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Intermediate vector mesons and unitary symmetry
Volume 8, number 5
INTERMEDIATE
PHYSICS
VECTOR
LETTERS
MESONS
AND
1 March 1964
UNITARY
SYMMETRY
*
S. O K U B O Department of Physics and Astronomy, Universily of Rochester, Rochester, N. Y.
Received 20 January 1964
The p u r p o s e of t h i s note i s to f o r m u l a t e a t h e o r y of the w e a k i n t e r a c t i o n in c o n n e c t i o n with the u n i t a r y s y m m e t r y m o d e l 1) b y i n t r o d u c i n g i n t e r m e d i a t e v e c t o r m e s o n s 2), both c h a r g e d and n e u t r a l . To t h i s end, we s h a l l a s s u m e f i r s t t h a t w e a k b a r y o n i c c u r r e n t s of the i n t e r a c t i o n a r e c o m p o n e n t s of s o m e u n i t a r y o c t e t s 3-5) and s e c o n d l y that the r e s u l t i n g e f f e c t i v e i n teraction for non-leptonic, strangeness-violating p r o c e s s e s i s a l s o a c o m p o n e n t 6-10) of a n o t h e r u n i tary octet. The underlying motivation for these ass u m p t i o n s i s due to the f a c t that t h e y a u t o m a t i c a l l y l e a d 11) to the A I = ½ r u l e f o r b o t h l e p t o n i c and nonleptonic strangeness-violating processes. Recently, D' E s p a g n a 12) h a s p r o p o s e d a t h e o r y s a t i s f y i n g s u c h conditions by introducing essentially five or six dist i n c t v e c t o r m e s o n s . In t h i s n o t e , we show t h a t the s a m e c a n be a c h i e v e d by u t i l i z i n g a m i n i m u m of f o u r d i s t i n c t v e c t o r m e s o n s . At the s a m e t i m e , we s h a l l note the p r e s e n c e of a s p e c i a l p r o p e r t y c h a r a c t e r i s tic of s u c h a t h e o r y . F i r s t of a l l , l e t u s c o n s t r u c t 13) two b a r y o n i c c u r r e n t s (F;Oba a n d (D).)ba out of the b a r y o n (and a n t i - b a r y o n ) o c t e t Bba (Bb a) w h e r e the l a t i n i n d i c e s r e f e r to the u n i t a r y s y m m e t r y a n d X i s an i n d e x s p e c i f y i n g the v e c t o r o r a x i a l v e c t o r n a t u r e of the current:
two a r b i t r a r y l i n e a r c o m b i n a t i o n s of Fba and Dba: .a
a
]b = ° ~ l l F b + a
a
Otl2D b,
a
(2)
a
Jb = c~21 F b + ~22Db , w h e r e otij (i,j = 1, 2, 3) a r e r e a l a r b i t r a r y n u m b e r s w h i c h m a y a s s u m e two d i f f e r e n t v a l u e s a c c o r d i n g to the v e c t o r and a x i a l - v e c t o r t y p e s of the c u r r e n t c o n s i d e r e d , r e s p e c t i v e l y . Note t h a t the C P - i n v a r i a n c e of t h e o r y d e m a n d s t h a t jb a and Jb a m u s t be self-conjugate, i.e.
(3) A f t e r t h e s e p r e l i m i n a r i e s , we s h a l l g i v e two d i f f e r e n t t h e o r i e s s a t i s f y i n g c o n d i t i o n s s t a t e d in the b e g i n n i n g of t h i s note. A s the f i r s t e x a m p l e , we i n t r o d u c e t h r e e c o m p l e x v e c t o r f i e l d W1, W2 and W 3 with c o m m o n m a s s m W w h e r e the f i r s t i s e l e c t r i c a l l y c h a r g e d w h i l e the two of the r e s t a r e n e u t r a l . T h e n , we a s s u m e the f o l l o w i n g i n t e r a c t i o n Hamiltonian: Hin t = H 1 + H2 + H 3 ,
(4a)
HI=g(l+J~+J~)Wl+g([+j21+J31)W 1 , (4b) H2= In eq. (1), the r e p e a t e d l a t i n i n d i c i e s (c a n d d) a r e u n d e r s t o o d to i m p l y a u t o m a t i c a l s u m m a t i o n s o v e r 1, 2 and 3, a s u s u a l a n d Qh a s s u m e s e i t h e r 7g o r 7 57p a c c o r d i n g to w h e t h e r the c u r r e n t u n d e r c o n s i d e r a t i o n i s of the v e c t o r o r a x i a l - v e c t o r t y p e . F o r s i m p l i c i t y , we s h a l l h e r e a f t e r o m i t the v e c t o r s u f f i x h f r o m a l l e x p r e s s i o n s . Two t r a c e l e s s t e n s o r s Fba , Dba s i g nify the u s u a l F a n d D c o u p l i n g s , r e s p e c t i v e l y . A c t u a l l y , we h a v e to a d d m e s o n c u r r e n t s to the r i g h t h a n d s : d e s of eq. (1), u n l e s s m e s o n s a r e bound 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 . H o w e v e r , a g a i n f o r s i m p l i c i t y , we s h a l l n e g l e c t t h e m . Now, l e t us i n t r o d u c e two o c t e t c u r r e n t s jb a and Jb a w h i c h a r e 362
+J
)W2 + g 0 2 + J
)W2,
(4c)
w h e r e l i s the l e p t o n i c c u r r e n t : l = be .7•(1
+75) e + ~-~t'vh(l+75 )U.
