IL N U O V O

CIMENTO

VOL. X X X I X ,

N. 2

16 S e t t e m b r e 1965

On the Connection between the Baryon and Meson Mass Spectra

(*).

P. G. o . FREUND

The Enrico Ferm, i Institute ]or Nuclear Studies and the Department o] Physics The University o/ Chicago - Chicago, Ill. ( r i c e v u t o il 12 L u g l i o 1965)

I R a r e c e n t p a p e r (J) we h a v e discussed in d e t a i l t h e c o n c e p t of u n i v e r s a l s t r o n g i n t e r a c t i o n s i n t h e l i g h t of M ~ s y m m e t r y . B y r e q u i r i n g t h e c o m p a t i b i l i t y of u n i v e r s a l i t y a n d of a b o u n d - s t a t e p i c t u r e of h a d r o n s , c o n c e r n i n g t h e i r p r e d i c t i o n s for magnetic moments, the remarkable mass formula

(l)

M~ = N MM_ _ N ( 3 0 0 -~ 50) MeV 2

h a s b e e n o b t a i n e d for t h e c e n t r a l masses of t r i a l i t y zero b a r r e n s m a d e o u t of n q u a r k s a n d ~ a n t i q u a r k s ( N = n ~ :~). I n p a r t i c u l a r t h i s gives t h e r e s u l t MB MM

(2)

3 2

for t h e r a t i o of t h e c e n t r a l b a r y o n a n d m e s o n masses. A pole m o d e l i n w h i c h a n o c t e t of u n i v e r s a l l y c o u p l e d s c a l a r m e s o n s d o m i n a t e s t h e s c a l a r f o r m f a c t o r s of h a d r o n s h a s b e e n s h o w n i n I to c o n n e c t t h e m a s s s p l i t t i n g s of m e s o n s a n d b a r y o n s ( a n d of all b a d r o n s ) . I n p a r t i c u l a r t h e f o r m u l a

K s-z~ 2

z----Y

2

3

1 ]

2

mM~0.9mM

h a s b e e n o b t a i n e d i n t h i s model. T h e f a c t o r 0.9 o n t h e r i g h t - h a n d side o r i g i n a t e d i n some d e t a i l e d f e a t u r e s of s t r o n g - i n t e r a c t i o n u n i v e r s a l i t y discussed i n I. F o r all o u r p u r p o s e s t h i s f a c t o r c a n b e i g n o r e d (since we s h a l l n e g l e c t t e r m s of order 1 0 %

(*) This work is supported by the U. S. Atomic Energy Commission. (1) p. G. O. FREUND: Phys. Rev. Left., |4, 1088; 15, E 176 (1965). This paper will be referred to hereafter as I. 49 - II Nuovo Cimento.

770

P.G.O.

FREUND

(,~ M~IM~) below) we can take simlaly K 2 __ ~2

E--~4 z

(3)

- - raM.

On the other hand, in a bound-state picture of hadrons, the Gell-Mann-Okubo (GMO) mass splittings are mainly determined by the GMO splitting of their constit u e n t quarks. We, therefore, again expect a relation between Z-2V and K~-~2. The question is what this relation is and whether it is compatible with (3) and the mass formula (2). We shall write a bound-state equation (2) for mesons incorporating the GM0 splitting of quarks

K+ K - y~ (P ÷ q, P -- q) = ig W (q~)q~(P) with P

P

~(P)=f W(q-)~o~(-~ + q, ~--q) dq ,

A8

(~s)s~.® (l),~,~o.

=

Here -~/Q is the central quark mass and A the GM0 term of quarks. We then find 2 2! the meson wave function ¢~(P) to obey the equation (up to terms of order M~/M(~.)

(4)

( p 2 M~)qJ~(P)- 2A* [As, ~o(P)]+~ = 0 ,

where MM is a suitably defined central meson mass. I n a completely similar way for baryons (5)

J ~ -A - s p~'.~r' " r ÷ 6o, (p 2 __ M 2~)q~a~,(p)__3A2(As:" ~' ~ ' + ~" ~,'~~"Asrv' )~,~,r,(P)

:

0.

