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Simple dynamical model for baryon resonances
V o l u m e 22, n u m b e r 5
SIMPLE
PHYSICS
DYNAMICAL
MODEL A. A. SLAVNOV
LETTERS
FOR
15 S e p t e m b e r 1966
BARYON
RESONANCES
and O. I. ZAVIALOV
Stehlov Mathematical Institute, Acad. Sci. Moscow, USSR Received 18 August 1966 The parameters of a number of baryon resonances are expressed in terms of few constants in the framework of the quark model.
Many p r o p e r t i e s of b a r y o n s w e r e s u c c e s s f u l l y e xpl a ine d in the f r a m e w o r k of the q u a r k m o d e l [1-3]. In the p r e s e n t p a p e r we p r o p o s e the s i m p l e d y n a m i c a l s c h e m e f o r the d e s c r i p t i o n of r e s o n a n c e s in m e s o n - b a r y o n s y s t e m s . T h i s s c h e m e e n a b l e s us to e x p r e s s m a s s e s , widths, i s o s p i n s and h y p e r c h a r g e s of a n u m b e r of b a r y o n r e s o n a n c e s in t e r m s of few c o n s t a n t s c h a r a c t e r i s i n g the q u a r k - m e s o n i n t e r a c t i o n . L e t us c o n s i d e r r e s o n a n c e s t a t e s in the m e s o n (octet 0 - ) - b a r y o n (octet ½+) s y s t e m s . The b a r y o n s a r e supposed to c o n s i s t of t h r e e q u a r k s , and the m e s o n s a r e t r e a t e d a s e l e m e n t a r y (i.e. the q u a r k s t r u c t u r e of m e s o n s is u n i m p o r t a n t f o r our p u r p o s e s ) . We a s s u m e that the m e s o n - b a r y o n f o r w a r d s c a t t e r i n g a m p l i t u d e can be e x p r e s s e d as follows R - h R 1 +fiR 2 + ~ R 3 + R
(1)
w h e r e R 1,2,3 a r e the s c a t t e r i n g a m p l i t u d e s f o r individual q u a r k s and 5,/3, ? a r e n o n z e r o c o e f f i c i e n t s , depending on b a r y o n - t a r g e t p r o p e r t i e s , /~ i s a function, d e s c r i b i n g the collectLve i n t e r a c t i o n of q u a r k s with an incident m e s o n . Due to the optical t h e o r e m the s a m e f o r m u l a is v a l i d f o r to ta l c r o s s s e c t i o n s . A s i m i l a r h y p o t h e s i s (in f a c t e v e n a m u c h b o l d e r one) has been a l r e a d y u s e d by s e v e r a l a u t h o r s [1-2]. Our b a s i c i d e a i s that if t h e r e is a r e s o n a n c e in the m e s o n - q u a r k s y s t e m then it will i n e v i t a b l y be p r e s e n t in the m e s o n s c a t t e r i n g by all b a r y o n s containing this quark. T h e r e f o r e by fL'~ing the p a r a m e t e r s of the m e s o n - q u a r k r e s o n a n c e , one can obtain a n u m b e r of p r e d i c t i o n s c o n c e r n i n g h y p e r c h a r g e s , i s o s p i n s , m a s s e s and widths of meson-baryon resonances. The m e s o n - q u a r k s c a t t e r i n g a m p l i t u d e c o n t a i n s t h r e e i n v a r i a n t a m p l i t u d e s (we m e a n unit a r y s t r u c t u r e only). 3 ® 8 = 3 + 6 " + 15 . 686
(2)
F o r r e a s o n s which will not be d i s c u s s e d in this p a p e r the r e s o n a n c e should be p r e s e n t in the a m plitude R(3). In the following we shall say a few w o r d s about the p o s s i b l e m e c h a n i s m of such a resonance. The m a s s of the quark is supposed to be l a r g e , so it looks r e a s o n a b l e to use the C h e w - L o w m o d el f o r the d e s c r i p t i o n of m e s o n - q u a r k s c a t t e r i n g . F o r e x a m p l e we can obtain the solution of the Low equation f o r the am p l i t u d e R(3) in a s t a n d a r d way. R(33)(z) - -
