The parity - violating internucleon potential and the field - current identity

Volume 29B, n u m b e r 4

THE

PHYSICS

LETTERS

12 May 1969

PARITY-VIOLATING INTERNUCLEON POTENTIAL THE FIELD-CURRENT IDENTITY*

AND

P . O L E S E N and J. S. R A O

Department of Physics and Astronomy, University of Rochester, Rochester, New York R e c e i v e d 24 M a r c h 1969

The parity-violating internucleon potential arising from p, ¢o and qb exchange is shown to be negligible using the field-current identity (vector-meson dominance). The implications of this result are discussed. In particular, it is pointed out that from an experimental point of view the process n + d ~ 3He + ~ should be more interesting than the process n + p --* d + ~. R e c e n t l y t h e r e h a s b e e n a g r e a t d e a l of i n t e r e s t in the p a r i t y - v i o l a t i n g i n t e r n u c l e o n p o t e n t i a l . One of t h e r e a s o n s f o r t h i s i n t e r e s t i s t h a t if s u f f i c i e n t l y a c c u r a t e e x p e r i m e n t a l d a t a a r e a v a i l a b l e it m i g h t be p o s s i b l e to d i s t i n g u i s h b e t w e e n t h e d i f f e r e n t w e a k H a m i l t o n i a n s p r o p o s e d to d e s c r i b e the n o n l e p t o n i c w e a k i n t e r a c t i o n s [1]. T h e o n e - p i o n e x c h a n g e c o n t r i b u t i o n , V~, h a s b e e n e v a l u a t e d by s e v e r a l a u t h o r s [2]. T h e a s s u m p t i o n t h a t h a s b e e n m a d e in c a l c u l a t i n g p a r i t y - v i o l a t i n g e f f e c t s in c o m p l e x n u c l e i and in p r o c e s s e s like n + p -* d + ~ and n + d ~ 3H + ~ i s t h a t t h e p o t e n t i a l V12 i s [3] v12 : v~ + Vp ÷ v ~ ÷ v ~ .

(1)

T h i s s e p a r a t i o n is n a t u r a l s i n c e the i s o s p i n p r o p e r t i e s of Vn i s IAII = 1 and of Vp, Vq~, V¢o i s l a l l = 0 , 2 . R e c e n t l y F i s h b a c h and T r a b e r t [4] h a v e u s e d t h e f i e l d - c u r r e n t i d e n t i t y [5] ( v e c t o r - m e s o n d o m i n a n c e ) to e s t i m a t e the p o t e n t i a l Vp f r o m p - e x c h a n g e . T h e s e a u t h o r s a c t u a l l y c o n c l u d e t h a t Vp h a s a c o n s i d e r a b l e m a g n i t u d e . In t h e p r e s e n t l e t t e r we show that Vp, V(~, and Vw a r e all c o m p l e t e l y n e g l i g i b l e . The d i f f e r e n c e b e t w e e n o u r r e s u l t and t h e r e s u l t f o r Vp r e p o r t e d in ref. 4 is due to the f a c t that we t a k e into a c c o u n t t h e c o v a r i a n t n a t u r e of the p a r i t y - v i o l a t i n g (weak) v e c t o r m e s o n - n u c l e o n v e r t e x . We m a k e the f o l l o w i n g two a s s u m t p i o n s : (a) The f i e l d - c u r r e n t i d e n t i t y [5] is valid. (b) In the c a l c u l a t i o n of the d i v e r g e n c e of a " c o v a r i a n t " t i m e - o r d e r e d p r o d u c t the S c h w i n g e r t e r m s c a n c e l the S e a g u l l t e r m s ( s e e r e f . 6). F o r a m o r e e x p l i c i t d e f i n i t i o n of t h i s a s s u m p t i o n s e e e q s . (2) and (6). In the c a l c u l a t i o n of the p, ¢ and w e x c h a n g e c o n t r i b u t i o n s to V12 one h a s to e v a l u a t e the w e a k , p a r i t y v i o l a t i n g v e r t e x w h e r e t h e v e c t o r m e s o n i s c o u p l e d to the n u c l e o n s . The a m p l i t u d e f o r t h i s v e r t e x is g i v e n by (v d e n o t e s the v e c t o r m e s o n , N i s t h e n u c l e o n , H P v i s the p a r i t y v i o l a t i n g p a r t of t h e w e a k Hamiltonian) w d ( S ( p ) ~ N(p')v) = =

(2) 1

= E,~ M xv = - i ~*( q 2 + m 2 ) ( 2 q o ) - -2 × [ f d 4 x exp(-iqx) + S~] , w h e r e q is t h e f o u r - m o m e n t u m c a r r i e d by the v e c t o r p a r t i c l e v (q = p ' - p ) , E~ i s i t s p o l a r i z a t i o n v e c t o r , and m v i t s m a s s . Sk i s t h e S e a g u l l t e r m i n t r o d u c e d to m a k e the v e r t e x f u n c t i o n A c o v a r i a n t . We s h a l l d i s c u s s t h e p,¢o and ~b c o n t r i b u t i o n s s e p a r a t e l y . T h e p - c o n t r i b u t i o n : In t h i s c a s e , one h a s

~

= (fp/m~)

v~(~)

,

w h e r e Pk is the p - f i e l d , a i s t h e i s o s p i n i n d e x and V~ is the i s o s p i n c u r r e n t . t h a t V~ i s a c o n s e r v e d c u r r e n t ) * Work supported in part by the U. S. Atomic Energy Commission.

