- Email: [email protected]
Internal symmetry implications of the Veneziano model in pseudoscalar-pseudoscalar scattering
~
Nuclear Physics B14 (1969) 437-457. North-Holland Publ. Comp., Amsterdm
INTERNAL SYMMETRY IMPLICATIONS OF THE VENEZIANO MODEL IN PSEUDOSCALAR-PSEUDOSCALA SCATTERING G. P. CANNING * $
Department of Theoretical Physics 12 Parks Road, Oxford, U.K. Received 4 February 1969 (Revised version received ] June 1969) Abstract: The leading order Veneziano model amplitudes for (JP) 0- × 0- -~ 0- × 0scattering** constrained by the absence of "exotic" resonances, is shown to predict the coupling constant relation of SU(3) coupled with the quark-model/SU(6) ideal singlet-octet mixing in the 1-, 2 +, nonets. The Adler self-consistency condition is found to hold (for the most part) both for pions and kaons fulfilling Lovelace's conjecture.
i. INTRODUCTION W e a r e c o n c e r n e d h e r e with two m a i n c o n s i d e r a t i o n s : r e l a t i o n s a m o n g t h e c o u p l i n g c o n s t a n t s of the 1 - a n d 2 + n o n e t s of r e s o n a n c e s to the 0 - n o n e t of e x t e r n a l p a r t i c l e s i m p l i e d b y the V e n e z i a n o m o d e l [1], a n d t h e m a s s r e l a t i o n s b e t w e e n the e x t e r n a l a n d i n t e r n a l p a r t i c l e s i m p l i e d by t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n [2] a s a p p l i e d to the V e n e z i a n o m o d e l by m e a n s of L o v e l a c e ' s [3] g a m m a f u n c t i o n d e n o m i n a t o r m e c h a n i s m . T h e l i n k b e t w e e n t h e s e two i s s h o w n e x p l i c i t l y in sect. 8, to follow f r o m q u a r k m o d e l d u a l i t y d i a g r a m s l i n k e d with a q u a r k - t y p e m a s s f o r m u l a , which r e p r o d u c e s o u r " i d e a l " m a s s e s of t a b l e 1. P r e v i o u s l y it h a s b e e n shown [5 - 8] that H a r a r i ' s [5] c o n j e c t u r e t a k e n with the a s s u m p t i o n of the a b s e n c e of exotic r e s o n a n c e s ( a s s u m p t i o n s that we a l s o m a k e ) i m p l y i n d e p e n d e n t l y of any m o d e l t h a t t h e t h r e e s e t s of t r a j e c t o r i e s p, ¢o, A2, f a n d q~, f ' a n d K*K N m u s t b e d e g e n e r a t e a m o n g s t t h e m s e l v e s , with in p a r t i c u l a r z e r o c o u p l i n g of f ' to ~u b u t not to KK, a n d t h a t the R e g g e c o u p l i n g c o n s t a n t s a r e r e l a t e d by U(3) q u a r k m o d e l n o n e t s y m m e t r y (and a r e e q u a l i n the c a s e of e x c h a n g e d e g e n e r a t e t r a j e c t o r i e s ) . By a c o n t i n u a t i o n (not s p e c i f i e d a p r i o r i ) f r o m t h e h i g h - e n e r g y r e g i o n ( f o r w a r d o r b a c k w a r d ) of one c h a n n e l into the low e n e r g y r e g i o n of the c o r t e s * Work supported by an S. R. C. Grant. ** While in preparation we were informed of the presentation by Kawarabayashi et al. [4] of a paper that contains results s i m i l a r to ours. Present Address: Dept. of Physics and Astronomy Tel-Aviv University, T e l - A v i v Israel.
438
G . P . CANNING
p o n d i n g c r o s s e d c h a n n e l , t h e s a m e r e l a t i o n s a r e f o u n d to h o l d f o r t h e a c t u a l c o u p l i n g s to t h e r e s o n a n t p a r t i c l e s , b u t t h e s e r e l a t i o n s a r e o n l y o b t a i n e d within the three sets. The Veneziano model by its explicit expression for resonance-Regge p o l e d u a l i t y s p e c i f i e s t h e n e c e s s a r y c o n t i n u a t i o n ; a n d a l s o , s i n c e it i n c o r p o r a t e s c r o s s i n g b e t w e e n a l l t h r e e c h a n n e l s , it a l l o w s u s to c o m p a r e t h e c o u p l i n g c o n s t a n t s in d i f f e r e n t c h a n n e l s , a n d t h e r e f o r e to o b t a i n r e l a t i o n s a m o n g t h e c o u p l i n g c o n s t a n t s of r e s o n a n c e s t h a t do not b e l o n g to t h e s a m e o n e of t h e t h r e e s e t s - f o r e x a m p l e in KTr -~ K~ we o b t a i n a r e l a t i o n gzv.K~.~ ¼gpnn g p K K ( t h i s w a s c o n s i d e r e d in o u r p r e v i o u s p r e p r i n t [9] a s a p r e l i m i n a r y to t h i s p a p e r ) ; a n d in p a r t i c u l a r t h e V e n e z i a n o m o d e l g i v e s u s t h e s e r e s u l t s d e s p i t e t h e n o n - d e g e n e r a c y of t h e t h r e e s e t s of t r a j e c t o r i e s (i.e. n o n - d e g e n e r a c y of t h e n o n e t m a s s e s ) . W e a r e a l s o e n a b l e d to c o n s i d e r s u c h r e a c t i o n s a s n~? -* ~V w h e r e t h e l a c k of e x o t i c c h a n n e l s l e a v e s u s with o t h e r w i s e l i t t l e to s a y .
2. M E T H O D W e a s s u m e H a r a r i ' s [5] c o n j e c t u r e : t h e P o m e r a n c h u k R e g g e t r a j e c t o r y i s a p a r a m e t r i s a t i o n of t h e d i f f r a c t i v e b a c k g r o u n d , a n d d o e s not c o r r e s p o n d to a " d u a l " t r a j e c t o r y of r e s o n a n c e s in t h e c r o s s e d c h a n n e l h a v i n g t h e q u a n t u m n u m b e r s of t h e v a c u u m . W e c o n s t r u c t a V e n e z i a n o m o d e l f o r t h a t p a r t of t h e a m p l i t u d e t h a t i s m a d e up of " d u a l " r e s o n a n c e s in t h e l o w - e n e r gy r e g i o n a n d R e g g e p o l e s in t h e h i g h - e n e r g y r e g i o n : o u r r e s u l t s a r e t h e n i n d e p e n d e n t of t h e e x a c t n a t u r e of t h e P o m e r a n c h u k c o n t r i b u t i o n . W e m a k e u s e of r e a c t i o n p r o c e s s e s , i n v o l v i n g two i d e n t i c a l (eg. Kn -~ Kn a n d Kff -~ K~) o r m e r e l y t i m e r e v e r s e d c h a n n e l s (eg. Kn -~ K~? a n d K ~ --* K~) w h i c h we l a b e l t h e s - a n d u - c h a n n e l s , t h e t h i r d ( a n d u s u a l l y t h e d i s t i n c t l y d i f f e r e n t ) c h a n n e l we l a b e l t h e t - c h a n n e l . W i t h t h e u s u a l d e f i n i t i o n s of t h e c r o s s e d c h a n n e l s , a n d in p a r t i c u l a r t h e isospin channel amplitudes:
where ample where obtain
s: a+b-~ c+d,
S a b , c d = (ab Is) (cdls) S (s) ,
u: a + ~ - - * b + d ,
Vae, bd
t: a+d
Tad, cb = (sdlt) ( c b l t ) ~ t ) ,
-~ c + b
,
= (aft[u) (bdlu) U (u) ,
t h e s u p e r f i x s, u , t i s u s e d h e r e f o r a n i s o s p i n , a n d w h e r e f o r e x (ab Is) i s a s h o r t h a n d f o r t h e SU(2) C l e b s c h - G o r d a n : ( a b I c s ) ~ , fl, a r e t h e I 3 c o m p o n e n t s of t h e p a r t i c l e s with isosp-i'n~a,/~:~we t h e c r o s s i n g m a t r i c e s [10] d e f i n e d by t h e e q u a t i o n s :
V (u) = X u s S (s) ,
T (t) : Xts S (s) .
U s i n g t h e )(us c r o s s i n g m a t r i x t o g e t h e r with t h e c o n d i t i o n t h a t s(S)(s, t, u) = U(U)(u, t, s) f o r u= s we f i n d S ( S ) ( u , t, s) in t e r m s of S(S')(s, t, u): t h e n c r o s s i n g to t h e T - c h a n n e l with Xts a n d e m p l o y i n g t h e r e l a t i o n f o u n d a b o v e , w e f i n d t h a t T (t) u n d e r s ~ u i s (+), (-) with a l t e r n a t i n g i s o s p i n t.
INTERNAL SYMMETRY IMPLICATIONS
439
We then c o n s t r u c t the t - c h a n n e l i s o s p i n amplitude T(t) having the a p p r o p r i a t e t - c h a n n e l r e s o n a n c e t r a j e c t o r i e s , and we c r o s s to the s - c h a n n e l u s ing X s t and i m p o s e the f u r t h e r condition that the s - c h a n n e l r e s o n a n c e t r a j e c t o r i e s a p p e a r in the c o r r e c t isospin channels. In p a r t i c u l a r we demand the a b s e n c e of "exotic" r e s o n a n c e s f r o m the a p p r o p r i a t e isospin r e a c t i o n channels. T h i s method of c o n s t r u c t i o n is p a r t i c u l a r l y useful in the c a s e of nons e i f - c o n j u g a t e p a r t i c l e s , w h e r e the u s e of SU(2) isospin index t e n s o r s is not t r i v i a l l y easy. Our p r e s c r i p t i o n is t h e r e f o r e to take all known r e s o n a n c e s in the s, t, u channels and f o r m f r o m t h e m all p o s s i b l e Veneziano t e r m s which a r e then to be combined using the isospin c r o s s i n g m a t r i c e s a s indicated above in o r d e r to obtain the m o s t g e n e r a l fit to the phenomenology. We employ the following notation f o r the m o s t g e n e r a l type of Veneziano t e r m having only single poles: r ( m - as(S)) r (n - at(t))
w h e r e the dash denotes the explicit non-dependence of the t e r m on the M a n d e l s t a m v a r i a b l e indicated by the o r d e r i n g s, t, u; and with m, n, p all (+) i n t e g e r s . In this notation the s u f f i c e s s, t a r e the p a r t i c l e s y m b o l s l a belling the r e s o n a n c e t r a j e c t o r i e s that a p p e a r in the s - and t - c h a n n e l s ; we d i f f e r e n t i a t e isospin d e g e n e r a t e t r a j e c t o r i e s only in the c h a n n e l s w h e r e they give r i s e to r e s o n a n c e s , in o r d e r to c l a r i f y the couplings involved. We s p e c i a l i z e to r e s o n a n c e t r a j e c t o r i e s having no spin z e r o p a r t i c l e s on t h e m (i.e. ones f o r which a h a s a (+) intercept), f o r which the leading t r a j e c t o r y t e r m s h a s m =n =p = 1. Our "leading t e r m " model is p r e c i s e l y to include only such t e r m s a s these. A "leading t e r m " s i g n a t u r e (+) in the s - c h a n n e l a p p e a r s a s the c o m b i n a tion: (ls~-l)+(sl+-ul
1)
and we i n t e r p r e t an u n s i g n a t u r e d t e r m like (s1 ~ - 1) a s a s u m of an (+) and (-) s i g n a t u r e exchange d e g e n e r a t e t r a j e c t o r y : (1 t l _ l ) = ½ [ ~ +
1
1)+~+_1
1)1+½[(1 - 1 _ 1 ) _ ~ _ _ 1
1)1.
The s i g n a t u r e d t e r m then h a s p o l e s at as+(S) =J>~l with r e s i d u e : 1
( J - 1)!
Of'
LI(I+(-)J)aJzJ +aJ - 1 zJ- I J{At T(_)J Au +(l+(-)J)B}+...~. ]
Here a
At,.
=-~qs
q' a ' s ,
m +o o
.-il ,
440
G . P . CANNING
B --
j_ no+
w h e r e we h a v e t a k e n as+(S ) = a°++ a ' s with u n i v e r s a l s l o p e a ' , a n d w h e r e t h e c o s i n e of t h e s c a t t e r i n g a n g l e i s d e f i n e d : c o s 0 s =_Zs -
Pc'Pc
lal
(c.m.) for process a+b ~ c+d.
Ipc I
t ~ ½ qsqsZ+ z-independbnt term . u ~ -2 qs qs z + z - i n d e p e n d e n t t e r m . On t a k i n g a p a r t i a l w a v e p r o j e c t i o n a t t h e r i g h t s i g n a t u r e p o i n t s we f i n d a r e s o n a n c e of s p i n J w h o s e p a r t i a l w a v e r e s i d u e i s
2 aJcj (J
zJ w h e r e Pj(z) = -c-j+ . . .
1)!a' -
,
w i t h "odd d a u g h t e r s " , s p i n J - 1 u n l e s s t h e m a s s c o n f i g u r a t i o n i s E E o r EU a n d a t = a u , a n d a l w a y s with " e v e n d a u g h t e r s " , s p i n J - 2 . A t w r o n g s~gnat u r e p o i n t s we f i n d a r e s o n a n c e of s p i n J - 1 , u n l e s s b y c h a n c e a ' E m ~ + a sO+ + a ~ + a o 1 = 0, w h i c h we c a l l a " m a s s r e c u r r e n c e " of t h e s p i n J - 1 p a r t i c l e at a s + ( S ) = J - 1 with m a s s i n c r e m e n t of a ' - 1 F i n a l l y we n o t e t h a t t h e c o m b i n a t i o n , (1 } _ 1 ) _ ( 1 },_ 1) h a s no l e a d i n g a s trajectory resonances, only the daughter terms. S i n c e t h e i n v a r i a n t i s o s p i n F e y n m a n n a m p l i t u d e c o r r e s p o n d i n g to a J s p i n p r o p a g a t o r a t t h e p o l e m ~ in t h e s - c h a n n e l i s [11]: v
gJ m2 gJ ) PJ{Zs) (qs q's )J 2 - J Cj ,
(s
i
el
w e c o m p a r e ~ i t h t h e p a r t i a l w a v e p r o j e c t i o n of t h e r e s i d u e a t as(S) =J (a right signature point):
2aJ c j
(J-1)!
a'
to o b t a i n ,
gJgJ
2a 'J-1
w h i c h i s i n d e p e n d e n t of t h e k i n e m a t i c f a c t o r s d e p e n d i n g on t h e e x t e r n a l p a r t i c l e m a s s e s (at l e a s t f o r l e a d i n g t r a j e c t o r i e s ) . H e r e t h e g j , g ] a r e t h e c o u p l i n g c o n s t a n t s of t h e s p i n J r e s o n a n c e , w h i c h b e i n g i s o s p i n i n v a r i a n t s a r e d e f i n e d in t e r m s of t h e p h y s i c a l c o u p l i n g to a d e c a y p r o c e s s :
INTERNAL SYMMETRY IMPLICATIONS
441
e gscYd 5= (~ dOi sv + 5 ) g s' c d ,
so that in the amplitude Sab ' cd = ( a b l s ) ( c d l s )
s(S) we have:
Sab, cd ~ ( a S I s ) ( c d } s ) g ~ d b g s c d , io e .