1-
z~33 T[
~3/~
[
d z ' p '3
~
- ~z f w
)
dz'p'3 B(z')
~ z' 2 (z +z')
(3) (The f i r s t index r e f e r s to u n i t a r y s t r u c t u r e , the second one is the spin index.) H e r e z is the m e s o n e n e r g y in l . s . , P i s the m e s o n m o m e n t u m , is the cut-off, )t33 is the e f f e c t i v e coupling constant in this channel. B(z') is an unknown function f i x ed by c r o s s - s y m m e t r y conditions. F o r z << w one can use the s o - c a l l e d " e f f e c t i v e r a d i u s app r o x i m a t i o n " . ( P r a c t i c a l l y the s a m e r e s u l t s a r e obtained by s i m p l y n e g l e c t i n g the left hand cut.) Then f o r z << w we w r i t e down the following ex p r e s s i o n f o r the phase shift ctg 5 =
(1 - rz)z ;t33(z2 _ ~2)~
(4)
r = ~33w/Tr is "the e f f e c t i v e r a d i u s of i n t e r a c t i o n " . When the p h ase shift p a s s e s through ½~, i.e. if 433 > 0, the r e s o n a n c e a p p e a r s . The width of the r e s o n a n c e is given by the f o r m u l a 3
F =
2~(1 - r 2 p 2 ) ~
(5)
r 3 + (rX/~) In (2nr/;~U) I t is not difficult to w r i t e down the m o d e l H a m i l tonian giving a p o s i t i v e ;t33 in a b r o k e n SU 3 theory.
Volume 22, number 5
PHYSICS LETTERS
15 September 1966
Table 1 Theory
Experiment [4]
Reaction
Isospin
Mass(GeV2) (squared)
Mass (MeV)
Width
~+P ~+~o
~ 1
1.47 2.00
1210 1410
114 46
~+~-
½
2.34
1530
30
Isospin
Mass (squared) (GeV2)
Mass (MeV)
N~(1236) Y~(1385)
a 1
1.53 1.91
1236 1383
120 ± 1.5 44 ~-5
~*(1530)
½
2.34
1530
7.5±1.7
Symbol
Width (MeV)
Table 2
Isospin
Theory Mass(GeV2) (squared)
Mass (MeV)
Width
K-N
1
2.76
1660
45
K-~ ° K-~ °
½ 0
3.29 3.63
1816 1900
13 <10
Reaction
Experiment [4] Symbol Isospin
Mass (squared) (GeV2)
(MeV)
Mass
Width (MeV)
Y~(1660)
I
2.76
1660
44 ~-5
~*(1816) ?
½ ?
3.29 ?
1816 ?
16 ?
Table 3
Reaction y
Isospin
Theory Mass(GeV2) (squared)
½
2.90
Experiment [4] Mass Width "(MeV) (MeV) Symbol Isospin 1695
128
N_~*(1688)
Mass (squared) (GeV2)
Mass (MeV)
Width '(MeV)
½
2.85
1688
125
2
~A
0
3.26
1810
53
Yo(1815)
0
3.29
1815
50
?
?
?
?
?
½
3.74
1933
140 ± 35
y~+
1
3.43
1850
37
~o
!2
3.77
1940
16
(The b r e a k i n g t e r m t r a n s f o r m s as the eighth component of an o c t e t . ) Thus the a s s u m p t i o n about the e x i s t e n c e of a r e s o n a n c e in the R(3) a m p l i t u d e s e e m s r a t h e r n a t u r a l . This m e a n s that for a ~ l - m e s o n - q u a r k s y s t e m the r e s o n a n c e o c c u r s in the i s o s p i n I=½ state, and for a K - m e s o n - q u a r k s y s t e m in the i s o s p i n I= 0 state. The p o s i t i o n of the r e s o n a n c e is determineel by the effective r a d i u s , Lhe width i s fixed by the f o r m u l a (5). The effective r a d i u s of a bound q u a r k n a t u r a l l y d i f f e r s f r o m that of a f r e e quark, and roughly speaking depends on the d i m e n s i o n s of the c o m posite p a r t i c l e . So we c l a i m that r = aM .
