a t it d o e s not h a v e I = 2 [9]. B o t h t h e A 2 a n d t h e A ~ will b e t a k e n a s m e m b e r s of SU(3) 8 's. T h e 10 ' s , ~ ' s , a n d 2 7 ' s of t h e qqcl(t m u l t i p l e t a r e a s s u m e d to l i e h i g h e r , p e r h a p s e v e n c l o s e to 2 GeV. E v e n a t this mass, such states will help provide the n e e d e d s a t u r a t i o n of b a r y o n - a n t i b a r y o n f i n i t e e n e r g y s u m r u l e s ( F E S R ) in e x o t i c c h a n n e l s [7]. T h e o r b i t a l s t r u c t u r e of t h e l o w e s t qqctq m u l t i p l e t i s a s s u m e d to i n v o l v e a r e l a t i v e S w a v e b e t w e e n a n y p a i r of q u a r k s , a n d w i l l b e d e n o t e d ( q q ~ ; a l l L -- 0). A n i m m e d i a t e c o r o l l a r y of t h i s a s s i g n m e n t i s t h e p r e d i c t i o n of a n u m b e r of p o s i t i v e - p a r i t y s t a t e s of J = 0, 1 a n d 2, a l l of w h i c h a r e " s u p e r f l u o u s " f r o m t h e s t a n d p o i n t of t h e q~; L p i c t u r e w i t h r o u g h l y h a r m o n i c f o r c e s . T h e l o w e s t qqqq s t a t e s s h o u l d b e narrow, a s a r e s u l t of t w o e m p i r i c a l r u l e s p r o p o s e d e a r l i e r to i n s u r e t h a t v e r t e x f u n c t i o n s s a t i s f y c o n s t r a i n t s i m p o s e d b y d u a l i t y [14]. T h e r u l e s a r e s t a t e m e n t s a b o u t t r i l i n e a r c o u p l i n g s , e x p r e s s e d in t e r m s of t h e q u a r k m o d e l ( t h o u g h n o t r e q u i r i n g quarks to exist): (I) E a c h p a i r of h a d r o n s a t a v e r t e x i s c o n n e c t e d by >i 1 q u a r k l i n e . (II) No q u a r k l i n e b e g i n s a n d e n d s in t h e s a m e hadron. The few couplings violating these rules, as est i m a t e d f r o m ~ ~ pv o r Z~ ~ K + p (if t h e Z~ e x i s t s ) , s e e m a t l e a s t a n o r d e r of m a g n i t u d e weaker than allowed couplings. 493
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I n d e p e n d e n t m o t i v a t i o n f o r r u l e (II) in the d e c a y s of the (qqctq; a l l L = 0) m e s o n s c o m e s f r o m the odd r e l a t i v e p a r i t y of e v e r y q~ p a i r in such m e s o n s . Only when a qCl p a i r had j P C = 0++ c o u l d it " v a n i s h into the v a c u u m " . Such a p a i r w o u l d h a v e to be a 3P 0 s t a t e , c o n t r a r y to o u r a s sumption. R u l e s (I) and (II) c o r r e s p o n d to SU(6)W s t a t e m e n t s . F o r e x a m p l e , t h e d e c a y s into a p a i r of SU(6) 35 m e s o n s of a 405 m u l t i p l e t ~ a r e d e s c r i b e d by t h r e e SU(6) w a m p l i t u d e s A405, A35 and A 1 [11]. Fig. 1 s h o w s how t h e s e a r e r e l a t e d to v a r i o u s q u a r k s t r u c t u r e s . R u l e (I) f o r b i d s the c o u p l i n g s in f i g s . (la) and ( l c ) , i m p l y i n g A405 = 0
(1)
and A 4 0 5 - 7 A 3 5 / ( 3 ) 1/2 - 8 A 1 / ( 1 5 ) 1/2 = 0 , (2) w h i l e r u l e (II) f o r b i d s the c o u p l i n g s in figs. (lb) and ( i c ) , i m p l y i n g A405 - (3)I/2A35 = 0
(3)
as well as eq. (2). Together, the rules forbid all decays of qqqcl states (called "M4" [14]) into p a i r s of qq ("M2") states. Allowed decays, illustrated in figs. (Id) and (le), include M4 --~B3B3 Such a multiplet is the c o r r e c t one for the qq(lq r e s o nances required by duality in baryon-antibaryon s c a t t e r i n g [7].
