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Mass formulae for the meson, baryon and isobar states in Schwinger's group, W3
Volume 9, number 2
PHYSICS LETTERS
A.A. Logunov, A.N. Tavkhelidze, I.T. Todorov, O.A. Khrustalev, Nuovo Cimento 30 (1963) 134; B.A.Arbuzov, A.A. Logunov, A.T. Filippov, O.A. Ktirustalev, Dubna preprint .R-1318: J. Exptl. Theor. Phys. (USSR) (to be published). 2) A. T. Filippov, Physics Letters 9 (1964) 78. 3) A. T. Filippov, Dubna preprint R-1493. 4) J. M. Charap and S. P. Fubini, Nuovo Cimento 14 (1959) 540.
MASS
FORMULAE
FOR THE MESON, IN SCHWINGER'S
1 April 1964
5) B.A.Arbuzov, A.T. Fflippov, O.A. Khrustalev,
Physics Letters 8 (1964) 205. 6) E.Predazzi and T.Regge, Nuovo Cimento 24 (1962) 518. 7) Bateman Manuscript Project, Higher Transcendental Functions, Volume I. A. Erdelyi, Editor (McGraw Hill, 1953) p.255.
BARYON AND ISOBAR GROUP, W3
STATES
E l i z a b e t h JOHNSON * and R. F. SAWYER **
Department of Physics, University of Wisconsin, Madison, Wisconsin Received 2 March 1964
In a r e c e n t note 1) Schwinger has p r o p o s e d a g e n e r a l i s a t i o n of SU3 s y m m e t r y which y i e l d s the m a i n r e s u l t s of SU3 in the s t r o n g i n t e r a c t i o n s 2) plus s o m e v e r y i n t e r e s t i n g f u r t h e r r e s u l t s . In Schwinger's paper mass relations are discussed for n i n e b a r y o n s , nine p s e u d o s c a l a r m e s o n s , and nine v e c t o r m e s o n s . The p r e s e n t note is devoted m a i n l y to the cons i d e r a t i o n of the ~+ b a r y o n r e s o n a n c e s in a model which is b a s i c a l l y S c h w i n g e r ' s (though m o d i fied somewhat). In W3 t h e r e will be 18 such obj e c t s , the c o n v e n t i o n a l 3) 10 plus an additional octet. We s h a l l d e r i v e m a s s r e l a t i o n s for the ent i r e m u l t i p l e t of 18. In addition we s h a l l p r o p o s e s o m e m o d i f i c a t i o n s of S c h w i n g e r ' s approach to baryon mass differences. Schwinger u s e s a v e r y definite m o d e l as a c o n t a i n e r for his group, W3. We s h a l l follow his s c h e m e exactly, though m a n y of the r e s u l t s would follow f r o m t h e o r i e s with the s a m e group, W3, and different f u n d a m e n t a l p a r t i c l e s . The b a s i c c o n s t i t u e n t s a r e two Sakata t r i p l e t s : tp with one unit of b a r y o n n u m b e r ; V with two u n i t s of b a r y o n n u m b e r . V1, V2, ~1, ~2 have T= ½, Y =-1 V3, gJ3 have T = 0 ,
(1)
Y =0
T h e s e i n t e r a c t through a n e u t r a l field, B, * Work supported in part by the U. S. Atomic Energy Commission. ** Work supported in part by the National Science Foundation. 212
coupled to ~ and ~ V in such a way that independent groups U 3 m a y act on ~ and V separately. Thus we have W 3 = U 3 @ U 3. Under the group w e use as notation for a representation (nl, where n I is the dimensionality of the V representation and n 2 that of the ~ representation. W e list the representations for the various particles : 1-, 0- m e s o n s : (1, 1) + (1, 8)(bound ~tp) b a r y o n s : (3, 3*)(bound V~) (2) I + i s o b a r s : (6*, 3*) If we allow an i n t e r m e d i a t e stage of Su3 s y m m e t r y in which ~ and V m u s t be t r a n s f o r m e d s i m u l t a n e o u s l y the above r e p r e s e n t a t i o n s d e c o m pose as n o r m a l d i r e c t products,
~
(3, 3*) ~
8 + 1
(6*, 3*) -. 10" + 8 (see footnote *). Our m a s s s p l i t t i n g m e c h a n i s m is e x t r e m e l y s i m p l e - m i n d e d (we c o n j e c t u r e that it is the m e c h a n i s m u s e d by Schwinger for the bosons). We take as p a r a m e t e r s the m a s s s p l i t t i n g s of the c o n s t i t u e n t s and weight them with the s q u a r e s of the app r o p r i a t e G l e b s c h - G o r d a n coefficients. In addition to this we take a t e r m which c o r r e s p o n d s to an SU3 i n v a r i a n t s c a t t e r i n g of the two constituents. We s h a l l t r e a t these c o n t r i b u t i o n s together as a perturbation. t Since our fundamental triplets have hypercharge reversed from the normal Sakata case and we have taken them to transform as 3 (rather than 3"), our representation, 1O*, has the same quantum numbers as the conventional 10.