(6)
W h e r e M i s the m a s s of the c o r r e s p o n d i n g b a r y o n and ~ is s o m e e m p i r i c a l coefficient. This m e a n s that in our model a i s the u n i v e r s a l c o n s t a n t c h a r a c t e r i z i n g the i n t e r a c t i o n of the m e s o n with all b a r y o n s . It i s c l e a r a l s o that for K and y - m e -
2"(1933)
sons the coefficients ~ a r e different, and t h e i r r a t i o should be a p p r o x i m a t e l y equal to the r a t i o of the c o r r e s p o n d i n g Compton wave lengths. ~r~
~ ~K/IK ~ ~ P y
•
(7)
Now let us c o n s i d e r r e s o n a n c e s in the ~ m e s o n b a r y o n s y s t e m . We have a l r e a d y m e n t i o n e d that for our s y s t e m a r e s o n a n c e o c c u r s in s t a t e s with I = ~, 1 i.e. in the s y s t e m s lr+n, r ' p , ~Op, ~On. Starting f r o m here we can at once d e t e r m i n e the highest p r o j e c t i o n of the i s o s p i n of the c o r r e s p o n d ing m e s o n - b a r y o n r e s o n a n c e . F o r e x a m p l e ~+ does not c o n t a i n the n - q u a r k , and t h e r e f o r e t h e r e is no "one quark" r e s o n a n c e in the v+~+ s y s t e m . So the highest p r o j e c t i o n o c c u r s i n the ~+~o s y s t e m , i.e. i s equal to 1. Analogously for the ~ s y s t e m the highest p r o j e c t i o n i s ½ and f o r the lrN s y s t e m it i s {. A c c o r d i n g to the f o r m u l a e (1) and (6) the m a s s of a r e s o n a n c e i s given by M.2 =M 2 + ~2+2/~
,
(8)
68"/
Volume 22, number 5
PHYSICS LETTERS
where M is the m a s s of the b a r y o n - t a r g e t . Choosing ~ = 3.51 ( G e V - 2 ) , ) ~ = 4.3 (GeV-2~) we obtain table 1, which c o n t a i n s the well-known decuplet. (Exept for the 12-hyperon, which of c o u r s e cannot be obtained in such r e a c t i o n s . ) Now we shall p a s s to the K m e s o n - b a r y o n s y s t e m s . The best a g r e e m e n t with the e x p e r i m e n t is achieved when 2 / ~ K = 1.64 GeV 2, ~K = 0.06 GeV -2. In complete analogy with the ~ - m e s o n case we obt a i n table 2. Table 2 c o n t a i n s one p r e d i c t e d r e s o nance, n a m e l y the analog of the ~ - h y p e r o n with a m a s s of about 1900 MeV and a width l e s s than 10 MeV. It looks p l a u s i b l e that the r e s o n a n c e s of the s a m e f a m i l y have the s a m e spin and p a r i t y . This is r e a l l y the case for table 1. Then the Y~(1660) and ~2"(1900) would have to p o s e s s the s a m e s p i n p a r i t y as ~*(1816)-(~-). ( E x p e r i m e n t a l l y the spin of the Y~(1660) >/ 3). P a s s i n g at l a s t to the 7 / m e s o n - b a r y o n s y s t e m we choose 2/o~7/ = 1.72 GeV2; ~T/= 0.2 GeV-2. The r e s u l t s a r e s u m m a r i z e d in table 3. So with the help of a m i n i m a l n u m b e r of p a r a m e t e r s we s u c c e e d in d e s c r i b i n g about 10 diff e r e n t r e s o n a n c e s . If the a p p r o x i m a t e c h a r a c t e r of the model is taken into account, the p a r a m e t e r s of the r e s o n a n c e s quoted in t a b l e s 1-3 a r e in s u r p r i s i n g l y good a g r e e m e n t with the e x p e r i m e n t . However it m u s t be pointed out that the width of the r e s o n a n c e ~*(1933) i s a b n o r m a l l y (from the point of view of our model) large. Secondly, the model p r e d i c t s an u n o b s e r v e d r e s o n a n c e with I = 1, s p i n - p a r i t y ~+ in the r e g i o n M ~ 1850 MeV. P r o b a b l y this r e s o n a n c e is m a s k e d by the neighb o u r i n g r e s o n a n c e Y~(1765). It should be noted also that the r e s o n a n c e s in the m e s o n - A s y s t e m which a r e not included in t a b l e s 1-2 have the s a m e q u a n t u m n u m b e r s as the resonances in the m e s o n - ~ system, and we identify them. They differ slightly in mass but the e r r o r s a r e comparable with the mean e r r o r of the model.