, (_)a+b+s s(S) ~ gg~D g s c d since
(~ I~ =(~ ~i ~ ~) : ¢~~ ~t ~+~~) ( ~+ ~+ ~ = ( a b l s ) (_)a+ b+ s Since in a c r o s s e d channel, e.g. the t-channel the s i g n a t u r e d c o m b i n a tion g i v e s a contribution to the 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 which f o r l a r g e t is given by: F(1-~(s))(ei~°~(S)+lj
o~(t) °~(s) .
the Regge r e s i d u e of the pole i.e. the i s o s p m i n v a r i a n t coupling constant of the R e g g e - t r a j e c t o r y to the e x t e r n a l p a r t i c l e s is given: ~(s) : ~[r(~(s))]
-1 ,
since F(1 - a(s)) = ~[F(~(s)) s i n ( ~ a ( s ) ) ] -1 and so the t r a j e c t o r y d e g e n e r a c y of the p a r t i c l e s in the 1-, 2 + nonets i m p l i e s on the Veneziano model a d e g e n e r a c y of the Regge t r a j e c t o r y coupling c o n s t a n t s , which is i m p o r t a n t in c o n s i d e r i n g the high e n e r g y i m p l i c a t i o n s of exchange d e g e n e r a c y and i s o s p i n - e x c h a n g e d e g e n e r a c y (see i n t r o d u c t o r y section). Note: even if e x t r a daughter t r a j e c t o r i e s in t e r m s with n = m , n - p >0, i.e. with daughter t r a j e c t o r i e s linked with daughter t r a j e c t o r i e s and not with leading t r a j e c t o r i e s (for which n ¢ m , n - p =0, m - p > 0 say), then our p r e d i c t i o n s on the coupling c o n s t a n t s of the leading t r a j e c t o r i e s " r e m a i n unchanged. We r e j e c t a l t o g e t h e r e x t r a leading t e r m s like n = m =p >1, which would r e d u c e c o n s i d e r a b l y the p r e d i c t i v e value of the model.
3. ~
-~ g~ SCATTERING
F i r s t we w r i t e down the well-known [3] Veneziano model a m p l i t u d e f o r
442
G.P. CANNING
the uu -~ ~~ r e a c t i o n s , f o r which the t h r e e c h a n n e l s s, t, u a r e i d e n t i c a l (but with note: z s ~ (+)t, z u ~ (+)t, z t ~ (+)s) and f o r which the i s o s p i n crossing matrices are:
Xus = ~
(i0xo -
3 5 3 1
,
Xts : ~
(i0)lO 3 -3
w h e n c e we 4ind S(2), S (0) is t ~ u ( + ) and S(1) is t ~ u ( - )
,
w h e n c e we obtain:
1 1) s(;) =g
0
o
-
1 _ 1) .~_(lf - p11) I _ ~g(-p 1 1 p11)
p
'
w h e r e we have t a k e n c a r e that t h e r e should be no r e s o n a n c e in S (2). W h e r e we h a v e b e e n c o n s t r a i n e d by the m o d e l to t a k e a p = a f i s o s p i n exchange degeneracy. In t h i s m o d e l , it is i m p o s s i b l e f o r a coupling to the f ' to b e i n t r o d u c e d , without a l s o a p ' ( e x p e r i m e n t a l l y u n o b s e r v e d ) i s o s p i n e x c h a n g e d e g e n e r a t e with the f'. F o r , w e r e it to c o u p l e , we would h a v e f o r the f ' coupling, t e r m s a p p e a r i n g in S(0) but not S(1) o r S(2), of the f o r m g , [ ( 1 1 _ 1) + (fl _ 1 1)]+ g .... I _1< f ' 1 - 1 ) + ( 1 ' - 1 1 ) ] ,
whiehtermsg'(fl,
o1, - 1), g_,,,1 tf p1 - 1) a p -
p e a r i n g in T(2) u n d e r the i s o s p i n c r o s s i n g g i v e n by X t s , i m p l y g' =g"= 0 in o r d e r f o r t h e r e to b e no I= 2 r e s o n a n c e s , thus p r e v e n t i n g the coupling of f ' in s(O). M o r e o v e r , given s u c h a p ' , its e x c h a n g e - d e g e n e r a t e p a r t n e r the A~ ( p r e d i c t e d by the coupling of the p ' in the KI{ -~ KK r e a c t i o n , f o r e x a m p l e ) would h a v e the m a s s of the f ' , so m a k i n g f ' i n s t e a d a m e m b e r of an octet. T h u s , if the f ' i s t r u l y ( m i x e d with f) a SU(3) s i n g l e t , then it cannot c o u p l e to g~ - a p r e d i c t i o n of the q u a r k m o d e l f o r f, f ' mixing. A s L o v e l a c e [3] p o i n t e d out, the A d l e r s e l f - c o n s i s t e n c y condition is s a t i s f i e d by the m a s s r e l a t i o n : 1 =(~ '(2rn 2 - 2rn 2) giving a '-1 = 1.14 GeV 2.
4. K~ -+ K~ S C A T T E R I N G F o r K~ ~ KTr we define the i s o s p i n r e a c t i o n c h a n n e l s :
S(½ ), S(~ ), U(½), U(½): K~ ~ K~ .
T(0), T(1): I ~ - ~ ~ , f o r which the i s o s p i n c r o s s i n g m a t r i c e s a r e :
X u s = 3(-12 4)
Xis = ~ ( ? '
2~/g~ -2
/
"
INTERNAL SYMMETRY IMPLICATIONS
443
W h e n c e we find T(1) is s ~ u (-) and T(0) is s ~ u (+); and a s s u m i n g h e r e in p a r t i c u l a r the a b s e n c e of I = ~ r e s o n a n c e s , we obtain: T(1) =g
E(1 1-1)-(*p
1
[(11 ,
T(0) = ~ C 6 g
=g
( 11 1) - p K*
K* 0
;
)(111)
f1- 1 + - f K *
]
'
( 1 1 - 1 ) 1 ( - 1P 1 ) -~g K* 1 S(½) = ~ g K* p w h e r e we have b e e n c o n s t r a i n e d to take a p = a f a s a b o v e and a K * = ~ K N e x c h a n g e d e g e n e r a c y , with only one independent coupling constant. We note that the p l a u s i b l e l e a d i n g t e r m ( 1 . _ 1 , 1) cannot a p p e a r in T(0) s i n c e u n d e r c r o s s i n g by X s t , it would give r i s e to a r e s o n a n c e in S(~). H e r e too the coupling to f ' of ~n is z e r o , and again b e c a u s e t h e r e is no i s o s p i n e x c h a n g e d e g e n e r a t e p ' p a r t n e r , which would be n e c e s s a r y to c a n cel out the r e s o n a n t t e r m like ( 1 . ~, _ 1) f r o m the S(~) channel. H e n c e we find, by c o n s i d e r i n g the r e s i d u e s at the p o l e s in the a p p r o p r i ate c h a n n e l s , the following coupling c o n s t a n t r e l a t i o n s : 2
3
(1-) g K * K : = : g p : : g p ~ (2+) g K N K : - 2 ~ g f ~ u
gfK~K
T h e f o r m e r , s e e appendix 1, is a p u r e SU(3) r e s u l t while the l a t t e r , see appendix 2, f o l l o w s f r o m SU(3) with the ideal f, f ' s i n g l e t o c t e t mixing p r e d i c t e d f r o m the z e r o coupling of f ' to v~. We note that the s a m e r a t i o is not found f o r the c o u p l i n g s of the 0 + d a u g h t e r octet, b e c a u s e the e x t e r n a l p a r t i c l e m a s s d i f f e r e n c e m a n i f e s t s itself in the s - and u - c h a n n e l s but not in the t - c h a n n e l . H e r e in the l i m i t a s we t a k e one pion f o u r - m o m e n t u m to z e r o , i.e. at the point s = u = m~,~ t = m 2, o r in the limit~as we takeo one k a o n f o u r - m o m e n t u m to z e r o , i.e. at the point s= u= m~, t= m~, we find that the A d l e r s e l f - c o n s i s t e n c y condition is s a t i s f i e d by the v a n i s h i n g of all the t e r m s owing to a z e r o in the a r g u m e n t of the g a m m a function d e n o m i n a t o r s if:
, 2
1 = a (m~+ m2K,- m 2 - m2K) i. e. ot ,-1 = 1.12 GeV 2 compare
444
G. P. CANNING c~'-1 = 1.215 GeV 2
f r o m OlK, = ~KN
~,-1 = 0.98 GeV 2
f r o m ~p
)
:
~f
.
5. KI~ -, KI{ S C A T T E R I N G F o r the KI{ ~ KK s c a t t e r i n g we define i s o s p i n r e a c t i o n c h a n n e l s : S (0), S(1); V (0), V(1): KK ~ KK T (0), T(1): KK ~ KK f o r which the i s o s p i n c r o s s i n g m a t r i c e s a r e : t
t
XUS = 3 ( 1 1 - 3 1 ) '
-i
3
Xts =3( 1 1 ) '
w h e n c e we find T (1) is s ~- u (+) and T(0) is s ~ u (-). A s s u m i n g then the a b s e n c e of any T - c h a n n e l r e s o n a n c e s (for which S = 2), we obtain:
r(0l=g
-
P
1 -( P
~
1
'
s(o/: iag,+gl (lo 1(; 0 + (1
- °1 1 )
1) 3
-
w h e r e we h a v e b e e n c o n s t r a i n e d to t a k e ~p =C~A2, ~ c o = ~ f , oz~o= ~ f ' - e x c h a n g e d e g e n e r a c y and n o n - d e g e n e r a c y of ~ f , with ~ f f o r T(0~r¢ 0 and the u s u a l i s o s p i n e x c h a n g e d e g e n e r a c y ~ = ~ ,x which c o u p l e d with the a b o v e i m p l i e s rnf = mA2 , moo = rno, and w h e r e we have m a d e u s e of the a b s e n c e of a p ' ( m ¢ ) and the p r e s e n c e ' o f a ~v(0+rn~), to d e t e r m i n e that only the d a u g h t e r c o m b i n a t i o n of the ~ t r a j e c t o r y s h o u l d a p p e a r in the S (1) channel. By c o n s i d e r i n g the r e s i d u e s at the p o l e s in the a p p r o p r i a t e c h a n n e l s , the following c o u p l i n g c o n s t a n t r e l a t i o n s a r e obtained: 2 • . 2 = g' (2+) g~_2KK" g~fKg:" g f ' K K (1-)
g2 . g2 . g2 = g' pKI{" wKI{" ~ p K I ~
-g+g":3g'+g+
3g"
: - 4g.
-g+g":3g'+g+ 3g" : - 4 g .
INTERNAL SYMMETRY IMPLICATIONS
445
F o r t h e (1-) o c t e t , s e e a p p e n d i x 1, we f i n d t h a t :
2 +2 3 2 g¢oKK g(pKV. = gpKK w h i c h i s an e x a c t SU(3) c o n d i t i o n s a t i s f i e d f o r a l l r e l a t i v e v a l u e s of g, g', g " . F o r t h e (2 +) o c t e t , s e e a p p e n d i x 2, we u s e t h e p r e v i o u s r e s u l t t h a t gf,u~ =0 to fix t h e r a t i o G/F=(1/~2) c o t 0 2 , a n d t h e n f o r SU(3) to h o l d we r e q u i r e a f u r t h e r c o n d i t i o n on t h e c o u p l i n g c o n s t a n t s : -g =g' +g" w h i c h t h e n d e t e r m i n e s t h a t c o t a 2 = v ~ a n d c o t 01 = ~f2 so t h a t t h e q u a r k m o d e l m i x i n g i s a g a i n s t r o n g l y i n d i c a t e d f o r b o t h t h e 1 - a n d 2 + o c t e t s with t h e c o u p l i n g c o n s t a n t s now in t h e r a t i o s : .~2 =1:1:2 gA2KI~".~2 ~fKI(" ~'f 'KI~ g 2 K i~ .
2 .g2 --= 1:1:2 "g~oKI~ " (pKK
i n d e p e n d e n t of t h e r e l a t i v e m a g n i t u d e of g', g'. W e n o t e t h a t t h e t e r m s l i k e ( 1 _pl 1) s a t i s f y t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n q u i t e w e l l , f o r in the l i m i t a s we t a k e one k a o n f o u r - m o m e n t u m to z e r o , i.e. a t t h e p o i n t s =u =t =m 2, t h e a r g u m e n t of t h e g a m m a f u n c t i o n d o m i n a t o r v a n i s h e s if:
1 =ot'(m2+m 2q~-2m2 )
i.e. if ot ' - 1 = 1 . 1 4 G e V 2 .
T h e c o n d i t i o n -(g+g') =g" d e r i v e d a b o v e , m a y t h e n b e s e e n a s an a p p r o x i m a t e c o n d i t i o n t h a t t h e o t h e r two t e r m s o n e , ( 1 _ 1 1), (+) a n d one, ( 1 _ 1 1), (-) a t t h a t p o i n t s h o u l d t e n d to c a n c e l . W e note'th~/t t h e a b o v e c o n d i t i'o~n i~" s a l s o s a t i s f i e d f o r t h e p a r t i c u l a r l y s i m p l e c h o i c e g' = - g , g" = 0, s a t i s f y i n g SU(2) with q u a r k m o d e l m i x i n g of s i n g l e t a n d o c t e t a n d a l s o t h e A d l e r c o n d i t i o n ; b u t s u c h a s i m p l i f i c a t i o n c a n n o t b e a p r e d i c t i o n of t h e V e n e z i a n o m o d e l . M o r e o v e r , t h e r e w o u l d t h e n b e a c o u p l i n g to t h e ~?v(0+) o d d ~d a u g h t e r , but not to t h e I = 1 m e m b e r of t h e o c t e t , t h e ~v(0 +) - a l t h o u g h t h i s n e e d not b e so if we r e g a r d t h e ~/v(0+m~p) a s t h e 7 ' of t h e (0+)p d a u g h t e r o c t e t - a n d in t h a t c a s e a c o u p l i n g to t h e ~v(0 +) w o u l d b e i n t r o d u c e d with t h e T + ~v(0 ) of i t s n o n e t in t e r m s l i k e :
(0 0
, 0)
0
0 0) inT (1), T (0)
ffV ??V
a n d we s h o u l d t h e n h a v e to s a y t h a t t h e n e a r i d e n t i t y of t h e ~ a n d nv(0+) masses is pure chance.
6. ~.~ -~ ~rW S C A T T E R I N G E T C . T u r n i n g now to 7rW -~ ~ s c a t t e r i n g a n d i t s r e l a t e d r e a c t i o n s , isospin reaction channels:
we d e f i n e
446
G. l?. CANNING S(1), U (1) : u~-~ ~ , T(O) : ~ u ~ ~
,
with T (0) = _ ~ / 3 S ( 1 ) . W e a t o n c e w r i t e down t h e a m p l i t u d e , S (1)
(1 1_1)+(12 gL.A 2 f
l 1)+(_~ l l)l -A 2 g 2
,
f r o m w h i c h we find: g f ~ gfTp?