(5)
It i s e a s y to s e e t h a t e q s . (4) g i v e r i s e to the IoN lowing e f f e c t i v e f o u r - f e r m i o n i n t e r a c t i o n f o r the non-leptonic, strangeness-violating process: * Research supported in part by the U. S. Atomic Energy Commission.
Volume 8, number 5 H'
PHYSICS
2 3 = T 3 + S2 ,
(6a) ,
Tb =
[(ja.j~)
_ 55
b(Jd.
a
.c
LETTERS
T h e n , we s e e t h a t t h e f o l l o w i n g n o n - l e p t o n i c , strangeness-violating four-fermion interaction results :
(6b)
H'
W
In this derivation we have discarted non-leptonic, strangeness-conserving interaction of the following form:
\ H"=
g2 [~2 .c j 3 . J 3 ] m w 2 t J c "]2 + c
"
(7)
We n o t e t h a t eq. (6a) h a s the d e s i r e d t e n s o r c h a r a c t e r m e n t i o n e d in t h e b e g i n n i n g of t h i s p a p e r . So f a r , we d i d not a s s u m e a n y r e l a t i o n s h i p b e t w e e n two o c t e t c u r r e n t s jb a and Jb a. One i n t e r e s t i n g c a s e i s when we h a v e
1 March 1964
2 3 =T3+T 2 ,
a at g ~2rt4a jc~ ,6a..d .c.. Tb= ~n---~WItVc" b ' - 7 bUc'Jd)].
(lla)
(llb)
In t h i s d e r i v a t i o n , we h a v e u t i l i z e d the f a c t t h a t the i . e . , ja a = 0 a n d a l s o we o m i t t e d the s t r a n g e n e s s - c o n s e r v i n g , n o n - l e p t o n i c i n t e r a c t i o n of the f o l l o w i n g f o r m :
tensorJb a i s t r a c e l e s s ,
H"
n_~_W)
.1 .1 , 1 ,2~.1 .3 .2 3 2[j~.j21+JI")I]+~t~--~W LJ3.Jl+J3"j21"
We find that our Hamiltonian, eqs. (lla, b), is (12) exactly the same as that of the previous one, eqs. (6a, b, c), provided that we assume eq. (8). The (8) only difference between the two theories lies in two Jba= o t ' j ~ , different forms of the non-leptonic, strangenessw h e r e ~ i s a r e a l c o n s t a n t . We note t h a n t h a t eq. (4b) conserving interaction H" as is seen from eq. (7) a n d eq. (8) g i v e e s s e n t i a l l y t h e s a m e t h e o r y a s h a s and (12). Hence, it would be quite difficult to exb e e n s u g g e s t e d by C a b i b b o 5) (and a l s o b y D' E s p a g perimentally distinguish two theories. At any rate, n a 12)) if we m a k e one m o r e a s s u m p t i o n t h a t t h e we have found a theory requiring only four distinct v e c t o r p a r t of jb a i s p u r e l y of the F - t y p e c o u p l i n g vector mesons since only two complex fields W1 and w h i l e the a x i a l v e c t o r p a r t of jb a i s a m i x t u r e of W2 are used in this case (see eqs. (10)). b o t h F a n d D t y p e s . One i n t e r e s t i n g r e s u l t of a s Now we shall gice an application of our theory s u m i n g eq. (8) i s t h a t the t e n s o r Tba b e c o m e s s e i f by utilizing the self-conjugate properties of the c o n j u g a t e , i . e . , Sba= (Tab)+ = Tba in eq. (6), s o t h a t interaction Hamiltonian eq. (9) or eq. (11a). To ileq. (6a) b e c o m e s lustrate our point, let us consider various decay modes of baryons into one baryon and one meson, H' 2 3 and suppose that the resulting effective Hamiltonian = T3 + T2 • (9) for these decays is given by non-derivative local We will shortly see that this special form of eq. (9) Yukawa interaction involving two baryons and one will in general 9) lead to additional constraints meson. Then, the unitary symmetry together with among various decay parameters of non-leptonic the special self-conjugate property of eq. (9) tells processes without assuming conditions like R-conus that they must be composed of a sum of the foljugation 7 - 1 0 ) . lowing i n t e r a c