(o)
{b)
(d)
(c)
(e)~/
LETTERS
7 D e c e m b e r 1970
w h e r e B 3 r e f e r s to any t h r e e - q u a r k b a r y o n ~J~, and M 4 ~ M2M 4. We now d i s c u s s the e x o t i c - I , Y M 4 s t a t e s . T h e p r e s e n c e of n o n - P o m e r a n c h u k i m a g i n a r y p a r t s in c e r t a i n e x o t i c b a r y o n - a n t i b a r y o n c h a n n e l s f o l l o w s f r o m r e a s o n a b l e a s s u m p t i o n s [7,15,16] ~ l W h e t h e r t h e s e i m a g i n a r y p a r t s a r e p r o v i d e d by r e s o n a n c e s , a s h a s b e e n s u g g e s t e d to h o l d in g e n e r a l [18], is quite a n o t h e r s t o r y , q u e s t i o n e d in t h e o r y [19] but u n c h e c k e d in p r a c t i c e . To f a c i l i t a t e the o b s e r v a n c e of such e x o t i c m e s o n s (if t h e y exist) one n e e d s p r o b a b l e m a s s e s , p r o d u c t i o n m e c h a n i s m s , and d e c a y p r o p e r t i e s . (a) P r o b a b l e m a s s e s o f M 4 m e s o n s w i t h e x o t i c I and Y
1. T h e f o l l o w i n g e m p i r i c a l r u l e h o l d s f o r the o r d i n a r y h a d r o n s , and it is a m u s i n g to s p e c u l a t e that it m i g h t h o l d f o r M 4 s t a t e s a s w e l l : R u l e o f t h u m b : If the s e l e c t i o n r u l e s (I) and (II) a l l o w two h a d r o n s to c o u p l e to any r e s o n a n c e , t h e y f o r m at l e a s t one such s t a t e b e t w e e n t h r e s h old and the p o i n t w h e r e k, the m a g n i t u d e of the CM 3 - m o m e n t u m , r e a c h e s 0.5 G e V / c . A s e x a m p l e s , a n+ and n+ a r e not a l l o w e d by r u l e (I) to f o r m r e s o n a n c e s , w h e r e a s s i n c e a n+ and a n - c a n do so, t h e y f o r m at l e a s t one s u c h s t a t e (the p and t h e or) b e l o w 1 GeV. T h e r u l e of t h u m b s e e m s to apply s e p a r a t e l y to e a c h i s o s p i n channel. Exceptions for conventional hadrons s e e m to r e f l e c t p u r e l y e x p e r i m e n t a l d i f f i c u l t i e s . F o r e x a m p l e , both SU(3) and the q u a r k m o d e l r e q u i r e an I = 0 p~ s t a t e in the v i c i n i t y of the B m e s o n (or l o w e r ) , and the t e m p o r a r y l a c k of such a s t a t e ( p r e d i c t e d by o u r " r u l e of t h u m b " ) is not disturbing. A c c o r d i n g to s e l e c t i o n r u l e s (I) and (II), the s y s t e m s A++fi, A-p, and t h e i r c h a r g e - c o n j u g a t e s ( t h e s e a r e the l o w e s t e x o t i c b a r y o n - a n t i b a r y o n c h a n n e l s ) a r e a l l o w e d to c o u p l e to M 4. By the " r u l e of t h u m b " , at l e a s t one such s t a t e s h o u l d l i e b e t w e e n the b o u n d s 2.2 G e V < M ( I =
2, Y = 0 ) < 2 . 5 G e V .
