- Email: [email protected]
Some remarks on higher mesons
A~
Nuclear Physics B2 (1967) 157-163. North-Holland Publ, Comp.. A m st er d am
SOME REMARKS o N HIGHER MESONS D. G. S U T H E R L A N D CERN,
Gene~,a
Received 17 May 1967
Abstract: A modeI based on exchange degeneracy and linear t r a j e c t o r i e s is given to c o r r e l a t e presently known meson resonances with I = 1. A critical discussion is given of some features of this model. Some r e m a r k s are made on the problem of the widths of the higher mesons in this scheme and on some related problems in higher meson spectroscopy.
It i s the p u r p o s e of t h i s note to a t t e m p t a f i t of a l l k n o w n I = 1 m e s o n s on s t r a i g h t - l i n e R e g g e t r a j e c t o r i e s . S e v e r a l p a p e r s [ 3 - 6 ] h a v e a p p e a r e d in r e c e n t m o n t h s in w h i c h p a r t i a l a t t e m p t s to u s e l i n e a r R e g g e t r a j e c t o r i e s a r e m a d e , but none of t h e m h a v e r e p r o d u c e d t h e s p l i t t i n g in t h e R - r e g i o n s e e n in t h e m i s s i n g m a s s e x p e r i m e n t [7]. S i n c e t h e y a l s o n e g l e c t e d p e a k s s u c h as 5 (r ef. [7]) and A1 w h i c h h a v e b e e n r e p o r t e d at l o w e r m e s s e s , one m a y f e e l t h a t t h i s c o u l d be r e m e d i e d if t h e y w e r e i n c l u d e d . T h e s e c o n d p u r p o s e i s , w i t h i n th e c o n t e x t of the m o d e l , to d i s c u s s , in w h a t we hope is a s e m i - q u a n t i t a t i v e w a y , the p r o b l e m of n a r r o w w i d t h s , w h i c h h a s aroused some interest recently. T o s o m e e x t e n t we f e e l t h a t , l a c k i n g a d e t a i l e d d y n a m i c a l m o d e l , o u r c a l c u l a t i o n s a r e v e r y t r i v i a l . O u r a p o l o g y i s only t h a t s i m i l a r a p p r o a c h e s s h o w p r o m i s i n g r e s u l t s in the b a r y o n s p e c t r u m [8]. One is thus e n c o u r a g e d to hope t h a t t r y i n g it f o r m e s o n s m a y e i t h e r p r o d u c e g o o d r e s u l t s , o r by f a l l i n g b a d l y , l e a d one e i t h e r to s o m e d y n a m i c a l u n d e r s t a n d i n g [9], if s u c c e s c o n t i n u e s f o r _ t h e b a r y o n s , o r to l o o k m o r e s u s p i c i o u s l y at t h e t h e o r y f o r b a r y o n s , w h i c h so f a r is c o m p l e t e l y c h e c k e d in o n l y r a t h e r f e w c a s e s . A f u r t h e r hope is t h a t th e p r o p o s e d t r a j e c t o r i e s m a y be of i n t e r e s t to R e g g e t h e o r e t i c i a n s , p a r t i c u l a r l y f o r c o n s i d e r a t i o n of u n n a t u r a l p a r i t y e x c h a n g e . F i n a l l y t h e c o n s i d e r a t i o n s on t h e w i d t h s m a y , w i t h a l i t t l e a d d i t i o n a l e x p e r * Only for the di-pion resonance at 1650 MeV is there e v i d e n c e [ i ] that I = 1 , not 2. for the charged resonances above 1.5 GeV. It should be noted that 1.7GeV is a very natural point at which higher I resonances might occur as can be seen from the following piece of numerological argument. In the baryon system, evidence for a 1--gresonance has recently been given by Cool et al. at 1860 MeV [21 . If one conjectures that the mass tn M at which new "non quark-antiquark" mesons appear is related to the mass mB at which new "non t h r e e - q u a r k " baryons appear by rn2 to__2 - rn2 m2 we find m M = 1620 MeV. We shall a s s u m e I= 1 for all higher M - "'~T B p' bumps, however.
158
D.G. SUT HER LA ND
i m e n t a l i n f o r m a t i o n , l e a d one to new s e l e c t i o n r u l e s of s t r o n g i n t e r a c t i o n d y n a m i c s , in t h e w a y t h a t t h e f a i l u r e of ~ to d e c a y to p~ h a s done. O u r t h e o r e t i c a l i n p u t w i l l b e t h e c o n c e p t of " e x c h a n g e d e g e n e r a c y " , i n t r o d u c e d b y A r n o l d [9], s u p p l e m e n t e d b y s t r a i g h t - l i n e R e g g e t r a j e c t o r i e s [8]. A r n o l d ' s i d e a i s t h a t r e s o n a n c e s s h o u l d o c c u r at i n t e r v a l s of AJ= 1, not of AJ= 2 a s in t h e u s u a l s c h e m e . T h i s i s v e r y n a t u r a l on a n y F e r m i - Y a n g m o d e l in w h i c h t h e m e s o n s a r e b o u n d s t a t e s of b a r y o n s a n d a n t i b a r y o n s , s i n c e t h e n m e s o n e x c h a n g e in t h e t - c h a n n e l s h o u l d d o m i n a t e o v e r d o u b l e b a r y o n e x c h a n g e in t h e u - c h a n n e l , g i v i n g a p r e d o m i n a n t l y d i r e c t p o t e n t i a l only. We a r e a w a r e t h a t t h i s i s o n l y a r a t h e r r o u g h a p p r o x i m a t i o n [10], a n d we s h a l l d i s c u s s t h e e r r o r s i n v o l v e d f o r t h e c a s e of t h e P - A 2 t r a j e c t o r y for illustration. O u r e x p e r i m e n t a l i n p u t w i l l c o n s i s t of t h e l o w - l y i n g r e s o n a n c e s ~ ( 1 3 9 ) 0 - - , p ( 7 6 8 ) 1 - + , A 2 ( 1 2 8 6 ) 2 + - , A l ( 1 0 8 0 ) l + - , B(1208)1++ a n d 5(962)0 + - , w h e r e o u r n o t a t i o n i s ( m a s s in M e V ) jPG. We t a k e o u r p a r a m e t e r s f r o m t h e r e s u l t s p r e s e n t e d b y F o c a c c i e t al. [11], w h e n t h e y h a v e o b s e r v e d the p e a k , s i n c e t h e i r r e s u l t s f o r the h i g h e r p e a k s a r e t h e o n e s we s h a l l b e c o n c e r n e d w i t h m o s t , a n d f r o m the R o s e n f e l d t a b l e s [12] in o t h e r c a s e s . We w i s h to e m p h a s i z e t h a t f o r o n l y ~, p a n d A 2 a r e t h e s e s p i n - p a r i t y a s s i g n m e n t s r e a s o n a b l e w e l l c o n f i r m e d * , a n d in t h e c a s e of 5 we h a v e to r e l y e n t i r e l y on theoretical conjecture [17-19] F o r t h e t r a j e c t o r i e s c o n s i s t i n g of (5, A2) a n d of (~, B) t r e a t m e n t s h a v e b e e n g i v e n p r e v i o u s l y [4, 5, 6, 11]. F o r t h e A1 a n d 5 t r a j e c t o r i e s , we h a v e no p a r t n e r s f r o m w h i c h to d e r i v e t h e t r a j e c t o r i e s . W e t h e n n e e d t h e f o l l o w i n g i n f o r m a t i o n : D e u t s c h m a n n et al. [20] h a v e p r e s e n t e d e v i d e n c e of a ~+ ~o r e s o n a n c e at 1910 M e V and, w h i l e G o l d h a b e r [21] a p p a r e n t l y f e e l s t h i s r e s o n a n c e to b e a s y e t u n c o n f i r m e d , we f e e l t h a t i t s s t a t i s t i c a l s i g n i f i c a n c e of f i v e s t a n d a r d d e v i a t i o n s , s h o u l d , f o r c o n s i s t e n c y w i t h t h e c r i t e r i a of r e f . [11], l e a d u s to t a k e it s e r i o u s l y . T h e m o s t n a t u r a l c h o i c e is t h e n 3 -+ if it i s to b e a s s o c i a t e d w i t h t h e 5 in a R e g g e t r a j e c t o r y . A s s e e n f r o m o u r p r e s e n t a t i o n of the r e s u l t s in t a b l e 1, we s t i l l h a v e not a c a n d i d a t e f o r R3(1748) (ref. [11]). F o r t h i s we p r o p o s e t h a t it s h o u l d b e a 2 -+ s t a t e , a s s o c i a t e d with t h e A1. T h i s we d e s c r i b e a s t r a j e c t o r y ( A 1 , R 3 ) I ; an a l t e r n a t i v e p r o p o s a l i s t h a t R 3 i s a 3 + - s t a t e [6]; t h i s we d e s c r i b e a s t r a j e c t o r y ( A 1 , R 3 ) 2 . S i n c e t h i s p r e d i c t s a 2 -+ s t a t e a t 1 4 5 0 M e V a n d , m o r e s e r i o u s l y , a 0-+ s t a t e a t 460 M e V , we a r e i n c l i n e d to f a v o u r t h e first assignment. L e t u s now m a k e s o m e g e n e r a l r e m a r k s a b o u t o u r s c h e m e . F i r s t of a l l , l e t u s m e n t i o n t h a t t h e v a l u e of 0.47 f o u n d f o r a (p, A 2 ) ( 0 ) a g r e e s n e i t h e r w i t h t h e v a l u e of 0.58 f o r t h e p - t r a j e c t o r y , n o r w i t h t h a t of 0.35 found f o r t h e A2 t r a j e c t o r y [10]. If we w e r e to c h o o s e C~p(0) = 0.58 a n d o l ~ 0 . 5 8 ) = 1, we w o u l d f i n d a 3-+ s t a t e a t 1820 M e V , not at 1650 MeV. H e n c e one m u s t a l l o w s o m e * Even for the A2 there have recently been some w o r r i e s about the spin parity a s signment [13] and about the possibility of two peaks in the 37T system [14-16]. ** The argument given in ref. [19] on the small rate depends on the existence of only Regge poles. Regge cuts of negative parity could spoil it (see ref. [36]): m e a s u r e m e n t of 5 production, if it is r e a l l y 0 +-, could be interesting as a test for Regge cuts.