t i o n s : R e t u r n i n g to the o r i g i n a l p r o b l e m , we s h a l l n e x t M 1 = T r (I3BPU) t u r n to the s e c o n d e x a m p l e of o u r t h e o r i e s . B e f o r e M2 = Tr (]3UBP) g o i n g into d e t a i l s , we n o t e t h a t the e x a m p l e p r e M3 = Tr (BPUB) s e n t e d j u s t in the a b o v e n e e d e d t h r e e c o m p l e x f i e l d s M4 = T r ( B P B U ) W1, W 2 a n d W3, a n d h e n c e t h a t we i n t r o d u c e d s i x (13) M5 =Tr (BBUP) d i s t i n c t v e c t o r m e s o n s . H o w e v e r , we s h a l l s h o w in M6 = Tr (BUPB) the b e l o w t h a t we c a n f o r m u l a t e a s e c o n d t h e o r y r e M 7 =Tr (BP) Tr(BU) q u i r i n g o n l y f o u r d i s t i n c t v e c t o r m e s o n s if we a s M 8 = T r (BU) T r ( B P ) s u m e eq. (8) f r o m t h e b e g i n n i n g . T h i s c a n b e M9 =Tr (BB) Tr(PU) a c h i e v e d w h e n we a s s u m e t h e f o l l o w i n g i n t e r a c t i o n H a m i l t o n i a n i n s t e a d of e q s . (4) : w h e r e we h a v e u s e d t h e m a t r i x n o t a t i o n s o t h a t M 1 f o r e x a m p l e s t a n d s f o r the f o l l o w i n g e x p r e s s i o n Hin t = H 1 + H 2 , (10a) -a b c d M1 = B b Bc P d Ua .1 .1 .2 3 -H 1 =g(l + 3 2 + a ' 3 3 )W1 + g ( [ + J 1 + ot . J l ) W 1 , (10b) * In this note, we use notations slightly different from 2 1 .3-that of ref. 3). For example, we have Bba = Nba, H2 = g(jll - ~ ' J 3 )W2 + g ( J l - ~ ' 3 2 )W2 " (10c) Bb a = Mba, Pb a =fb a in t e r m s of the notation of ref. 3). 363
Volume 8, number 5
PHYSICS
a n d w h e r e the t e n s o r Pba r e p r e s e n t s the p s e u d o scalar octet. Furthermore, the matrix U is a spur i o n m a t r i x of the f o r m : 0 0 0 U=(O O1 ~ ) '
(14)
w h i c h r e f l e c t s the s p e c i a l f o r m of o u r o r i g i n a l i n t e r a c t i o n H a m i l t o n i a n eq. (9)(or eq. ( l l a ) ) . M o r e o v e r , e a c h t e r m of eq. (13) a c t u a l l y c o n s i s t s of a s u m of s c a l a r and p s e u d o - s c a l a r t e r m s . A l t h o u g h we h a v e n i n e f o r m s in eq. (13), we h a v e one l i n e a r r e l a t i o n * a m o n g t h e m a n d h e n c e only e i g h t of M/ (i = 1, . . . , 9) a r e l i n e a r l y i n d e p e n d e n t . Now, if we t a k e a c c o u n t of the C P - i n v a r i a n c e of t h e o r y , t h e n we find that the s c a l a r and p s e u d o - s c a l a r p a r t s of o u r i n t e r a c t i o n s M i (i = 1, . . . , 9) m u s t e n t e r in the following combinations, separately: Scalar coupling : M1-M5, M3-M6, M7-M 8 P s e u d o - s c a l a r c o u p l i n g : M I + M 5 , M3+M6, M7+M8, M2' M4' M9 (15) T h u s , we s e e t h a t the s c a l a r p a r t of o u r e f f e c t i v e Hamiltonian contains only three arbitrary constants. T h e n , t h i s g i v e s one c o n s t r a i n t a m o n g the s c a l a r p a r t s of v a r i o u s d e c a y m e t r i x e l e m e n t s , a p a r t f r o m 1 c o n s e q u e n c e s due to the AI = ~ r u l e : M o r e p r e c i s e l y , when we w r i t e the m a t r i x e l e m e n t f o r the d e c a y process A - B + C as M(A-B+C)=[E~-ArZ~ A B then we o b t a i n **
LETTERS
1 March 1964
w h i l e the l e f t s i d e g i v e s - 1 . 5 0 . In t h i s e s t i m a t i o n , we a d o p t e d the n e g a t i v e s i g n s f o r b o t h Ms(Z+~p+zr
O) and
MS('-- ~ A+~-)