(4)
S i m i l a r l y , the A~-+~ and Y~+p s y s t e m s i m p l y the bounds 2.35 G e V < M ( I = Fig. 1. (a)-(c). Tensor structure of couplings to pairs of 35 mesons of a 405 meson; quark lines represent tensor indices: (a) Coupling proportional to A405; pure S wave; (b) Coupling proportional to A405 - (3) -1/2 A35; S and D waves; (c) Coupling ~A405 - 7 A35/(3)I/2 - 8 AI/(15)I/2; S, D, and G waves. (d) Coupling of M4 to B3B3. (e) Coupling of Ni4 to M2M4.
494
3/2,
IY 1 = 1)<2.6 G e V .
A p p l y i n g t h e s a m e r u l e to the ~ - A~. s y s t e m , which is a l l o w e d to f o r m an I = 2 M 4 s t a t e s i n c e t h e A~ is a s s u m e d to be an M4, one would e x p e c t
$$ The subscript denotes the number of quarks plus antiquarks. M is a meson, B a baryon. "~$~"See, however, ref. [17].
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PHYSICS
an I = 2, Y = 0 state:~ b e t w e e n the b o u n d s 1.45 GeV < M(I= 2, Y = 0 ) < I . 9 G e V . 2. Dependence o f m a s s on SU(3) r e p r e s e n t a tion. T h e s i m p l e s t s u c h d e p e n d e n c e w h i c h c a n a p p e a r in an SU(6) m a s s f o r m u l a c o m e s in t h r o u g h the C a s i m i r o p e r a t o r C~3) [20], which is 0 f o r 1, 6 f o r 8, 12 f o r 10 and ~0, and 16 f o r 27. T h e s i m p l e s t SU(3) b r e a k i n g t e r m is of the f o r m ~y2 - I(I+l). F o r s t a t e s of J = 2 we c a n t h e n g u e s s at r e a s o n a b l e p a r a m e t e r s in the s a m p l e m a s s formula M 2 = M 2 + aC(23) + b[~Y 2 - I(I+ 1)]
Y = 0) ~ 1.7 G e V .
(6)
H o w e v e r , the d e p e n d e n c e on C~3) n e e d not be l i n e a r [11], o t h e r SU(6) b r e a k i n g c o u l d be p r e s ent, and t h i s e x o t i c s t a t e c o u l d e a s i l y l i e s o m e what h i g h e r . O u r only p o i n t is that with the s i m p l e s t p o s s i b l e a s s u m p t i o n s , it is not d i f f i c u l t to p u s h the e x o t i c m e s o n s q u i t e high in m a s s , w h i l e c o n t i n u i n g to "count s t a t e s " by t h e b a d l y b r o k e n s y m m e t r y SU(6). If t h e J = 2 SU(3) s i n g l e t w e r e i n d e e d to l i e a s low a s 1 GeV, of c o u r s e , nonet s y m m e t r y w o u l d b e p o o r f o r the 1 and 8 J = 2 M 4 s t a t e s . T h e A 2 s p l i t t i n g , then, n e e d not i m p l y that of a l l its n o n e t p a r t n e r s . T h e A 2 - A~. d e g e n e r a c y , if not accidental, would then follow from considerat i o n s o t h e r than t h e p r e s e n t e l e m e n t a r y " t a x o nomic" ones. $ If this state belonged to ( q q ~ ; all L = 0) its JP would be 2+; a 0 + state cannot couple by J, P conservation, and the 1 + state has G = -.