HIGtlER MESONS
159
Table 1 Resonances and trajectories. Trajectory
(p, a2)
(Tr, B)
(5,S)
(A1.R3) 1
(A1,R3) 2
p(760)1 -+
g(159)0--
5(962)0 +-
A1(1080)1 +-
Al(1080)l e-
A2(1286)2~-
B(1210)1 +F
1355 1.+
R3(1748)2 -+
1450 2 -~
R1(1650)3 -+
R2(1705)2--
1652 2+.
2220 3 ÷-
R3(1748)3 +-
S(1950)4 e-
2090 3 ÷+ S(1910)3 -+
2610 4 -+
2000 4 .+
T(2206)5 -~
2410 4 - -
2140 4 ÷-
2220 5 ~-
U(2440)6 e-
2700 5 ~+
2350 5 -+
2430 6 -+
V(2650) 7-+
2540 6+-
~(0)
0.47
-0.01
-1.00
0.39
-0.23
Or' (GeV/c) -2
0.92
0.69
1.08
0.53
1.05
c u r v a t u r e in the t r a j e c t o r y . T h e b a r y o n t r a j e c t o r i e s , on the o t h e r hand, s e e m to fit r a t h e r w e l l with l i n e a r t r a j e c t o r i e s , at l e a s t f o r m 2 > 1 GeV 2. A f u r t h e r r e m a r k is that the p o s i t i o n s of the h i g h e r r e s o n a n c e s a r e v e r y s e n s i t i v e to t h o s e of the input. T h u s if ruA2 is c h o s e n a s 1305 MeV [12] i n s t e a d of 1286 MeV [11], t h e n we find ru T = 2260, not 2206, MeV. A f u r t h e r f e a t u r e of o u r s c h e m e w h i c h we find s o m e w h a t d i s q u i e t i n g is the fact t h a t the v a l u e s of c~' w h i c h we find f o r the u n n a t u r a l p a r i t y m e s o n t r a j e c t o r i e s d i f f e r s s u b s t a n t i a l l y f r o m t h o s e of the n a t u r a l p a r i t y m e s o n t r a j e c t o r i e s , w h e r e a s for the b a r y o n t r a j e c t o r i e s the v a l u e s of c~' a g r e e w i t h i n 10% for a l l t h r e e e s t a b l i s h e d t r a j e c t o r i e s . A f u r t h e r p o i n t is t h a t ~ ( A 1 - R 3) (0) is 0.39 f o r o u r f i r s t s o l u t i o n . T h i s m a y be i n t e r e s t i n g f o r e x p l a i n i n g r e a c t i o n s i n v o l v i n g u n n a t u r a l p a r i t y e x c h a n g e , b u t c o n t r a d i c t s , on the o t h e r h a n d , a s u g g e s t i o n by D u r a n d [22] that np c h a r g e e x c h a n g e m a y be e x p l a i n e d by h a v i n g ~A1 ( 0 ) < 0. F r o m t a b l e 1 it will be s e e n that we have r e s o n a n c e s c l o s e to the o b s e r v e d p e a k s at S, T , U a n d a l l t h r e e p e a k s in the R - r e g i o n . I n a d d i t i o n we h a v e a f o u r t h p e a k in the R - r e g i o n at 1652 MeV with j P G = 2+-. T h i s is r a t h e r g r a t i f y i n g in view of the r e p o r t e d e x i s t e n c e of a 3~ r e s o n a n c e at 1640 MeV [21]. Of c o u r s e , we a l s o p r e d i c t o t h e r r e s o n a n c e s , at 2090 MeV a n d at 1355 MeV w h i c h have not y e t b e e n o b s e r v e d . A l t h o u g h one c a n a r g u e t h a t the f o r m e r m a y have e s c a p e d d e t e c t i o n b e c a u s e of a l a r g e width, the f a i l u r e to d e t e c t the s e c o n d in the e x p e r i m e n t of W e h m a n n et al. [23] is r a t h e r d i s t u r b i n g . We s h o u l d a l s o p o i n t out that two r e s o n a n c e s a r e p r e d i c t e d in the v i c i n i t y of S, t h r e e in the v i c i n i t y of T , a n d t h r e e in the v i c i n ity of U. T h i s m a y be r e l e v a n t for the o b s e r v a t i o n in ~p (total c r o s s - s e c t i o n s ) of r e s o n a n c e s n e a r the T a n d U of ref. [11], b u t with m u c h l a r g e r w i d t h s of a b o u t 90 MeV [24]. C l e a r l y o u r s c h e m e owes s o m e t h i n g to the q u a r k m o d e l [17]. T h o u g h the i d e a of e x c h a n g e d e g e n e r a c y d e p e n d s o n l y on a b a r y o n - a n t i b a r y o n m o d e l f o r the m e s o n s , the r e s t r i c t i o n to only s i n g l e t a n d o c t e t r e p r e s e n t a t i o n s a n d to
160
D.G. SUT HER LA ND
the spin values of the trajectory we employ are harder to justify if nucleons and nucleon resonances and their antiparticles are the relevant baryons. Our reason for trying straight-line trajectories, rather than the more concrete model of ref. [17], is simply that the observation by Deutschmann et al. [20] of a 1910 MeV resonance decaying to ~+vo is incompatible with that scheme, so that some modification, at least, is required, and the only guide available is the success of the baryon trajectory scheme. We now discuss the problem of the widths of these mesons. This has seemed rather hard to understand since the widths of all the resonances seen in the missing mass spectrometer above 1.5 GeV are ~ 38 MeV. This contradicts a general feeling that the widths of resonances should increase with their mass. Indeed the baryons do seem to show some slow increase with mass, at least as fas as present analysis is concerned. We shall try to estimate how serious this problem is within our framework. It should be borne in mind that bubble chamber data [21, i] on peaks in the R-region suggest much larger widths. If these resonances are the same as those seen by Focacci et al [11], we feel that the higher statistics and better resolution of the latter experiment should lead us to take their narrow width seriously. Our discussion will be based on a very simple expression for the partial width F l for the decay of a meson of mass M into a quasi-two-body state with relative orbital angular momentum l,
Fl :yq--
(qR) 2I
rn [ 1 . 3 . 5 . . .