From eqs. (4) and (5) we then readily calculate K (6)

_

~2

:

_

2A2

~/~,

Z 2 _ _ ~ / ' 2 = __ 6Z12 ~ / ~ ,

whence (7)

K 2- - r~~

1

E2__~z

3

This relation is the main result of the present paper. With E ÷ ~ P ~ 2M B, (7) and (3') again lead to the mass formula (2) and again universality and bound picture are

(~) N. N. ]~OGOL~UBOV,V. IV~ATVEEV, N. VAN HIEU, D. STOIANOV, B. STRUMINSKY, A. TAVKHELIDZE a n d V. SHELEST: D u b n a p r e p r i n t s D-2075 a n d P-214A.

ON THE CONNECTION BETWEEN THE BARYON AND MESON ~ASS SPECTRA

771

c o m p a t i b l e (*). E x p e r i m e n t a l l y ( K 2 - z ~ ) / ( Z z - - . 4 P~) ~ 0.27 i n r e s o n a b l e a g r e e m e n t w i t h (7). T h e d i s c r e p a n c y o f ~ 2 0 % is o f t h e s a m e o r d e r a s t h e d i s c r e p a n c y b e t w e e n the bound-state model prediction /%--3 and /~p--2.79. Both discrepancies are e x p e c t e d t o b e o f o r d e r M ~~/ M t ~2. H e n c e t y. p i c a l l y M B2/ M ( 2~ O.1. F u r t h e r m o r e f r o m (6) o n e cart e s t i m a t e t h e G M O s p l i t t i n g o f t h e q u a r k s . W e flint

s 2 - - d" = - - ~/,3 ~42

( w h e r e s(d) is t h e I =

0 ([--1)

quark).

s--d

K 2 _ jr2 .... 2

0.12 G-eV ~

Hence

M = 64 - - " M e V ,

w h i c h f o r e.g. M,~ : : 4 M v is ~ 16 M e V . T h u s t h e G M O s p l i t t i n g of q u a r k s is v e r y small which may well be the reasorl for the success of SU 3 symmetry i~ spite of the apparent large GMO splitting of the low-lying hadron multiplets. W e n o w s t i l l h a v e to m a k e a f e w g e n e r a l r e m a r k s o n o u r m e t h o d . 1) T h e f a c t t h a t u n i v e r s a l i t y a n d b o u n d - s t a t e m o d e l s a r e i n t i m a t e l y c o n n e c t e d w a s k n o w n f o r a l o n g t i m e (3) e v e n t h r o u g h i t h a s n o t b e e n r e a l i z e d p r e v i o u s l y t h a t t h i s c o n n e c t i o n f i x e s t h e m a s s s p e c t r u m of h a d r o n s ( b o t h t h e c e n t r a l m a s s e s a n d t h e GM(:) s p l i t t i n g s ) . 2) I t is a m u s i n g t h a t tile m a s s s p e c t r u m of h a d r o n s is d e t e r m i n e d b y a r e l a t i v e l y h ) w (( e f f e c t i v e v i r t u a l q u a r k m a s s ~) of ~ 300 M e V . 3) I f t h e m a s s e s of all h a d r o a s a r e c o n n e c t e d , t h e n p a s t e x p e r i e n c e t e n d s t o suggest that some (presumably aoneompact) supersymmetry might be at work p l a c i n g all h a d r o n s i n a ( p r e s u m a b l y i n f i u i t e - d i m e n s i o n a l (~)) r e p r e s e n t a t i o n (**).

(*) In the bouml state model all mass formulae (for fcrmions, bosons (parafermiaos.~) alike) arc mass squared formulae. (~) ]={. S. FEVNMAN: l'roceeding,~ of the ~lix-en,-Provenee Conf. on Elementary 1'articles vol. 2 (Saelay, 1961 ), p. 207. I wish to t h a n k J..1. SAKURAIfor calling m y attention to this reference. (4) Infinite-dimensional unitary representations for the group 15I~ have been constructed in P. (]. O. FItEITND: Phys. Rev. Left., 14, 803 (1965). See also the more recent work of Y. I)OTHAN, M. (IEL.L-MANN and Y. NE'EMAN: Caltech prepriut. (**) It is amusing t h a t iu h[~ one can build up an.tiq~larks out of 11 quarks, provided baryon conservation is good only modulo 12.