7 December 1970
(b) Production m e c h a n i s m s T h e b e s t way to p r o d u c e M 4 s t a t e s s h o u l d be v i a " a l l o w e d " c o u p l i n g s [14]. T h e B3B3 c o u p l i n g s of such s t a t e s m a y be i n v e s t i g a t e d in b a r y o n - e x c h a n g e r e a c t i o n s , which i n c l u d e ,pp ~ n n i h i l a t i o n s [12] and b a c k w a r d m e s o n - b a r y o n : s c a t t e r i n g [21,14]. A f u r t h e r m e c h a n i s m , c o n s i s t e n t with the r u l e s of r e f . [14] and m e n t i o n e d p r e v i o u s l y by A r n o l d [22], is the u t i l i z a t i o n of t h e a l l o w e d (M2M4M4) and (M4B3B3) c o u p l i n g s t h r o u g h p r o c e s s e s of the f o r m M2 B3 ~ M4 (forward) B3 ,
(5)
u s i n g the f o l l o w i n g inputs: (i) T h e A~ b e l o n g s to an o c t e t , with M(A~) 1.3 GeV. (ii) T h e v a l u e of b is a p p r o x i m a t e l y the s a m e a s that which l e a d s to the K** - A2, K* - p, o r K - 7r m a s s d i f f e r e n c e . One w o u l d be s u r p r i s e d if SU(3) w e r e b e t t e r f o r qqq~ m a s s e s t h a n f o r qcl m a s s e s . (In the c a s e of n u c l e i , i s o s p i n is m o r e b a d l y b r o k e n in the m a s s e s of m o r e c o m p l e x systems.) (iii) T h e m a s s of the J = 2 u n i t a r y s i n g l e t e x c e e d s 1 GeV o r so. A l t h o u g h f o r b i d d e n by r u l e s (I) and (II) to c o u p l e to uTr, such a s t a t e b e l o w 1 GeV w o u l d s e e m h a r d to m i s s u n l e s s the s e l e c t i o n r u l e s w e r e c o n s i d e r a b l y b e t t e r t h a n we b e l i e v e t h e m to be. M o r e o v e r , a J = 2 s t a t e b e l o w 1 GeV w o u l d h a v e to lie on a R e g g e t r a j e c t o r y of a b n o r m a l l y g r e a t s l o p e , a p o s s i b l e if u n l i k e l y circumstance. G i v e n t h e s e t h r e e i n p u t s , one then p r e d i c t s t h a t the J= 2, I= 2, Y= O p a r t n e r of t h e A ~ i n s 405 m u l t i p l e t w o u l d l i e b e l o w M ( J = 2, I = 2 ,
LETTERS
(7)
i n v o l v i n g M 4 e x c h a n g e . An SU(3) 1 M 4 t r a j e c t o r y c o u l d l i e quite high without h a v i n g b e e n i d e n t i f i e d p r e v i o u s l y ~ . F o r e x a m p l e , the l a r g e C2(3) d e p e n d e n c e of the s h o r t d i s c u s s e d a b o v e c o u l d l e a d to an i n t e r c e p t ~(0) of n e a r l y 1. On the o t h e r hand, the h i g h e s t - l y i n g I : 1, Y = 0 M 4 t r a j e c t o r y w o u l d be that of the A2, with i n t e r c e p t p e r h a p s h a l f a unit l o w e r . In s h o r t , p r o c e s s e s i n v o l v i n g exchange o f no quantum n u m b e r s c o u l d be a g o o d way to p r o d u c e q q ~ s t a t e s at high e n e r g y . T h e d o m i n a n c e of I = Y = 0 e x c h a n g e in r e a c t i o n (7) w o u l d h a v e several effects: 1. A 0 v e r s u s A ~ production. In c o m p a r i n g , for example, v
g+ n ~ A 0 p
(8)
with ~-p ~A~p
,
(9)