in eq. (18) s i n c e t h e y g i v e a b e t t e r a g r e e m e n t than o t h e r c h o i c e s . H o w e v e r , a s we s e e , the a g r e e m e n t i s r a t h e r g o o d . Of c o u r s e , we m u s t r e m e m b e r f i r s t that the v i o l a t i o n of the s t r i c t u n i t a r y s y m m e t r y m a y be q u i t e l a r g e and s e c o n d l y t h a t in our d e r i v a t i o n we a s s u m e d t h a t the e f f e c t i v e H a m i l t o n i a n c o n s i s t of a m i x t u r e of s c a l a r a n d p s e u d o s c a l a r c o u p l i n g s a l o n e . If we i n c l u d e v e c t o r a n d a x i a l v e c t o r p a r t s , then we d o n ' t g e t any r e l a t i o n s any m o r e of t h i s k i n d u n l e s s we a s s u m e R - i n v a r i a n c e . We a l s o n o t e t h a t e v e n if the i n t e r a c t i o n c o n s i s t s o n l y of v e c t o r a n d a x i a l - v e c t o r c o u p l i n g s i n s t e a d of s c a l a r a n d p s e u d o - s c a l a r o n e s , we d o n ' t o b t a i n any r e l a t i o n s a g a i n o t h e r than t h o s e i m p l i e d 1 by the AI = ~ r u l e . A n o t h e r c o n s e q u e n c e of o u r I-Iamiltonian eq. (9) i s c o n c e r n e d to KlO - 2n d e c a y . In t h i s c a s e , we c a n show in t h e s a m e w a y t h a t the e f f e c t i v e l o c a l Yukawa H a m i l t o n i a n r e s p o n s i b l e f o r t h i s d e c a y m u s t be of d e r i v a t i v e t y p e b e c a u s e n o n - d e r i v a t i v e H a m i l t o n i a n s c a n b e s h o w n to v a n i s h by a s i m i l a r r e a s o n i n g . H o w e v e r , t h i s e x a m p l e d o e s not g i v e a n y u s e f u l i n f o r m a t i o n at a l l . F i n a l l y , we r e m a r k t h a t o u r H a m i l t o n i a n eq. (4) o r (9) d o e s not p o s s e s s t h e s o - c a l l e d b a r y o n l e p t o n s y m m e t r y 13).
]½ f f ( A ) [ M s + Y 5 M p ] U ( B ) , C (16)
Ms(A~p+g- ) + 2Ms(-~--A+Tr-) =ff3MsC2;+-p+~° ) . F r o m the known d e c a y r a t e s a n d a s y m m e t r y (17) p a r a m e t e r s of t h e s e d e c a y s , we c a n c a l c u l a t e M s. W h e n we n o r m a l i z e Ms(A ~ p + n - ) = 1 f o r s i m p l i c i t y , t h e n we c o m p u t e
The a u t h o r w o u l d l i k e to e x p r e s s h i s g r a t i t u d e to D r s . R i a z u d d i n a n d A. Z i m m e r m a n f o r m a n y i l luminating conversations.
References
1) M.Gell-Mann, Phys. Rev. 125 (1962) 1067; Y.Neeman, Nuclear Phys.26 (1961) 222. 2) E . g . T . D . Lee and C.N.Yang, Phys. Rev. 119 (1960) 1410. M s ( ~ + - p + n °) = + 0 . 8 4 , Ms(Z--A+n-) : ~ 1.25 , 3) S.Okubo, P r o g r . T h e o r e t . P h y s . (Kyote) 27 (1962) 949. 4) N. Cabibbo and R.Gatto, Nuovo Cimento 21 (1962) 872. t h u s e n a b l i n g u s to c h e c k the v a l i d i t y of eq. (17). [18/'' 5) N. Cabibbo, Phys. Rev. Letters 10 (1963} 531. We f i n d the r i g h t - h a n d s i d e of eq. (17) to be - 1 . 4 5 , 6) E. Eberle and S. Iwao, Physics Letters 6 (1963) 302. 7) H.Sugawara, preprint. 8) K. Fujii and D. Ito, preprint. * This can be shown by using a method analogous to that 9) B.W. Lee, preprint. used in the appendix of ref. 3). 10) N . P . Chang, preprint. ** Eq. (17} has been also noted by M.Gell-Mann and by 11) E . g . B . D ' E s p a g n a and J . P r e n k t i , Nuovo Cimento 24 B. W. Lee and S. L. Glashow. The author would like to (1962) 497. e x p r e s s his gratitude to P r o f e s s o r J. Prentki for in12) B.D'Espagna, Physics Letters 7 (1963) 209. forming him of the existence of these p a p e r s and also for 13) R. E . Marshak, C. Ryan, T . K . Radha and K. Raman, pointing out an e r r o r included in the original manuscript. P h y s . R e v . Letters 11 (1963) 396.