(2/-1)] 2 '
w h e r e q i s the m o m e n t u m a v a i l a b l e in the d e c a y , R is s o m e " r a d i u s of i n t e r a c t i o n " a n d Y is the r e d u c e d width, w h i c h is a p r i o r i u n l i k e l y to v a r y w i d e l y f r o m one r e s o n a n c e to the n e x t * . We s h a l l show that f o r the h i g h e r m e s o n r e s o n a n c e s t h i s f o r m u l a l e a d s , on n o r m a l i z i n g y to the width of the p, to a r e a s o n a b l e e x p l a n a t i o n of the w i d t h s . F o r R we t a k e ~r¢ M e V - 1 , s i n c e t h i s is the m a s s a p p e a r i n g in the e l e c t r o m a g n e t i c f o r m f a c t o r s . Such a c h o i c e s e e m s r e a s o n a b l e , a s it g i v e s a width of 100 MeV f o r foIf we t a k e the U to b e 5-+, for i n s t a n c e , we find t h a t a r e a s o n a b l e e s t i m a t e of its w i d t h s is 60 MeV, if we k e e p a l l q u a s i - t w o - b o d y s t a t e s i m p l i e d by o u r m o d e l . One m a y , of c o u r s e , a n t i c i p a t e that t h e r e w i l l be s o m e f u r t h e r c o n t r i b u t i o n to the w i d t h f r o m m a n y p a r t i c l e s t a t e s , b u t p r e s u m a b l y it s h o u l d be of the s a m e o r d e r a s that f r o m the q u a s i - t w o - b o d y s t a t e s . M o s t of the c o n t r i b u t i o n to o u r e s t i m a t e c o m e s f r o m the d e c a y into p + fo or ¢o + A 2 in d - w a v e , a n d if we w e r e to d e c r e a s e the v a l u e of 7 for t h e s e m o d e s by 2, a g r e e m e n t with ref. [11] w o u l d be o b t a i n e d . S i n c e ~ c a n c e r t a i n l y b e e x p e c t e d to v a r y by 10 f r o m one r e s o n a n c e to a n o t h e r w i t h o u t i m p l y i n g a n y * We are unable to give any really satisfactory justification for this paramctrization, which is cssentially a "relativized" version of the formula given in Chapter VIII of Blatt and Weisskopf [25]. A similar expression was suggested by Layson [26] for baryon widths. This formula employs nothing much more than the idea that strong forces are of short range, which is almost the only thing we know. Strictly speaking, the formula given here is only correct if qR is small, but the correction terms tend to decrease F l and for our purposes are not important.
HIGHER MESONS
161
p a r t i c u l a r s e l e c t i o n r u l e (cf., t h e A 2 ~ K K w i t h fo ~ ) , it i s c l e a r t h a t we cannot conclude that anything very surprising is happening. T h e r e a s o n why t h e w i d t h d o e s not i n c r e a s e v e r y r a p i d l y in t h i s s c h e m e is that the spacing between resonances is decreasing rapidly while the spin is increasing rapidly. There is thus rather little phase space available for d e c a y into n e a r b y s t a t e s w i t h low o r b i t a l a n g u l a r m o m e n t u m w h i l e the s t a t e s w i t h l a r g e p h a s e s p a c e h a v e h i g h a n g u l a r m o m e n t u m b a r r i e r s . It i s v e r y i n t e r e s t i n g h e r e to r e m a r k t h e r e c e n t o b s e r v a t i o n by B a r t k e et al. [27] of a r e s o n a n c e at 3690 M e V w i t h F = 50=~ 30 M e V in N+7~, w h i c h l e n d s s u p p o r t to t h i s p i c t u r e f o r b a r y o n s a l s o . In the r e g i o n of S s i m i l a r e s t i m a t e s show t h a t S ( I 9 5 0 ) 4 + - s h o u l d h a v e a n a r r o w w i d t h c o m p a t i b l e w i t h the l i m i t of 35 M e V [11]. F o r S(1910)3 -+, h o w e v e r , an a p p r e c i a b l y l a r g e r w i d t h of a b o u t 150 M e V , w i t h ~ , ~A, a n d pp e a c h h a v i n g c o m p a r a b l e p r o b a b i l i t y i s e x p e c t e d , w h i c h m a y w e l l i m p l y t h a t t h e r e s o n a n c e o b s e r v e d b y F o c a c c i et al. i s the 4+- s t a t e . So f a r o u r r e s u l t s a r e in f a i r a c c o r d w i t h e x p e r i m e n t e v e n if we do not c h a n g e y a p p r e c i a b l y f r o m t h e v a l u e f o u n d f o r t h e p. R e a s o n a b l e r e s u l t s a l s o e n s u e f o r t h e R1(1650)3 -+ f o r w h i c h we a n t i c i p a t e a w i d t h of 15 M e V f o r ~ , a l t h o u g h it a p p e a r s t h a t p+p s h o u l d h a v e a w i d t h of 40 M e V s i n c e it o c c u r s in a p - w a v e . F o r t h e 2 + - s t a t e p r o p o s e d at 1650 M e V we a n t i c i p a t e c o m p a r a b l e w i d t h s of a b o u t 60 M e V e a c h f o r ~fo, ~P a n d ~B. T h i s i s in f a i r a c c o r d w i t h t h e t o t a l w i d t h of 100 M e V q u o t e d in r e f . [21], b u t d i s a g r e e s s o m e w h a t w i t h t h e p r e l i m i n a r y e v i d e n c e g i v e n t h e r e t h a t ~fo d o m i n a t e s o v e r ~p. H o w e v e r , if we c o n s i d e r the 2 -+ a n d 2+- s t a t e s in t h e R - r e g i o n , we f i n d t h a t , k e e p i n g y a t i t s v a l u e d e d u c e d f r o m p, we w o u l d e x p e c t t h a t e a c h r e s o n a n c e s h o u l d h a v e a w i d t h of a b o u t 200 M e V f o r e a c h of t h e m o d e s 0 - + 1 - in a p w a v e a n d 0-+2 + in an s w a v e . T h i s i n d i c a t e s t h a t )~ s h o u l d b e ~ of t h e v a l u e of y d e d u c e d f r o m the p if we i d e n t i f y t h e s e r e s o n a n c e s w i t h t h e R 2 a n d R 3 of r e f . [11]. L e t u s c o m p a r e o u r r e s u l t s on w i d t h s w i t h t h a t of p r e v i o u s t r e a t m e n t s . F r e u n d [5] s u g g e s t s t h a t a s y m p t o t i c a l l y the p a r t i a l w i d t h t o two p s e u d o s c a l a r s , f o r i n s t a n c e , s h o u l d t e n d to i n f i n i t y a s 2 2 J J ( J - ~ ) ; t h i s r e s u l t a p p e a r s to d i s a g r e e w i t h o u r s i m p l e m o d e l , b u t it s h o u l d b e n o t e d t h a t F r e u n d ' s w i d t h s a r e p r o p o r t i o n a l to (t3(E))2, w h e r e ~(t) i s t h e r e d u c e d r e s i due f u n c t i o n , w h i c h he, s o m e w h a t a r b i t r a r i l y in o u r v i e w , s e t s c o n s t a n t a b o v e t=~np2. It s e e m s t h a t no r e l i a b l e e s t i m a t e s of [3(t) a t high t - v a l u e s a r e a v a i l a b l e but in v i e w of i t s r a p i d v a r i a t i o n at l o w e r t - v a l u e s we f e e l it m a y w e l l d e c r e a s e r a p i d l y at l a r g e I. T h i s w o u l d b e r e f l e c t e d in a L a g r a n g i a n t h e o r y of t h e d e c a y of t h e s e h i g h e r m e s o n s b y t h e f a c t t h a t t h e c o u p l i n g c o n s t a n t s h o u l d d e c r e a s e r a p i d l y w i t h i n c r e a s i n g r e s o n a n c e m a s s , s o a s to i n c o r p o r a t e t h e c e n t r i f u g a l b a r r i e r e f f e c t f o u n d in a b o u n d a r y c o n d i t i o n m o d e l of t h e s e m e s o n s . A p r o p o s a l c l o s e l y r e l a t e d to o u r s w a s m a d e i n d e p e n d e n t l y b y D a l i t z [17] a t t h e B e r k e l e y C o n f e r e n c e . He p r o p o s e d u s i n g a s m a l l e r r a d i u s t h a n t h a t u s e d h e r e , b u t d i d not m a k e n u m e r i c a l c o m p a r i s o n s in t h e R - r e g i o n . It i s c l e a r f r o m o u r d i s c u s s i o n t h a t no v e r y s t r o n g o v e r l a p f u n c t i o n e f f e c t [17] between resonances with widely different spin is necessary, but that the l a r g e s t s u p p r e s s i o n of y s e e m s n e c e s s a r y f o r t h e d e c a y s of 2 - - s t a t e s i n t o
162
D.G. SUTHERLAND
