CP violating neutral leptonic currents and KL,So → μμ decays

CP violating neutral leptonic currents and KL,So → μμ decays

!8.C.4~ Nuclear Physics B40 (1972) 541-561. North-Holland Publishing Company L ;--A Cp V I O L A T I N G AND NEUTRAL LEPTONIC CURRENTS K °L,...
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!8.C.4~

Nuclear Physics

B40

(1972) 541-561.

North-Holland Publishing Company

L ;--A

Cp V I O L A T I N G AND

NEUTRAL

LEPTONIC

CURRENTS

K °L,S - - ' jug D E C A Y S *

/ S. O K U B O and M. B A C E Department of Physics and Astronomy, University of Rochester, Rochester, New York R e c e i v e d 12 N o v e m b e r 1971 (Revised m a n u s c r i p t r e c e i v e d 13 J a n u a r y 1972)

A b s t r a c t : On the basis of a CP violating model with s t r o n g s e l f - i n t e r a c t i o n among the i n t e r m e d i a t e v e c t o r m e s o n s , we p r e d i c t the e x i s t e n c e of CP violating neutral leptonic c u r r e n t s . This model can explain the puzzling decay modes K ~ , S -" tt/J.. We also d i s c u s s v a r i o u s r a r e decay modes such as K~ ~ 7r°eg K~ ~ ~7+~"°e P and 7r+~-e6 as well as K+ + 7r+ ee- andTr +~ .'

1. N E U T R A L

LEPTONIC

CURRENT

A N D K °L , S ~ p p D E C A Y S

A r e c e n t e x p e r i m e n t [1] h a s g i v e n u s a s u r p r i s i n g l y s m a l l u p p e r b o u n d o f o r t h e d e c a y r a t e K L ~ p p . I m p r o v i n g t h e m e t h o d o r i g i n a l l y u s e d by L e e a n d C h r i s t [2], G a i l l a r d h a s d e m o n s t r a t e d [3] f i r s t t h a t the e x p e r i m e n t a l d e c a y r a t e i s i n c o n s i s t e n t w i t h t h e s u p e r - w e a k t h e o r y a n d s e c o n d t h a t the K~ ~ p ~ r a t e m u s t s a t i s f y a l o w e r b o u n d F(K~-

pfi) >/ 1.6 x 10 .7 .

(1)

F(K~ ~ all) o S u c h a r e l a t i v e l y l a r g e d e c a y r a t e f o r K S ~ ~zfi i s e x t r e m e l y d i f f i c u l t to u n d e r s t a n d [4, 5] if the d e c a y p r o c e e d s o n l y v i a v i r t u a l two p h o t o n e x c h a n g e . T h e p u r p o s e of t h i s n o t e i s to s h o w t h a t s u c h a l a r g e d e c a y r a t e is c o n s i s t e n t w i t h the t h e o r y of C P v i o l a t i o n w h i c h h a s b e e n p r o p o s e d e l s e w h e r e [6, 7], w h e r e w e h a v e a t r i p l e t o r o c t e t of i n t e r m e d i a t e v e c t o r b o s o n s w h i c h h a v e s t r o n g i n t e r a c t i o n s a m o n g t h e m s e l v e s but a r e c o u p l e d w e a k l y ( o r e l e c t r o m a g n e t i c a l l y ) to a l l o t h e r p a r t i c l e s . A s we s h a l l s h o w in t h e n e x t s e c t i o n , o u r t h e o r y p r e d i c t s t h e e x i s t e n c e of an e f f e c t i v e C P = -1 n e u t r a l s e m i - l e p t o n i c i n t e r a c t i o n of t h e f o r m

* Work supported in p a r t by the U. S. Atomic E n e r g y C o m m i s s i o n .

S. Okubo, M. Bac/e, CP violating model

542

H2(x) = ~1i G

sin Oji7)(x) {#(x) yX(1 + y5 ) p(x)

+ g(X)Yx(l+Ys)e(x) - Y/j(X)Yx(l+Y5)uia(x) - Ue(X)yx(l+ y5)Ue(X)},

(2)

w h e r e diT)(x)-- is a s t r a n g e n e s s - c h a n g i n g CP = -1 n e u t r a l h a d r o n i c c u r r e n t of the f o r m JiT)(x) = f ? A i 7 ) ( x ) + , Vi7)(x).