364
INTERMEDIATE
PHYSICS
VECTOR
LETTERS
MESONS
AND
1 March 1964
UNITARY
SYMMETRY
*
S. O K U B O Department of Physics and Astronomy, Universily of Rochester, Rochester, N. Y.
Received 20 January 1964
The p u r p o s e of t h i s note i s to f o r m u l a t e a t h e o r y of the w e a k i n t e r a c t i o n in c o n n e c t i o n with the u n i t a r y s y m m e t r y m o d e l 1) b y i n t r o d u c i n g i n t e r m e d i a t e v e c t o r m e s o n s 2), both c h a r g e d and n e u t r a l . To t h i s end, we s h a l l a s s u m e f i r s t t h a t w e a k b a r y o n i c c u r r e n t s of the i n t e r a c t i o n a r e c o m p o n e n t s of s o m e u n i t a r y o c t e t s 3-5) and s e c o n d l y that the r e s u l t i n g e f f e c t i v e i n teraction for non-leptonic, strangeness-violating p r o c e s s e s i s a l s o a c o m p o n e n t 6-10) of a n o t h e r u n i tary octet. The underlying motivation for these ass u m p t i o n s i s due to the f a c t that t h e y a u t o m a t i c a l l y l e a d 11) to the A I = ½ r u l e f o r b o t h l e p t o n i c and nonleptonic strangeness-violating processes. Recently, D' E s p a g n a 12) h a s p r o p o s e d a t h e o r y s a t i s f y i n g s u c h conditions by introducing essentially five or six dist i n c t v e c t o r m e s o n s . In t h i s n o t e , we show t h a t the s a m e c a n be a c h i e v e d by u t i l i z i n g a m i n i m u m of f o u r d i s t i n c t v e c t o r m e s o n s . At the s a m e t i m e , we s h a l l note the p r e s e n c e of a s p e c i a l p r o p e r t y c h a r a c t e r i s tic of s u c h a t h e o r y . F i r s t of a l l , l e t u s c o n s t r u c t 13) two b a r y o n i c c u r r e n t s (F;Oba a n d (D).)ba out of the b a r y o n (and a n t i - b a r y o n ) o c t e t Bba (Bb a) w h e r e the l a t i n i n d i c e s r e f e r to the u n i t a r y s y m m e t r y a n d X i s an i n d e x s p e c i f y i n g the v e c t o r o r a x i a l v e c t o r n a t u r e of the current:
two a r b i t r a r y l i n e a r c o m b i n a t i o n s of Fba and Dba: .a
a
]b = ° ~ l l F b + a
a
Otl2D b,
a
(2)
a
Jb = c~21 F b + ~22Db , w h e r e otij (i,j = 1, 2, 3) a r e r e a l a r b i t r a r y n u m b e r s w h i c h m a y a s s u m e two d i f f e r e n t v a l u e s a c c o r d i n g to the v e c t o r and a x i a l - v e c t o r t y p e s of the c u r r e n t c o n s i d e r e d , r e s p e c t i v e l y . Note t h a t the C P - i n v a r i a n c e of t h e o r y d e m a n d s t h a t jb a and Jb a m u s t be self-conjugate, i.e.
(3) A f t e r t h e s e p r e l i m i n a r i e s , we s h a l l g i v e two d i f f e r e n t t h e o r i e s s a t i s f y i n g c o n d i t i o n s s t a t e d in the b e g i n n i n g of t h i s note. A s the f i r s t e x a m p l e , we i n t r o d u c e t h r e e c o m p l e x v e c t o r f i e l d W1, W2 and W 3 with c o m m o n m a s s m W w h e r e the f i r s t i s e l e c t r i c a l l y c h a r g e d w h i l e the two of the r e s t a r e n e u t r a l . T h e n , we a s s u m e the f o l l o w i n g i n t e r a c t i o n Hamiltonian: Hin t = H 1 + H2 + H 3 ,
(4a)
HI=g(l+J~+J~)Wl+g([+j21+J31)W 1 , (4b) H2= In eq. (1), the r e p e a t e d l a t i n i n d i c i e s (c a n d d) a r e u n d e r s t o o d to i m p l y a u t o m a t i c a l s u m m a t i o n s o v e r 1, 2 and 3, a s u s u a l a n d Qh a s s u m e s e i t h e r 7g o r 7 57p a c c o r d i n g to w h e t h e r the c u r r e n t u n d e r c o n s i d e r a t i o n i s of the v e c t o r o r a x i a l - v e c t o r t y p e . F o r s i m p l i c i t y , we s h a l l h e r e a f t e r o m i t the v e c t o r s u f f i x h f r o m a l l e x p r e s s i o n s . Two t r a c e l e s s t e n s o r s Fba , Dba s i g nify the u s u a l F a n d D c o u p l i n g s , r e s p e c t i v e l y . A c t u a l l y , we h a v e to a d d m e s o n c u r r e n t s to the r i g h t h a n d s : d e s of eq. (1), u n l e s s m e s o n s a r e bound 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 . H o w e v e r , a g a i n f o r s i m p l i c i t y , we s h a l l n e g l e c t t h e m . Now, l e t us i n t r o d u c e two o c t e t c u r r e n t s jb a and Jb a w h i c h a r e 362
+J
)W2 + g 0 2 + J
)W2,
(4c)
w h e r e l i s the l e p t o n i c c u r r e n t : l = be .7•(1
+75) e + ~-~t'vh(l+75 )U.