=
- ~-~ 2 g~2u~ •
S i m i l a r l y we f i n d b y c o n s i d e r i n g t h e r e a c t i o n s uT?' ~ ~77' a n d u~ ~ n ~ ' t h a t : gf~
gf~'7?' = - ~ f 3 4 2 ~ v ' '
gf~
gf~v'
= - ~3 gA2~7? g A 2 ~ ? , .
If w e c o m p a r e t h e s e r e l a t i o n s w i t h t h e SU(3) p r e d i c t i o n s , s e e a p p e n d i x 2, t a k i n g F= G a n d c o t 82 = ~/2 ( d e d u c e d in p a r t i c u l a r f r o m g f , u u =0), we f i n d o n l y t h a t I F : J 2 a n d we a r e u n a b l e to m a k e a n y p r e d i c t i o n on ~, ~' m i x i n g . W e c o m p a r e t h i s with t h e uu -~ ~u s c a t t e r i n g , a n d n o t e t h a t t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n f o r uB -~ u~ w o u l d h o l d o n l y if ~ h a d t h e s a m e m a s s a s t h e u. ( C o m p a r e o u r n e c e s s a r y a s s u m p t i o n t h a t m w = m p a n d m A 2 -: mf in t h e KI4 -~ KI~ s c a t t e r i n g . ) H o w e v e r , a s s h o w n in t a b l e 1, w h e r e we p r o p o s e t h e i d e a l m a s s r e l a t i o n s t h a t t h i s m o d e l s e e m s to r e q u i r e , if we n e v e r t h e l e s s s e t m~/= mu ~ 0, t h e n t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n i s s a t i s f i e d f o r t h e (0-) o c t e t a s a w h o l e , i . e . i n c l u d i n g t h e 77. But we w o u l d t h e n b e u n a b l e to o b t a i n t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n f o r t h e r e a c t i o n s a b o v e t h a t i n c l u d e t h e ~?'; i n d e e d on t h e q u a r k m o d e l u s i n g d u a l i t y d i a g r a m s t h e s e r e a c t i o n s w o u l d not be included.
7. KTT -~ K ~ S C A T T E R I N G E T C . S i n c e in t h e r e a c t i o n s K~-~ K~, K~? -~ K ~ e t c . , t h e K* a n d t h e K N c a n c o u p l e , t h e l a c k of t h e c o n d i t i o n of a p u r e s i g n a t u r e d t r a j e c t o r y in b o t h c h a n n e l s ( a s w a s t h e c a s e f o r nV -~ ~ s c a t t e r i n g ) , a l l o w s u s to i n t r o d u c e at l e a s t two i n d e p e n d e n t c o u p l i n g c o n s t a n t s . F o r e x a m p l e with i s o s p i n r e a c 1 1 t i o n c h a n n e l s d e f i n e d : S(~), U(~) : Kn ~ KT?, T ( 1 ) : K K --* ~T/we h a v e f o r
T(1) = ~ S(~) g
i(11 1)+(11 )1 K* A 2
A 2 K* 1
-K*
'
INTERNAL SYMMETRY IMPLICATIONS
447
so t h a t , e x c e p t b y t h e r a t h e r c o m p l i c a t e d p r o c e d u r e of u s i n g r a t i o s of c o u p l i n g c o n s t a n t s to r e s o n a n c e s in d i f f e r e n t o c t e t s , it i s v e r y d i f f i c u l t to o b t a i n m u c h i n f o r m a t i o n with r e g a r d to t h e c o u p l i n g c o n s t a n t s in s u c h r e a c tions. But w e r e we to s h u t o u r e y e s to t h e ~, ~ m a s s d i f f e r e n c e , a n d d e c l a r e t h a t t h e t e r m with c o u p l i n g c o n s t a n t g s a t i s f i e d t h e A d l e r c o n d i t i o n f o r t h e p i o n , k a o n a n d e t a b u t t h a t t h e one with g ' d i d not, a n d a s a r e s u l t s e t g ' = 0 , t h e n we w o u l d o b t a i n t h e g o o d SU(3), q u a r k m o d e l m i x i n g r e s u l t ; gA2~ gA2K~
=2 ~
gKNK~ gKNK~.
M o r e o v e r , we n o t e t h a t in t h i s r e a c t i o n t h e r e a r e no o d d d a u g h t e r s of t h e A 2 p r e s e n t in T(1), o w i n g to t h e EU m a s s c o n f i g u r a t i o n , so t h a t we c a n n o t a p p e a l to n e c e s s a r y d a u g h t e r t e r m s to b e a d d e d in o r d e r to s a t i s f y t h e Adler condition. A r g u i n g a l o n g t h e s a m e l i n e s we o b t a i n f o r t h e K~ -~ K~ s c a t t e r i n g t h e a m p l i t u d e : T(0) = - ~/2S(½) g
I(1 1 1)+(_~ 1 1)] K* f
K*
+g
,[(1 1 1)+(_~ 10l K* f ' -
.,,'1 1 1)
K*
+ g ~K*
K*
a n d t h e n if we s e t g " = 0, g ' = 0 by r e a s o n of t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n ( a p p l i e d f o r t h e i d e a l m a s s e s ) we find: g f 77~?gfKI~ :
-~
2 2 g~:NK 77,
w h i c h t h u s l e a d s to a g o o d SU(3) q u a r k m o d e l r e s u l t f o r t h e c o u p l i n g c o n stants, and which at the same time satisfies the Adler condition for both t h e k a o n s a n d t h e eta. S i m i l a r l y f o r K~?' --* KT?': T(0) = - ~/2 S(½) g
I(1K* f1' - 1)+( -1f ' K1* 1 )]
,
so that g f ' 7' 7' g f ' K I ~ = - "/~ 2 t,~2KNK B, , w h e r e a g a i n on t h e g r o u n d s of t h e A d l e r s e l f c o n s i s t e n c y c o n d i t i o n a p p l i e d w i t h t h e i d e a l m a s s e s , w e h a v e r e j e c t e d t h e t e r m c o n t a i n i n g a c o u p l i n g to t h e f a n d t h e t e r m (1K. 1 . 1). A g a i n we s a t i s f y t h e A d l e r c o n d i t i o n f o r k a o n s f o r o u r i d e a l m a s s e s a n d t h e c o u p l i n g c o n s t a n t s a r e in a c c o r d with t h e q u a r k m o d e l SU(3). F o r KT? -~ KT?', a p p l y i n g t h e s a m e p h i l o s o p h y , we o b t a i n : T(0) : - ~ S ( ~ ) : g
*-K*
1
,
w h i c h s a t i s f i e s t h e A d l e r c o n d i t i o n b o t h f o r t h e k a o n s a n d t h e e t a . A n d we o b t a i n t h e s a m e r e s u l t f o r K~ -~ KV', w h e r e a g a i n we h a v e no q u a r k m o d e l p a r t i c l e f o r ~ ' to c o u p l e to.
448
G.P. CANNING
8. D U A L I T Y DIAGRAMS. Q U A R K M O D E L MASS F O R M U L A We now show how the duality d i a g r a m s of R o s n e r [12] and H a r a r i [ 1 3 ] t o g e t h e r with a s i m p l e q u a r k - t y p e m a s s f o r m u l a f o r the m e s o n s , which g i v e s the ideal m a s s e s of t a b l e 1, p r e d i c t that the t e r m s that s a t i s f y the A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n a r e just t h o s e that a r e r e q u i r e d to give the q u a r k m o d e l nonet U(3) c o u p l i n g c o n s t a n t s [14]. It is c l e a r t h a t if we take only t h o s e V e n e z i a n o t e r m s that have a c o r r e s p o n d i n g duality d i a g r a m , then R o s n e t ' s p r e s c r i p t i o n a u t o m a t i c a l l y g i v e s the U(3) coupling c o n s t a n t r e l a t i o n s , and we find that they a r e i n d e e d the t e r m s we r e q u i r e to be p r e s e n t in o r d e r to give t h e s e r e s u l t s . F o r m e s o n s r e p r e s e n t e d by a ql~12 b o u n d state, the s, t duality d i a g r a m f o r the r e a c t i o n s r e l a t e d by c r o s s i n g : s : a+b ~ ~+a t : a+c
: a+
~B+8 d ~
~+b
is w r i t t e n :
.
q,\
t
/q,_° a - qlq2
'oL
......
,. . . . .
4
b ~ q4~1 c ~ q2~3 d ~ q3q4
b
q3 d q.
"s"%q2q~
q~
w h e r e q l , q2, q3, q4 a r e c h o s e n f r o m the set p, n, X of q u a r k s such that q l q2 etc. c o r r e s p o n d to the e x t e r n a l p a r t i c l e s a etc. (in s o m e c h a r g e " state): p r o v i d e d t h a t it is p o s s i b l e to d r a w such a d i a g r a m , t h e n the s c h a n n e l r e s o n a n c e " s " ~ cl2q4 and the t - c h a n n e l r e s o n a n c e "t" ~ q 3 q l will not be exotic. T h i s c o r r e s p o n d s to the V e n e z i a n o t e r m (1s 1 _ 1) and by i n t e r c h a n g i n g p a r t i c l e s c and d we obtain an s, u duality d i a g r a m and by i n t e r c h a n g i n g b and d we obtain a t, u duality d i a g r a m , f o r which the c o r r e s p o n d i n g V e n e ziano t e r m s ( s l - u1 1), (_1 ul 1) m a y be w r i t t e n . We now a s s u m e the following q u a r k - t y p e m a s s f o r m u l a f o r a m e s o n m a s s M c o m p o s e d of ql~12 q u a r k s : ~2(q1~t2 ) = m 2 ( q l q 2) + M 2 = m 2 ( q l ) + m 2 ( ~ 2) + M o2 ,
INTERNAL
SYMMETRY
IMPLICATIONS
449
DUALITY DIAGRAMS: We have for simplicity used throughout the z e r o c h a r g e states for the m e s o n s ; and it should be noted that nfi contains both isosinglet and i s o v e c t o r states.
~- .... 1..... T---,,u,.~;
• ---
s
~
I-
....
- - 2
/
:
|- -
- "u"~XX
\
-o,~'
/<---.. "uO,A2
-~,f' L 1 (~ - o
(i)
(ll)
I)
K
s---.-
I.
. . .
I
-
....
T---
(i _ i D
~
t
~
F.
"t"~,.~
° '~K* ,KN
"t,~*,KN
1 1 (K,A2 -1)
(iii) n
K
it
(iv)
/ n
ii
(-A2K .I)
K
,,:~:
_
/
\ "~K*.KN
'~,K* ,KIt
(v)
II
(ir,tf,-l), (vi)
Ii
(-f,K,1)
(vii)
1 1 (K,-K,I)
~K~L
450
G.P. CANNING
w h e r e M 2 is the z e r o - p o i n t ( r e l a t i v e to m 2 =o) m a s s of the nonet, equal to the m a s s of the d e g e n e r a t e i s o v e c t o r and i s o s i n g l e t (a d e g e n e r a c y d e m a n d e d by the V e n e z i a n o m o d e l ) , and h e r e ( s e e L o v e l a c e [3]) M o2 is set to be ½ f o r the 1- nonet and 1 + ~ f o r the 2 + n o n e t (and by definition 0 f o r the 0- honer): and m 2 ( q l ) = m 2 ( ~ l ) is set to be z e r o f o r n o n - s t r a n g e q u a r k s and ¼ f o r strange quarks. T h i s m a s s f o r m u l a , linked with the a l l o w e d V e n e z i a n o t e r m s d e t e r m i n e d by d r a w i n g the duality d i a g r a m s , n e c e s s a r i l y g i v e s the A d l e r s e l f - c o n s i s t e n c y condition. F o r e x a m p l e , in r e a c t i o n a + b -~ E+ d, w h e r e we m a y d r a w a n s , ! duality d i a g r a m c o r r e s p o n d i n g to the V e n e z i a n o t e r m (s1 ~ - 1) w h e r e " s " ~ Cl2q4 and "t" ~ ~3ql , the t r a j e c t o r y f u n c t i o n s a r e d e t e r m i n e d by t h e i r p a s s i n g t h r o u g h the a p p r o p r i a t e 1- m e s o n s t a t e s : 1 = c~°+ 0/'M2(~2q4 ) = t~S ~ ° + M 20+ m 2 = 0 /o8 + l-+m2(~]2q4 2
(~12q4)
) ,
i.e.:
0/° :½-m2(~12q4 ) w h e r e 0/s(S ) =- 0/°+ 0/'s, a l s o
0/~ : ½ - m 2 (Ul3ql) ; and at the A d l e r p o i n t s , a s we take the s e v e r a l (0-) m e s o n f o u r - m o m e n t a to zero, i.e.:
tPa~O: (i) pd
(ii)
s=m 2 ,
o:
t:m ,
t=m
2
Pb-* 0 :
t=m2d,
s=m2
:m2
and so given that rn 2 = m 2(qi~2) = m 2(qi) + m 2(92) etc. , we find the argument of the gamma function denominator of (1 ~ _ i) for condition (i) 1 - 0/s(S) - 0/t(t) = I - ½+m2(~12q4) - m2(Cllq4) -
=0
½+ m 2 (:-13q1) - m 2(q2cI3)
INTERNAL SYMMETRY IMPLICATIONS a n d f o r condition
451
(ii) 1 - as(S) - c~t(t) = 1 - ½ + rn2(ffl2q4 ) - m2(q4q 3) -½ + rn2(~3q 1) - m2(ql~2) =
0
.