one m i g h t e x p e c t the c r o s s s e c t i o n f o r r e a c t i o n (9) to l e v e l off at high e n e r g y if c o n s i d e r a b l e A~. w e r e b e i n g p r o d u c e d . An e x i s t i n g c o m p i l a t i o n [23] s h o w s a d i f f e r e n c e b e t w e e n the two p r o c e s s e s t e n d i n g in t h i s d i r e c t i o n , but with l a r g e e r r o r s . 2. P r o m i n e n c e o f G = - states in ~ -p ~ M~p at high energy. T h i s p r e d i c t i o n f o l l o w s f r o m the G = + n a t u r e of the d o m i n a n t M 4 e x c h a n g e . If the s t a t e s s e e n in h i g h - e n e r g y m i s s i n g - m a s s e x p e r i m e n t s of the f o r m [24] n - p ~ X - p a r e i n d e e d M 4 ' s , a s h a s b e e n s u g g e s t e d [9,14], and if t h e y c o n t i n u e to a l t e r n a t e in G - p a r i t y a s in the s e q u e n c e p - A 2 - g, the d o m i n a n t p e a k s at high e n e r g y should be the S(1930), the U(2382) and the X-(2800), with the T(2195) and X-(2620) s u p pressed. 3. E x o t i c s t a t e s . T h e p r o c e s s (7), f o l l o w e d by the d e c a y M 4-~ M~ (exotic I, Y) M 2 ,
(10)
~"Such a trajectory, called X by Arnold, has been postulated in order to explain certain features of interactions of conventional hadrons [22]. 495
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PHYSICS
LETTERS
7 December 1970
Table 1 "Better" and "worse" meson exchange reactions for M4 production at high energy. "Better" (No quantum numbers exchanged)
"Worse" (Charge or strangeness exchanged) 17r-p
K ~ p ~ p X -~
K~p
- A~
X--
- - n X ~ ~+
[~ Xprr ~
~Tr-X +~ ~ A pn ~ pp ' p X ~
pp ~ A Z * * ~
~,K- Z*~ ~ 7r-p
~ pX-
~-p
~
[~ pp'n's h o u l d be a good way to p r o d u c e e x o t i c m e s o n s . T h e a b s e n c e of such s t a t e s b e l o w 1450 MeV in n - p ~ n+p ( M M ) - - [13] s h o u l d t h e n be c o n s i d e r e d good evidence a g a i n s t such s t a t e s in t h i s m a s s r e g i o n , p a r t i c u l a r l y if the n+ ( M M ) - - m a s s in t h i s r e a c t i o n e m b r a c e s the r a n g e up to 1.9 GeV or soS. 4. " B e t t e r " and " w o r s e " e x p e r i m e n t s . If I = Y : 0 M 4 e x c h a n g e i n d e e d d o m i n a t e s r e a c t i o n (7), some meson exchange reactions are "better" for M 4 production, while others are "worse". Some c o m p a r i s o n s a r e m a d e in t a b l e 1. 5. P o m e r a n c h u k o n exchange. It is p o s s i b l e f o r the P o m e r a n c h u k o n (whose q u a r k s t r u c t u r e is u n c e r t a i n ) to p l a y a r o l e in r e a c t i o n (7). H o w e v e r , (i) it would not be s p e c i f i c to M 4 p r o d u c t i o n (one c o u l d t e s t f o r it in p r i n c i p l e by looking f o r e v i d e n c e of M6, M8, e t c . ) , and (ii) it w o u l d be e x p e c t e d to h a v e a s h a r p l y p e a k e d t d i s t r i b u t i o n , at l e a s t f r o m o u r p a s t e x p e r i e n c e . If the t d i s t r i b u t i o n a s s o c i a t e d with M 4 e x c h a n g e is any f l a t t e r t h a n that a s s o c i a t e d with P o m e r a n c h u k o n e x c h a n g e , the p r o d u c t i o n of M 4 r e s o n a n c e s at m o d e r a t e t (as in the c o n f i g u r a t i o n of r e f s . [13] and [24]) m a y win out o v e r d i f f r a c t i v e b a c k g r o u n d . (c) Decay P r o p e r t i e s If the l o w e s t e x o t i c m e s o n s lie b e l o w b a r y o n antibaryon threshold, as seems quite reasonable f r o m the a r g u m e n t in the s e c o n d p a r t of s e c t i o n (a), the M2M 4 d e c a y s of t h e s e m e s o n s w i l l s t i l l be a l l o w e d . In p r i n c i p l e , t h e y c o u l d a c q u i r e c o n s i d e r a b l e width t h r o u g h t h e s e c e d a y s , u n l e s s all c o u p l i n g s i n v o l v i n g M 4 ' s t u r n out to be w e a k $$. ~" This is because the "rule of thumb" would predict an M~ resonance in the M~-~"+ channel below 1.9 GeV if an M~- lay below 1.45 GeV. ~$ Ref, [25] predicts a normal strength for these couplings. 496