2++0 - a n d 0 - + 1 - s t a t e s . T h i s e f f e c t w i l l only b e m a d e w o r s e by d e c r e a s i n g the r a d i u s of i n t e r a c t i o n and, s i n c e no s t r o n g effect is o b s e r v e d in the d e c a y of 2 + ~ 0 - + 0 - s u c h a s fo ~ 7rTr, the j u m p in s p i n it not e n o u g h to m a k e u s f e e l t h a t t h i s s h o u l d b e a r e a l l y l a r g e effect. A f a c t o r of 10 s u p p r e s s i o n i s not y e t e n o u g h , h o w e v e r , to m a k e u s f e e l that a n y t h i n g r e a l l y r e m a r k a b l e i s h a p p e n i n g , s u c h a s o c c u r r e d in the s u p p r e s s i o n of ~o ~pTr w h e r e a f a c t o r of 100 s u p p r e s s i o n w a s i n v o l v e d ~ . It s e e m s that an i m p r o v e d b o u n d m u s t be s e t on the p a r t i a l w i d t h s of t h e s e r e s o n a n c e s , e i t h e r by i m p r o v e m e n t of the t o t a l w i d t h m e a s u r e m e n t , o r , p e r h a p s m o r e p r o b a b l y , by f i n d i n g t h a t the r e l e v a n t m o d e s a r e only a s m a l l f r a c t i o n of the t o t a l width, b e f o r e a n y t h i n g r e a l l y s u r p r i s i n g a n d s i g n i f i c a n t c a n be c o n c l u d e d . We m a k e s o m e b r i e f r e m a r k s on two o t h e r p r o b l e m s . T h e f i r s t c o n c e r n s the ~p d a t a of G i a c o m e l l i et al. F o r p e a k s w h i c h t h e y o b s e r v e in the T - a n d U - r e g i o n : G i a c o m e U i et al. d e t e r m i n e (J -t-1~)FNN ~ 20 MeV, w h e r e J is the s p i n of the r e s o n a n c e . T a k i n g f o r U, f o r i n s t a n c e , the m o s t c o n s e r v a t i v e s p i n a n d o r b i t a l a n g u l a r m o m e n t u m a v a i l a b l e in o u r s c h e m e , n a m e l y J = 4, I = 4, we d e d u c e YNN = 10yp. T h i s m a y l e n d s o m e s u p p o r t to a F e r m i - Y a n g m o d e l of the m e s o n s . O u r s e c o n d r e m a r k is on the L - m e s o n [28]. Z a l e w s k i [29] i n f o r m s u s t h a t a n a t t e m p t to fit the b r a n c h i n g r a t i o s of t h i s r e s o n a n c e l e a d s to s t r o n g d i s a g r e e m e n t with SU(3). T h i s m a y m e a n t h a t , j u s t a s in the c a s e of the R - r e g i o n , w h i c h f i r s t e s t i m a t e s f r o m the m i s s i n g m a s s e x p e r i m e n t g a v e a s a b r o a d p e a k of w i d t h ~ 100 MeV, s e v e r a l n a r r o w e r r e s o n a n c e s , i n c l u d i n g the 3 - K* s u g g e s t e d by A h m a d z e d a h [6], w i l l be found in t h i s r e g i o n . A n a l y s i s of s p i n a n d p a r i t y a s s u m i n g o n l y one r e s o n a n c e w i l l t h e n give misleading results. F i n a l l y , l e t u s s t a t e that we dot not b e l i e v e that t h i s m o d e l w i l l give a c o m p l e t e a n s w e r to the m e s o n s p e c t r u m . We p r e s e n t it only a s a m o d e l to show what m a y be l e a r n t f r o m c u r r e n t e x p e r i m e n t e v i d e n c e a n d to show s i m p l e e f f e c t s s u c h a s the e a s e with w h i c h s e v e r a l r e s o n a n c e s h a v i n g m a s s e s v e r y c l o s e t o g e t h e r c a n a r i s e , a n d t h a t t h e r e is not y e t a n y t h i n g d e c i s i v e l y d i s t u r b i n g a b o u t the n a r r o w w i d t h s o b s e r v e d . I t s s u c c e s s o r , in the c a s e of its f a i l u r e , the i m p l i c a t i o n s f o r the b a r y o n s , we l e a v e to the r e a d e r to j u d g e ~.
] It is d e a r that we are not committing the first basic e r r o r of strong coupling theo r y d i s c u s s e d by Low (ref.[37]), and we feel that to take factors of 10too s e r i ously would be to commit the second. The p e s s i m i s m of this last paragraph has recently received support from the appearance of a paper by Vanderhagen et al. [30], in which ap-Tr- resonance is r e ported at 1320 MeV. (The flexibility of numerology is well illustrated by changing m to the average mass of the pseudoscalars, ~ 400 MeV, and using a linear instead of a quadratic relation in the first footnote.) It should also be pointed out that we have taken a rather naive approach to Regge r e c u r r e n c e s and have not worried about possible conspiration t r a j e c t o r i e s , which have led Halpern [31] to propose a 0- assignment for the 6(962), or daughter t r a j e c t o r i e s [32]. For completeness we should perhaps also mention other suggestions on the 6, of 0- by Wu and Tuan [33], using group-theory, and by Logan et al. [34] who suggest it may be the p' introduced by F i s e h e r and HSgaasen [35].
HIGHER
MESONS
163
The author would like to thank Dr. B. Magli~ for provoking him to write this. He is grateful for discussions with Dr. Chan Hong-Mo and Dr. N. Cottingham and to several members of the Theoretical Study Division at CERN for encouragement.
REFERENCES [1] D . J . C r e n n e l l et al., P h y s . R e v . L e t t e r s 18 (1967) 323. [2] R. L. Cool et al., P h y s . R e v . L e t t e r s 17 (1966) 102. [3] D . C l i n e , Nuovo C i m e n t o 45A (1966) 750. [4] S . M i n a m i , Nuovo C i m e n t o 46A (1966) 544. [5] P . G . O . F r e u n d , P h y s . R e v . , to be p u b l i s h e d . [ 6 ] A . A h m a d z e d a h , Nuovo C i m e n t o 46A (1966) 415. [7] W . K i e n z l e et al., P h y s . L e t t e r s 19 (1965) 438. [81 L . V a n Hove, R a p p o r t e u r ' s t a l k at B e r k e l e y Conf. (1966). [9] R . C . A r n o l d , P h y s . R e v . L e t t e r s 14 (1965) 657. [10] V . B a r g e r and L . D u r a n d III, W i s c o n s i n p r e p r i n t (1967); V . B a r g e r and M . O l s s o n , P h y s . Rev. L e t t e r s 18 (1967) 294. [11] M . N . F o c a c c i et al., P h y s . R e v . L e t t e r s 17 (1966) 890; 1209 (E). [12] A . H . R o s e n f e l d et al., R e v s . Mod. P h y s . 39 (1967) 1. [ 13] D . R . O . M o r r i s o n , p r i v a t e c o m m u n i c a t i o n . [14] B . L e v r a t et al.. P h y s . L e t t e r s 22 (1966) 714. [15] K . K r u s e et al.. B e r k e l e y Conf. (1966) u n p u b l i s h e d . [16] A. F . G a r f i n k e l et al.. A m . P h y s . Soc. 12 (1967) 9. [17] R . H . D a l i t z . R a p p o r t e u r ' s t a l k at B e r k e l e y Conf. (1966). [18] M. L~vy, U n i v e r s i t y of P a r i s p r e p r i n t (1967). [19] D . G . S u t h e r l a n d , Nuovo C i m e n t o 46A (1966) 551. [20] M . D e u t s c h m a n n et al., P h y s . L e t t e r s 18 (1965) 351. [211 G. G o l d h a b e r , R a p p o r t e u r ' s t a l k at B e r k e l e y Conf. (1966). [22] L . D u r a n d III, P h y s . R e v . L e t t e r s 18 (1966) 58. [23] A . W e h m a n n et al., P h y s . Rev. L e t t e r s 17 (1966) 1113. [24] C . G i a c o m e l l i , s e m i n a r g i v e n at C E R N on w o r k p e r f o r m e d at B r o o k h a v e n (1967). [25] J . B l a t t and V . F . W e i s s k o p f , T h e o r e t i c a l n u c l e a r p h y s i c s ( J . W i l e y and Sons, New Y o r k , 1952). [26] W . M . L a y s o n , Nuovo C i m e n t o 20 (1961) 1207. [27] J . B a a r t k e et al., P h y s . L e t t e r s 24B (1967) 118. [28] J . B a r t s c h et al., P h y s . L e t t e r s 22 (1966) 357. [29] K. Z a l e w s k i , p r i v a t e c o m m u n i c a t i o n . [30] R . V a n d e r h a g e n et ai., E e o l e P o l y t e e h n i q u e p r e p r i n t , P a r i s (1967), s u b m i t t e d to Phys. Letters. [31] M . B . H a l p e r n , P r i n c e t o n p r e p r i n t (1967). [32] D . Z . F r e e d m a n a n d J . M . W a n g , P h y s . R e v . 153 (1967) 1596. [33] S . F . T u a n a n d T . T . W u , P h y s . R e v . L e t t e r s 18 (1967) 349. [ 3 4 ] R . K . Logan, J . B e a u p r e a n d L . S e r t o r i o , P h y s . Rev. L e t t e r s 18 (1967) 259. [35] W . F i s c h e r and H . H b g a a s e n , P h y s . L e t t e r s 22 (1966) 516. [361 V . N . G r i b o v , Int. Conf. h i g h e n e r g y p h y s i c s , B e r k e l e y , Calif. (1966). [37] F . E . L o w , B r a n d e i s L e c t u r e Notes, 1965 (Gordon and B r e a c h , 1966) p. 587.