T h e v a n i s h i n g of the a r g u m e n t of the g a m m a function d e n o m i n a t o r g i v e s the z e r o r e q u i r e d f o r the V e n e z i a n o t e r m . We now take s o m e e x a m p l e s , showing how the a b s e n c e of a duality d i a g r a m a l l o w s us to r e j e c t the t e r m s that we found "unwanted" in the v a r i o u s r e a c t i o n s we c o n s i d e r e d . C o n s i d e r KI~ --* KK s c a t t e r i n g with r e a c t i o n c h a n n e l s defined: s, u: KK --* KK, t: KK -~ KK. T h e r e e x i s t h e r e only two s, u duality d i a g r a m s (i), (ii) ( t h e r e being no t - c h a n n e l r e s o n a n c e m a d e up of the exotic X~ q u a r k s ) , c o r r e s p o n d i n g to the V e n e z i a n o t e r m s ( 1 1 1), ( 1 1 1) r u l i n g out the t e r m s ( ; - 1 1) and ( 1 _ 1 1) o t h e r w i s e allowed, w~ich h a v e n o n - v a n i s h i n g d e n o m i h a t 6 r g a m m a ' fudction a r g u m e n t s at the A d l e r point (and which m i g h t a l s o l e a d to c o u p l i n g c o n s t a n t r e l a t i o n s i n c o n s i s t e n t with SU(3)). S i m i l a r l y f o r KTr -~ K~ with r e a c t i o n c h a n n e l s defined: s: KTr ~ K~?, u: Kr/--* KTr, t: KK --* 7rU, we find only s, t and u, t duality d i a g r a m s (iii), (iv) 1 A2 1 - 1), ( - A 2 K* 1) and no d i a g r a m c o r r e s p o n d i n g to V e n e z i a n o t e r m s (K* c o r r e s p o n d i n g to ( 1 . _ 1 . 1) o t h e r w i s e allowed, which is a g a i n the t e r m that is not z e r o at the A d l e r point. We a l s o find e x a c t l y the s a m e situation in the r e a c t i o n K~' --* K~7', w h e r e only s, t and u, t duality d i a g r a m s (v), (vi) c o r r e s p o n d i n g to ( 1 . fl, _ 1), ( _ 1 1 - 1 ) and none c o r r e s p o n d i n g to ( 1 . _ 1 , 1) a r e found f K* K K " C o n t r a s t the r e a c t i o n K~? ~ KV' f o r which only one s, u duality d i a g r a m (vii) m a y be d r a w n c o r r e s p o n d i n g to (1K. - 1 . 1) - a r e s u l t that is n e c e s s a r ily o b t a i n e d if we a s s u m e that t h e r e is no q u a r k m o d e l m e s o n to couple to. 9. CONCLUSION We h a v e found that ( 0 - ) x (0-) ~ ( 0 - ) x (0-) m e s o n s c a t t e r i n g on a f i r s t o r d e r V e n e z i a n o m o d e l g i v e s a s t r o n g indication of a nonet 9 = 8 + 1 SU(3) s t r u c t u r e f o r the i n t e r n a l s y m m e t r y of the m e s o n s with a r e q u i r e d m a s s n o n - d e g e n e r a c y b e t w e e n the p h y s i c a l s i n g l e t / o c t e t m i x e d i s o s c a l a r s . T h e p r e d i c t i o n s a r e s t r o n g e s t in the r e a c t i o n s with m o s t i s o s p i n c h a n n e l s : eg. 7rTr ---* n n and Kn ---* KTr (the l a t t e r h a v i n g b e e n c o n s i d e r e d in o u r p r e v i o u s p r e p r i n t [9] a s a p r e l i m i n a r y to this p a p e r ) , w h e r e we find that the A d l e r c o n dition h o l d s well f o r both p i o n s and kaons. In o t h e r r e a c t i o n s , eg. KK ~ KK we a r e not so c o m p l e t e l y c o n s t r a i n e d by the V e n e z i a n o m o d e l , but if we allow o u r s e l v e s to r e j e c t any " u n n e c e s s a r y " t e r m s on the g r o u n d s that t h e y do not s a t i s f y the A d l e r condition f o r o u r i d e a l i s e d m a s s e s in t a b l e 3, then we a l w a y s obtain S U ( 3 ) / q u a r k m o d e l c o u p ling c o n s t a n t r e l a t i o n s , and the A d l e r condition is s a t i s f i e d f o r the whole p s e u d o s c a l a r o c t e t (with p h y s i c a l ~).
452
G . P . CANNII'~G
W e n o t e t h a t we a r e not o b l i g e d to t a k e q u a r k m o d e l m i x i n g f o r t h e 0 n o n e t , n o r a r e ~ e o b l i g e d to t a k e t h e e t a m a s s to b e d e g e n e r a t e with t h e p i o n m a s s : only w h e n we i m p o s e t h e A d l e r s e l f c o n s i s t e n c y c o n d i t i o n on t h e l e a d i n g o r d e r t e r m s a n d u s e t h e i d e a l m a s s e s do we o b t a i n s u c h r e s u l t s a n d a s shown in s e c t i o n 8. t h i s i s e q u i v a l e n t to a s s u m i n g t h e v a l i d i t y of t h e q u a r k m o d e l , f o r t h e r e we show t h a t by t a k i n g o n l y t h o s e V e n e z i a n o t e r m s for which there exists a duality diagram (necessarily quark model) and by assuming a quark type mass formula, the Adler self consistency condition follows automatically for the whole 0- nonet and the nonet coupling constant relations then follow. In d e r i v i n g in p a r t i c u l a r t h e c o u p l i n g c o n s t a n t r e l a t i o n s ( a s f a r a s c a n b e d e t e r m i n e d , r a t h e r b e t t e r in a c c o r d with e x p e r i m e n t t h a n t h e m a s s r e l a t i o n s ) we h a v e at m o s t a s s u m e d SU(2) × U(1) s y m m e t r y a n d in p a r t i c u l a r we h a v e not h a d to a s s u m e t h e c o m p l e t e SU(3) o c t e t m a s s d e g e n e r a c i e s a s . w a s done in p r e v i o u s b o o t s t r a p [15] a n d F. E. S . R . [16] c a l c u l a t i o n s of t h e o c t e t c o u p l i n g c o n s t a n t s , a l t h o u g h we h a v e h a d to a s s u m e t h e d e g e n e r a c i e s of t h e p h y s i c a l i s o s c a l a r a n d i s o v e c t o r p a r t i c l e m a s s e s . W e now l i s t , t a b l e 1, o u r i d e a l m a s s e s in u n i t s of a '-1 =1.1 GeV2: t h e s e a r e t h e m a s s r e l a t i o n s r e q u i r e d b y t h e V e n e z i a n o m o d e l , l i n k e d with t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n f o r t h e (0-) o c t e t , a n d a s a r e s u l t of w h i c h we a r e a b l e to f i n d u n a m b i g u o u s l y t h e S U ( 3 ) / q u a r k m o d e l n o n e t c o u p l i n g c o n s t a n t r e l a t i o n s d e r i v e d p r e v i o u s l y . In t h e t a b l e , t h e s y m b o l on t h e l e f t i s t h e u s u a l j P C s y m b o l , a n d t h e p a r t i c l e n a m e s a r e e n t e r e d , with, w h e r e a p p r o p r i a t e , c o n v e n t i o n a l n o n - s y s t e m a t i c s y m b o l s i n c l u d e d in b r a c k e t s . Table 1 Ideal m a s s e s
0-+, 1++
i - - , 2 ++
? 0 ++
? 1+-
rrA, ~ a (~r)(27)
KA (K)
rl}k (? Xo) PN, ~ g)) (00)
rrA, ~A (A1) (? D) KN (K*)
KA
~0~ (¢p) rrl,~, rlbl (57) (s*)
~A (?E) ~N, V~ (A2) (f)
KN
KN
V~
(K**)
(f')
KA
~o~ (?)
r/~ (2) PA' gOA (B) (7 H)
The predicted p a r t i c l e s in the above table a r e given the conventional s y s t e m a t i c names: IG: rg ~ 1-, p ~ 1+ , ~ ~ 0+, ~O ~ 0-; P(-)J: N/A according as + / - , (subscript) and the p r i m e denotes the heavy i s o s c a l a r s of the nonets. Below in b r a c k e t s are written the n o n - s y s t e m a t i c names (if different) of the p a r t i c l e s listed in the tables that we associate with our predictions.
INTERNAL SYMMETRY IMPLICATIONS
4~53
The final two rows have been included to show what we expect to find f r o m (0') x (0-) ~ (0 +) × (0-) scattering. I wish to thank the n u m e r o u s m e m b e r s of the D e p a r t m e n t for t h e i r valuable c o m m e n t s and suggestions.
APPEND~
1
The coupling for (0-)× (0-) to (1-) octet and singlet [17] a n t i - s y m m e t r i c , (L = 1) and so we have, omitting the couplings of the 0- singlets, only one type of coupling, symbolically: G(8-aV ]-8-px-Sp) ~ Gtr(V8 [P8, P81 where the s u p e r f i x e s P, V stand for p s e u d o s c a l a r and v e c t o r and where the suffix a denotes a n t i s y m m e t r i c ; and where V8, P8 a r e the usual m a t r i x r e p r e s e n t a t i o n s for m e s o n v e c t o r and p s e u d o s c a l a r octets. We define the mixing angle for the SU(3) singlet and octet to be, X = X 8 sin0 +X 1 cos0 X ' = X 8 cosa - X 1 sin6 where the X' has the higher m a s s c o m p a r e d with the X. Then we obtain the following coupling constants, tabulated in table 2. APPENDIX 2 We tabulate the coupling constants for ( 0 - ) x (0-) to (2 +) octet and singlet, for easy c o m p a r i s o n [18]. H e r e the coupling is s y m m e t r i c (L = 2 coupling) and we t h e r e f o r e have the following couplings which in SU(3) may have in p r i n c i p l e independent coupling constants (the s u p e r f i x e s P, T standing for p s e u d o s c a l a r , t e n s o r and the suffix S standing for s y m m e t r i c ) F(8-TI 8-p x 8 P) ~ F t r ( T 8 { P 8 ,
P8 }) '
G(1TI 8_Px8P) ~ G t r ( T l { P 8 , P8} ) , j ( s T I _8Pxl_ P) ~ J t r ( T 8 { P 8, P1}) , I (1T ] I_P×I p) ~ I t r ( T l { P 1 , P1} ) , where T1, T8, P1, P8 a r e the usual m a t r i x r e p r e s e n t a t i o n s of SU(3) n o r m a l i s e d so that t r ( P 2) = 1 etc. with only the octet states c o r r e s p o n d i n g to traceless matrices.
G. P. CANNING
454
Table 2 (1- ] 0- x 0-) SU(3) coupling constants (isospin invariant)
Resonance
External particle state
Coupling constant ideal w - q~ general mixing
p
~
~-C
p
KK
-,/-2
K*
K~
- ~ ~/-3 a
cO
KK
-~-~sin 01 C
- ~ 12 C
(fl
KK
-~-~cos 01 G
-G
1
G
We define the m i x i n g a n g l e s f o r the o c t e t s (0-), (2 +) a s in appendix 1, and we w r i t e f o r e x a m p l e : 7/~ s i n 0 o P8(78 ) + c o s 0 o P l ( 7 1 ) , w h e r e we i n d i c a t e by the a r g u m e n t , that only the c o m p o n e n t s of the m a t r i c e s c o r r e s p o n d i n g to the e n t r i e s of 78, 7/1 in the p a r t i c l e m a t r i c e s a r e to h a v e the n o n - z e r o n u m e r i c a l v a l u e specified. We t a b u l a t e , t a b l e 3, the i s o s p i n i n v a r i a n t coupling c o n s t a n t s , f i r s t a s s u m i n g the g e n e r a l SU(3) f o r m , s e c o n d l y a s s u m i n g F=G, c o t 0 2 = ~/2 ( q u a r k m o d e l nonet s t r u c t u r e f o r (2+)) and then with F=G =J=I, c o t 0 2 = c o t 0 o = ~-2 (nonet s t r u c t u r e f o r both (2 +) and (0-)).
Part. State
lrI]
~'
7r~
~u
7777'
77'7/'
Res.
A2
A2
f
f'
f
f
Final
Oo]
02 - G s i n 02]
constants
02cos2 %]
- J s i n 02sin 2 8 o + I cos 8 2 s i n 280]
cos e
2eos 2 8o
V sin 2 eo + ~ I
( F / 3 ~/2) cos 20 0
2 eo
(~/~-(F/~ sin 02oos 2 8o + c
sin
+ ~ J cos2%-(I/3
(F/6 ~ )
cos 200
since o
V~ s i n 280
+ ½J s i n 2 8o + ~ - I
(F/3 ~ ) sin 2 Oo
0
-~/-32~F
F= G 82 ~ 35 0
(isospin invariant)
(isospin invariant)
+ J s i n 02c os 2 8 o - ½ 1 cos 8 2sin 2 0o]
0 2 s i n 2 8o + ½ G cos 02sin 20 o
02sin2%+Ieos
(1/~)[-(F/2V~)sin
+J sin
(11v%[:-(F/~/2) sin 02sin 2 0o + G cos 02sin 2 0o
-[(F/~)cos
-[(F/'/2)sinO2+Gcos02]
(1/vr3[(F/vf2)cos 8o-JsinOo]
(1/x/3[(F/~)sin 8o+J cos
general
Coupling
Table 3 (2+I 0- × 0-) SU(3) coupling constants
0
(F/V~)
F = G= J = I 0o ~ 35 °° 82 ~ 35
5n
P~
-~Fsin20 e +½ ~/2 J c o s 200+ (I/6 sin 20 o
(1/ff3) [ - (F/2~/2cosO2sin20o-½GsinO2sin20 o
- ~ ~J
- J c o s a 2sin 2 00 - / s i n 02sin 20o]
F /2vz2 -½F
1F - (~/3/2 ~f2) F
( 1 / ~ ) [ - (F/2~-2)sinOo+J eos0o]
(1/yr3) [ - (F/2~f'2) cos 0 o - J sin 0 o]
K~
KT?
K~'
KN
KN
KN
+F / ~ 2
(1/~f3)[ +½ F e o s 0 2 +V~G sin 021
Kg
-F
35 °
F = G=J=I
KK
+ F/2
(1/#r3) [ +½ F s i n 02-~/-20osO2G ]
sin 2 0 o- ~ 1sin2
- ½Fcos 200
(1/'~-) [ - (F/~/2) cos 02cos 20o-G sin02cos 200
0o
+ ~v~ J s i n 20 o - ~ I c o s 200
+J cos 02cos 2 00 + ½/sin 02sin 200]
- ~ 2 F s i n 2 0o
(1/~-)[ - (F/~2)cos 02sin2 00 - G sin 02sin 2 00
F=G 02 ~ 35 °
+J cos 02sin 2 0o - / s i n 02cos 200 ]
genera[
K~
7/'7/'
Part. State
Coupling constants (isospin invariant)
A2
f'
Res.
Final
Table 3 (continued)
p
¢91
INTERNAL SYMMETRY IMPLICATIONS
457
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G. Veneziano, Nuovo Cimento 57A (1968) 190. S. L. Adler, Phys. Rev. 137 (1965) B1022. C. Lovelace, Phys. Letters 28B (1968) 264. K. Kawarabayashi, S. Kitakado and H. Yabuki, Phys. Letters 28B (1969) 432. H. Harari, Phys. Rev. Letters 20 (1968) 1395. J. L. Rosner, Phys. Rev. Letters 21 (1968) 950. C. B. Chiu, J. Finkelstein, Phys. Letters 27B (1968) 510. H. J, Lipkin, Nucl. Phys. B9 (1969) 349. G . P . C a n n i n g , Oxford preprint 42/68. P. C a r r u t h e r s , Introduction to unitary s y m m e t r y (Interscienee Publishers, 1966). [11] M. D. Scadron, Phys. Rev. 165 (1968) 1640; R. L. Sugar and J. D. Sullivan, Phys. Rev. 166 (1968) 1515. [12] J. L. Rosner, Tel-Aviv University preprint TUAP-70-69. [13] H. Harari, WeizmannInstitute preprint. [14] G.P.Canning, Oxford preprint '13/69. [15] Chan Hong-Mo, P. C. Calies and J. E. Paton, Phys. Rev. Letters 11 (1963) 521. [16] C. Schmid and J. Yellin, Phys. Letters 27B (1968) 19. [17] M. Gourdin, Unitary symmetries (North Holland, Amsterdam, 1967). [18] S. L. Glashow and R. H. Socolow, Phys. Rev. Letters 15 (1965) 329.