Volume 9, number 2
PHYSICS
T h e SU 3 i n v a r i a n t s c a t t e r i n g t e r m a r i s e s n a t u r a l l y a s a p e r t u r b a t i o n of the m e s o n s t a t e s in ~ c h w i n g e r ' s m o d e l , s i n c e i t i s only a p a r t i c u l a r m e c h a n i s m (B exchange) which l e a v e s s i n g l e t and o c t e t u n s p l i t . G r a p h s in which ~ and ~ a n n i h i l a t e i n t o B ' s a r e p a r t of our p e r t u r b a t i o n . F o r the b a r y o n c a s e the SU3 i n v a r i a n t V ~ s c a t t e r i n g t e r m i s an a d d i t i o n a l p a r a m e t e r which h a s b e e n i n t r o d u c e d in o r d e r to p u r i f y the s t a t e s (in the SU 3 s e n s e ) s u f f i c i e n t l y to obtain an a p p r o x i m a t e G e l l - M a n n - O k u b o r e l a t i o n 4). We obtain the following ( m a s s ) 2 r e l a t i o n f o r the nine v e c t o r m e s o n s t a t e s . (~-p)(cv-p) = 4(K*-p)(q~+c~-2K*)
LETTERS
1 April 1964
r e l a t i o n b e t w e e n 6M(3) and 8M(1) i m p l i e d by a definite t r a n s f o r m a t i o n p r o p e r t y of the s y m m e t r y breaking Hamiltonian is irrelevant. W e f o r m the e i g h t e e n ~+ s t a t e s f r o m ~/"S(6"), w h e r e S(6") i s a 6* r e p r e s e n t a t i o n of the group a c t i n g on V. S(6") could be c o m p o s e d of VVV for e x a m p l e 5). It would be quite difficult to d e a l with a l l t h e s e c o n s t i t u e n t s i n d i v i d u a l l y so we i n s t e a d s i m p l y a s s i g n a m a s s s p l i t t i n g between the t h r e e i s o t o p i c m u l t i p l e t s in S(6") in a c c o r d with the G e l l - M a n n Okubo r u l e (equal s p a c i n g in t h i s c a s e ) ,
(3)
(Eq. 4 of Schwinger) and a s i m i l a r equation for the nine 0" states. For the baryons we obtain, 2N+2.~-
Z- 3A=~
q(Z+A-N-X)(2;-A) (yo.+A_N_X)
(4)
We w i s h to m a k e the s a m e point about a l l such equations: if the ninth m e m b e r of the m u l t i p l e t i s s t r o n g l y s p l i t off f r o m the eight, then the G e l l Mann- Okubo r e l a t i o n s follow a u t o m a t i c a l l y . Thus putting in the e x p e r i m e n t a l a b s e n c e of a ninth 0m e s o n at low m a s s , we p r e d i c t the G e l l - M a n n Okubo r e l a t i o n for the 0" octet. Putting in Y * at 1405 MeV we p r e d i c t a s u i t a b l e s m a l l n u m b e r f o r 2 N + 2 ~ - x - 3 A . We cannot u s e (4) to p r e d i c t the Y * m a s s ; the n u m b e r i s too s e n s i t i v e to s m a l l c h a n g e s in the left hand s i d e (which would be z e r o if the G e l l - M a n n - O k u b o r e l a t i o n s w e r e e x a c t l y satisfied) t. F o r the v e c t o r m e s o n s , h o w e v e r , the d e p a r t u r e f r o m G e l l - M a n n - O k u b o i s s u f f i c i e n t to l e a d to a r e a l nonet m a s s f o r m u l a , eq. (3). In c o n t r a