Nuclear Physics B2 (1967) 157-163. North-Holland Publ, Comp.. A m st er d am
SOME REMARKS o N HIGHER MESONS D. G. S U T H E R L A N D CERN,
Gene~,a
Received 17 May 1967
Abstract: A modeI based on exchange degeneracy and linear t r a j e c t o r i e s is given to c o r r e l a t e presently known meson resonances with I = 1. A critical discussion is given of some features of this model. Some r e m a r k s are made on the problem of the widths of the higher mesons in this scheme and on some related problems in higher meson spectroscopy.
It i s the p u r p o s e of t h i s note to a t t e m p t a f i t of a l l k n o w n I = 1 m e s o n s on s t r a i g h t - l i n e R e g g e t r a j e c t o r i e s . S e v e r a l p a p e r s [ 3 - 6 ] h a v e a p p e a r e d in r e c e n t m o n t h s in w h i c h p a r t i a l a t t e m p t s to u s e l i n e a r R e g g e t r a j e c t o r i e s a r e m a d e , but none of t h e m h a v e r e p r o d u c e d t h e s p l i t t i n g in t h e R - r e g i o n s e e n in t h e m i s s i n g m a s s e x p e r i m e n t [7]. S i n c e t h e y a l s o n e g l e c t e d p e a k s s u c h as 5 (r ef. [7]) and A1 w h i c h h a v e b e e n r e p o r t e d at l o w e r m e s s e s , one m a y f e e l t h a t t h i s c o u l d be r e m e d i e d if t h e y w e r e i n c l u d e d . T h e s e c o n d p u r p o s e i s , w i t h i n th e c o n t e x t of the m o d e l , to d i s c u s s , in w h a t we hope is a s e m i - q u a n t i t a t i v e w a y , the p r o b l e m of n a r r o w w i d t h s , w h i c h h a s aroused some interest recently. T o s o m e e x t e n t we f e e l t h a t , l a c k i n g a d e t a i l e d d y n a m i c a l m o d e l , o u r c a l c u l a t i o n s a r e v e r y t r i v i a l . O u r a p o l o g y i s only t h a t s i m i l a r a p p r o a c h e s s h o w p r o m i s i n g r e s u l t s in the b a r y o n s p e c t r u m [8]. One is thus e n c o u r a g e d to hope t h a t t r y i n g it f o r m e s o n s m a y e i t h e r p r o d u c e g o o d r e s u l t s , o r by f a l l i n g b a d l y , l e a d one e i t h e r to s o m e d y n a m i c a l u n d e r s t a n d i n g [9], if s u c c e s c o n t i n u e s f o r _ t h e b a r y o n s , o r to l o o k m o r e s u s p i c i o u s l y at t h e t h e o r y f o r b a r y o n s , w h i c h so f a r is c o m p l e t e l y c h e c k e d in o n l y r a t h e r f e w c a s e s . A f u r t h e r hope is t h a t th e p r o p o s e d t r a j e c t o r i e s m a y be of i n t e r e s t to R e g g e t h e o r e t i c i a n s , p a r t i c u l a r l y f o r c o n s i d e r a t i o n of u n n a t u r a l p a r i t y e x c h a n g e . F i n a l l y t h e c o n s i d e r a t i o n s on t h e w i d t h s m a y , w i t h a l i t t l e a d d i t i o n a l e x p e r * Only for the di-pion resonance at 1650 MeV is there e v i d e n c e [ i ] that I = 1 , not 2. for the charged resonances above 1.5 GeV. It should be noted that 1.7GeV is a very natural point at which higher I resonances might occur as can be seen from the following piece of numerological argument. In the baryon system, evidence for a 1--gresonance has recently been given by Cool et al. at 1860 MeV [21 . If one conjectures that the mass tn M at which new "non quark-antiquark" mesons appear is related to the mass mB at which new "non t h r e e - q u a r k " baryons appear by rn2 to__2 - rn2 m2 we find m M = 1620 MeV. We shall a s s u m e I= 1 for all higher M - "'~T B p' bumps, however.
158
D.G. SUT HER LA ND
i m e n t a l i n f o r m a t i o n , l e a d one to new s e l e c t i o n r u l e s of s t r o n g i n t e r a c t i o n d y n a m i c s , in t h e w a y t h a t t h e f a i l u r e of ~ to d e c a y to p~ h a s done. O u r t h e o r e t i c a l i n p u t w i l l b e t h e c o n c e p t of " e x c h a n g e d e g e n e r a c y " , i n t r o d u c e d b y A r n o l d [9], s u p p l e m e n t e d b y s t r a i g h t - l i n e R e g g e t r a j e c t o r i e s [8]. A r n o l d ' s i d e a i s t h a t r e s o n a n c e s s h o u l d o c c u r at i n t e r v a l s of AJ= 1, not of AJ= 2 a s in t h e u s u a l s c h e m e . T h i s i s v e r y n a t u r a l on a n y F e r m i - Y a n g m o d e l in w h i c h t h e m e s o n s a r e b o u n d s t a t e s of b a r y o n s a n d a n t i b a r y o n s , s i n c e t h e n m e s o n e x c h a n g e in t h e t - c h a n n e l s h o u l d d o m i n a t e o v e r d o u b l e b a r y o n e x c h a n g e in t h e u - c h a n n e l , g i v i n g a p r e d o m i n a n t l y d i r e c t p o t e n t i a l only. We a r e a w a r e t h a t t h i s i s o n l y a r a t h e r r o u g h a p p r o x i m a t i o n [10], a n d we s h a l l d i s c u s s t h e e r r o r s i n v o l v e d f o r t h e c a s e of t h e P - A 2 t r a j e c t o r y for illustration. O u r e x p e r i m e n t a l i n p u t w i l l c o n s i s t of t h e l o w - l y i n g r e s o n a n c e s ~ ( 1 3 9 ) 0 - - , p ( 7 6 8 ) 1 - + , A 2 ( 1 2 8 6 ) 2 + - , A l ( 1 0 8 0 ) l + - , B(1208)1++ a n d 5(962)0 + - , w h e r e o u r n o t a t i o n i s ( m a s s in M e V ) jPG. We t a k e o u r p a r a m e t e r s f r o m t h e r e s u l t s p r e s e n t e d b y F o c a c c i e t al. [11], w h e n t h e y h a v e o b s e r v e d the p e a k , s i n c e t h e i r r e s u l t s f o r the h i g h e r p e a k s a r e t h e o n e s we s h a l l b e c o n c e r n e d w i t h m o s t , a n d f r o m the R o s e n f e l d t a b l e s [12] in o t h e r c a s e s . We w i s h to e m p h a s i z e t h a t f o r o n l y ~, p a n d A 2 a r e t h e s e s p i n - p a r i t y a s s i g n m e n t s r e a s o n a b l e w e l l c o n f i r m e d * , a n d in t h e c a s e of 5 we h a v e to r e l y e n t i r e l y on theoretical conjecture [17-19] F o r t h e t r a j e c t o r i e s c o n s i s t i n g of (5, A2) a n d of (~, B) t r e a t m e n t s h a v e b e e n g i v e n p r e v i o u s l y [4, 5, 6, 11]. F o r t h e A1 a n d 5 t r a j e c t o r i e s , we h a v e no p a r t n e r s f r o m w h i c h to d e r i v e t h e t r a j e c t o r i e s . W e t h e n n e e d t h e f o l l o w i n g i n f o r m a t i o n : D e u t s c h m a n n et al. [20] h a v e p r e s e n t e d e v i d e n c e of a ~+ ~o r e s o n a n c e at 1910 M e V and, w h i l e G o l d h a b e r [21] a p p a r e n t l y f e e l s t h i s r e s o n a n c e to b e a s y e t u n c o n f i r m e d , we f e e l t h a t i t s s t a t i s t i c a l s i g n i f i c a n c e of f i v e s t a n d a r d d e v i a t i o n s , s h o u l d , f o r c o n s i s t e n c y w i t h t h e c r i t e r i a of r e f . [11], l e a d u s to t a k e it s e r i o u s l y . T h e m o s t n a t u r a l c h o i c e is t h e n 3 -+ if it i s to b e a s s o c i a t e d w i t h t h e 5 in a R e g g e t r a j e c t o r y . A s s e e n f r o m o u r p r e s e n t a t i o n of the r e s u l t s in t a b l e 1, we s t i l l h a v e not a c a n d i d a t e f o r R3(1748) (ref. [11]). F o r t h i s we p r o p o s e t h a t it s h o u l d b e a 2 -+ s t a t e , a s s o c i a t e d with t h e A1. T h i s we d e s c r i b e a s t r a j e c t o r y ( A 1 , R 3 ) I ; an a l t e r n a t i v e p r o p o s a l i s t h a t R 3 i s a 3 + - s t a t e [6]; t h i s we d e s c r i b e a s t r a j e c t o r y ( A 1 , R 3 ) 2 . S i n c e t h i s p r e d i c t s a 2 -+ s t a t e a t 1 4 5 0 M e V a n d , m o r e s e r i o u s l y , a 0-+ s t a t e a t 460 M e V , we a r e i n c l i n e d to f a v o u r t h e first assignment. L e t u s now m a k e s o m e g e n e r a l r e m a r k s a b o u t o u r s c h e m e . F i r s t of a l l , l e t u s m e n t i o n t h a t t h e v a l u e of 0.47 f o u n d f o r a (p, A 2 ) ( 0 ) a g r e e s n e i t h e r w i t h t h e v a l u e of 0.58 f o r t h e p - t r a j e c t o r y , n o r w i t h t h a t of 0.35 found f o r t h e A2 t r a j e c t o r y [10]. If we w e r e to c h o o s e C~p(0) = 0.58 a n d o l ~ 0 . 5 8 ) = 1, we w o u l d f i n d a 3-+ s t a t e a t 1820 M e V , not at 1650 MeV. H e n c e one m u s t a l l o w s o m e * Even for the A2 there have recently been some w o r r i e s about the spin parity a s signment [13] and about the possibility of two peaks in the 37T system [14-16]. ** The argument given in ref. [19] on the small rate depends on the existence of only Regge poles. Regge cuts of negative parity could spoil it (see ref. [36]): m e a s u r e m e n t of 5 production, if it is r e a l l y 0 +-, could be interesting as a test for Regge cuts.