nX ° [~pl~
I n d e e d , the only M 4 ' s a s n a r r o w a s the A½ m a y be the l o w e s t o n e s , which we h a v e a s s u m e d to be n o n - e x o t i c . T h e s e s t a t e s w o u l d be n a r r o w s i n c e t h e y w o u l d h a v e to d e c a y by a v i o l a t i o n of the s e l e c t i o n r u l e s of r e f . [14]. A c r u c i a l t e s t of the s e l e c t i o n r u l e s would be to f o l l o w the s u b s e q u e n t d e c a y s of any c a n d i d a t e s f o r M4 s t a t e s , s u c h a s t h o s e p r o d u c e d in m i s s i n g m a s s e x p e r i m e n t s ( w h e t h e r of e x o t i c [13] o r o r d i n a r y [24] I and Y). T h e s e d e c a y s s h o u l d o c c u r v i a the c a s c a d e M4~
~4M2
--~M~' (narrow) M_~ •
(11)
( v i o l a t i o n of s e l e c t i o n r u l e s ) T h e n a r r o w M 4 s t a t e s w o u l d be the ground slale o f exotic m a t t e r , with o r d i n a r y I, Y which w o u l d not b e t r a y t h e i r q q q q n a t u r e . At any p o i n t in (11), if an M 4 s t i l l lay a b o v e B3B 3 t h r e s h o l d , t h e " c h a i n " c o u l d be " b r o k e n " by M 4 ~ B3B 3. Both m u l t i m e s o n and BB f i n a l s t a t e s w i l l be v i s i b l e in f o r t h c o m i n g e x p e r i m e n t s at S e r p u k h o v [26]. P r o c e s s e s such a s M 2 B 3 ~ M4 ~5
L B~
(12) (exotic) M 2
a r e a l l o w e d by t h e s e l e c t i o n r u l e s , and m a y be w o r t h looking f o r . Some further remarks: (d) T h e L - e x c i t e d s t a t e s of q q ~ s y s t e m s m a y p o s s e s s c o n s i d e r a b l e d e g e n e r a c y , g i v i n g r i s e to b e h a v i o r s i m i l a r to that s u g g e s t e d in r e f . [27]. (e) In the a b s e n c e of e x o t i c p a r t n e r s of the A~.,
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implied by the present discussion, one would be t e m p t e d to a d o p t a m o d e l s u c h a s t h a t of r e f . [28], w h i c h i n t r o d u c e s e x t r a d e g r e e s of f r e e d o m i n t o t h e q(l; L p i c t u r e w i t h o u t r e q u i r i n g s t a t e s of i n c r e a s i n g I a n d Y. (f) A s c o m p a n i o n s of t h e A½ a t low m a s s e s w o u l d b e n a r r o w b u t not n e c e s s a r i l y d e g e n e r a t e w i t h b r o a d e r s t a t e s , t h e y m i g h t b e v e r y h a r d to s e e . H o w e v e r , c l a i m s f o r s o m e of t h e s e l o w l y i n g " s h a r p s t a t e s " a l r e a d y do e x i s t [29,30]. A c o m b i n a t i o n of h i g h s t a t i s t i c s a n d h i g h r e s o l u t i o n w i l l b e n e e d e d to c o n f i r m s u c h s t a t e s . T h e y must be distinguished from statistical fluctuations, which are also typically narrow~. Facilities such as the CERN Omega spectrometer