s t to S c h w i n g e r ' s m o d e l o u r s p r e d i c t s no f u r t h e r r e l a t i o n s f o r the b a r y o n m a s s e s ; the p a r a m e t e r s m a y be f i t t e d to the e x p e r i m e n t a l m a s s e s . However in the c o n s t i t u e n t t h e o r y one of the p a r a m e t e r s in the b a r y o n c a s e (the ~ m a s s splitting) i s the s a m e a s one in the b o s o n c a s e . T h i s l e a d s to the s i n g l e r e l a t i o n
My.- M~ = MK* - Mp
(5)
which i s s a t i s f i e d within the p r e c i s i o n of our considerations. A s t r i k i n g p r o p e r t y of the above m a s s s p l i t t i n g m e c h a n i s m i s that no p r o p e r t i e s of the f u n d a m e n t a l s y m m e t r y b r e a k i n g i n t e r a c t i o n w e r e invoked to obtain the G e l l - M a n n - O k u b o r u l e . F o r e a c h c o n s t i t u e n t only a s i n g l e m a s s d i f f e r e n c e , 5M(3)-SM(1), e n t e r e d the f o r m u l a , s o that the
ST=I ,
Y=0,
5m=0
ST = ½ ,
Y=-I,
5m = b
ST=O ,
Y = -2 ,
5m=2b
(6)
In the 18 t h e r e a r e two YI* and two ~z_*. In a d dition t h e r e a r e the N*(10) and ~-(10) f r o m the r e p r e s e n t a t i o n ten ; and N½*(8), Yo8(8) f r o m eight. Again we t a k e an SU 3 i n v a r i a n t s c a t t e r i n g for the ~JS s t a t e ; with a p a r a m e t e r ~. The r e s u l t s in t h i s c a s e a r e b e s t d i s p l a y e d in t e r m s of the o r i g inal p a r a m e t e r s . The p a r a m e t e r , a, i s the m a s s s p l i t t i n g , g / 3 - ~ l . The YI* and ~---1" m a s s e s will be 2 given by the e i g e n v a l u e s of the two by two m a t r i c e s l i s t e d below. State
5m
N~*(10)
- 0
2
Yz*(zo, 8)
a+2b
f~-(10)
-
N*(8)
~ x
2
70*(8)
~ x+b
For the ten states we automatically obtain equal spacing i f ;~ is large 3). We can obtain equal s p a c i n g for any value of X V we c h o o s e a = b. T a k i n g YI*(8) to be the 1660 MeV s t a t e 6) the obs e r v e d equal s p a c i n g of the d e c u p l e t 7) i m p l i e s the m a s s e s f o r the o t h e r octet m e m b e r s , N * - ~ 1510 MeV,
t"o* -~ 1660 MeV,
2
. ~ -. 1810 MeV.
(6)
2
t t Only ff the first order mechanism were exact we would be allowed to regard as exact our replacement of the mass splitting parameters by the observed masses on the left hand side of (4).
213
Volume 9, number 2
PHYSICS LETTERS
Since t h e s e c o n s i d e r a t i o n s w e r e a ll quite approximate we should allow p e r h a p s 50 MeV d e v i ations f r o m the above. We s u g g e s t a s e a r c h f o r these states. We have p r o f i t e d f r o m a n u m b e r of c o n v e r s a tions with Dr. V. B a r g e r . References 1) J. S, Schwinger, "A Ninth Baryon?", Harvard University, preprint. 2) M. Gell-Mann, Phys. Rev. 125 (1962) 1067; Y.Ne'eman, Nuclear Phys.26 (1961) 222; J.Wess, Nuovo Cimento 15 (1960) 52.