(5)
It i s e a s y to s e e t h a t e q s . (4) g i v e r i s e to the IoN lowing e f f e c t i v e f o u r - f e r m i o n i n t e r a c t i o n f o r the non-leptonic, strangeness-violating process: * Research supported in part by the U. S. Atomic Energy Commission.
Volume 8, number 5 H'
PHYSICS
2 3 = T 3 + S2 ,
(6a) ,
Tb =
[(ja.j~)
_ 55
b(Jd.
a
.c
LETTERS
T h e n , we s e e t h a t t h e f o l l o w i n g n o n - l e p t o n i c , strangeness-violating four-fermion interaction results :
(6b)
H'
W
In this derivation we have discarted non-leptonic, strangeness-conserving interaction of the following form:
\ H"=
g2 [~2 .c j 3 . J 3 ] m w 2 t J c "]2 + c
"
(7)
We n o t e t h a t eq. (6a) h a s the d e s i r e d t e n s o r c h a r a c t e r m e n t i o n e d in t h e b e g i n n i n g of t h i s p a p e r . So f a r , we d i d not a s s u m e a n y r e l a t i o n s h i p b e t w e e n two o c t e t c u r r e n t s jb a and Jb a. One i n t e r e s t i n g c a s e i s when we h a v e
1 March 1964
2 3 =T3+T 2 ,
a at g ~2rt4a jc~ ,6a..d .c.. Tb= ~n---~WItVc" b ' - 7 bUc'Jd)].
(lla)
(llb)
In t h i s d e r i v a t i o n , we h a v e u t i l i z e d the f a c t t h a t the i . e . , ja a = 0 a n d a l s o we o m i t t e d the s t r a n g e n e s s - c o n s e r v i n g , n o n - l e p t o n i c i n t e r a c t i o n of the f o l l o w i n g f o r m :
tensorJb a i s t r a c e l e s s ,
H"
n_~_W)
.1 .1 , 1 ,2~.1 .3 .2 3 2[j~.j21+JI")I]+~t~--~W LJ3.Jl+J3"j21"
We find that our Hamiltonian, eqs. (lla, b), is (12) exactly the same as that of the previous one, eqs. (6a, b, c), provided that we assume eq. (8). The (8) only difference between the two theories lies in two Jba= o t ' j ~ , different forms of the non-leptonic, strangenessw h e r e ~ i s a r e a l c o n s t a n t . We note t h a n t h a t eq. (4b) conserving interaction H" as is seen from eq. (7) a n d eq. (8) g i v e e s s e n t i a l l y t h e s a m e t h e o r y a s h a s and (12). Hence, it would be quite difficult to exb e e n s u g g e s t e d by C a b i b b o 5) (and a l s o b y D' E s p a g perimentally distinguish two theories. At any rate, n a 12)) if we m a k e one m o r e a s s u m p t i o n t h a t t h e we have found a theory requiring only four distinct v e c t o r p a r t of jb a i s p u r e l y of the F - t y p e c o u p l i n g vector mesons since only two complex fields W1 and w h i l e the a x i a l v e c t o r p a r t of jb a i s a m i x t u r e of W2 are used in this case (see eqs. (10)). b o t h F a n d D t y p e s . One i n t e r e s t i n g r e s u l t of a s Now we shall gice an application of our theory s u m i n g eq. (8) i s t h a t the t e n s o r Tba b e c o m e s s e i f by utilizing the self-conjugate properties of the c o n j u g a t e , i . e . , Sba= (Tab)+ = Tba in eq. (6), s o t h a t interaction Hamiltonian eq. (9) or eq. (11a). To ileq. (6a) b e c o m e s lustrate our point, let us consider various decay modes of baryons into one baryon and one meson, H' 2 3 and suppose that the resulting effective Hamiltonian = T3 + T2 • (9) for these decays is given by non-derivative local We will shortly see that this special form of eq. (9) Yukawa interaction involving two baryons and one will in general 9) lead to additional constraints meson. Then, the unitary symmetry together with among various decay parameters of non-leptonic the special self-conjugate property of eq. (9) tells processes without assuming conditions like R-conus that they must be composed of a sum of the foljugation 7 - 1 0 ) . lowing i n t e r a c t i o n s : R