Nuclear Physics B14 (1969) 437-457. North-Holland Publ. Comp., Amsterdm
INTERNAL SYMMETRY IMPLICATIONS OF THE VENEZIANO MODEL IN PSEUDOSCALAR-PSEUDOSCALA SCATTERING G. P. CANNING * $
Department of Theoretical Physics 12 Parks Road, Oxford, U.K. Received 4 February 1969 (Revised version received ] June 1969) Abstract: The leading order Veneziano model amplitudes for (JP) 0- × 0- -~ 0- × 0scattering** constrained by the absence of "exotic" resonances, is shown to predict the coupling constant relation of SU(3) coupled with the quark-model/SU(6) ideal singlet-octet mixing in the 1-, 2 +, nonets. The Adler self-consistency condition is found to hold (for the most part) both for pions and kaons fulfilling Lovelace's conjecture.
i. INTRODUCTION W e a r e c o n c e r n e d h e r e with two m a i n c o n s i d e r a t i o n s : r e l a t i o n s a m o n g t h e c o u p l i n g c o n s t a n t s of the 1 - a n d 2 + n o n e t s of r e s o n a n c e s to the 0 - n o n e t of e x t e r n a l p a r t i c l e s i m p l i e d b y the V e n e z i a n o m o d e l [1], a n d t h e m a s s r e l a t i o n s b e t w e e n the e x t e r n a l a n d i n t e r n a l p a r t i c l e s i m p l i e d by t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n [2] a s a p p l i e d to the V e n e z i a n o m o d e l by m e a n s of L o v e l a c e ' s [3] g a m m a f u n c t i o n d e n o m i n a t o r m e c h a n i s m . T h e l i n k b e t w e e n t h e s e two i s s h o w n e x p l i c i t l y in sect. 8, to follow f r o m q u a r k m o d e l d u a l i t y d i a g r a m s l i n k e d with a q u a r k - t y p e m a s s f o r m u l a , which r e p r o d u c e s o u r " i d e a l " m a s s e s of t a b l e 1. P r e v i o u s l y it h a s b e e n shown [5 - 8] that H a r a r i ' s [5] c o n j e c t u r e t a k e n with the a s s u m p t i o n of the a b s e n c e of exotic r e s o n a n c e s ( a s s u m p t i o n s that we a l s o m a k e ) i m p l y i n d e p e n d e n t l y of any m o d e l t h a t t h e t h r e e s e t s of t r a j e c t o r i e s p, ¢o, A2, f a n d q~, f ' a n d K*K N m u s t b e d e g e n e r a t e a m o n g s t t h e m s e l v e s , with in p a r t i c u l a r z e r o c o u p l i n g of f ' to ~u b u t not to KK, a n d t h a t the R e g g e c o u p l i n g c o n s t a n t s a r e r e l a t e d by U(3) q u a r k m o d e l n o n e t s y m m e t r y (and a r e e q u a l i n the c a s e of e x c h a n g e d e g e n e r a t e t r a j e c t o r i e s ) . By a c o n t i n u a t i o n (not s p e c i f i e d a p r i o r i ) f r o m t h e h i g h - e n e r g y r e g i o n ( f o r w a r d o r b a c k w a r d ) of one c h a n n e l into the low e n e r g y r e g i o n of the c o r t e s * Work supported by an S. R. C. Grant. ** While in preparation we were informed of the presentation by Kawarabayashi et al. [4] of a paper that contains results s i m i l a r to ours. Present Address: Dept. of Physics and Astronomy Tel-Aviv University, T e l - A v i v Israel.
438
G . P . CANNING
p o n d i n g c r o s s e d c h a n n e l , t h e s a m e r e l a t i o n s a r e f o u n d to h o l d f o r t h e a c t u a l c o u p l i n g s to t h e r e s o n a n t p a r t i c l e s , b u t t h e s e r e l a t i o n s a r e o n l y o b t a i n e d within the three sets. The Veneziano model by its explicit expression for resonance-Regge p o l e d u a l i t y s p e c i f i e s t h e n e c e s s a r y c o n t i n u a t i o n ; a n d a l s o , s i n c e it i n c o r p o r a t e s c r o s s i n g b e t w e e n a l l t h r e e c h a n n e l s , it a l l o w s u s to c o m p a r e t h e c o u p l i n g c o n s t a n t s in d i f f e r e n t c h a n n e l s , a n d t h e r e f o r e to o b t a i n r e l a t i o n s a m o n g t h e c o u p l i n g c o n s t a n t s of r e s o n a n c e s t h a t do not b e l o n g to t h e s a m e o n e of t h e t h r e e s e t s - f o r e x a m p l e in KTr -~ K~ we o b t a i n a r e l a t i o n gzv.K~.~ ¼gpnn g p K K ( t h i s w a s c o n s i d e r e d in o u r p r e v i o u s p r e p r i n t [9] a s a p r e l i m i n a r y to t h i s p a p e r ) ; a n d in p a r t i c u l a r t h e V e n e z i a n o m o d e l g i v e s u s t h e s e r e s u l t s d e s p i t e t h e n o n - d e g e n e r a c y of t h e t h r e e s e t s of t r a j e c t o r i e s (i.e. n o n - d e g e n e r a c y of t h e n o n e t m a s s e s ) . W e a r e a l s o e n a b l e d to c o n s i d e r s u c h r e a c t i o n s a s n~? -* ~V w h e r e t h e l a c k of e x o t i c c h a n n e l s l e a v e s u s with o t h e r w i s e l i t t l e to s a y .
2. M E T H O D W e a s s u m e H a r a r i ' s [5] c o n j e c t u r e : t h e P o m e r a n c h u k R e g g e t r a j e c t o r y i s a p a r a m e t r i s a t i o n of t h e d i f f r a c t i v e b a c k g r o u n d , a n d d o e s not c o r r e s p o n d to a " d u a l " t r a j e c t o r y of r e s o n a n c e s in t h e c r o s s e d c h a n n e l h a v i n g t h e q u a n t u m n u m b e r s of t h e v a c u u m . W e c o n s t r u c t a V e n e z i a n o m o d e l f o r t h a t p a r t of t h e a m p l i t u d e t h a t i s m a d e up of " d u a l " r e s o n a n c e s in t h e l o w - e n e r gy r e g i o n a n d R e g g e p o l e s in t h e h i g h - e n e r g y r e g i o n : o u r r e s u l t s a r e t h e n i n d e p e n d e n t of t h e e x a c t n a t u r e of t h e P o m e r a n c h u k c o n t r i b u t i o n . W e m a k e u s e of r e a c t i o n p r o c e s s e s , i n v o l v i n g two i d e n t i c a l (eg. Kn -~ Kn a n d Kff -~ K~) o r m e r e l y t i m e r e v e r s e d c h a n n e l s (eg. Kn -~ K~? a n d K ~ --* K~) w h i c h we l a b e l t h e s - a n d u - c h a n n e l s , t h e t h i r d ( a n d u s u a l l y t h e d i s t i n c t l y d i f f e r e n t ) c h a n n e l we l a b e l t h e t - c h a n n e l . W i t h t h e u s u a l d e f i n i t i o n s of t h e c r o s s e d c h a n n e l s , a n d in p a r t i c u l a r t h e isospin channel amplitudes:
where ample where obtain
s: a+b-~ c+d,
S a b , c d = (ab Is) (cdls) S (s) ,
u: a + ~ - - * b + d ,
Vae, bd
t: a+d
Tad, cb = (sdlt) ( c b l t ) ~ t ) ,
-~ c + b
,
= (aft[u) (bdlu) U (u) ,
t h e s u p e r f i x s, u , t i s u s e d h e r e f o r a n i s o s p i n , a n d w h e r e f o r e x (ab Is) i s a s h o r t h a n d f o r t h e SU(2) C l e b s c h - G o r d a n : ( a b I c s ) ~ , fl, a r e t h e I 3 c o m p o n e n t s of t h e p a r t i c l e s with isosp-i'n~a,/~:~we t h e c r o s s i n g m a t r i c e s [10] d e f i n e d by t h e e q u a t i o n s :
V (u) = X u s S (s) ,
T (t) : Xts S (s) .
U s i n g t h e )(us c r o s s i n g m a t r i x t o g e t h e r with t h e c o n d i t i o n t h a t s(S)(s, t, u) = U(U)(u, t, s) f o r u= s we f i n d S ( S ) ( u , t, s) in t e r m s of S(S')(s, t, u): t h e n c r o s s i n g to t h e T - c h a n n e l with Xts a n d e m p l o y i n g t h e r e l a t i o n f o u n d a b o v e , w e f i n d t h a t T (t) u n d e r s ~ u i s (+), (-) with a l t e r n a t i n g i s o s p i n t.
INTERNAL SYMMETRY IMPLICATIONS
439
We then c o n s t r u c t the t - c h a n n e l i s o s p i n amplitude T(t) having the a p p r o p r i a t e t - c h a n n e l r e s o n a n c e t r a j e c t o r i e s , and we c r o s s to the s - c h a n n e l u s ing X s t and i m p o s e the f u r t h e r condition that the s - c h a n n e l r e s o n a n c e t r a j e c t o r i e s a p p e a r in the c o r r e c t isospin channels. In p a r t i c u l a r we demand the a b s e n c e of "exotic" r e s o n a n c e s f r o m the a p p r o p r i a t e isospin r e a c t i o n channels. T h i s method of c o n s t r u c t i o n is p a r t i c u l a r l y useful in the c a s e of nons e i f - c o n j u g a t e p a r t i c l e s , w h e r e the u s e of SU(2) isospin index t e n s o r s is not t r i v i a l l y easy. Our p r e s c r i p t i o n is t h e r e f o r e to take all known r e s o n a n c e s in the s, t, u channels and f o r m f r o m t h e m all p o s s i b l e Veneziano t e r m s which a r e then to be combined using the isospin c r o s s i n g m a t r i c e s a s indicated above in o r d e r to obtain the m o s t g e n e r a l fit to the phenomenology. We employ the following notation f o r the m o s t g e n e r a l type of Veneziano t e r m having only single poles: r ( m - as(S)) r (n - at(t))
w h e r e the dash denotes the explicit non-dependence of the t e r m on the M a n d e l s t a m v a r i a b l e indicated by the o r d e r i n g s, t, u; and with m, n, p all (+) i n t e g e r s . In this notation the s u f f i c e s s, t a r e the p a r t i c l e s y m b o l s l a belling the r e s o n a n c e t r a j e c t o r i e s that a p p e a r in the s - and t - c h a n n e l s ; we d i f f e r e n t i a t e isospin d e g e n e r a t e t r a j e c t o r i e s only in the c h a n n e l s w h e r e they give r i s e to r e s o n a n c e s , in o r d e r to c l a r i f y the couplings involved. We s p e c i a l i z e to r e s o n a n c e t r a j e c t o r i e s having no spin z e r o p a r t i c l e s on t h e m (i.e. ones f o r which a h a s a (+) intercept), f o r which the leading t r a j e c t o r y t e r m s h a s m =n =p = 1. Our "leading t e r m " model is p r e c i s e l y to include only such t e r m s a s these. A "leading t e r m " s i g n a t u r e (+) in the s - c h a n n e l a p p e a r s a s the c o m b i n a tion: (ls~-l)+(sl+-ul
1)
and we i n t e r p r e t an u n s i g n a t u r e d t e r m like (s1 ~ - 1) a s a s u m of an (+) and (-) s i g n a t u r e exchange d e g e n e r a t e t r a j e c t o r y : (1 t l _ l ) = ½ [ ~ +
1
1)+~+_1
1)1+½[(1 - 1 _ 1 ) _ ~ _ _ 1
1)1.
The s i g n a t u r e d t e r m then h a s p o l e s at as+(S) =J>~l with r e s i d u e : 1
( J - 1)!
Of'
LI(I+(-)J)aJzJ +aJ - 1 zJ- I J{At T(_)J Au +(l+(-)J)B}+...~. ]
Here a
At,.
=-~qs
q' a ' s ,
m +o o
.-il ,
440
G . P . CANNING
B --
j_ no+
w h e r e we h a v e t a k e n as+(S ) = a°++ a ' s with u n i v e r s a l s l o p e a ' , a n d w h e r e t h e c o s i n e of t h e s c a t t e r i n g a n g l e i s d e f i n e d : c o s 0 s =_Zs -
Pc'Pc
lal
(c.m.) for process a+b ~ c+d.
Ipc I
t ~ ½ qsqsZ+ z-independbnt term . u ~ -2 qs qs z + z - i n d e p e n d e n t t e r m . On t a k i n g a p a r t i a l w a v e p r o j e c t i o n a t t h e r i g h t s i g n a t u r e p o i n t s we f i n d a r e s o n a n c e of s p i n J w h o s e p a r t i a l w a v e r e s i d u e i s
2 aJcj (J
zJ w h e r e Pj(z) = -c-j+ . . .
1)!a' -
,
w i t h "odd d a u g h t e r s " , s p i n J - 1 u n l e s s t h e m a s s c o n f i g u r a t i o n i s E E o r EU a n d a t = a u , a n d a l w a y s with " e v e n d a u g h t e r s " , s p i n J - 2 . A t w r o n g s~gnat u r e p o i n t s we f i n d a r e s o n a n c e of s p i n J - 1 , u n l e s s b y c h a n c e a ' E m ~ + a sO+ + a ~ + a o 1 = 0, w h i c h we c a l l a " m a s s r e c u r r e n c e " of t h e s p i n J - 1 p a r t i c l e at a s + ( S ) = J - 1 with m a s s i n c r e m e n t of a ' - 1 F i n a l l y we n o t e t h a t t h e c o m b i n a t i o n , (1 } _ 1 ) _ ( 1 },_ 1) h a s no l e a d i n g a s trajectory resonances, only the daughter terms. S i n c e t h e i n v a r i a n t i s o s p i n F e y n m a n n a m p l i t u d e c o r r e s p o n d i n g to a J s p i n p r o p a g a t o r a t t h e p o l e m ~ in t h e s - c h a n n e l i s [11]: v
gJ m2 gJ ) PJ{Zs) (qs q's )J 2 - J Cj ,
(s
i
el
w e c o m p a r e ~ i t h t h e p a r t i a l w a v e p r o j e c t i o n of t h e r e s i d u e a t as(S) =J (a right signature point):
2aJ c j
(J-1)!
a'
to o b t a i n ,
gJgJ
2a 'J-1
w h i c h i s i n d e p e n d e n t of t h e k i n e m a t i c f a c t o r s d e p e n d i n g on t h e e x t e r n a l p a r t i c l e m a s s e s (at l e a s t f o r l e a d i n g t r a j e c t o r i e s ) . H e r e t h e g j , g ] a r e t h e c o u p l i n g c o n s t a n t s of t h e s p i n J r e s o n a n c e , w h i c h b e i n g i s o s p i n i n v a r i a n t s a r e d e f i n e d in t e r m s of t h e p h y s i c a l c o u p l i n g to a d e c a y p r o c e s s :
INTERNAL SYMMETRY IMPLICATIONS
441
e gscYd 5= (~ dOi sv + 5 ) g s' c d ,
so that in the amplitude Sab ' cd = ( a b l s ) ( c d l s )
s(S) we have:
Sab, cd ~ ( a S I s ) ( c d } s ) g ~ d b g s c d , io e .