1 April 1964
3) R. Behrends, J. Dreitlein, C. Fronsdal and B.W. Lee, Revs. Mod. Phys. 34 (1962) 1. S. L. Glashow and J. J. Sakurai, Nuovo Cimente 25 (1962) 337. 4) M. Cell-Mann, Phys.Rev. 125 (1962) 1067; S. Okubo, Progress of Theoretical Physics, 27 (1962) 949. 5) The composition of the representation, 6, is discussed in: M. ]keda, S. Ogawa and Y. Ohnuki, P r o g ress of Theoretical Physics 27 (1962) 949. 6) M. Taher-Zadeh et al., Phys.Rev. Letters 11 (1963) 470. 7) Barnes et al., Phys.Rev. Letters 12 (1964) 204.
* * * * *
QUASI-ELEMENTARY MASSLESS BOSONS ASSOCIATED WITH THE QUANTUM ELECTRODYNAMICS OF JOHNSON, BAKER AND WILLEY* Y . NAMBU The Enrico Fermi Institute f o r Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois
Received 2 February 1964
R e c e n t l y J o h n s o n et al. 1) h a v e p r o p o s e d a new f o r m u l a t i o n of quantum e l e c t r o d y n a m i c s b a s e d on a n o n - p e r t u r b a t i v e p r o c e d u r e . T h e y c l a i m that the r e n o r m a l i s a t i o n c o n s t a n t s as well as the e l e c t r o n s e l f - e n e r g y can be m a d e finite, thus e l i m inating the d i v e r g e n c e p r o b l e m s **. We c o n s i d e r h e r e one p a r t i c u l a r a s p e c t of this t h e o r y , n a m e l y the p r o b l e m of m a s s l e s s b o s o n s r e s u l t i n g f r o m the spon t an eo u s breakdown of i n v a r i a n c e s . We a r g u e that they e x i s t (with c e r t a i n q u a l i f i c a t i o n s ) and a r e coupled to the e l e c t r o n with a s t r e n g t h c o m p a r a b l e to the e l e c t r o m a g n e t i c coupling, and h e n c e the t h e o r y in this p a r t i c u l a r c o n t e c t is not c o n s i s t e n t with the known p r o p e r t i e s of the e l e c t ron. The k ey points of the t h e o r y a r e as follows. 1. The Landau gauge is c h o s e n f o r the photon G r e e n ' s function so as to m a k e the e l e c t r o n and v e r t e x r e n o r m a l i s a t i o n c o n s ta n t s Z 2 and Z 1 (=Z2) finite (in the l o w e s t o r d e r ) . 2. The b a r e m a s s m o of the e l e c t r o n is z e r o . 3. The u n r e n o r m a l i z e d e l e c t r o n G r e e n ' s function * Work supported by the U. S. Atomic Energy Commission. ** There is, however, a certain difficulty concerning Z 3 . I thank Dr. Johnson for a private communication. 214
S (p) is obtained in the l o w e s t o r d e r " H a r t r e e Fock" approximation -
eo 2
(1) D~v(k ) : ( ~
- k~k~/~2) 1 / k 2 ,
1
w h e r e e o = Z ~ e is the b a r e c h a r g e . They find a s e l f - c o n s i s t e n t solution with the a s y m p t o t i c b eh av i o u r S - 1 0 ) = ~.p - i ~ 0 2)
~q,2 = _ ~ )
: m, ~ ( ~ ) ~ 0 2 ) -a ,
a ~ 3eo2/16~2 = 3 a o / ~
(2)
.
The b a s i c L a g r a n g i a n in the t h e o r y of Jo h n so n et al. has no m a s s t e r m n o r any c o n s t a n t which can supply a s c a l e of m a s s . It f o l l o w s e a s i l y that the equations of m o t i o n a r e i n v a r i a n t under i) V5t r a n s f o r m a t i o n ~(x) ~ e~p [iT~t]~(x) and ii) s c a l e t r a n s f o r m a t i o n ~(x) k ~ (kx), A ~(x) -. ~ A ~(kx). Th e solution with n o n - v a n i s h i n g rn r e q u i r e s a spontaneous b r eak d o w n of t h e s e i n v a r i a n c e s , and one should ex p ect , a c c o r d i n g to the. well known a r g u m e n t s 2), m a s s l e s s bosons ( c o l l e c t i v e e l e c t r o n - p o s i t r o n bound s t a t e s ) to e m e r g e as a r e s u l t T h e s e b o s o n s a r e p s e u d o s c a l a r and s c a l a r , r e -
PHYSICS LETTERS
A.A. Logunov, A.N. Tavkhelidze, I.T. Todorov, O.A. Khrustalev, Nuovo Cimento 30 (1963) 134; B.A.Arbuzov, A.A. Logunov, A.T. Filippov, O.A. Ktirustalev, Dubna preprint .R-1318: J. Exptl. Theor. Phys. (USSR) (to be published). 2) A. T. Filippov, Physics Letters 9 (1964) 78. 3) A. T. Filippov, Dubna preprint R-1493. 4) J. M. Charap and S. P. Fubini, Nuovo Cimento 14 (1959) 540.