688
15 September 1966
In c o n c l u s i o n we should like to e m p h a s i z e that the model p r e d i c t s h y p e r c h a r g e s and i s o s p i n s of the r e s o n a n c e s . F u r t h e r m o r e all m a s s e s of a given f a m i l y a r e fixed by one c o n s t a n t (12), which in addition d e t e r m i n e s the r a t i o of widths since the c o n s t a n t ~ e n t e r s the f o r m u l a (5) p r a c t i c a l l y a s a m u l t i p l i e r . (The G e l l - M a a n - O k u b o f o r m u l a for an octet for example c o n t a i n s t h r e e a r b i t r a r y c o n s t a n t s . ) The coefficients for d i f f e r e n t m e s o n s a r e connected by the "control" r e l a t i o n (7) which is fulfilled with quite r e a s o n a b l e a c c u r a c y . The unusual f e a t u r e of the model i s the p r e diction of the [2*-hyperon (iV/ ~ 1900 MeV), which in a c e r t a i n s e n s e drops out f r o m the t r a d i t i o n a l G e l l - M a i m - O k u b o s c h e m e of u n i t a r y m u l t i p l e t s . Namely, if the N½(1518), Y~(1520), Y*(1660)and ~*(1816) a r e c o m b i n e d in the octet ~ - , then the hypothetical [2"(1900) has no p a r t n e r s to f o r m a multiplet. On the other hand if the Y~(1660), -*(1816) and [2* a r e c o m b i n e d in a decuplet (in our s c h e m e it is m o r e n a t u r a l ) , then the G e l l Mann-Okubo m a s s f o r m u l a fails. I n f o r m a t i o n on the existence of such a h y p e r o n would be a dec i s i v e a r g u m e n t in favour (or against) the p r o posed model. The a u t h o r s would like to thank A c a d e m i c i a n N. N. Bogolubov, and P r o f e s s o r s A. M. Baldin and B. V. Medvedev for helpful d i s c u s s i o n .
References 1. E.M. Levin and L.L.Franefurt, JETP Letters 2 (1966) 105. 2. H.J. Lipkin and F. Seheck, Phys. Rev. Letters 16 (1966) 71. 3. N.N. Bogolubov, B.V. Struminsky and A. N. Tavchelidze, Dubna preprint D-1968 (1965). 4. A.H.Rosenfeld and A. Barbaro-Galieri et al., UCRL-8030 (1966).
SIMPLE
PHYSICS
DYNAMICAL
MODEL A. A. SLAVNOV
LETTERS
FOR
15 S e p t e m b e r 1966
BARYON
RESONANCES
and O. I. ZAVIALOV
Stehlov Mathematical Institute, Acad. Sci. Moscow, USSR Received 18 August 1966 The parameters of a number of baryon resonances are expressed in terms of few constants in the framework of the quark model.
Many p r o p e r t i e s of b a r y o n s w e r e s u c c e s s f u l l y e xpl a ine d in the f r a m e w o r k of the q u a r k m o d e l [1-3]. In the p r e s e n t p a p e r we p r o p o s e the s i m p l e d y n a m i c a l s c h e m e f o r the d e s c r i p t i o n of r e s o n a n c e s in m e s o n - b a r y o n s y s t e m s . T h i s s c h e m e e n a b l e s us to e x p r e s s m a s s e s , widths, i s o s p i n s and h y p e r c h a r g e s of a n u m b e r of b a r y o n r e s o n a n c e s in t e r m s of few c o n s t a n t s c h a r a c t e r i s i n g the q u a r k - m e s o n i n t e r a c t i o n . L e t us c o n s i d e r r e s o n a n c e s t a t e s in the m e s o n (octet 0 - ) - b a r y o n (octet ½+) s y s t e m s . The b a r y o n s a r e supposed to c o n s i s t of t h r e e q u a r k s , and the m e s o n s a r e t r e a t e d a s e l e m e n t a r y (i.e. the q u a r k s t r u c t u r e of m e s o n s is u n i m p o r t a n t f o r our p u r p o s e s ) . We a s s u m e that the m e s o n - b a r y o n f o r w a r d s c a t t e r i n g a m p l i t u d e can be e x p r e s s e d as follows R - h R 1 +fiR 2 + ~ R 3 + R
(1)