, (_)a+b+s s(S) ~ gg~D g s c d since
(~ I~ =(~ ~i ~ ~) : ¢~~ ~t ~+~~) ( ~+ ~+ ~ = ( a b l s ) (_)a+ b+ s Since in a c r o s s e d channel, e.g. the t-channel the s i g n a t u r e d c o m b i n a tion g i v e s a contribution to the 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 which f o r l a r g e t is given by: F(1-~(s))(ei~°~(S)+lj
o~(t) °~(s) .
the Regge r e s i d u e of the pole i.e. the i s o s p m i n v a r i a n t coupling constant of the R e g g e - t r a j e c t o r y to the e x t e r n a l p a r t i c l e s is given: ~(s) : ~[r(~(s))]
-1 ,
since F(1 - a(s)) = ~[F(~(s)) s i n ( ~ a ( s ) ) ] -1 and so the t r a j e c t o r y d e g e n e r a c y of the p a r t i c l e s in the 1-, 2 + nonets i m p l i e s on the Veneziano model a d e g e n e r a c y of the Regge t r a j e c t o r y coupling c o n s t a n t s , which is i m p o r t a n t in c o n s i d e r i n g the high e n e r g y i m p l i c a t i o n s of exchange d e g e n e r a c y and i s o s p i n - e x c h a n g e d e g e n e r a c y (see i n t r o d u c t o r y section). Note: even if e x t r a daughter t r a j e c t o r i e s in t e r m s with n = m , n - p >0, i.e. with daughter t r a j e c t o r i e s linked with daughter t r a j e c t o r i e s and not with leading t r a j e c t o r i e s (for which n ¢ m , n - p =0, m - p > 0 say), then our p r e d i c t i o n s on the coupling c o n s t a n t s of the leading t r a j e c t o r i e s " r e m a i n unchanged. We r e j e c t a l t o g e t h e r e x t r a leading t e r m s like n = m =p >1, which would r e d u c e c o n s i d e r a b l y the p r e d i c t i v e value of the model.
3. ~
-~ g~ SCATTERING
F i r s t we w r i t e down the well-known [3] Veneziano model a m p l i t u d e f o r
442
G.P. CANNING
the uu -~ ~~ r e a c t i o n s , f o r which the t h r e e c h a n n e l s s, t, u a r e i d e n t i c a l (but with note: z s ~ (+)t, z u ~ (+)t, z t ~ (+)s) and f o r which the i s o s p i n crossing matrices are:
Xus = ~
(i0xo -
3 5 3 1
,
Xts : ~
(i0)lO 3 -3
w h e n c e we 4ind S(2), S (0) is t ~ u ( + ) and S(1) is t ~ u ( - )
,
w h e n c e we obtain:
1 1) s(;) =g
0
o
-
1 _ 1) .~_(lf - p11) I _ ~g(-p 1 1 p11)
p
'
w h e r e we have t a k e n c a r e that t h e r e should be no r e s o n a n c e in S (2). W h e r e we h a v e b e e n c o n s t r a i n e d by the m o d e l to t a k e a p = a f i s o s p i n exchange degeneracy. In t h i s m o d e l , it is i m p o s s i b l e f o r a coupling to the f ' to b e i n t r o d u c e d , without a l s o a p ' ( e x p e r i m e n t a l l y u n o b s e r v e d ) i s o s p i n e x c h a n g e d e g e n e r a t e with the f'. F o r , w e r e it to c o u p l e , we would h a v e f o r the f ' coupling, t e r m s a p p e a r i n g in S(0) but not S(1) o r S(2), of the f o r m g , [ ( 1 1 _ 1) + (fl _ 1 1)]+ g .... I _1< f ' 1 - 1 ) + ( 1 ' - 1 1 ) ] ,
whiehtermsg'(fl,
o1, - 1), g_,,,1 tf p1 - 1) a p -
p e a r i n g in T(2) u n d e r the i s o s p i n c r o s s i n g g i v e n by X t s , i m p l y g' =g"= 0 in o r d e r f o r t h e r e to b e no I= 2 r e s o n a n c e s , thus p r e v e n t i n g the coupling of f ' in s(O). M o r e o v e r , given s u c h a p ' , its e x c h a n g e - d e g e n e r a t e p a r t n e r the A~ ( p r e d i c t e d by the coupling of the p ' in the KI{ -~ KK r e a c t i o n , f o r e x a m p l e ) would h a v e the m a s s of the f ' , so m a k i n g f ' i n s t e a d a m e m b e r of an octet. T h u s , if the f ' i s t r u l y ( m i x e d with f) a SU(3) s i n g l e t , then it cannot c o u p l e to g~ - a p r e d i c t i o n of the q u a r k m o d e l f o r f, f ' mixing. A s L o v e l a c e [3] p o i n t e d out, the A d l e r s e l f - c o n s i s t e n c y condition is s a t i s f i e d by the m a s s r e l a t i o n : 1 =(~ '(2rn 2 - 2rn 2) giving a '-1 = 1.14 GeV 2.
4. K~ -+ K~ S C A T T E R I N G F o r K~ ~ KTr we define the i s o s p i n r e a c t i o n c h a n n e l s :
S(½ ), S(~ ), U(½), U(½): K~ ~ K~ .
T(0), T(1): I ~ - ~ ~ , f o r which the i s o s p i n c r o s s i n g m a t r i c e s a r e :
X u s = 3(-12 4)
Xis = ~ ( ? '
2~/g~ -2
/
"
INTERNAL SYMMETRY IMPLICATIONS
443
W h e n c e we find T(1) is s ~ u (-) and T(0) is s ~ u (+); and a s s u m i n g h e r e in p a r t i c u l a r the a b s e n c e of I = ~ r e s o n a n c e s , we obtain: T(1) =g
E(1 1-1)-(*p
1
[(11 ,
T(0) = ~ C 6 g
=g
( 11 1) - p K*
K* 0
;
)(111)
f1- 1 + - f K *
]
'
( 1 1 - 1 ) 1 ( - 1P 1 ) -~g K* 1 S(½) = ~ g K* p w h e r e we have b e e n c o n s t r a i n e d to take a p = a f a s a b o v e and a K * = ~ K N e x c h a n g e d e g e n e r a c y , with only one independent coupling constant. We note that the p l a u s i b l e l e a d i n g t e r m ( 1 . _ 1 , 1) cannot a p p e a r in T(0) s i n c e u n d e r c r o s s i n g by X s t , it would give r i s e to a r e s o n a n c e in S(~). H e r e too the coupling to f ' of ~n is z e r o , and again b e c a u s e t h e r e is no i s o s p i n e x c h a n g e d e g e n e r a t e p ' p a r t n e r , which would be n e c e s s a r y to c a n cel out the r e s o n a n t t e r m like ( 1 . ~, _ 1) f r o m the S(~) channel. H e n c e we find, by c o n s i d e r i n g the r e s i d u e s at the p o l e s in the a p p r o p r i ate c h a n n e l s , the following coupling c o n s t a n t r e l a t i o n s : 2
3
(1-) g K * K : = : g p : : g p ~ (2+) g K N K : - 2 ~ g f ~ u
gfK~K
T h e f o r m e r , s e e appendix 1, is a p u r e SU(3) r e s u l t while the l a t t e r , see appendix 2, f o l l o w s f r o m SU(3) with the ideal f, f ' s i n g l e t o c t e t mixing p r e d i c t e d f r o m the z e r o coupling of f ' to v~. We note that the s a m e r a t i o is not found f o r the c o u p l i n g s of the 0 + d a u g h t e r octet, b e c a u s e the e x t e r n a l p a r t i c l e m a s s d i f f e r e n c e m a n i f e s t s itself in the s - and u - c h a n n e l s but not in the t - c h a n n e l . H e r e in the l i m i t a s we t a k e one pion f o u r - m o m e n t u m to z e r o , i.e. at the point s = u = m~,~ t = m 2, o r in the limit~as we takeo one k a o n f o u r - m o m e n t u m to z e r o , i.e. at the point s= u= m~, t= m~, we find that the A d l e r s e l f - c o n s i s t e n c y condition is s a t i s f i e d by the v a n i s h i n g of all the t e r m s owing to a z e r o in the a r g u m e n t of the g a m m a function d e n o m i n a t o r s if:
, 2
1 = a (m~+ m2K,- m 2 - m2K) i. e. ot ,-1 = 1.12 GeV 2 compare
444
G. P. CANNING c~'-1 = 1.215 GeV 2
f r o m OlK, = ~KN
~,-1 = 0.98 GeV 2
f r o m ~p
)
:
~f
.
5. KI~ -, KI{ S C A T T E R I N G F o r the KI{ ~ KK s c a t t e r i n g we define i s o s p i n r e a c t i o n c h a n n e l s : S (0), S(1); V (0), V(1): KK ~ KK T (0), T(1): KK ~ KK f o r which the i s o s p i n c r o s s i n g m a t r i c e s a r e : t
t
XUS = 3 ( 1 1 - 3 1 ) '
-i
3
Xts =3( 1 1 ) '
w h e n c e we find T (1) is s ~- u (+) and T(0) is s ~ u (-). A s s u m i n g then the a b s e n c e of any T - c h a n n e l r e s o n a n c e s (for which S = 2), we obtain:
r(0l=g
-
P
1 -( P
~
1
'
s(o/: iag,+gl (lo 1(; 0 + (1
- °1 1 )
1) 3
-
w h e r e we h a v e b e e n c o n s t r a i n e d to t a k e ~p =C~A2, ~ c o = ~ f , oz~o= ~ f ' - e x c h a n g e d e g e n e r a c y and n o n - d e g e n e r a c y of ~ f , with ~ f f o r T(0~r¢ 0 and the u s u a l i s o s p i n e x c h a n g e d e g e n e r a c y ~ = ~ ,x which c o u p l e d with the a b o v e i m p l i e s rnf = mA2 , moo = rno, and w h e r e we have m a d e u s e of the a b s e n c e of a p ' ( m ¢ ) and the p r e s e n c e ' o f a ~v(0+rn~), to d e t e r m i n e that only the d a u g h t e r c o m b i n a t i o n of the ~ t r a j e c t o r y s h o u l d a p p e a r in the S (1) channel. By c o n s i d e r i n g the r e s i d u e s at the p o l e s in the a p p r o p r i a t e c h a n n e l s , the following c o u p l i n g c o n s t a n t r e l a t i o n s a r e obtained: 2 • . 2 = g' (2+) g~_2KK" g~fKg:" g f ' K K (1-)
g2 . g2 . g2 = g' pKI{" wKI{" ~ p K I ~
-g+g":3g'+g+
3g"
: - 4g.
-g+g":3g'+g+ 3g" : - 4 g .
INTERNAL SYMMETRY IMPLICATIONS
445
F o r t h e (1-) o c t e t , s e e a p p e n d i x 1, we f i n d t h a t :
2 +2 3 2 g¢oKK g(pKV. = gpKK w h i c h i s an e x a c t SU(3) c o n d i t i o n s a t i s f i e d f o r a l l r e l a t i v e v a l u e s of g, g', g " . F o r t h e (2 +) o c t e t , s e e a p p e n d i x 2, we u s e t h e p r e v i o u s r e s u l t t h a t gf,u~ =0 to fix t h e r a t i o G/F=(1/~2) c o t 0 2 , a n d t h e n f o r SU(3) to h o l d we r e q u i r e a f u r t h e r c o n d i t i o n on t h e c o u p l i n g c o n s t a n t s : -g =g' +g" w h i c h t h e n d e t e r m i n e s t h a t c o t a 2 = v ~ a n d c o t 01 = ~f2 so t h a t t h e q u a r k m o d e l m i x i n g i s a g a i n s t r o n g l y i n d i c a t e d f o r b o t h t h e 1 - a n d 2 + o c t e t s with t h e c o u p l i n g c o n s t a n t s now in t h e r a t i o s : .~2 =1:1:2 gA2KI~".~2 ~fKI(" ~'f 'KI~ g 2 K i~ .
2 .g2 --= 1:1:2 "g~oKI~ " (pKK
i n d e p e n d e n t of t h e r e l a t i v e m a g n i t u d e of g', g'. W e n o t e t h a t t h e t e r m s l i k e ( 1 _pl 1) s a t i s f y t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n q u i t e w e l l , f o r in the l i m i t a s we t a k e one k a o n f o u r - m o m e n t u m to z e r o , i.e. a t t h e p o i n t s =u =t =m 2, t h e a r g u m e n t of t h e g a m m a f u n c t i o n d o m i n a t o r v a n i s h e s if:
1 =ot'(m2+m 2q~-2m2 )
i.e. if ot ' - 1 = 1 . 1 4 G e V 2 .
T h e c o n d i t i o n -(g+g') =g" d e r i v e d a b o v e , m a y t h e n b e s e e n a s an a p p r o x i m a t e c o n d i t i o n t h a t t h e o t h e r two t e r m s o n e , ( 1 _ 1 1), (+) a n d one, ( 1 _ 1 1), (-) a t t h a t p o i n t s h o u l d t e n d to c a n c e l . W e note'th~/t t h e a b o v e c o n d i t i'o~n i~" s a l s o s a t i s f i e d f o r t h e p a r t i c u l a r l y s i m p l e c h o i c e g' = - g , g" = 0, s a t i s f y i n g SU(2) with q u a r k m o d e l m i x i n g of s i n g l e t a n d o c t e t a n d a l s o t h e A d l e r c o n d i t i o n ; b u t s u c h a s i m p l i f i c a t i o n c a n n o t b e a p r e d i c t i o n of t h e V e n e z i a n o m o d e l . M o r e o v e r , t h e r e w o u l d t h e n b e a c o u p l i n g to t h e ~?v(0+) o d d ~d a u g h t e r , but not to t h e I = 1 m e m b e r of t h e o c t e t , t h e ~v(0 +) - a l t h o u g h t h i s n e e d not b e so if we r e g a r d t h e ~/v(0+m~p) a s t h e 7 ' of t h e (0+)p d a u g h t e r o c t e t - a n d in t h a t c a s e a c o u p l i n g to t h e ~v(0 +) w o u l d b e i n t r o d u c e d with t h e T + ~v(0 ) of i t s n o n e t in t e r m s l i k e :
(0 0
, 0)
0
0 0) inT (1), T (0)
ffV ??V
a n d we s h o u l d t h e n h a v e to s a y t h a t t h e n e a r i d e n t i t y of t h e ~ a n d nv(0+) masses is pure chance.
6. ~.~ -~ ~rW S C A T T E R I N G E T C . T u r n i n g now to 7rW -~ ~ s c a t t e r i n g a n d i t s r e l a t e d r e a c t i o n s , isospin reaction channels:
we d e f i n e
446
G. l?. CANNING S(1), U (1) : u~-~ ~ , T(O) : ~ u ~ ~
,
with T (0) = _ ~ / 3 S ( 1 ) . W e a t o n c e w r i t e down t h e a m p l i t u d e , S (1)
(1 1_1)+(12 gL.A 2 f
l 1)+(_~ l l)l -A 2 g 2
,
f r o m w h i c h we find: g f ~ gfTp?