HIGtlER MESONS
159
Table 1 Resonances and trajectories. Trajectory
(p, a2)
(Tr, B)
(5,S)
(A1.R3) 1
(A1,R3) 2
p(760)1 -+
g(159)0--
5(962)0 +-
A1(1080)1 +-
Al(1080)l e-
A2(1286)2~-
B(1210)1 +F
1355 1.+
R3(1748)2 -+
1450 2 -~
R1(1650)3 -+
R2(1705)2--
1652 2+.
2220 3 ÷-
R3(1748)3 +-
S(1950)4 e-
2090 3 ÷+ S(1910)3 -+
2610 4 -+
2000 4 .+
T(2206)5 -~
2410 4 - -
2140 4 ÷-
2220 5 ~-
U(2440)6 e-
2700 5 ~+
2350 5 -+
2430 6 -+
V(2650) 7-+
2540 6+-
~(0)
0.47
-0.01
-1.00
0.39
-0.23
Or' (GeV/c) -2
0.92
0.69
1.08
0.53
1.05
c u r v a t u r e in the t r a j e c t o r y . T h e b a r y o n t r a j e c t o r i e s , on the o t h e r hand, s e e m to fit r a t h e r w e l l with l i n e a r t r a j e c t o r i e s , at l e a s t f o r m 2 > 1 GeV 2. A f u r t h e r r e m a r k is that the p o s i t i o n s of the h i g h e r r e s o n a n c e s a r e v e r y s e n s i t i v e to t h o s e of the input. T h u s if ruA2 is c h o s e n a s 1305 MeV [12] i n s t e a d of 1286 MeV [11], t h e n we find ru T = 2260, not 2206, MeV. A f u r t h e r f e a t u r e of o u r s c h e m e w h i c h we find s o m e w h a t d i s q u i e t i n g is the fact t h a t the v a l u e s of c~' w h i c h we find f o r the u n n a t u r a l p a r i t y m e s o n t r a j e c t o r i e s d i f f e r s s u b s t a n t i a l l y f r o m t h o s e of the n a t u r a l p a r i t y m e s o n t r a j e c t o r i e s , w h e r e a s for the b a r y o n t r a j e c t o r i e s the v a l u e s of c~' a g r e e w i t h i n 10% for a l l t h r e e e s t a b l i s h e d t r a j e c t o r i e s . A f u r t h e r p o i n t is t h a t ~ ( A 1 - R 3) (0) is 0.39 f o r o u r f i r s t s o l u t i o n . T h i s m a y be i n t e r e s t i n g f o r e x p l a i n i n g r e a c t i o n s i n v o l v i n g u n n a t u r a l p a r i t y e x c h a n g e , b u t c o n t r a d i c t s , on the o t h e r h a n d , a s u g g e s t i o n by D u r a n d [22] that np c h a r g e e x c h a n g e m a y be e x p l a i n e d by h a v i n g ~A1 ( 0 ) < 0. F r o m t a b l e 1 it will be s e e n that we have r e s o n a n c e s c l o s e to the o b s e r v e d p e a k s at S, T , U a n d a l l t h r e e p e a k s in the R - r e g i o n . I n a d d i t i o n we h a v e a f o u r t h p e a k in the R - r e g i o n at 1652 MeV with j P G = 2+-. T h i s is r a t h e r g r a t i f y i n g in view of the r e p o r t e d e x i s t e n c e of a 3~ r e s o n a n c e at 1640 MeV [21]. Of c o u r s e , we a l s o p r e d i c t o t h e r r e s o n a n c e s , at 2090 MeV a n d at 1355 MeV w h i c h have not y e t b e e n o b s e r v e d . A l t h o u g h one c a n a r g u e t h a t the f o r m e r m a y have e s c a p e d d e t e c t i o n b e c a u s e of a l a r g e width, the f a i l u r e to d e t e c t the s e c o n d in the e x p e r i m e n t of W e h m a n n et al. [23] is r a t h e r d i s t u r b i n g . We s h o u l d a l s o p o i n t out that two r e s o n a n c e s a r e p r e d i c t e d in the v i c i n i t y of S, t h r e e in the v i c i n i t y of T , a n d t h r e e in the v i c i n ity of U. T h i s m a y be r e l e v a n t for the o b s e r v a t i o n in ~p (total c r o s s - s e c t i o n s ) of r e s o n a n c e s n e a r the T a n d U of ref. [11], b u t with m u c h l a r g e r w i d t h s of a b o u t 90 MeV [24]. C l e a r l y o u r s c h e m e owes s o m e t h i n g to the q u a r k m o d e l [17]. T h o u g h the i d e a of e x c h a n g e d e g e n e r a c y d e p e n d s o n l y on a b a r y o n - a n t i b a r y o n m o d e l f o r the m e s o n s , the r e s t r i c t i o n to only s i n g l e t a n d o c t e t r e p r e s e n t a t i o n s a n d to
160
D.G. SUT HER LA ND
the spin values of the trajectory we employ are harder to justify if nucleons and nucleon resonances and their antiparticles are the relevant baryons. Our reason for trying straight-line trajectories, rather than the more concrete model of ref. [17], is simply that the observation by Deutschmann et al. [20] of a 1910 MeV resonance decaying to ~+vo is incompatible with that scheme, so that some modification, at least, is required, and the only guide available is the success of the baryon trajectory scheme. We now discuss the problem of the widths of these mesons. This has seemed rather hard to understand since the widths of all the resonances seen in the missing mass spectrometer above 1.5 GeV are ~ 38 MeV. This contradicts a general feeling that the widths of resonances should increase with their mass. Indeed the baryons do seem to show some slow increase with mass, at least as fas as present analysis is concerned. We shall try to estimate how serious this problem is within our framework. It should be borne in mind that bubble chamber data [21, i] on peaks in the R-region suggest much larger widths. If these resonances are the same as those seen by Focacci et al [11], we feel that the higher statistics and better resolution of the latter experiment should lead us to take their narrow width seriously. Our discussion will be based on a very simple expression for the partial width F l for the decay of a meson of mass M into a quasi-two-body state with relative orbital angular momentum l,
Fl :yq--
(qR) 2I
rn [ 1 . 3 . 5 . . .