w h e r e R 1,2,3 a r e the s c a t t e r i n g a m p l i t u d e s f o r individual q u a r k s and 5,/3, ? a r e n o n z e r o c o e f f i c i e n t s , depending on b a r y o n - t a r g e t p r o p e r t i e s , /~ i s a function, d e s c r i b i n g the collectLve i n t e r a c t i o n of q u a r k s with an incident m e s o n . Due to the optical t h e o r e m the s a m e f o r m u l a is v a l i d f o r to ta l c r o s s s e c t i o n s . A s i m i l a r h y p o t h e s i s (in f a c t e v e n a m u c h b o l d e r one) has been a l r e a d y u s e d by s e v e r a l a u t h o r s [1-2]. Our b a s i c i d e a i s that if t h e r e is a r e s o n a n c e in the m e s o n - q u a r k s y s t e m then it will i n e v i t a b l y be p r e s e n t in the m e s o n s c a t t e r i n g by all b a r y o n s containing this quark. T h e r e f o r e by fL'~ing the p a r a m e t e r s of the m e s o n - q u a r k r e s o n a n c e , one can obtain a n u m b e r of p r e d i c t i o n s c o n c e r n i n g h y p e r c h a r g e s , i s o s p i n s , m a s s e s and widths of meson-baryon resonances. The m e s o n - q u a r k s c a t t e r i n g a m p l i t u d e c o n t a i n s t h r e e i n v a r i a n t a m p l i t u d e s (we m e a n unit a r y s t r u c t u r e only). 3 ® 8 = 3 + 6 " + 15 . 686
(2)
F o r r e a s o n s which will not be d i s c u s s e d in this p a p e r the r e s o n a n c e should be p r e s e n t in the a m plitude R(3). In the following we shall say a few w o r d s about the p o s s i b l e m e c h a n i s m of such a resonance. The m a s s of the quark is supposed to be l a r g e , so it looks r e a s o n a b l e to use the C h e w - L o w m o d el f o r the d e s c r i p t i o n of m e s o n - q u a r k s c a t t e r i n g . F o r e x a m p l e we can obtain the solution of the Low equation f o r the am p l i t u d e R(3) in a s t a n d a r d way. R(33)(z) - -
1-
z~33 T[
~3/~
[
d z ' p '3
~
- ~z f w
)
dz'p'3 B(z')
~ z' 2 (z +z')
(3) (The f i r s t index r e f e r s to u n i t a r y s t r u c t u r e , the second one is the spin index.) H e r e z is the m e s o n e n e r g y in l . s . , P i s the m e s o n m o m e n t u m , is the cut-off, )t33 is the e f f e c t i v e coupling constant in this channel. B(z') is an unknown function f i x ed by c r o s s - s y m m e t r y conditions. F o r z << w one can use the s o - c a l l e d " e f f e c t i v e r a d i u s app r o x i m a t i o n " . ( P r a c t i c a l l y the s a m e r e s u l t s a r e obtained by s i m p l y n e g l e c t i n g the left hand cut.) Then f o r z << w we w r i t e down the following ex p r e s s i o n f o r the phase shift ctg 5 =
(1 - rz)z ;t33(z2 _ ~2)~
(4)
r = ~33w/Tr is "the e f f e c t i v e r a d i u s of i n t e r a c t i o n " . When the p h ase shift p a s s e s through ½~, i.e. if 433 > 0, the r e s o n a n c e a p p e a r s . The width of the r e s o n a n c e is given by the f o r m u l a 3
F =
2~(1 - r 2 p 2 ) ~
(5)
r 3 + (rX/~) In (2nr/;~U) I t is not difficult to w r i t e down the m o d e l H a m i l tonian giving a p o s i t i v e ;t33 in a b r o k e n SU 3 theory.
Volume 22, number 5
PHYSICS LETTERS
15 September 1966
Table 1 Theory
Experiment [4]
Reaction
Isospin
Mass(GeV2) (squared)
Mass (MeV)
Width
~+P ~+~o
~ 1
1.47 2.00
1210 1410
114 46
~+~-
½
2.34
1530
30
Isospin
Mass (squared) (GeV2)
Mass (MeV)
N~(1236) Y~(1385)
a 1
1.53 1.91
1236 1383
120 ± 1.5 44 ~-5
~*(1530)
½
2.34
1530
7.5±1.7
Symbol
Width (MeV)
Table 2
Isospin
Theory Mass(GeV2) (squared)
Mass (MeV)
Width
K-N
1
2.76
1660
45
K-~ ° K-~ °
½ 0
3.29 3.63
1816 1900
13 <10
Reaction
Experiment [4] Symbol Isospin
Mass (squared) (GeV2)
(MeV)
Mass
Width (MeV)
Y~(1660)
I
2.76
1660
44 ~-5
~*(1816) ?