MASS
FORMULAE
FOR THE MESON, IN SCHWINGER'S
1 April 1964
5) B.A.Arbuzov, A.T. Fflippov, O.A. Khrustalev,
Physics Letters 8 (1964) 205. 6) E.Predazzi and T.Regge, Nuovo Cimento 24 (1962) 518. 7) Bateman Manuscript Project, Higher Transcendental Functions, Volume I. A. Erdelyi, Editor (McGraw Hill, 1953) p.255.
BARYON AND ISOBAR GROUP, W3
STATES
E l i z a b e t h JOHNSON * and R. F. SAWYER **
Department of Physics, University of Wisconsin, Madison, Wisconsin Received 2 March 1964
In a r e c e n t note 1) Schwinger has p r o p o s e d a g e n e r a l i s a t i o n of SU3 s y m m e t r y which y i e l d s the m a i n r e s u l t s of SU3 in the s t r o n g i n t e r a c t i o n s 2) plus s o m e v e r y i n t e r e s t i n g f u r t h e r r e s u l t s . In Schwinger's paper mass relations are discussed for n i n e b a r y o n s , nine p s e u d o s c a l a r m e s o n s , and nine v e c t o r m e s o n s . The p r e s e n t note is devoted m a i n l y to the cons i d e r a t i o n of the ~+ b a r y o n r e s o n a n c e s in a model which is b a s i c a l l y S c h w i n g e r ' s (though m o d i fied somewhat). In W3 t h e r e will be 18 such obj e c t s , the c o n v e n t i o n a l 3) 10 plus an additional octet. We s h a l l d e r i v e m a s s r e l a t i o n s for the ent i r e m u l t i p l e t of 18. In addition we s h a l l p r o p o s e s o m e m o d i f i c a t i o n s of S c h w i n g e r ' s approach to baryon mass differences. Schwinger u s e s a v e r y definite m o d e l as a c o n t a i n e r for his group, W3. We s h a l l follow his s c h e m e exactly, though m a n y of the r e s u l t s would follow f r o m t h e o r i e s with the s a m e group, W3, and different f u n d a m e n t a l p a r t i c l e s . The b a s i c c o n s t i t u e n t s a r e two Sakata t r i p l e t s : tp with one unit of b a r y o n n u m b e r ; V with two u n i t s of b a r y o n n u m b e r . V1, V2, ~1, ~2 have T= ½, Y =-1 V3, gJ3 have T = 0 ,
(1)
Y =0
T h e s e i n t e r a c t through a n e u t r a l field, B, * Work supported in part by the U. S. Atomic Energy Commission. ** Work supported in part by the National Science Foundation. 212
coupled to ~ and ~ V in such a way that independent groups U 3 m a y act on ~ and V separately. Thus we have W 3 = U 3 @ U 3. Under the group w e use as notation for a representation (nl, where n I is the dimensionality of the V representation and n 2 that of the ~ representation. W e list the representations for the various particles : 1-, 0- m e s o n s : (1, 1) + (1, 8)(bound ~tp) b a r y o n s : (3, 3*)(bound V~) (2) I + i s o b a r s : (6*, 3*) If we allow an i n t e r m e d i a t e stage of Su3 s y m m e t r y in which ~ and V m u s t be t r a n s f o r m e d s i m u l t a n e o u s l y the above r e p r e s e n t a t i o n s d e c o m pose as n o r m a l d i r e c t products,
~
(3, 3*) ~
8 + 1
(6*, 3*) -. 10" + 8 (see footnote *). Our m a s s s p l i t t i n g m e c h a n i s m is e x t r e m e l y s i m p l e - m i n d e d (we c o n j e c t u r e that it is the m e c h a n i s m u s e d by Schwinger for the bosons). We take as p a r a m e t e r s the m a s s s p l i t t i n g s of the c o n s t i t u e n t s and weight them with the s q u a r e s of the app r o p r i a t e G l e b s c h - G o r d a n coefficients. In addition to this we take a t e r m which c o r r e s p o n d s to an SU3 i n v a r i a n t s c a t t e r i n g of the two constituents. We s h a l l t r e a t these c o n t r i b u t i o n s together as a perturbation. t Since our fundamental triplets have hypercharge reversed from the normal Sakata case and we have taken them to transform as 3 (rather than 3"), our representation, 1O*, has the same quantum numbers as the conventional 10.