=
- ~-~ 2 g~2u~ •
S i m i l a r l y we f i n d b y c o n s i d e r i n g t h e r e a c t i o n s uT?' ~ ~77' a n d u~ ~ n ~ ' t h a t : gf~
gf~'7?' = - ~ f 3 4 2 ~ v ' '
gf~
gf~v'
= - ~3 gA2~7? g A 2 ~ ? , .
If w e c o m p a r e t h e s e r e l a t i o n s w i t h t h e SU(3) p r e d i c t i o n s , s e e a p p e n d i x 2, t a k i n g F= G a n d c o t 82 = ~/2 ( d e d u c e d in p a r t i c u l a r f r o m g f , u u =0), we f i n d o n l y t h a t I F : J 2 a n d we a r e u n a b l e to m a k e a n y p r e d i c t i o n on ~, ~' m i x i n g . W e c o m p a r e t h i s with t h e uu -~ ~u s c a t t e r i n g , a n d n o t e t h a t t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n f o r uB -~ u~ w o u l d h o l d o n l y if ~ h a d t h e s a m e m a s s a s t h e u. ( C o m p a r e o u r n e c e s s a r y a s s u m p t i o n t h a t m w = m p a n d m A 2 -: mf in t h e KI4 -~ KI~ s c a t t e r i n g . ) H o w e v e r , a s s h o w n in t a b l e 1, w h e r e we p r o p o s e t h e i d e a l m a s s r e l a t i o n s t h a t t h i s m o d e l s e e m s to r e q u i r e , if we n e v e r t h e l e s s s e t m~/= mu ~ 0, t h e n t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n i s s a t i s f i e d f o r t h e (0-) o c t e t a s a w h o l e , i . e . i n c l u d i n g t h e 77. But we w o u l d t h e n b e u n a b l e to o b t a i n t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n f o r t h e r e a c t i o n s a b o v e t h a t i n c l u d e t h e ~?'; i n d e e d on t h e q u a r k m o d e l u s i n g d u a l i t y d i a g r a m s t h e s e r e a c t i o n s w o u l d not be included.
7. KTT -~ K ~ S C A T T E R I N G E T C . S i n c e in t h e r e a c t i o n s K~-~ K~, K~? -~ K ~ e t c . , t h e K* a n d t h e K N c a n c o u p l e , t h e l a c k of t h e c o n d i t i o n of a p u r e s i g n a t u r e d t r a j e c t o r y in b o t h c h a n n e l s ( a s w a s t h e c a s e f o r nV -~ ~ s c a t t e r i n g ) , a l l o w s u s to i n t r o d u c e at l e a s t two i n d e p e n d e n t c o u p l i n g c o n s t a n t s . F o r e x a m p l e with i s o s p i n r e a c 1 1 t i o n c h a n n e l s d e f i n e d : S(~), U(~) : Kn ~ KT?, T ( 1 ) : K K --* ~T/we h a v e f o r
T(1) = ~ S(~) g
i(11 1)+(11 )1 K* A 2
A 2 K* 1
-K*
'
INTERNAL SYMMETRY IMPLICATIONS
447
so t h a t , e x c e p t b y t h e r a t h e r c o m p l i c a t e d p r o c e d u r e of u s i n g r a t i o s of c o u p l i n g c o n s t a n t s to r e s o n a n c e s in d i f f e r e n t o c t e t s , it i s v e r y d i f f i c u l t to o b t a i n m u c h i n f o r m a t i o n with r e g a r d to t h e c o u p l i n g c o n s t a n t s in s u c h r e a c tions. But w e r e we to s h u t o u r e y e s to t h e ~, ~ m a s s d i f f e r e n c e , a n d d e c l a r e t h a t t h e t e r m with c o u p l i n g c o n s t a n t g s a t i s f i e d t h e A d l e r c o n d i t i o n f o r t h e p i o n , k a o n a n d e t a b u t t h a t t h e one with g ' d i d not, a n d a s a r e s u l t s e t g ' = 0 , t h e n we w o u l d o b t a i n t h e g o o d SU(3), q u a r k m o d e l m i x i n g r e s u l t ; gA2~ gA2K~
=2 ~
gKNK~ gKNK~.
M o r e o v e r , we n o t e t h a t in t h i s r e a c t i o n t h e r e a r e no o d d d a u g h t e r s of t h e A 2 p r e s e n t in T(1), o w i n g to t h e EU m a s s c o n f i g u r a t i o n , so t h a t we c a n n o t a p p e a l to n e c e s s a r y d a u g h t e r t e r m s to b e a d d e d in o r d e r to s a t i s f y t h e Adler condition. A r g u i n g a l o n g t h e s a m e l i n e s we o b t a i n f o r t h e K~ -~ K~ s c a t t e r i n g t h e a m p l i t u d e : T(0) = - ~/2S(½) g
I(1 1 1)+(_~ 1 1)] K* f
K*
+g
,[(1 1 1)+(_~ 10l K* f ' -
.,,'1 1 1)
K*
+ g ~K*
K*
a n d t h e n if we s e t g " = 0, g ' = 0 by r e a s o n of t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n ( a p p l i e d f o r t h e i d e a l m a s s e s ) we find: g f 77~?gfKI~ :
-~
2 2 g~:NK 77,
w h i c h t h u s l e a d s to a g o o d SU(3) q u a r k m o d e l r e s u l t f o r t h e c o u p l i n g c o n stants, and which at the same time satisfies the Adler condition for both t h e k a o n s a n d t h e eta. S i m i l a r l y f o r K~?' --* KT?': T(0) = - ~/2 S(½) g
I(1K* f1' - 1)+( -1f ' K1* 1 )]
,
so that g f ' 7' 7' g f ' K I ~ = - "/~ 2 t,~2KNK B, , w h e r e a g a i n on t h e g r o u n d s of t h e A d l e r s e l f c o n s i s t e n c y c o n d i t i o n a p p l i e d w i t h t h e i d e a l m a s s e s , w e h a v e r e j e c t e d t h e t e r m c o n t a i n i n g a c o u p l i n g to t h e f a n d t h e t e r m (1K. 1 . 1). A g a i n we s a t i s f y t h e A d l e r c o n d i t i o n f o r k a o n s f o r o u r i d e a l m a s s e s a n d t h e c o u p l i n g c o n s t a n t s a r e in a c c o r d with t h e q u a r k m o d e l SU(3). F o r KT? -~ KT?', a p p l y i n g t h e s a m e p h i l o s o p h y , we o b t a i n : T(0) : - ~ S ( ~ ) : g
*-K*
1
,
w h i c h s a t i s f i e s t h e A d l e r c o n d i t i o n b o t h f o r t h e k a o n s a n d t h e e t a . A n d we o b t a i n t h e s a m e r e s u l t f o r K~ -~ KV', w h e r e a g a i n we h a v e no q u a r k m o d e l p a r t i c l e f o r ~ ' to c o u p l e to.
448
G.P. CANNING
8. D U A L I T Y DIAGRAMS. Q U A R K M O D E L MASS F O R M U L A We now show how the duality d i a g r a m s of R o s n e r [12] and H a r a r i [ 1 3 ] t o g e t h e r with a s i m p l e q u a r k - t y p e m a s s f o r m u l a f o r the m e s o n s , which g i v e s the ideal m a s s e s of t a b l e 1, p r e d i c t that the t e r m s that s a t i s f y the A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n a r e just t h o s e that a r e r e q u i r e d to give the q u a r k m o d e l nonet U(3) c o u p l i n g c o n s t a n t s [14]. It is c l e a r t h a t if we take only t h o s e V e n e z i a n o t e r m s that have a c o r r e s p o n d i n g duality d i a g r a m , then R o s n e t ' s p r e s c r i p t i o n a u t o m a t i c a l l y g i v e s the U(3) coupling c o n s t a n t r e l a t i o n s , and we find that they a r e i n d e e d the t e r m s we r e q u i r e to be p r e s e n t in o r d e r to give t h e s e r e s u l t s . F o r m e s o n s r e p r e s e n t e d by a ql~12 b o u n d state, the s, t duality d i a g r a m f o r the r e a c t i o n s r e l a t e d by c r o s s i n g : s : a+b ~ ~+a t : a+c
: a+
~B+8 d ~
~+b
is w r i t t e n :
.
q,\
t
/q,_° a - qlq2
'oL
......
,. . . . .
4
b ~ q4~1 c ~ q2~3 d ~ q3q4
b
q3 d q.
"s"%q2q~
q~
w h e r e q l , q2, q3, q4 a r e c h o s e n f r o m the set p, n, X of q u a r k s such that q l q2 etc. c o r r e s p o n d to the e x t e r n a l p a r t i c l e s a etc. (in s o m e c h a r g e " state): p r o v i d e d t h a t it is p o s s i b l e to d r a w such a d i a g r a m , t h e n the s c h a n n e l r e s o n a n c e " s " ~ cl2q4 and the t - c h a n n e l r e s o n a n c e "t" ~ q 3 q l will not be exotic. T h i s c o r r e s p o n d s to the V e n e z i a n o t e r m (1s 1 _ 1) and by i n t e r c h a n g i n g p a r t i c l e s c and d we obtain an s, u duality d i a g r a m and by i n t e r c h a n g i n g b and d we obtain a t, u duality d i a g r a m , f o r which the c o r r e s p o n d i n g V e n e ziano t e r m s ( s l - u1 1), (_1 ul 1) m a y be w r i t t e n . We now a s s u m e the following q u a r k - t y p e m a s s f o r m u l a f o r a m e s o n m a s s M c o m p o s e d of ql~12 q u a r k s : ~2(q1~t2 ) = m 2 ( q l q 2) + M 2 = m 2 ( q l ) + m 2 ( ~ 2) + M o2 ,
INTERNAL
SYMMETRY
IMPLICATIONS
449
DUALITY DIAGRAMS: We have for simplicity used throughout the z e r o c h a r g e states for the m e s o n s ; and it should be noted that nfi contains both isosinglet and i s o v e c t o r states.
~- .... 1..... T---,,u,.~;
• ---
s
~
I-
....
- - 2
/
:
|- -
- "u"~XX
\
-o,~'
/<---.. "uO,A2
-~,f' L 1 (~ - o
(i)
(ll)
I)
K
s---.-
I.
. . .
I
-
....
T---
(i _ i D
~
t
~
F.
"t"~,.~
° '~K* ,KN
"t,~*,KN
1 1 (K,A2 -1)
(iii) n
K
it
(iv)
/ n
ii
(-A2K .I)
K
,,:~:
_
/
\ "~K*.KN
'~,K* ,KIt
(v)
II
(ir,tf,-l), (vi)
Ii
(-f,K,1)
(vii)
1 1 (K,-K,I)
~K~L
450
G.P. CANNING
w h e r e M 2 is the z e r o - p o i n t ( r e l a t i v e to m 2 =o) m a s s of the nonet, equal to the m a s s of the d e g e n e r a t e i s o v e c t o r and i s o s i n g l e t (a d e g e n e r a c y d e m a n d e d by the V e n e z i a n o m o d e l ) , and h e r e ( s e e L o v e l a c e [3]) M o2 is set to be ½ f o r the 1- nonet and 1 + ~ f o r the 2 + n o n e t (and by definition 0 f o r the 0- honer): and m 2 ( q l ) = m 2 ( ~ l ) is set to be z e r o f o r n o n - s t r a n g e q u a r k s and ¼ f o r strange quarks. T h i s m a s s f o r m u l a , linked with the a l l o w e d V e n e z i a n o t e r m s d e t e r m i n e d by d r a w i n g the duality d i a g r a m s , n e c e s s a r i l y g i v e s the A d l e r s e l f - c o n s i s t e n c y condition. F o r e x a m p l e , in r e a c t i o n a + b -~ E+ d, w h e r e we m a y d r a w a n s , ! duality d i a g r a m c o r r e s p o n d i n g to the V e n e z i a n o t e r m (s1 ~ - 1) w h e r e " s " ~ Cl2q4 and "t" ~ ~3ql , the t r a j e c t o r y f u n c t i o n s a r e d e t e r m i n e d by t h e i r p a s s i n g t h r o u g h the a p p r o p r i a t e 1- m e s o n s t a t e s : 1 = c~°+ 0/'M2(~2q4 ) = t~S ~ ° + M 20+ m 2 = 0 /o8 + l-+m2(~]2q4 2
(~12q4)
) ,
i.e.:
0/° :½-m2(~12q4 ) w h e r e 0/s(S ) =- 0/°+ 0/'s, a l s o
0/~ : ½ - m 2 (Ul3ql) ; and at the A d l e r p o i n t s , a s we take the s e v e r a l (0-) m e s o n f o u r - m o m e n t a to zero, i.e.:
tPa~O: (i) pd
(ii)
s=m 2 ,
o:
t:m ,
t=m
2
Pb-* 0 :
t=m2d,
s=m2
:m2
and so given that rn 2 = m 2(qi~2) = m 2(qi) + m 2(92) etc. , we find the argument of the gamma function denominator of (1 ~ _ i) for condition (i) 1 - 0/s(S) - 0/t(t) = I - ½+m2(~12q4) - m2(Cllq4) -
=0
½+ m 2 (:-13q1) - m 2(q2cI3)
INTERNAL SYMMETRY IMPLICATIONS a n d f o r condition
451
(ii) 1 - as(S) - c~t(t) = 1 - ½ + rn2(ffl2q4 ) - m2(q4q 3) -½ + rn2(~3q 1) - m2(ql~2) =
0
.