½ ?
3.29 ?
1816 ?
16 ?
Table 3
Reaction y
Isospin
Theory Mass(GeV2) (squared)
½
2.90
Experiment [4] Mass Width "(MeV) (MeV) Symbol Isospin 1695
128
N_~*(1688)
Mass (squared) (GeV2)
Mass (MeV)
Width '(MeV)
½
2.85
1688
125
2
~A
0
3.26
1810
53
Yo(1815)
0
3.29
1815
50
?
?
?
?
?
½
3.74
1933
140 ± 35
y~+
1
3.43
1850
37
~o
!2
3.77
1940
16
(The b r e a k i n g t e r m t r a n s f o r m s as the eighth component of an o c t e t . ) Thus the a s s u m p t i o n about the e x i s t e n c e of a r e s o n a n c e in the R(3) a m p l i t u d e s e e m s r a t h e r n a t u r a l . This m e a n s that for a ~ l - m e s o n - q u a r k s y s t e m the r e s o n a n c e o c c u r s in the i s o s p i n I=½ state, and for a K - m e s o n - q u a r k s y s t e m in the i s o s p i n I= 0 state. The p o s i t i o n of the r e s o n a n c e is determineel by the effective r a d i u s , Lhe width i s fixed by the f o r m u l a (5). The effective r a d i u s of a bound q u a r k n a t u r a l l y d i f f e r s f r o m that of a f r e e quark, and roughly speaking depends on the d i m e n s i o n s of the c o m posite p a r t i c l e . So we c l a i m that r = aM .
(6)
W h e r e M i s the m a s s of the c o r r e s p o n d i n g b a r y o n and ~ is s o m e e m p i r i c a l coefficient. This m e a n s that in our model a i s the u n i v e r s a l c o n s t a n t c h a r a c t e r i z i n g the i n t e r a c t i o n of the m e s o n with all b a r y o n s . It i s c l e a r a l s o that for K and y - m e -
2"(1933)
sons the coefficients ~ a r e different, and t h e i r r a t i o should be a p p r o x i m a t e l y equal to the r a t i o of the c o r r e s p o n d i n g Compton wave lengths. ~r~
~ ~K/IK ~ ~ P y
•
(7)
Now let us c o n s i d e r r e s o n a n c e s in the ~ m e s o n b a r y o n s y s t e m . We have a l r e a d y m e n t i o n e d that for our s y s t e m a r e s o n a n c e o c c u r s in s t a t e s with I = ~, 1 i.e. in the s y s t e m s lr+n, r ' p , ~Op, ~On. Starting f r o m here we can at once d e t e r m i n e the highest p r o j e c t i o n of the i s o s p i n of the c o r r e s p o n d ing m e s o n - b a r y o n r e s o n a n c e . F o r e x a m p l e ~+ does not c o n t a i n the n - q u a r k , and t h e r e f o r e t h e r e is no "one quark" r e s o n a n c e in the v+~+ s y s t e m . So the highest p r o j e c t i o n o c c u r s i n the ~+~o s y s t e m , i.e. i s equal to 1. Analogously for the ~ s y s t e m the highest p r o j e c t i o n i s ½ and f o r the lrN s y s t e m it i s {. A c c o r d i n g to the f o r m u l a e (1) and (6) the m a s s of a r e s o n a n c e i s given by M.2 =M 2 + ~2+2/~
,
(8)
68"/
Volume 22, number 5
PHYSICS LETTERS
where M is the m a s s of the b a r y o n - t a r g e t . Choosing ~ = 3.51 ( G e V - 2 ) , ) ~ = 4.3 (GeV-2~) we obtain table 1, which c o n t a i n s the well-known decuplet. (Exept for the 12-hyperon, which of c o u r s e cannot be obtained in such r e a c t i o n s . ) Now we shall p a s s to the K m e s o n - b a r y o n s y s t e m s . The best a g r e e m e n t with the e x p e r i m e n t is achieved when 2 / ~ K = 1.64 GeV 2, ~K = 0.06 GeV -2. In complete analogy with the ~ - m e s o n case we obt a i n table 2. Table 2 c o n t a i n s one p r e d i c t e d r e s o nance, n a m e l y the analog of the ~ - h y p e r o n with a m a s s of about 1900 MeV and a width l e s s than 10 MeV. It looks p l a u s i b l e that the r e s o n a n c e s of the s a m e f a m i l y have the s a m e spin and p a r i t y . This is r e a l l y the case for