s h o u l d b e h e l p f u l in t h i s r e s p e c t . T o s u m m a r i z e , we h a v e s u g g e s t e d t h a t t h e e x p e r i m e n t a l p r o b l e m of t h e A 2 s p l i t t i n g a n d t h e t h e o r e t i c a l p r o b l e m of d u a l i t y in b a r y o n - a n t i b a r y o n s y s t e m s m a y b o t h b e s o l v e d in a q u a l i t a t i v e w a y by t h e e x i s t e n c e of qqqq s t a t e s s u b j e c t to c e r t a i n s e l e c t i o n r u l e s I $ . T h i s w o u l d b e e n c o u r a g i n g f o r m o d e l s of n - p o i n t f u n c t i o n s i n c o r p o r a t i n g d u a l i t y g r a p h s [32], w h i c h r e q u i r e such states for BB scattering$'~I. One should be cautious, however, about assuming that they s o l v e t h e B B s a t u r a t i o n p r o b l e m exactly. E v e n f o r t h e b e s t - k n o w n of p r o c e s s e s , t h e quantitative c o n c e p t of " d u a l i t y " ( a c c u r a t e r e s o n a n c e s a t u r a t i o n of n o n - P o m e r a n c h u k F E S R ) s e e m s to h a v e i t s l i m i t a t i o n s [35]. T h i s w o r k w a s d o n e in p a r t a t C a l t e c h , a t t h e U n i v e r s i t y of C a l i f o r n i a i n B e r k e l e y , a n d a t C E R N . I a m g r a t e f u l to M. G e l l - M a n n , G . F . C h e w a n d S. M a n d e l s t a m , a n d J. P r e n t k i f o r t h e h o s p i t a l i t y of t h e i r r e s p e c t i v e i n s t i t u t i o n s , a n d to R. A r n o l d , M. B a n d e r , G. F. C h e w , t h e C h u n g k i n g R e s t a u r a n t , P. G. O. F r e u n d , M. G e l l - M a n n , G. G o l d h a b e r , D. H o r n , M. J a c o b , W. K i e n z l e , J. M a n d u l a , S. M e s h k o v , N. P . S a m i o s , J. W e y e r s a n d G. Z w e i g f o r h e l p f u l a d v i c e . "~The author thanks D. R. O. M o r r i s o n and N. P. Samios for reminding him of this i m p o r t a n t point. "~'~'Similar (but more limited) r u l e s following f r o m SU(6)W were noted much e a r l i e r [31]. They forbid, e.g., the state in eq. (6) to decay to 7r+~+. "~'~"~The Utower n of q~ excitations does not close ]33,34] {see, e.g., eq. (12)).
References [1] M. Gell-Mann, Phys. L e t t e r s 8 (1964) 2]4; G. Zweig, CERN r e p o r t s TH-401 and TH-412 (1964), unpublished.
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[2] G. Goldhaber, in Proe. 13th Intern. Conf. on Highe n e r g ) , p h y s i c s , Berkeley, 1966, ed. M.AlstonGarnjost (Berkeley, University of California P r e s s , 1967) p. 103: R . H . D a l i t z , Ibid., p. 215. G. Zweig, in P r o c . 1968 Philadelphia Meson Conf., eds. C . B a l t a y and A. Rosenfeld (New York, Benjamin, 1968) p. 485. [3] B. L e v r a t et al., Phys. L e t t e r s 22 (1966) 714: M:N. F o e a c c i e t a l . , Phys. Rev. Letters 17 (1966i 890. [4] A. B a r b a r o - G a l t i e r i , in P r e c . 1970 Philadelphia Meson Conf., eds. C . B a l t a y and A. Rosenfeld (New York, Columbia University P r e s s , 1.970)~ [5] G. Goldhaber, in Proc. 15th Intern. Conf. on Highenergy physics, Kiev, USSR, 1970, to be published. [6] Y.Dothan and D. Horn, Phys. Rev. D1 (19701 916. [7] J. Rosner, Phys. Rev. Letters 21 (1968~ 950, 1422(E). [8] K. L a s s i l a and P. Ruuskanen, Phys. Rev. [.etters 19 (1967) 762; S. Meshkov, in P r o e . 