Volume 9, number 2
PHYSICS
T h e SU 3 i n v a r i a n t s c a t t e r i n g t e r m a r i s e s n a t u r a l l y a s a p e r t u r b a t i o n of the m e s o n s t a t e s in ~ c h w i n g e r ' s m o d e l , s i n c e i t i s only a p a r t i c u l a r m e c h a n i s m (B exchange) which l e a v e s s i n g l e t and o c t e t u n s p l i t . G r a p h s in which ~ and ~ a n n i h i l a t e i n t o B ' s a r e p a r t of our p e r t u r b a t i o n . F o r the b a r y o n c a s e the SU3 i n v a r i a n t V ~ s c a t t e r i n g t e r m i s an a d d i t i o n a l p a r a m e t e r which h a s b e e n i n t r o d u c e d in o r d e r to p u r i f y the s t a t e s (in the SU 3 s e n s e ) s u f f i c i e n t l y to obtain an a p p r o x i m a t e G e l l - M a n n - O k u b o r e l a t i o n 4). We obtain the following ( m a s s ) 2 r e l a t i o n f o r the nine v e c t o r m e s o n s t a t e s . (~-p)(cv-p) = 4(K*-p)(q~+c~-2K*)
LETTERS
1 April 1964
r e l a t i o n b e t w e e n 6M(3) and 8M(1) i m p l i e d by a definite t r a n s f o r m a t i o n p r o p e r t y of the s y m m e t r y breaking Hamiltonian is irrelevant. W e f o r m the e i g h t e e n ~+ s t a t e s f r o m ~/"S(6"), w h e r e S(6") i s a 6* r e p r e s e n t a t i o n of the group a c t i n g on V. S(6") could be c o m p o s e d of VVV for e x a m p l e 5). It would be quite difficult to d e a l with a l l t h e s e c o n s t i t u e n t s i n d i v i d u a l l y so we i n s t e a d s i m p l y a s s i g n a m a s s s p l i t t i n g between the t h r e e i s o t o p i c m u l t i p l e t s in S(6") in a c c o r d with the G e l l - M a n n Okubo r u l e (equal s p a c i n g in t h i s c a s e ) ,
(3)
(Eq. 4 of Schwinger) and a s i m i l a r equation for the nine 0" states. For the baryons we obtain, 2N+2.~-
Z- 3A=~
q(Z+A-N-X)(2;-A) (yo.+A_N_X)
(4)
We w i s h to m a k e the s a m e point about a l l such equations: if the ninth m e m b e r of the m u l t i p l e t i s s t r o n g l y s p l i t off f r o m the eight, then the G e l l Mann- Okubo r e l a t i o n s follow a u t o m a t i c a l l y . Thus putting in the e x p e r i m e n t a l a b s e n c e of a ninth 0m e s o n at low m a s s , we p r e d i c t the G e l l - M a n n Okubo r e l a t i o n for the 0" octet. Putting in Y * at 1405 MeV we p r e d i c t a s u i t a b l e s m a l l n u m b e r f o r 2 N + 2 ~ - x - 3 A . We cannot u s e (4) to p r e d i c t the Y * m a s s ; the n u m b e r i s too s e n s i t i v e to s m a l l c h a n g e s in the left hand s i d e (which would be z e r o if the G e l l - M a n n - O k u b o r e l a t i o n s w e r e e x a c t l y satisfied) t. F o r the v e c t o r m e s o n s , h o w e v e r , the d e p a r t u r e f r o m G e l l - M a n n - O k u b o i s s u f f i c i e n t to l e a d to a r e a l nonet m a s s f o r m u l a , eq. (3). In c o n t r a