(3) F r o m eq. (2) we get (using

233

Volume 29B, number 4

q k M 2 = (2qo)-2-

m9

PHYSICS LETTERS

(q2+m2)f d4x exp(-iqx)6(Xo)
12 May 1969 P~ (0)]IN(P)} + qxSk Hw

(4)

The e q u a l - t i m e c o m m u t a t o r in eq. (4) i s , in g e n e r a l , of the f o r m

HPV(0)' ~3(x) + S.T.

(5)

w h e r e H.PV(0) ' is r e l a t e d to HPV(0) by an SU(2) r o t a t i o n and S.T. s t a n d s f o r p o s s i b l e Schwinger t e r m s . The only~thing we know about~S. T. and the Seagull S k i s that t h e s e c o n t r i b u t i o n s c a n c e l in qk MkPbY a s s u m p t i o n (b). Hence eq. (4) b e c o m e s i

fp

q~ M p = (2qo)-~ ~_~ (q2 + m2 )(N(p')]HPV(0) ' IN(p)).

(6)

P F r o m C P - c o n s e r v a t i o n the r i g h t - h a n d side of eq. (6) v a n i s h e s in the SU(2) limit. Hence

q k M p = 0,

(7)

so M k is o r t h o g o n a l to q~tThe w- and ¢ - c o n t r i b u t i o n s : One h a s M~:

i(2qo )-½ ( q 2 + rn2)

I f d4x exp(-iqx)(N(p')]T(llk(x)HPwV(O)lN(p)}+ Ski ,

(8)

w h e r e ~ s t a n d s for co and 4. F r o m the f i e l d - c u r r e n t identity [5] one has

ll~t : A l l Yp. + Bll Ntt ,

(9)

w h e r e A ~ and Bll a r e c o n s t a n t s and IT# i s the h y p e r c h a r g e c u r r e n t and Ntt the b a r y o n n u m b e r c u r r e n t . T h e r e f o r e we obtain a s b e f o r e {the c o m m u t a t o r h a s no S.T. ' s s i n c e t h e s e t e r m s a r e c a n c e l l e d by qkSk) qk M ~ = (2qo) -½ (q2 + m 2 ) f

d4x exp(_iqx)6(Xo)(N(p,)[[llk(x),

HPV(oIJIN(p)>.

(10)

Now II X i s the sum of the eighth component of an octet and a u n i t a r y singlet. The s i n g l e t o b v i o u s l y c o m m u t e s with HPwV, and s i n c e HPVwc o n s e r v e s h y p e r c h a r g e , Yk a l s o c o m m u t e s with HPv.w Hence

[~k(x), HPwV(O)]xo:O : 0 ,

(11)

and

q~ M~= q~ M~= O .

(12)

$

Hence M p MXw, and MkCare o r t h o g o n a l to qk- T h i s is t r u e independent of any m o d e l f o r H P v. Since the g e n e r a l , c o v a r i a n t e x p r e s s i o n f o r M k i s w M k = i~(p') [7xhA(q 2) + okuq u hE(q 2 ) + iqkhp(q2)]75 kau([), it follows f r o m qkMk = 0 that Mk is n e g l i g i b l e f o r m o m e n t u m t r a n s f e r s of p h y s i c a l i n t e r e s t . Hence the p , co and ~ c o n t r i b u t i o n s to V12 can s a f e l y be i g n o r e d a s c o m p a r e d to the o n e - p i o n exchange contribution

y~. The p h y s i c a l i m p l i c a t i o n s of the vanishing of the v e c t o r - m e s o n c o n t r i b u t i o n s a r e many. We mention a few i m p o r t a n t points: (1) In the c a s e of c o m p l e x nuclei, w h e r e e x p e r i m e n t a l data e x i s t (with l a r g e e r r o r s ) it a p p e a r e d that the e x t r a - c u r r e n t m o d e l (the weak H a m i l t o n i a n involving n e u t r a l c u r r e n t s [7]) was not in a g r e e m e n t with e x p e r i m e n t . H o w e v e r , with the n u m e r i c a l u n i m p o r t a n c e of the v e c t o r - m e s o n exchange p o t e n t i a l ( i m p l i e d by o u r a s s u m p t i o n s ) the d i s a g r e e m e n t with e x p e r i m e n t s f o r the e x t r a - c u r r e n t m o d e l is e n t i r e l y inconclusive. In fact, if one t a k e s the e x i s t i n g e x p e r i m e n t a l evidence v e r y s e r i o u s l y , the e x t r a - c u r r e n t m o d e l a p p e a r s to have a s l i g h t l y b e t t e r a g r e e m e n t with e x p e r i m e n t s than the s t a n d a r d model involving only c h a r g e d c u r r e n t s . H o w e v e r , b e c a u s e of the e x p e r i m e n t a l u n c e r t a i n t y we do not wish to m a k e any s t a t e m e n t in f a v o r of any m o d e l f o r the weak Hamiltonian. (2) P e r h a p s of m o r e i n t e r e s t f r o m an e x p e r i m e n t a l point of view i s the effect of o u r r e s u l t in the two reactionsn+p --.d + 7 a n d n + d--* 3H + 7. It h a s been shown [ 3 ] t h a t in the r e a c t i o n n +p--* d + 7 t h e 234

