Partial conservation and the 2+ mesons

Volume 24B, number 6

PHYSICS

LETTERS

20 March 1967

P A R T I A L C O N S E R V A T I O N AND THE

2 + MESONS

W. KROLIKOWSKI Institute of Theoretical Physics, University of Warsaw, Warsaw and Institute for Nuclear Research, Warsaw Received 15 February 1967

A partial

conservation

relation

involving the 2+ mesons

is conjectured.

D e l b u r g o et al. [1] d e v e l o p e d the i m p o r t a n t i d e a that the r e c e n t l y d i s c o v e r e d 2 + m e s o n s h a v e s o m e s t r e s s t e n s o r s a s t h e i r s o u r c e s . The s t r e s s t e n s o r s u n d e r c o n s i d e r a t i o n w e r e o b t a i n e d a s the U(3) g e n e r a l i z a t i o n of the u s u a l s t r e s s t e n s o r O#v(X) o r a l t e r n a t i v e l y i t s f r e e p a r t i c l e p a r t . Of c o u r s e , only the c o n s e r v e d , t r a c e l e s s p a r t s of the s t r e s s t e n s o r s ought to be c o n s i d e r e d to get s y m m e t r i c a l , c o n s e r v e d , t r a c e l e s s t e n s o r c u r r e n t s that m a y be i d e n t i f i e d with s o u r c e s of the s p i n 2 + m e s o n s . T h i s c o n j e c t u r e f o r the 2 + m e s o n s is s t r i c t l y p a r a l l e l to the b a s i c i d e a p r o p a g a t e d f o r the 1- m e s o n s by S a k u r a i [2] that t h e s e m e s o n s h a v e the c u r r e n t s of the U(3) g e n e r a t o r s as t h e i r s o u r c e s . O b v i o u s l y , only c o n s e r v e d v e c t o r c u r r e n t s m a y be i d e n t i f i e d w i t h s o u r c e s of the s p i n 1- m e s o n s . T h u s , we a r g u e d s o m e t i m e ago [3] that t h i s f a c t i n f e r s an e s s e n t i a l d i f f e r e n c e b e t w e e n the c u r r e n t s of g e n e r a t o r s and the s o u r c e c u r r e n t s , if the g e n e r a t e d g r o u p is not a s t r i c t s y m m e t r y g r o u p . Due to this a r g u m e n t we p r o p o s e d [3] i n s t e a d of the S a k u r a i h y p o t h e s i s a n o t h e r link b e t w e e n the 1- m e s o n s and the U(3) g e n e r a t o r s . N a m e l y we c o n j e c t u r e d that the e l e c t r i c c u r r e n t (equal to the w e l l known l i n e a r c o m b i n a t i o n of the c u r r e n t s of the "3" and "8" g e n e r a t o r s ) is p r o p o r t i o n a l to a s u p e r p o s i t i o n of the n e u t r a l 1- m e s o n s pO and t(~(~ + co)w , the l a t t e r b e i n g the f a m i l i a r ¢o-~b m i x t u r e r e p r e s e n t i n g the "8" v e c t o r m e s o n . T h i s c o n j e c u r e w a s c a l l e d the p a r t i a l c o n s e r v a t i o n of a t e n s o r c u r r e n t ( P C T C ) [ 3 , 4 ] , s i n c e the e l e c t r i c c u r r e n t c a n be i d e n t i c a l l y e x p r e s s e d a s the d i v e r g e n c e of an a n t i s y m m e t r i c t e n s o r c u r r e n t . In the p r e s e n t note we d i s c u s s the 2 + m e s o n s a l o n g the s a m e line a s s u m i n g the h y p o t h e s i s that the s t r e s s t e n s o r Opv(X) (being a s y m m e t r i c a l , c o n s e r v e d but not t r a c e l e s s t e n s o r ) is p r o p o r t i o n a l to a s u p e r p o s i t i o n of the n e u t r a l 2 + m e s o n f (1254 MeV) and a n e u t r a l 0 + m e s o n we c a l l cr (if s u c h a m e s o n e x i s t s ) . T h i s h y p o t h e s i s m a y be a l s o c o n s i d e r e d as a p a r t i a l c o n s e r v a t i o n r e l a t i o n , s i n c e the s t r e s s t e n s o r 0 # v ( x ) c a n be i d e n t i c a l l y e x p r e s s e d as the d i v e r g e n c e of a t e n s o r of the t h i r d rank. We w r i t e down this h y p o t h e s i s as f o l l o w s