e t u r n i n g to the o r i g i n a l p r o b l e m , we s h a l l n e x t M 1 = T r (I3BPU) t u r n to the s e c o n d e x a m p l e of o u r t h e o r i e s . B e f o r e M2 = Tr (]3UBP) g o i n g into d e t a i l s , we n o t e t h a t the e x a m p l e p r e M3 = Tr (BPUB) s e n t e d j u s t in the a b o v e n e e d e d t h r e e c o m p l e x f i e l d s M4 = T r ( B P B U ) W1, W 2 a n d W3, a n d h e n c e t h a t we i n t r o d u c e d s i x (13) M5 =Tr (BBUP) d i s t i n c t v e c t o r m e s o n s . H o w e v e r , we s h a l l s h o w in M6 = Tr (BUPB) the b e l o w t h a t we c a n f o r m u l a t e a s e c o n d t h e o r y r e M 7 =Tr (BP) Tr(BU) q u i r i n g o n l y f o u r d i s t i n c t v e c t o r m e s o n s if we a s M 8 = T r (BU) T r ( B P ) s u m e eq. (8) f r o m t h e b e g i n n i n g . T h i s c a n b e M9 =Tr (BB) Tr(PU) a c h i e v e d w h e n we a s s u m e t h e f o l l o w i n g i n t e r a c t i o n H a m i l t o n i a n i n s t e a d of e q s . (4) : w h e r e we h a v e u s e d t h e m a t r i x n o t a t i o n s o t h a t M 1 f o r e x a m p l e s t a n d s f o r the f o l l o w i n g e x p r e s s i o n Hin t = H 1 + H 2 , (10a) -a b c d M1 = B b Bc P d Ua .1 .1 .2 3 -H 1 =g(l + 3 2 + a ' 3 3 )W1 + g ( [ + J 1 + ot . J l ) W 1 , (10b) * In this note, we use notations slightly different from 2 1 .3-that of ref. 3). For example, we have Bba = Nba, H2 = g(jll - ~ ' J 3 )W2 + g ( J l - ~ ' 3 2 )W2 " (10c) Bb a = Mba, Pb a =fb a in t e r m s of the notation of ref. 3). 363
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PHYSICS
a n d w h e r e the t e n s o r Pba r e p r e s e n t s the p s e u d o scalar octet. Furthermore, the matrix U is a spur i o n m a t r i x of the f o r m : 0 0 0 U=(O O1 ~ ) '
(14)
w h i c h r e f l e c t s the s p e c i a l f o r m of o u r o r i g i n a l i n t e r a c t i o n H a m i l t o n i a n eq. (9)(or eq. ( l l a ) ) . M o r e o v e r , e a c h t e r m of eq. (13) a c t u a l l y c o n s i s t s of a s u m of s c a l a r and p s e u d o - s c a l a r t e r m s . A l t h o u g h we h a v e n i n e f o r m s in eq. (13), we h a v e one l i n e a r r e l a t i o n * a m o n g t h e m a n d h e n c e only e i g h t of M/ (i = 1, . . . , 9) a r e l i n e a r l y i n d e p e n d e n t . Now, if we t a k e a c c o u n t of the C P - i n v a r i a n c e of t h e o r y , t h e n we find that the s c a l a r and p s e u d o - s c a l a r p a r t s of o u r i n t e r a c t i o n s M i (i = 1, . . . , 9) m u s t e n t e r in the following combinations, separately: Scalar coupling : M1-M5, M3-M6, M7-M 8 P s e u d o - s c a l a r c o u p l i n g : M I + M 5 , M3+M6, M7+M8, M2' M4' M9 (15) T h u s , we s e e t h a t the s c a l a r p a r t of o u r e f f e c t i v e Hamiltonian contains only three arbitrary constants. T h e n , t h i s g i v e s one c o n s t r a i n t a m o n g the s c a l a r p a r t s of v a r i o u s d e c a y m e t r i x e l e m e n t s , a p a r t f r o m 1 c o n s e q u e n c e s due to the AI = ~ r u l e : M o r e p r e c i s e l y , when we w r i t e the m a t r i x e l e m e n t f o r the d e c a y process A - B + C as M(A-B+C)=[E~-ArZ~ A B then we o b t a i n **
LETTERS
1 March 1964
w h i l e the l e f t s i d e g i v e s - 1 . 5 0 . In t h i s e s t i m a t i o n , we a d o p t e d the n e g a t i v e s i g n s f o r b o t h Ms(Z+~p+zr
O) and
MS('-- ~ A+~-)