Volume 29B, n u m b e r 4

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12 May 1969

a s y m m e t r i c p o l a r i z a t i o n P of t h e p h o t o n a r i s e s s o l e l y b e c a u s e of t h e e f f e c t of l A I I = 0, 2 p a r i t y v i o l a t i n g i n t e r n u c l e o n p o t e n t i a l . If o n e d o m i n a t e s t h i s p a r t of t h e p o t e n t i a l b y v e c t o r - m e s o n e x c h a n g e i t f o l l o w s that P=0inn+p-* d + ~. H o w e v e r , a s M o s k a l e v [3] h a s s h o w n , in t h e p r o c e s s n + d--* 3 H + y t h e p o l a r i z a t i o n P of t h e p h o t o n i s a r e s u l t of b o t h t y p e s of p o t e n t i a l s . C o n s e q u e n t l y , e v e n if t h e v e c t o r - m e s o n p o t e n t i a l i s n e g l i g i b l e , o n e s t i l l h a s P ¢ 0. C o n s e q u e n t l y , t h e p r o c e s s n + d --* 3H + y s h o u l d b e m o r e i n t e r e s t i n g f r o m a n e x p e r i m e n t a l p o i n t of v i e w t h a n t h e p r o c e s s n + p ~ d + y. (3) If o n e g i v e s u p t h e a s s u m p t i o n (b) i n g e n e r a l S e a g u l l a n d S c h w i n g e r t e r m s c o m b i n e t o g i v e a n a r b i t r a r y s c a l a r c o n t r i b u t i o n to q~ M k. In t h a t c a s e o n e c a n n o t m a k e a n y n u m e r i c a l c o n c l u s i o n s . A n e x p e r i m e n t a l i n v e s t i g a t i o n of t h e t w o p r o c e s s e s d i s c u s s e d in (2) m a y b e of h e l p i n i n v e s t i g a t i n g ( f r o m a n e x p e r i m e n t a l p o i n t of v i e w ) w h e t h e r S e a g u l l r e a l l y c a n c e l S c h w i n g e r t e r m s ! W e t h a n k E. F i s h b a c h , R. E. M a r s h a k , V. S. M a t h u r a n d S. O k u b o f o r i n t e r e s t i n g d i s c u s s i o n s . A f t e r t h e c o m p l e t i o n of t h i s w o r k we r e c e i v e d a t h e s i s b y M. F e u e r ( H a r v a r d U n i v e r s i t y T h e s i s , 1969), wherein similar results were obtained with respect to the p-exchange potential.

References 1. N. Cabibbo, Phys. Rev. L e t t e r s 10 {1963} 531; B.drEspagnat, Phys. L e t t e r s 1 {1963} 209: R. F. Dashen, S.C. F r a u t s c h i , M. Gell-Mann and Y. Rara~ in The eightfold way, cds. M. Gell-Mann and Y. Ne'eman (W.A. Benjamin, Inc., New York 1964} p.254; T.D. Lee, Phys. Rev. 171 (1968) 1731; G. Segr~, Phys. Rev. 173 (1968} 1730. 2. B . H . J . McKellar, Phys. L e t t e r s 26B (1967) 107; E. Fishbach, Phys. Rev. 174 (19 68) 1694. 3. F . C . M i c h e l , Phys. Rev. 133 (1964) B329; B . H . J . McKellar, Phys. Rev. L e t t e r s 20 (1968) 1542; D. Tadic, Phys. Rev. 174 (1968) 1694: G.S.Danilov, Phys. L e t t e r s 18 {1965} 40: A . N . M o s k a l e v , J E T P L e t t e r s 8 (1968} 27. 4. E. Fishbach and K. T r a b e r t , P r e p r i n t (1969). 5. M. G e l l - M a n n a n d F. Z a c h a r i a s e n , Phys. Rev. 124 (1961) 953; N. M. Kroll, T.D. Lee and B. Zumino, Phys. Rev. 157 {1967} 1376. 6. L.S. Brown, Phys. Rev. 150 (1966} 1338; D. G. Boulware and L. S. Brown, Phys. Rev. 156 {1967} 1724; R. P. Feynman, P r o c . 1967 Intern. Conf. on P a r t i c l e s and fields (Interscience, New York, 1967} p. 111. 7. B.d~Espagnat, Phys. L e t t e r s 1 (1963} 209.

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