Ogv(x) = a f cogv(x) + a~ OpO v - Ol~V[~ co(x) ,

(1)

rn 2cr w h e r e ~o~v(x ) and co(x) a r e f i e l d s of the f and ~ m e s o n s , and af and a~ d e n o t e s o m e c o n s t a n t s o r " g e n t l e " f u n c t i o n s o f G . F o r the f i e l d cot.Lv(x) we h a v e the c o n s t r a i n t s

¢P~v = covg ,

Ovq~t~v = 0 ,

T a k i n g m a t r i x e l e m e n t s of (1) b e t w e e n e n e r g y - m o m e n t u m

cog~ = 0 .

(2)

e i g e n s t a t e s we o b t a i n

(B I Oi.zv(x) I A) = a f ( B I CO/Iv ( x ) I A ) + a~ OUr q2 _ q uqv(B[co(x)I A)=

(3) = af (BI Jt~v(x)

A) + act O#v q 2 + rnf2

q2 - q~qv rn~2

q2+

rn 2

305

Volume 24B, number 6

PHYSICS LETTERS

20 March 1967

w h e r e q = PB - PA is the e n e r g y - m o m e n t u m t r a n s f e r , w h e r e a s

j ~,(x) = (-D + m2)~ ~ ( x ) ,

(4)

j(x) = ( - D + m2)~(x)

denote the s o u r c e c u r r e n t s of the f and cr m e s o n s , r e s p e c t i v e l y . The l a s t e q u a l i t y in eq. (3) is v a l i d in the c a s e when the a s y m p t o t i c f i e l d s of the f and cr m e s o n s do not c o n t r i b u t e , a s e . g . for s t a t e s with e q u a l n u m b e r s of p a r t i c l e s ( a m o n g o t h e r s for o n e - p a r t i c l e s t a t e s ) . F o r o n e - p a r t i c l e s t a t e s with s p i n ½ we c a n w r i t e the l e f t - h a n d - s i d e of eq. (3) a s follows [5, 1]

(p'lo~u

(0)IP> = u ( P ' ) [ ( Y # P v , + Y u P ~ ) ~ G ( 1 ) ( q 2) + PgPu4-~ G(2)(q 2) + ( 6 , u q 2 -

q,qu)(l/m)G(3)(q2)]u(p)(2~)-~5

i

w h e r e q =p'-p. P =p'+p, p,2 = p 2 = - m 2 a n d ~ ( p ) u ( p ) = 1. F r o m eq.(5) we get G(1)(0)+G(2)(0) = 1, s i n c e p=0 = m/(27r)3, w h e r e = u(p')u(p) 63( p ' - p ) = 63( p' - p ) and H = f d3x Ooo(X) is the total e n e r g y . In a s i m i l a r way we e x p r e s s ~ ' ij~u(0)[p> in t e r m s of the f o r m - f a c t o r s s u b s t i t u t e d for

G(i)(q 2)

(gf/m) K(i)(q 2)

(i = 1 , 2 , 3 ) , w h e r e K(1)(-m 2) + K(2)(-rn 2) = 1. F i n a l l y , we put ( p ' ]j(0)]p) : ga K(q2) g(p ') u (p) (27r)-3 ,

where

K(-m 2)

G(i)(q2)

-

(6)

= 1. T h e n we o b t a i n f o r m eq. (3)

af

gf g(i)(q 2)

2 2 q +mf

m

af (i=1,2) ,

G(3)(q 2) = q2+rn'--~

gfK(3)(q 2) m

a¢:r mg e K (q2) + q2+ m2 cr

m2 (x

(7)