(2/-1)] 2 '
w h e r e q i s the m o m e n t u m a v a i l a b l e in the d e c a y , R is s o m e " r a d i u s of i n t e r a c t i o n " a n d Y is the r e d u c e d width, w h i c h is a p r i o r i u n l i k e l y to v a r y w i d e l y f r o m one r e s o n a n c e to the n e x t * . We s h a l l show that f o r the h i g h e r m e s o n r e s o n a n c e s t h i s f o r m u l a l e a d s , on n o r m a l i z i n g y to the width of the p, to a r e a s o n a b l e e x p l a n a t i o n of the w i d t h s . F o r R we t a k e ~r¢ M e V - 1 , s i n c e t h i s is the m a s s a p p e a r i n g in the e l e c t r o m a g n e t i c f o r m f a c t o r s . Such a c h o i c e s e e m s r e a s o n a b l e , a s it g i v e s a width of 100 MeV f o r foIf we t a k e the U to b e 5-+, for i n s t a n c e , we find t h a t a r e a s o n a b l e e s t i m a t e of its w i d t h s is 60 MeV, if we k e e p a l l q u a s i - t w o - b o d y s t a t e s i m p l i e d by o u r m o d e l . One m a y , of c o u r s e , a n t i c i p a t e that t h e r e w i l l be s o m e f u r t h e r c o n t r i b u t i o n to the w i d t h f r o m m a n y p a r t i c l e s t a t e s , b u t p r e s u m a b l y it s h o u l d be of the s a m e o r d e r a s that f r o m the q u a s i - t w o - b o d y s t a t e s . M o s t of the c o n t r i b u t i o n to o u r e s t i m a t e c o m e s f r o m the d e c a y into p + fo or ¢o + A 2 in d - w a v e , a n d if we w e r e to d e c r e a s e the v a l u e of 7 for t h e s e m o d e s by 2, a g r e e m e n t with ref. [11] w o u l d be o b t a i n e d . S i n c e ~ c a n c e r t a i n l y b e e x p e c t e d to v a r y by 10 f r o m one r e s o n a n c e to a n o t h e r w i t h o u t i m p l y i n g a n y * We are unable to give any really satisfactory justification for this paramctrization, which is cssentially a "relativized" version of the formula given in Chapter VIII of Blatt and Weisskopf [25]. A similar expression was suggested by Layson [26] for baryon widths. This formula employs nothing much more than the idea that strong forces are of short range, which is almost the only thing we know. Strictly speaking, the formula given here is only correct if qR is small, but the correction terms tend to decrease F l and for our purposes are not important.
HIGHER MESONS
161
p a r t i c u l a r s e l e c t i o n r u l e (cf., t h e A 2 ~ K K w i t h fo ~ ) , it i s c l e a r t h a t we cannot conclude that anything very surprising is happening. T h e r e a s o n why t h e w i d t h d o e s not i n c r e a s e v e r y r a p i d l y in t h i s s c h e m e is that the spacing between resonances is decreasing rapidly while the spin is increasing rapidly. There is thus rather little phase space available for d e c a y into n e a r b y s t a t e s w i t h low o r b i t a l a n g u l a r m o m e n t u m w h i l e the s t a t e s w i t h l a r g e p h a s e s p a c e h a v e h i g h a n g u l a r m o m e n t u m b a r r i e r s . It i s v e r y i n t e r e s t i n g h e r e to r e m a r k t h e r e c e n t o b s e r v a t i o n by B a r t k e et al. [27] of a r e s o n a n c e at 3690 M e V w i t h F = 50=~ 30 M e V in N+7~, w h i c h l e n d s s u p p o r t to t h i s p i c t u r e f o r b a r y o n s a l s o . In the r e g i o n of S s i m i l a r e s t i m a t e s show t h a t S ( I 9 5 0 ) 4 + - s h o u l d h a v e a n a r r o w w i d t h c o m p a t i b l e w i t h the l i m i t of 35 M e V [11]. F o r S(1910)3 -+, h o w e v e r , an a p p r e c i a b l y l a r g e r w i d t h of a b o u t 150 M e V , w i t h ~ , ~A, a n d pp e a c h h a v i n g c o m p a r a b l e p r o b a b i l i t y i s e x p e c t e d , w h i c h m a y w e l l i m p l y t h a t t h e r e s o n a n c e o b s e r v e d b y F o c a c c i et al. i s the 4+- s t a t e . So f a r o u r r e s u l t s a r e in f a i r a c c o r d w i t h e x p e r i m e n t e v e n if we do not c h a n g e y a p p r e c i a b l y f r o m t h e v a l u e f o u n d f o r t h e p. R e a s o n a b l e r e s u l t s a l s o e n s u e f o r t h e R1(1650)3 -+ f o r w h i c h we a n t i c i p a t e a w i d t h of 15 M e V f o r ~ , a l t h o u g h it a p p e a r s t h a t p+p s h o u l d h a v e a w i d t h of 40 M e V s i n c e it o c c u r s in a p - w a v e . F o r t h e 2 + - s t a t e p r o p o s e d at 1650 M e V we a n t i c i p a t e c o m p a r a b l e w i d t h s of a b o u t 60 M e V e a c h f o r ~fo, ~P a n d ~B. T h i s i s in f a i r a c c o r d w i t h t h e t o t a l w i d t h of 100 M e V q u o t e d in r e f . [21], b u t d i s a g r e e s s o m e w h a t w i t h t h e p r e l i m i n a r y e v i d e n c e g i v e n t h e r e t h a t ~fo d o m i n a t e s o v e r ~p. H o w e v e r , if we c o n s i d e r the 2 -+ a n d 2+- s t a t e s in t h e R - r e g i o n , we f i n d t h a t , k e e p i n g y a t i t s v a l u e d e d u c e d f r o m p, we w o u l d e x p e c t t h a t e a c h r e s o n a n c e s h o u l d h a v e a w i d t h of a b o u t 200 M e V f o r e a c h of t h e m o d e s 0 - + 1 - in a p w a v e a n d 0-+2 + in an s w a v e . T h i s i n d i c a t e s t h a t )~ s h o u l d b e ~ of t h e v a l u e of y d e d u c e d f r o m the p if we i d e n t i f y t h e s e r e s o n a n c e s w i t h t h e R 2 a n d R 3 of r e f . [11]. L e t u s c o m p a r e o u r r e s u l t s on w i d t h s w i t h t h a t of p r e v i o u s t r e a t m e n t s . F r e u n d [5] s u g g e s t s t h a t a s y m p t o t i c a l l y the p a r t i a l w i d t h t o two p s e u d o s c a l a r s , f o r i n s t a n c e , s h o u l d t e n d to i n f i n i t y a s 2 2 J J ( J - ~ ) ; t h i s r e s u l t a p p e a r s to d i s a g r e e w i t h o u r s i m p l e m o d e l , b u t it s h o u l d b e n o t e d t h a t F r e u n d ' s w i d t h s a r e p r o p o r t i o n a l to (t3(E))2, w h e r e ~(t) i s t h e r e d u c e d r e s i due f u n c t i o n , w h i c h he, s o m e w h a t a r b i t r a r i l y in o u r v i e w , s e t s c o n s t a n t a b o v e t=~np2. It s e e m s t h a t no r e l i a b l e e s t i m a t e s of [3(t) a t high t - v a l u e s a r e a v a i l a b l e but in v i e w of i t s r a p i d v a r i a t i o n at l o w e r t - v a l u e s we f e e l it m a y w e l l d e c r e a s e r a p i d l y at l a r g e I. T h i s w o u l d b e r e f l e c t e d in a L a g r a n g i a n t h e o r y of t h e d e c a y of t h e s e h i g h e r m e s o n s b y t h e f a c t t h a t t h e c o u p l i n g c o n s t a n t s h o u l d d e c r e a s e r a p i d l y w i t h i n c r e a s i n g r e s o n a n c e m a s s , s o a s to i n c o r p o r a t e t h e c e n t r i f u g a l b a r r i e r e f f e c t f o u n d in a b o u n d a r y c o n d i t i o n m o d e l of t h e s e m e s o n s . A p r o p o s a l c l o s e l y r e l a t e d to o u r s w a s m a d e i n d e p e n d e n t l y b y D a l i t z [17] a t t h e B e r k e l e y C o n f e r e n c e . He p r o p o s e d u s i n g a s m a l l e r r a d i u s t h a n t h a t u s e d h e r e , b u t d i d not m a k e n u m e r i c a l c o m p a r i s o n s in t h e R - r e g i o n . It i s c l e a r f r o m o u r d i s c u s s i o n t h a t no v e r y s t r o n g o v e r l a p f u n c t i o n e f f e c t [17] between resonances with widely different spin is necessary, but that the l a r g e s t s u p p r e s s i o n of y s e e m s n e c e s s a r y f o r t h e d e c a y s of 2 - - s t a t e s i n t o
162
D.G. SUTHERLAND