s t to S c h w i n g e r ' s m o d e l o u r s p r e d i c t s no f u r t h e r r e l a t i o n s f o r the b a r y o n m a s s e s ; the p a r a m e t e r s m a y be f i t t e d to the e x p e r i m e n t a l m a s s e s . However in the c o n s t i t u e n t t h e o r y one of the p a r a m e t e r s in the b a r y o n c a s e (the ~ m a s s splitting) i s the s a m e a s one in the b o s o n c a s e . T h i s l e a d s to the s i n g l e r e l a t i o n
My.- M~ = MK* - Mp
(5)
which i s s a t i s f i e d within the p r e c i s i o n of our considerations. A s t r i k i n g p r o p e r t y of the above m a s s s p l i t t i n g m e c h a n i s m i s that no p r o p e r t i e s of the f u n d a m e n t a l s y m m e t r y b r e a k i n g i n t e r a c t i o n w e r e invoked to obtain the G e l l - M a n n - O k u b o r u l e . F o r e a c h c o n s t i t u e n t only a s i n g l e m a s s d i f f e r e n c e , 5M(3)-SM(1), e n t e r e d the f o r m u l a , s o that the
ST=I ,
Y=0,
5m=0
ST = ½ ,
Y=-I,
5m = b
ST=O ,
Y = -2 ,
5m=2b
(6)
In the 18 t h e r e a r e two YI* and two ~z_*. In a d dition t h e r e a r e the N*(10) and ~-(10) f r o m the r e p r e s e n t a t i o n ten ; and N½*(8), Yo8(8) f r o m eight. Again we t a k e an SU 3 i n v a r i a n t s c a t t e r i n g for the ~JS s t a t e ; with a p a r a m e t e r ~. The r e s u l t s in t h i s c a s e a r e b e s t d i s p l a y e d in t e r m s of the o r i g inal p a r a m e t e r s . The p a r a m e t e r , a, i s the m a s s s p l i t t i n g , g / 3 - ~ l . The YI* and ~---1" m a s s e s will be 2 given by the e i g e n v a l u e s of the two by two m a t r i c e s l i s t e d below. State
5m
N~*(10)
- 0
2
Yz*(zo, 8)
a+2b
f~-(10)
-
N*(8)
~ x
2
70*(8)
~ x+b
For the ten states we automatically obtain equal spacing i f ;~ is large 3). We can obtain equal s p a c i n g for any value of X V we c h o o s e a = b. T a k i n g YI*(8) to be the 1660 MeV s t a t e 6) the obs e r v e d equal s p a c i n g of the d e c u p l e t 7) i m p l i e s the m a s s e s f o r the o t h e r octet m e m b e r s , N * - ~ 1510 MeV,
t"o* -~ 1660 MeV,
2
. ~ -. 1810 MeV.
(6)
2
t t Only ff the first order mechanism were exact we would be allowed to regard as exact our replacement of the mass splitting parameters by the observed masses on the left hand side of (4).
213
Volume 9, number 2
PHYSICS LETTERS
Since t h e s e c o n s i d e r a t i o n s w e r e a ll quite approximate we should allow p e r h a p s 50 MeV d e v i ations f r o m the above. We s u g g e s t a s e a r c h f o r these states. We have p r o f i t e d f r o m a n u m b e r of c o n v e r s a tions with Dr. V. B a r g e r . References 1) J. S, Schwinger, "A Ninth Baryon?", Harvard University, preprint. 2) M. Gell-Mann, Phys. Rev. 125 (1962) 1067; Y.Ne'eman, Nuclear Phys.26 (1961) 222; J.Wess, Nuovo Cimento 15 (1960) 52.