2

H e n c e we get

m mf af =

gf[K(1)(0) + g(2)(0)]

(8)

On the o t h e r hand, s i n c e the s t r e s s t e n s o r 0~v(x) is a s o u r c e of the g r a v i t a t i o n a l f i e l d , it follows f r o m (3) that af m u l t i p l i e d by the g r a v i t a t i o n a l c o n s t a n t is the t r a n s i t i o n a m p l i t u d e for the v i r t u a l p r o c e s s f ~ g r a v i t o n . If K(1)(q 2) + K]2)(q 2) w e a k l y d e p e n d s on q2, t h e n K(1)(0) + K(2)(0) ~ 1. In t h i s c a s e the f o r m u l a (8) shows that the c o u p l i n g c o n s t a n t s gf of the f m e s o n s to d i f f e r e n t p a r t i c l e s a r e a p p r o x i m a t e l y p r o p o r t i o n a l to m a s s e s m of t h e s e p a r t i c l e s . T h e n , if we a s s u m e that the f m e s o n is a Regge pole lying on the P o m e r a n c h u k t r a j e c t o r y , we o b t a i n a p p r o x i m a t e l y the f o r m u l a d e r i v e d in ref. 1 for total c r o s s s e c t i o n s in high e n e r g y l i m i t ~ab - -

~cd

ma mb :

-

-

(9)

mc md

We can s e e f r o m (5) and (7) that the e n e r g y - m o m e n t u m d i s t r i b u t i o n i n s i d e p h y s i c a l p a r t i c l e s d i s p l a y s p o l e s at the m o m e n t u m t r a n s f e r q2 e q u a l to - m .2 and - m 2. So, we find h e r e a n a n a l o g y with the e l e c t r i c c h a r g e d i s t ~ i b u t i o ~ w h i c h ~ c c o r d i n g to the p a r t i a l conser~vation of a t e n s o r c u r r e n t [3] has p o l e s at q2 e q u a l to - m ~ , - m ~ a n d -m~o. T h i s f o r m a l a n a l o g y is a l s o s e e n , if we c o n s i d e r the m e a s u r i n g of both d i s t r i b u t i o n s . The c h a r g e d i s t r i b u t i o n can be m e a s u r e d by e l e c t r o m a g n e t i c s c a t t e r i n g , s i n c e the e l e c t r i c c u r r e n t is a s o u r c e for the e l e c t r o m a g n e t i c field. S i m i l a r l y the e n e r g y - m o m e n t u m d i s t r i b u t i o n could in p r i n c i p l e be m e a s u r e d by g r a v i t a t i o n a l s c a t t e r i n g , b e c a u s e the s t r e s s t e n s o r Ogv(x ) is a s o u r c e of the g r a v i t a t i o n a l field. H o w e v e r , any r e a s o n a b l e r e a l i z a t i o n of s u c h m e a s u r e m e n t s e e m s b e y o n d o u r e x p e r i m e n t a l t e c h n i q u e . In this s i t u a t i o n one m a y w o n d e r w h e t h e r t h e r e a r e o t h e r w a y s f o r m e a s u r i n g the e n e r g y - m o m e n t u m d i s t r i b u t i o n i n s i d e p h y s i c a l p a r t i c l e s . P e r h a p s , s u c h a m e a s u r e m e n t m a y be b a s e d on i n e r t i a l p r o p e r t i e s of the e n e r g y . 1. 2. 3. 4.

R. Delburgo, A. Salam and J. Strathdee, Trieste preprint IC/66/115. J . J . Sakurai, Annals of Physics 11 {1960} 1. W. Kr61ikowski, Nuovo Cimento 42 {1966) 435; 44 {1966) 745; 46 {1966) 106. R. F. Dashen and M. Gell-Mann, 1966 Coral Gables Conference; S.Fubini, G. Segr~ and J. D. Waleeka, Annals of Physics, to be published; M. Ademollo, R. Gatto, G. Longhi and G. Veniziano, Phys. Letters 22 {1966) 521. 5. H. Pagels, Phys. Rev. 144 (1965} B1250.

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