2++0 - a n d 0 - + 1 - s t a t e s . T h i s e f f e c t w i l l only b e m a d e w o r s e by d e c r e a s i n g the r a d i u s of i n t e r a c t i o n and, s i n c e no s t r o n g effect is o b s e r v e d in the d e c a y of 2 + ~ 0 - + 0 - s u c h a s fo ~ 7rTr, the j u m p in s p i n it not e n o u g h to m a k e u s f e e l t h a t t h i s s h o u l d b e a r e a l l y l a r g e effect. A f a c t o r of 10 s u p p r e s s i o n i s not y e t e n o u g h , h o w e v e r , to m a k e u s f e e l that a n y t h i n g r e a l l y r e m a r k a b l e i s h a p p e n i n g , s u c h a s o c c u r r e d in the s u p p r e s s i o n of ~o ~pTr w h e r e a f a c t o r of 100 s u p p r e s s i o n w a s i n v o l v e d ~ . It s e e m s that an i m p r o v e d b o u n d m u s t be s e t on the p a r t i a l w i d t h s of t h e s e r e s o n a n c e s , e i t h e r by i m p r o v e m e n t of the t o t a l w i d t h m e a s u r e m e n t , o r , p e r h a p s m o r e p r o b a b l y , by f i n d i n g t h a t the r e l e v a n t m o d e s a r e only a s m a l l f r a c t i o n of the t o t a l width, b e f o r e a n y t h i n g r e a l l y s u r p r i s i n g a n d s i g n i f i c a n t c a n be c o n c l u d e d . We m a k e s o m e b r i e f r e m a r k s on two o t h e r p r o b l e m s . T h e f i r s t c o n c e r n s the ~p d a t a of G i a c o m e l l i et al. F o r p e a k s w h i c h t h e y o b s e r v e in the T - a n d U - r e g i o n : G i a c o m e U i et al. d e t e r m i n e (J -t-1~)FNN ~ 20 MeV, w h e r e J is the s p i n of the r e s o n a n c e . T a k i n g f o r U, f o r i n s t a n c e , the m o s t c o n s e r v a t i v e s p i n a n d o r b i t a l a n g u l a r m o m e n t u m a v a i l a b l e in o u r s c h e m e , n a m e l y J = 4, I = 4, we d e d u c e YNN = 10yp. T h i s m a y l e n d s o m e s u p p o r t to a F e r m i - Y a n g m o d e l of the m e s o n s . O u r s e c o n d r e m a r k is on the L - m e s o n [28]. Z a l e w s k i [29] i n f o r m s u s t h a t a n a t t e m p t to fit the b r a n c h i n g r a t i o s of t h i s r e s o n a n c e l e a d s to s t r o n g d i s a g r e e m e n t with SU(3). T h i s m a y m e a n t h a t , j u s t a s in the c a s e of the R - r e g i o n , w h i c h f i r s t e s t i m a t e s f r o m the m i s s i n g m a s s e x p e r i m e n t g a v e a s a b r o a d p e a k of w i d t h ~ 100 MeV, s e v e r a l n a r r o w e r r e s o n a n c e s , i n c l u d i n g the 3 - K* s u g g e s t e d by A h m a d z e d a h [6], w i l l be found in t h i s r e g i o n . A n a l y s i s of s p i n a n d p a r i t y a s s u m i n g o n l y one r e s o n a n c e w i l l t h e n give misleading results. F i n a l l y , l e t u s s t a t e that we dot not b e l i e v e that t h i s m o d e l w i l l give a c o m p l e t e a n s w e r to the m e s o n s p e c t r u m . We p r e s e n t it only a s a m o d e l to show what m a y be l e a r n t f r o m c u r r e n t e x p e r i m e n t e v i d e n c e a n d to show s i m p l e e f f e c t s s u c h a s the e a s e with w h i c h s e v e r a l r e s o n a n c e s h a v i n g m a s s e s v e r y c l o s e t o g e t h e r c a n a r i s e , a n d t h a t t h e r e is not y e t a n y t h i n g d e c i s i v e l y d i s t u r b i n g a b o u t the n a r r o w w i d t h s o b s e r v e d . I t s s u c c e s s o r , in the c a s e of its f a i l u r e , the i m p l i c a t i o n s f o r the b a r y o n s , we l e a v e to the r e a d e r to j u d g e ~.
] It is d e a r that we are not committing the first basic e r r o r of strong coupling theo r y d i s c u s s e d by Low (ref.[37]), and we feel that to take factors of 10too s e r i ously would be to commit the second. The p e s s i m i s m of this last paragraph has recently received support from the appearance of a paper by Vanderhagen et al. [30], in which ap-Tr- resonance is r e ported at 1320 MeV. (The flexibility of numerology is well illustrated by changing m to the average mass of the pseudoscalars, ~ 400 MeV, and using a linear instead of a quadratic relation in the first footnote.) It should also be pointed out that we have taken a rather naive approach to Regge r e c u r r e n c e s and have not worried about possible conspiration t r a j e c t o r i e s , which have led Halpern [31] to propose a 0- assignment for the 6(962), or daughter t r a j e c t o r i e s [32]. For completeness we should perhaps also mention other suggestions on the 6, of 0- by Wu and Tuan [33], using group-theory, and by Logan et al. [34] who suggest it may be the p' introduced by F i s e h e r and HSgaasen [35].
HIGHER
MESONS
163
The author would like to thank Dr. B. Magli~ for provoking him to write this. He is grateful for discussions with Dr. Chan Hong-Mo and Dr. N. Cottingham and to several members of the Theoretical Study Division at CERN for encouragement.
REFERENCES [1] D . J . C r e n n e l l et al., P h y s . R e v . L e t t e r s 18 (1967) 323. [2] R. L. Cool et al., P h y s . R e v . L e t t e r s 17 (1966) 102. [3] D . C l i n e , Nuovo C i m e n t o 45A (1966) 750. [4] S . M i n a m i , Nuovo C i m e n t o 46A (1966) 544. [5] P . G . O . F r e u n d , P h y s . R e v . , to be p u b l i s h e d . [ 6 ] A . A h m a d z e d a h , Nuovo C i m e n t o 46A (1966) 415. [7] W . K i e n z l e et al., P h y s . L e t t e r s 19 (1965) 438. [81 L . V a n Hove, R a p p o r t e u r ' s t a l k at B e r k e l e y Conf. (1966). [9] R . C . A r n o l d , P h y s . R e v . L e t t e r s 14 (1965) 657. [10] V . B a r g e r and L . D u r a n d III, W i s c o n s i n p r e p r i n t (1967); V . B a r g e r and M . O l s s o n , P h y s . Rev. L e t t e r s 18 (1967) 294. [11] M . N . F o c a c c i et al., P h y s . R e v . L e t t e r s 17 (1966) 890; 1209 (E). [12] A . H . R o s e n f e l d et al., R e v s . Mod. P h y s . 39 (1967) 1. [ 13] D . R . O . M o r r i s o n , p r i v a t e c o m m u n i c a t i o n . [14] B . L e v r a t et al.. P h y s . L e t t e r s 22 (1966) 714. [15] K . K r u s e et al.. B e r k e l e y Conf. (1966) u n p u b l i s h e d . [16] A. F . G a r f i n k e l et al.. A m . P h y s . Soc. 12 (1967) 9. [17] R . H . D a l i t z . R a p p o r t e u r ' s t a l k at B e r k e l e y Conf. (1966). [18] M. L~vy, U n i v e r s i t y of P a r i s p r e p r i n t (1967). [19] D . G . S u t h e r l a n d , Nuovo C i m e n t o 46A (1966) 551. [20] M . D e u t s c h m a n n et al., P h y s . L e t t e r s 18 (1965) 351. [211 G. G o l d h a b e r , R a p p o r t e u r ' s t a l k at B e r k e l e y Conf. (1966). [22] L . D u r a n d III, P h y s . R e v . L e t t e r s 18 (1966) 58. [23] A . W e h m a n n et al., P h y s . Rev. L e t t e r s 17 (1966) 1113. [24] C . G i a c o m e l l i , s e m i n a r g i v e n at C E R N on w o r k p e r f o r m e d at B r o o k h a v e n (1967). [25] J . B l a t t and V . F . W e i s s k o p f , T h e o r e t i c a l n u c l e a r p h y s i c s ( J . W i l e y and Sons, New Y o r k , 1952). [26] W . M . L a y s o n , Nuovo C i m e n t o 20 (1961) 1207. [27] J . B a a r t k e et al., P h y s . L e t t e r s 24B (1967) 118. [28] J . B a r t s c h et al., P h y s . L e t t e r s 22 (1966) 357. [29] K. Z a l e w s k i , p r i v a t e c o m m u n i c a t i o n . [30] R . V a n d e r h a g e n et ai., E e o l e P o l y t e e h n i q u e p r e p r i n t , P a r i s (1967), s u b m i t t e d to Phys. Letters. [31] M . B . H a l p e r n , P r i n c e t o n p r e p r i n t (1967). [32] D . Z . F r e e d m a n a n d J . M . W a n g , P h y s . R e v . 153 (1967) 1596. [33] S . F . T u a n a n d T . T . W u , P h y s . R e v . L e t t e r s 18 (1967) 349. [ 3 4 ] R . K . Logan, J . B e a u p r e a n d L . S e r t o r i o , P h y s . Rev. L e t t e r s 18 (1967) 259. [35] W . F i s c h e r and H . H b g a a s e n , P h y s . L e t t e r s 22 (1966) 516. [361 V . N . G r i b o v , Int. Conf. h i g h e n e r g y p h y s i c s , B e r k e l e y , Calif. (1966). [37] F . E . L o w , B r a n d e i s L e c t u r e Notes, 1965 (Gordon and B r e a c h , 1966) p. 587.