1970 Philadelphia Meson Conf., op. cit. [9] R. Arnold and J. Uretsky, Phys. Rev. Letters 23 (1969) 444: S.F. Tuan, Phys. Rev. L e t t e r s 23 (1969) 1198. [10] Y. Dothan, M. Gell-Mann and ¥. Nereman, Phys. L e t t e r s 17 (1965) 148. [11] D.Horn, J. Coyne, S. Meshkov and J. C a r t e r , Phys. Rev. 147 (1966) 980. [12] J. Rosner, in P r o c . 1970 Philadelphia Meson Conf., op. cit. [13] C. Lechanoine, r e p o r t on the CERN-Serpukhov m i s s i n g - m a s s s p e c t r o m e t e r collaboration to the 1970 Kiev Conference, op. cir. [14] P. G. O. Freund, R. Waltz and J. Rosner, Nucl. Phys. B13 (1969) 237. [15] H.J. Lipkin, Phys. Rev. L e t t e r s 16 (1966) 1015: R.H. Capps, Phys. Rev. 168 (1968)173]. [16] J. Rosner, C . R e b b i a n d R. Slansky, Phys. Rev. 188 (1969) 2367. [17] M.Kugler, Phys. L e t t e r s 32B (1970) 107. [18] P . G . O . Freund, Phys. Rev. L e t t e r s 20 (1969) 235: H . H a r a r i , Phys. Rev. L e t t e r s 20 (1969) 1395. [19] H.J. Lipkin, Nucl. Phys. B9 (1969) 349: D . P . R o y and M.Suzuki, Phys. L e t t e r s 28B (1969) 558; M. Kugler, Phys. Rev. 180 (1969) 1538: J. Mandula, J. Weyers and G. Zweig, Phys. Rev. L e t t e r s 23 (1969) 266: R . H . C a p p s , Phys. Rev. 185 (1969)2008: H.J. Lipkin, Phys. L e t t e r s 32B (1970) 301. [20] M . A . B . B ~ g a n d V. Singh, Phys. Rev. L e t t e r s 13 (1964) 418. [21] M. J a c o b a n d J. Weyers, C E R N p r e p r i n t TH-1169, June, 1970, to be published; J. Rosner, in Proc. 8th E a s t e r n Theoretical P h y s i c s Conf., Syracuse, New York, October, 1969, ed. F . R o h r l i c h (Syracuse, P h y s i c s D e p a r t ment of Syracuse University, 1969) p. 231; J. Rosner, invited talk at the Washington Meeting of the A m e r i c a n Physical Society, April, 1970, Caltech r e p o r t CALT-68-254, unpublished. [22] R. Arnold, private communication, Argonne p r e print, Aug. 1970, to be published. [23] J. T. C a r r o l l et al., Production of A 2 mesons, W i s c o n s i n - T o r o n t o preprint, to be published. 497
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[24] M. Martin, in P r o c . 1970 Philadelphia Meson Conf., op. cit. [25] P. H. Frampton and P. G. O. Freund, Enrico F e r m i Institute p r e p r i n t EFI-70-44, August 1970, to be published. [26] W. Kienzle, private communication. [27] L. -H. Chan, R. Slansky and D. Sutherland, Phys. Rev. Letters 25 (1970) 482. [28] D.Horn, Nuovo Cimento 62A (1969) 581. [29] P a r t i c l e Data Group, Phys. L e t t e r s 33B {1970) 1. [30] A. Astier, in P r o c . 1970 Kiev Conference, op. cit. [31] D.Horn, H . J . Lipkin and S. Meshkov, Phys. Rev. L e t t e r s 17 (1966) 1200.
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[32] M. Imachi et al.~ P r o g . Theor. Phys. (Kyoto) 40 (1968) 353; H . H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 562; J. Rosner, Phys. Rev. L e t t e r s 22 (1969) 689; D. E. Neville, Phys. Rev. L e t t e r s 22 (1969) 494; Chan Hong-Mo and J . E . P a t o n , Nucl. Phys. B10 (1969) 516; Z. Koba and H. B. Nielsen, Nucl. Phys. B10 (1969) 633. [33] D. P. Roy and M. Suzuki, Phys. L e t t e r s 28B {1969) 508. [34] S. Mandelstam, Phys. Rev. D1 (1970) 1720. [35] R.Dolen, D.Horn and C.Schmid, Phys. Rev. 166 (1968) 1768.