T h e v a n i s h i n g of the a r g u m e n t of the g a m m a function d e n o m i n a t o r g i v e s the z e r o r e q u i r e d f o r the V e n e z i a n o t e r m . We now take s o m e e x a m p l e s , showing how the a b s e n c e of a duality d i a g r a m a l l o w s us to r e j e c t the t e r m s that we found "unwanted" in the v a r i o u s r e a c t i o n s we c o n s i d e r e d . C o n s i d e r KI~ --* KK s c a t t e r i n g with r e a c t i o n c h a n n e l s defined: s, u: KK --* KK, t: KK -~ KK. T h e r e e x i s t h e r e only two s, u duality d i a g r a m s (i), (ii) ( t h e r e being no t - c h a n n e l r e s o n a n c e m a d e up of the exotic X~ q u a r k s ) , c o r r e s p o n d i n g to the V e n e z i a n o t e r m s ( 1 1 1), ( 1 1 1) r u l i n g out the t e r m s ( ; - 1 1) and ( 1 _ 1 1) o t h e r w i s e allowed, w~ich h a v e n o n - v a n i s h i n g d e n o m i h a t 6 r g a m m a ' fudction a r g u m e n t s at the A d l e r point (and which m i g h t a l s o l e a d to c o u p l i n g c o n s t a n t r e l a t i o n s i n c o n s i s t e n t with SU(3)). S i m i l a r l y f o r KTr -~ K~ with r e a c t i o n c h a n n e l s defined: s: KTr ~ K~?, u: Kr/--* KTr, t: KK --* 7rU, we find only s, t and u, t duality d i a g r a m s (iii), (iv) 1 A2 1 - 1), ( - A 2 K* 1) and no d i a g r a m c o r r e s p o n d i n g to V e n e z i a n o t e r m s (K* c o r r e s p o n d i n g to ( 1 . _ 1 . 1) o t h e r w i s e allowed, which is a g a i n the t e r m that is not z e r o at the A d l e r point. We a l s o find e x a c t l y the s a m e situation in the r e a c t i o n K~' --* K~7', w h e r e only s, t and u, t duality d i a g r a m s (v), (vi) c o r r e s p o n d i n g to ( 1 . fl, _ 1), ( _ 1 1 - 1 ) and none c o r r e s p o n d i n g to ( 1 . _ 1 , 1) a r e found f K* K K " C o n t r a s t the r e a c t i o n K~? ~ KV' f o r which only one s, u duality d i a g r a m (vii) m a y be d r a w n c o r r e s p o n d i n g to (1K. - 1 . 1) - a r e s u l t that is n e c e s s a r ily o b t a i n e d if we a s s u m e that t h e r e is no q u a r k m o d e l m e s o n to couple to. 9. CONCLUSION We h a v e found that ( 0 - ) x (0-) ~ ( 0 - ) x (0-) m e s o n s c a t t e r i n g on a f i r s t o r d e r V e n e z i a n o m o d e l g i v e s a s t r o n g indication of a nonet 9 = 8 + 1 SU(3) s t r u c t u r e f o r the i n t e r n a l s y m m e t r y of the m e s o n s with a r e q u i r e d m a s s n o n - d e g e n e r a c y b e t w e e n the p h y s i c a l s i n g l e t / o c t e t m i x e d i s o s c a l a r s . T h e p r e d i c t i o n s a r e s t r o n g e s t in the r e a c t i o n s with m o s t i s o s p i n c h a n n e l s : eg. 7rTr ---* n n and Kn ---* KTr (the l a t t e r h a v i n g b e e n c o n s i d e r e d in o u r p r e v i o u s p r e p r i n t [9] a s a p r e l i m i n a r y to this p a p e r ) , w h e r e we find that the A d l e r c o n dition h o l d s well f o r both p i o n s and kaons. In o t h e r r e a c t i o n s , eg. KK ~ KK we a r e not so c o m p l e t e l y c o n s t r a i n e d by the V e n e z i a n o m o d e l , but if we allow o u r s e l v e s to r e j e c t any " u n n e c e s s a r y " t e r m s on the g r o u n d s that t h e y do not s a t i s f y the A d l e r condition f o r o u r i d e a l i s e d m a s s e s in t a b l e 3, then we a l w a y s obtain S U ( 3 ) / q u a r k m o d e l c o u p ling c o n s t a n t r e l a t i o n s , and the A d l e r condition is s a t i s f i e d f o r the whole p s e u d o s c a l a r o c t e t (with p h y s i c a l ~).
452
G . P . CANNII'~G
W e n o t e t h a t we a r e not o b l i g e d to t a k e q u a r k m o d e l m i x i n g f o r t h e 0 n o n e t , n o r a r e ~ e o b l i g e d to t a k e t h e e t a m a s s to b e d e g e n e r a t e with t h e p i o n m a s s : only w h e n we i m p o s e t h e A d l e r s e l f c o n s i s t e n c y c o n d i t i o n on t h e l e a d i n g o r d e r t e r m s a n d u s e t h e i d e a l m a s s e s do we o b t a i n s u c h r e s u l t s a n d a s shown in s e c t i o n 8. t h i s i s e q u i v a l e n t to a s s u m i n g t h e v a l i d i t y of t h e q u a r k m o d e l , f o r t h e r e we show t h a t by t a k i n g o n l y t h o s e V e n e z i a n o t e r m s for which there exists a duality diagram (necessarily quark model) and by assuming a quark type mass formula, the Adler self consistency condition follows automatically for the whole 0- nonet and the nonet coupling constant relations then follow. In d e r i v i n g in p a r t i c u l a r t h e c o u p l i n g c o n s t a n t r e l a t i o n s ( a s f a r a s c a n b e d e t e r m i n e d , r a t h e r b e t t e r in a c c o r d with e x p e r i m e n t t h a n t h e m a s s r e l a t i o n s ) we h a v e at m o s t a s s u m e d SU(2) × U(1) s y m m e t r y a n d in p a r t i c u l a r we h a v e not h a d to a s s u m e t h e c o m p l e t e SU(3) o c t e t m a s s d e g e n e r a c i e s a s . w a s done in p r e v i o u s b o o t s t r a p [15] a n d F. E. S . R . [16] c a l c u l a t i o n s of t h e o c t e t c o u p l i n g c o n s t a n t s , a l t h o u g h we h a v e h a d to a s s u m e t h e d e g e n e r a c i e s of t h e p h y s i c a l i s o s c a l a r a n d i s o v e c t o r p a r t i c l e m a s s e s . W e now l i s t , t a b l e 1, o u r i d e a l m a s s e s in u n i t s of a '-1 =1.1 GeV2: t h e s e a r e t h e m a s s r e l a t i o n s r e q u i r e d b y t h e V e n e z i a n o m o d e l , l i n k e d with t h e A d l e r s e l f - c o n s i s t e n c y c o n d i t i o n f o r t h e (0-) o c t e t , a n d a s a r e s u l t of w h i c h we a r e a b l e to f i n d u n a m b i g u o u s l y t h e S U ( 3 ) / q u a r k m o d e l n o n e t c o u p l i n g c o n s t a n t r e l a t i o n s d e r i v e d p r e v i o u s l y . In t h e t a b l e , t h e s y m b o l on t h e l e f t i s t h e u s u a l j P C s y m b o l , a n d t h e p a r t i c l e n a m e s a r e e n t e r e d , with, w h e r e a p p r o p r i a t e , c o n v e n t i o n a l n o n - s y s t e m a t i c s y m b o l s i n c l u d e d in b r a c k e t s . Table 1 Ideal m a s s e s
0-+, 1++
i - - , 2 ++
? 0 ++
? 1+-
rrA, ~ a (~r)(27)
KA (K)
rl}k (? Xo) PN, ~ g)) (00)
rrA, ~A (A1) (? D) KN (K*)
KA
~0~ (¢p) rrl,~, rlbl (57) (s*)
~A (?E) ~N, V~ (A2) (f)
KN
KN
V~
(K**)
(f')
KA
~o~ (?)
r/~ (2) PA' gOA (B) (7 H)
The predicted p a r t i c l e s in the above table a r e given the conventional s y s t e m a t i c names: IG: rg ~ 1-, p ~ 1+ , ~ ~ 0+, ~O ~ 0-; P(-)J: N/A according as + / - , (subscript) and the p r i m e denotes the heavy i s o s c a l a r s of the nonets. Below in b r a c k e t s are written the n o n - s y s t e m a t i c names (if different) of the p a r t i c l e s listed in the tables that we associate with our predictions.
INTERNAL SYMMETRY IMPLICATIONS
4~53
The final two rows have been included to show what we expect to find f r o m (0') x (0-) ~ (0 +) × (0-) scattering. I wish to thank the n u m e r o u s m e m b e r s of the D e p a r t m e n t for t h e i r valuable c o m m e n t s and suggestions.
APPEND~
1
The coupling for (0-)× (0-) to (1-) octet and singlet [17] a n t i - s y m m e t r i c , (L = 1) and so we have, omitting the couplings of the 0- singlets, only one type of coupling, symbolically: G(8-aV ]-8-px-Sp) ~ Gtr(V8 [P8, P81 where the s u p e r f i x e s P, V stand for p s e u d o s c a l a r and v e c t o r and where the suffix a denotes a n t i s y m m e t r i c ; and where V8, P8 a r e the usual m a t r i x r e p r e s e n t a t i o n s for m e s o n v e c t o r and p s e u d o s c a l a r octets. We define the mixing angle for the SU(3) singlet and octet to be, X = X 8 sin0 +X 1 cos0 X ' = X 8 cosa - X 1 sin6 where the X' has the higher m a s s c o m p a r e d with the X. Then we obtain the following coupling constants, tabulated in table 2. APPENDIX 2 We tabulate the coupling constants for ( 0 - ) x (0-) to (2 +) octet and singlet, for easy c o m p a r i s o n [18]. H e r e the coupling is s y m m e t r i c (L = 2 coupling) and we t h e r e f o r e have the following couplings which in SU(3) may have in p r i n c i p l e independent coupling constants (the s u p e r f i x e s P, T standing for p s e u d o s c a l a r , t e n s o r and the suffix S standing for s y m m e t r i c ) F(8-TI 8-p x 8 P) ~ F t r ( T 8 { P 8 ,
P8 }) '
G(1TI 8_Px8P) ~ G t r ( T l { P 8 , P8} ) , j ( s T I _8Pxl_ P) ~ J t r ( T 8 { P 8, P1}) , I (1T ] I_P×I p) ~ I t r ( T l { P 1 , P1} ) , where T1, T8, P1, P8 a r e the usual m a t r i x r e p r e s e n t a t i o n s of SU(3) n o r m a l i s e d so that t r ( P 2) = 1 etc. with only the octet states c o r r e s p o n d i n g to traceless matrices.
G. P. CANNING
454
Table 2 (1- ] 0- x 0-) SU(3) coupling constants (isospin invariant)
Resonance
External particle state
Coupling constant ideal w - q~ general mixing
p
~
~-C
p
KK
-,/-2
K*
K~
- ~ ~/-3 a
cO
KK
-~-~sin 01 C
- ~ 12 C
(fl
KK
-~-~cos 01 G
-G
1
G
We define the m i x i n g a n g l e s f o r the o c t e t s (0-), (2 +) a s in appendix 1, and we w r i t e f o r e x a m p l e : 7/~ s i n 0 o P8(78 ) + c o s 0 o P l ( 7 1 ) , w h e r e we i n d i c a t e by the a r g u m e n t , that only the c o m p o n e n t s of the m a t r i c e s c o r r e s p o n d i n g to the e n t r i e s of 78, 7/1 in the p a r t i c l e m a t r i c e s a r e to h a v e the n o n - z e r o n u m e r i c a l v a l u e specified. We t a b u l a t e , t a b l e 3, the i s o s p i n i n v a r i a n t coupling c o n s t a n t s , f i r s t a s s u m i n g the g e n e r a l SU(3) f o r m , s e c o n d l y a s s u m i n g F=G, c o t 0 2 = ~/2 ( q u a r k m o d e l nonet s t r u c t u r e f o r (2+)) and then with F=G =J=I, c o t 0 2 = c o t 0 o = ~-2 (nonet s t r u c t u r e f o r both (2 +) and (0-)).
Part. State
lrI]
~'
7r~
~u
7777'
77'7/'
Res.
A2
A2
f
f'
f
f
Final
Oo]
02 - G s i n 02]
constants
02cos2 %]
- J s i n 02sin 2 8 o + I cos 8 2 s i n 280]
cos e
2eos 2 8o
V sin 2 eo + ~ I
( F / 3 ~/2) cos 20 0
2 eo
(~/~-(F/~ sin 02oos 2 8o + c
sin
+ ~ J cos2%-(I/3
(F/6 ~ )
cos 200
since o
V~ s i n 280
+ ½J s i n 2 8o + ~ - I
(F/3 ~ ) sin 2 Oo
0
-~/-32~F
F= G 82 ~ 35 0
(isospin invariant)
(isospin invariant)
+ J s i n 02c os 2 8 o - ½ 1 cos 8 2sin 2 0o]
0 2 s i n 2 8o + ½ G cos 02sin 20 o
02sin2%+Ieos
(1/~)[-(F/2V~)sin
+J sin
(11v%[:-(F/~/2) sin 02sin 2 0o + G cos 02sin 2 0o
-[(F/~)cos
-[(F/'/2)sinO2+Gcos02]
(1/vr3[(F/vf2)cos 8o-JsinOo]
(1/x/3[(F/~)sin 8o+J cos
general
Coupling
Table 3 (2+I 0- × 0-) SU(3) coupling constants
0
(F/V~)
F = G= J = I 0o ~ 35 °° 82 ~ 35
5n
P~
-~Fsin20 e +½ ~/2 J c o s 200+ (I/6 sin 20 o
(1/ff3) [ - (F/2~/2cosO2sin20o-½GsinO2sin20 o
- ~ ~J
- J c o s a 2sin 2 00 - / s i n 02sin 20o]
F /2vz2 -½F
1F - (~/3/2 ~f2) F
( 1 / ~ ) [ - (F/2~-2)sinOo+J eos0o]
(1/yr3) [ - (F/2~f'2) cos 0 o - J sin 0 o]
K~
KT?
K~'
KN
KN
KN
+F / ~ 2
(1/~f3)[ +½ F e o s 0 2 +V~G sin 021
Kg
-F
35 °
F = G=J=I
KK
+ F/2
(1/#r3) [ +½ F s i n 02-~/-20osO2G ]
sin 2 0 o- ~ 1sin2
- ½Fcos 200
(1/'~-) [ - (F/~/2) cos 02cos 20o-G sin02cos 200
0o
+ ~v~ J s i n 20 o - ~ I c o s 200
+J cos 02cos 2 00 + ½/sin 02sin 200]
- ~ 2 F s i n 2 0o
(1/~-)[ - (F/~2)cos 02sin2 00 - G sin 02sin 2 00
F=G 02 ~ 35 °
+J cos 02sin 2 0o - / s i n 02cos 200 ]
genera[
K~
7/'7/'
Part. State
Coupling constants (isospin invariant)
A2
f'
Res.
Final
Table 3 (continued)
p
¢91
INTERNAL SYMMETRY IMPLICATIONS
457
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G. Veneziano, Nuovo Cimento 57A (1968) 190. S. L. Adler, Phys. Rev. 137 (1965) B1022. C. Lovelace, Phys. Letters 28B (1968) 264. K. Kawarabayashi, S. Kitakado and H. Yabuki, Phys. Letters 28B (1969) 432. H. Harari, Phys. Rev. Letters 20 (1968) 1395. J. L. Rosner, Phys. Rev. Letters 21 (1968) 950. C. B. Chiu, J. Finkelstein, Phys. Letters 27B (1968) 510. H. J, Lipkin, Nucl. Phys. B9 (1969) 349. G . P . C a n n i n g , Oxford preprint 42/68. P. C a r r u t h e r s , Introduction to unitary s y m m e t r y (Interscienee Publishers, 1966). [11] M. D. Scadron, Phys. Rev. 165 (1968) 1640; R. L. Sugar and J. D. Sullivan, Phys. Rev. 166 (1968) 1515. [12] J. L. Rosner, Tel-Aviv University preprint TUAP-70-69. [13] H. Harari, WeizmannInstitute preprint. [14] G.P.Canning, Oxford preprint '13/69. [15] Chan Hong-Mo, P. C. Calies and J. E. Paton, Phys. Rev. Letters 11 (1963) 521. [16] C. Schmid and J. Yellin, Phys. Letters 27B (1968) 19. [17] M. Gourdin, Unitary symmetries (North Holland, Amsterdam, 1967). [18] S. L. Glashow and R. H. Socolow, Phys. Rev. Letters 15 (1965) 329.