table 1. Then the Y~(1660) and ~2"(1900) would have to p o s e s s the s a m e s p i n p a r i t y as ~*(1816)-(~-). ( E x p e r i m e n t a l l y the spin of the Y~(1660) >/ 3). P a s s i n g at l a s t to the 7 / m e s o n - b a r y o n s y s t e m we choose 2/o~7/ = 1.72 GeV2; ~T/= 0.2 GeV-2. The r e s u l t s a r e s u m m a r i z e d in table 3. So with the help of a m i n i m a l n u m b e r of p a r a m e t e r s we s u c c e e d in d e s c r i b i n g about 10 diff e r e n t r e s o n a n c e s . If the a p p r o x i m a t e c h a r a c t e r of the model is taken into account, the p a r a m e t e r s of the r e s o n a n c e s quoted in t a b l e s 1-3 a r e in s u r p r i s i n g l y good a g r e e m e n t with the e x p e r i m e n t . However it m u s t be pointed out that the width of the r e s o n a n c e ~*(1933) i s a b n o r m a l l y (from the point of view of our model) large. Secondly, the model p r e d i c t s an u n o b s e r v e d r e s o n a n c e with I = 1, s p i n - p a r i t y ~+ in the r e g i o n M ~ 1850 MeV. P r o b a b l y this r e s o n a n c e is m a s k e d by the neighb o u r i n g r e s o n a n c e Y~(1765). It should be noted also that the r e s o n a n c e s in the m e s o n - A s y s t e m which a r e not included in t a b l e s 1-2 have the s a m e q u a n t u m n u m b e r s as the resonances in the m e s o n - ~ system, and we identify them. They differ slightly in mass but the e r r o r s a r e comparable with the mean e r r o r of the model.
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15 September 1966
In c o n c l u s i o n we should like to e m p h a s i z e that the model p r e d i c t s h y p e r c h a r g e s and i s o s p i n s of the r e s o n a n c e s . F u r t h e r m o r e all m a s s e s of a given f a m i l y a r e fixed by one c o n s t a n t (12), which in addition d e t e r m i n e s the r a t i o of widths since the c o n s t a n t ~ e n t e r s the f o r m u l a (5) p r a c t i c a l l y a s a m u l t i p l i e r . (The G e l l - M a a n - O k u b o f o r m u l a for an octet for example c o n t a i n s t h r e e a r b i t r a r y c o n s t a n t s . ) The coefficients for d i f f e r e n t m e s o n s a r e connected by the "control" r e l a t i o n (7) which is fulfilled with quite r e a s o n a b l e a c c u r a c y . The unusual f e a t u r e of the model i s the p r e diction of the [2*-hyperon (iV/ ~ 1900 MeV), which in a c e r t a i n s e n s e drops out f r o m the t r a d i t i o n a l G e l l - M a i m - O k u b o s c h e m e of u n i t a r y m u l t i p l e t s . Namely, if the N½(1518), Y~(1520), Y*(1660)and ~*(1816) a r e c o m b i n e d in the octet ~ - , then the hypothetical [2"(1900) has no p a r t n e r s to f o r m a multiplet. On the other hand if the Y~(1660), -*(1816) and [2* a r e c o m b i n e d in a decuplet (in our s c h e m e it is m o r e n a t u r a l ) , then the G e l l Mann-Okubo m a s s f o r m u l a fails. I n f o r m a t i o n on the existence of such a h y p e r o n would be a dec i s i v e a r g u m e n t in favour (or against) the p r o posed model. The a u t h o r s would like to thank A c a d e m i c i a n N. N. Bogolubov, and P r o f e s s o r s A. M. Baldin and B. V. Medvedev for helpful d i s c u s s i o n .
References 1. E.M. Levin and L.L.Franefurt, JETP Letters 2 (1966) 105. 2. H.J. Lipkin and F. Seheck, Phys. Rev. Letters 16 (1966) 71. 3. N.N. Bogolubov, B.V. Struminsky and A. N. Tavchelidze, Dubna preprint D-1968 (1965). 4. A.H.Rosenfeld and A. Barbaro-Galieri et al., UCRL-8030 (1966).