in eq. (18) s i n c e t h e y g i v e a b e t t e r a g r e e m e n t than o t h e r c h o i c e s . H o w e v e r , a s we s e e , the a g r e e m e n t i s r a t h e r g o o d . Of c o u r s e , we m u s t r e m e m b e r f i r s t that the v i o l a t i o n of the s t r i c t u n i t a r y s y m m e t r y m a y be q u i t e l a r g e and s e c o n d l y t h a t in our d e r i v a t i o n we a s s u m e d t h a t the e f f e c t i v e H a m i l t o n i a n c o n s i s t of a m i x t u r e of s c a l a r a n d p s e u d o s c a l a r c o u p l i n g s a l o n e . If we i n c l u d e v e c t o r a n d a x i a l v e c t o r p a r t s , then we d o n ' t g e t any r e l a t i o n s any m o r e of t h i s k i n d u n l e s s we a s s u m e R - i n v a r i a n c e . We a l s o n o t e t h a t e v e n if the i n t e r a c t i o n c o n s i s t s o n l y of v e c t o r a n d a x i a l - v e c t o r c o u p l i n g s i n s t e a d of s c a l a r a n d p s e u d o - s c a l a r o n e s , we d o n ' t o b t a i n any r e l a t i o n s a g a i n o t h e r than t h o s e i m p l i e d 1 by the AI = ~ r u l e . A n o t h e r c o n s e q u e n c e of o u r I-Iamiltonian eq. (9) i s c o n c e r n e d to KlO - 2n d e c a y . In t h i s c a s e , we c a n show in t h e s a m e w a y t h a t the e f f e c t i v e l o c a l Yukawa H a m i l t o n i a n r e s p o n s i b l e f o r t h i s d e c a y m u s t be of d e r i v a t i v e t y p e b e c a u s e n o n - d e r i v a t i v e H a m i l t o n i a n s c a n b e s h o w n to v a n i s h by a s i m i l a r r e a s o n i n g . H o w e v e r , t h i s e x a m p l e d o e s not g i v e a n y u s e f u l i n f o r m a t i o n at a l l . F i n a l l y , we r e m a r k t h a t o u r H a m i l t o n i a n eq. (4) o r (9) d o e s not p o s s e s s t h e s o - c a l l e d b a r y o n l e p t o n s y m m e t r y 13).
]½ f f ( A ) [ M s + Y 5 M p ] U ( B ) , C (16)
Ms(A~p+g- ) + 2Ms(-~--A+Tr-) =ff3MsC2;+-p+~° ) . F r o m the known d e c a y r a t e s a n d a s y m m e t r y (17) p a r a m e t e r s of t h e s e d e c a y s , we c a n c a l c u l a t e M s. W h e n we n o r m a l i z e Ms(A ~ p + n - ) = 1 f o r s i m p l i c i t y , t h e n we c o m p u t e
The a u t h o r w o u l d l i k e to e x p r e s s h i s g r a t i t u d e to D r s . R i a z u d d i n a n d A. Z i m m e r m a n f o r m a n y i l luminating conversations.
References
1) M.Gell-Mann, Phys. Rev. 125 (1962) 1067; Y.Neeman, Nuclear Phys.26 (1961) 222. 2) E . g . T . D . Lee and C.N.Yang, Phys. Rev. 119 (1960) 1410. M s ( ~ + - p + n °) = + 0 . 8 4 , Ms(Z--A+n-) : ~ 1.25 , 3) S.Okubo, P r o g r . T h e o r e t . P h y s . (Kyote) 27 (1962) 949. 4) N. Cabibbo and R.Gatto, Nuovo Cimento 21 (1962) 872. t h u s e n a b l i n g u s to c h e c k the v a l i d i t y of eq. (17). [18/'' 5) N. Cabibbo, Phys. Rev. Letters 10 (1963} 531. We f i n d the r i g h t - h a n d s i d e of eq. (17) to be - 1 . 4 5 , 6) E. Eberle and S. Iwao, Physics Letters 6 (1963) 302. 7) H.Sugawara, preprint. 8) K. Fujii and D. Ito, preprint. * This can be shown by using a method analogous to that 9) B.W. Lee, preprint. used in the appendix of ref. 3). 10) N . P . Chang, preprint. ** Eq. (17} has been also noted by M.Gell-Mann and by 11) E . g . B . D ' E s p a g n a and J . P r e n k t i , Nuovo Cimento 24 B. W. Lee and S. L. Glashow. The author would like to (1962) 497. e x p r e s s his gratitude to P r o f e s s o r J. Prentki for in12) B.D'Espagna, Physics Letters 7 (1963) 209. forming him of the existence of these p a p e r s and also for 13) R. E . Marshak, C. Ryan, T . K . Radha and K. Raman, pointing out an e r r o r included in the original manuscript. P h y s . R e v . Letters 11 (1963) 396.
364