1 April 1964
3) R. Behrends, J. Dreitlein, C. Fronsdal and B.W. Lee, Revs. Mod. Phys. 34 (1962) 1. S. L. Glashow and J. J. Sakurai, Nuovo Cimente 25 (1962) 337. 4) M. Cell-Mann, Phys.Rev. 125 (1962) 1067; S. Okubo, Progress of Theoretical Physics, 27 (1962) 949. 5) The composition of the representation, 6, is discussed in: M. ]keda, S. Ogawa and Y. Ohnuki, P r o g ress of Theoretical Physics 27 (1962) 949. 6) M. Taher-Zadeh et al., Phys.Rev. Letters 11 (1963) 470. 7) Barnes et al., Phys.Rev. Letters 12 (1964) 204.
* * * * *
QUASI-ELEMENTARY MASSLESS BOSONS ASSOCIATED WITH THE QUANTUM ELECTRODYNAMICS OF JOHNSON, BAKER AND WILLEY* Y . NAMBU The Enrico Fermi Institute f o r Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois
Received 2 February 1964
R e c e n t l y J o h n s o n et al. 1) h a v e p r o p o s e d a new f o r m u l a t i o n of quantum e l e c t r o d y n a m i c s b a s e d on a n o n - p e r t u r b a t i v e p r o c e d u r e . T h e y c l a i m that the r e n o r m a l i s a t i o n c o n s t a n t s as well as the e l e c t r o n s e l f - e n e r g y can be m a d e finite, thus e l i m inating the d i v e r g e n c e p r o b l e m s **. We c o n s i d e r h e r e one p a r t i c u l a r a s p e c t of this t h e o r y , n a m e l y the p r o b l e m of m a s s l e s s b o s o n s r e s u l t i n g f r o m the spon t an eo u s breakdown of i n v a r i a n c e s . We a r g u e that they e x i s t (with c e r t a i n q u a l i f i c a t i o n s ) and a r e coupled to the e l e c t r o n with a s t r e n g t h c o m p a r a b l e to the e l e c t r o m a g n e t i c coupling, and h e n c e the t h e o r y in this p a r t i c u l a r c o n t e c t is not c o n s i s t e n t with the known p r o p e r t i e s of the e l e c t ron. The k ey points of the t h e o r y a r e as follows. 1. The Landau gauge is c h o s e n f o r the photon G r e e n ' s function so as to m a k e the e l e c t r o n and v e r t e x r e n o r m a l i s a t i o n c o n s ta n t s Z 2 and Z 1 (=Z2) finite (in the l o w e s t o r d e r ) . 2. The b a r e m a s s m o of the e l e c t r o n is z e r o . 3. The u n r e n o r m a l i z e d e l e c t r o n G r e e n ' s function * Work supported by the U. S. Atomic Energy Commission. ** There is, however, a certain difficulty concerning Z 3 . I thank Dr. Johnson for a private communication. 214
S (p) is obtained in the l o w e s t o r d e r " H a r t r e e Fock" approximation -
eo 2
(1) D~v(k ) : ( ~
- k~k~/~2) 1 / k 2 ,
1
w h e r e e o = Z ~ e is the b a r e c h a r g e . They find a s e l f - c o n s i s t e n t solution with the a s y m p t o t i c b eh av i o u r S - 1 0 ) = ~.p - i ~ 0 2)
~q,2 = _ ~ )
: m, ~ ( ~ ) ~ 0 2 ) -a ,
a ~ 3eo2/16~2 = 3 a o / ~
(2)
.
The b a s i c L a g r a n g i a n in the t h e o r y of Jo h n so n et al. has no m a s s t e r m n o r any c o n s t a n t which can supply a s c a l e of m a s s . It f o l l o w s e a s i l y that the equations of m o t i o n a r e i n v a r i a n t under i) V5t r a n s f o r m a t i o n ~(x) ~ e~p [iT~t]~(x) and ii) s c a l e t r a n s f o r m a t i o n ~(x) k ~ (kx), A ~(x) -. ~ A ~(kx). Th e solution with n o n - v a n i s h i n g rn r e q u i r e s a spontaneous b r eak d o w n of t h e s e i n v a r i a n c e s , and one should ex p ect , a c c o r d i n g to the. well known a r g u m e n t s 2), m a s s l e s s bosons ( c o l l e c t i v e e l e c t r o n - p o s i t r o n bound s t a t e s ) to e m e r g e as a r e s u l t T h e s e b o s o n s a r e p s e u d o s c a l a r and s c a l a r , r e -