(3)

H e r e , F3 and Y a r e r e a l c o n s t a n t s of the o r d e r 10 -2 ~ 10 -3 , and

V~a)(x)(A(pa)(x)),---

Ca = 1 , 2 , . . .

vector) hadronic currents.

8) a r e the o c t e t of s t a n d a r d v e c t o r ( a x i a l In the q u a r k m o d e l , we have

v(x~)(x)= ½ig(x)~,xx~q(x).

d(x~)(~) : ½i~(x) ~,x~,Sx: (~),

Note that H2(x) has CP = -1 p a r i t y , and that the m a g n i t u d e of the c o n s t a n t s and ~' is c o m p a r a b l e to the fine s t r u c t u r e c o n s t a n t a = e2/47r. F o r c o m p a r i s o n , we w r i t e down [8] the u s u a l CP = +1 Cabibbo H a m i l tonian f o r s e m i - l e p t o n i c d e c a y s //l(X)

-1

= ~

iG

{cos O(V i 1+i2 )(x)+ A i 1+i2 )(x)) + sin 0( Vi 4+i5 )(x)+ A i 4+i5 )(x))}

× {/J:(x) yX(1 + },5) up(x) + ~(x) YX(1 + ~5 ) Ue(X)} + h e r m i t i a n conjugate.

(4)

The o v e r a l l m u l t i p l i c a t i v e f a c t o r - i is due to o u r n o r m a I i z a t i o n of Vx and A x. Although the s e c o n d - o r d e r effect of Hl(X) can be a CP c o n s e r v i n g n e u t r a l l e p t o n i c d e c a y of h a d r o n s , its m a g n i t u d e is f a r s m a l l e r in g e n e r a l than

Ha(x). In this s e c t i o n we a c c e p t the e x i s t e n c e of the new CP violating n e u t r a l leptonic i n t e r a c t i o n , eq. (2), and i n v e s t i g a t e its p h y s i c a l i m p l i c a t i o n s p h e n o m e n o l o g i c a l l y . We m a y r e m a r k that an a n a l o g o u s H a m i l t o n i a n has been p r o p o s e d s o m e time ago by O a k e s [9] with d i f f e r e n t m o t i v a t i o n and with a l a r g e r coupling p a r a m e t e r but without c o n t a i n i n g e l e c t r o n and muon c u r rents. To begin, we note t h a t H 2 ( x ) d o e s give a n o n - z e r o d e c a y m a t r i x e l e m e n t o f o r K~ -~ t~fi, but not f o r K 2 ~ /~ta. Notice that this c o n c l u s i o n is just the r e v e r s e [10] of the u s u a l one w h e r e the e f f e c t i v e H a m i l t o n i a n is CP c o n s e r v i n g . H e r e a f t e r , K~ and K~ imply CP = +1 and CP = -1 c o m b i n a t i o n s of the K o and 1~o c o m p I e x , r e s p e c t i v e l y . Then, we c o m p u t e

F(-)(K~~ ~#) _:~2[i riK+ ~ufi)"~K:-jSL1

4

- \ink/-]

(5) '

S. Okubo, M. Bac~e, CP ~;iolating tnodel

543

w h e r e F(±)(K~ ~ ~]J) i s the d e c a y r a t e of K o1 into the f i n a l t~y s t a t e w i t h C P p a r i t y ±1, r e s p e c t i v e l y . Since the C P c o n s e r v i n g e l e c t r o m a g n e t i c d e + o - . c a y r a t e F( )(K 1 ~ ~,.F.) i s e x p e c t e d to be m u c h s m a l l e r [11] and we n e g l e c t a s m a l l c o n t r i b u t i o n to K ~ - K ~ m i x i n g f o r a s i m i l a r r e a s o n , we can a p proximate

so t h a t eq. (5) g i v e s

F(K~ ~ t~i~)

0.43fi 2 × 10 - 2 .

F(K~s ~ all)

(6)

C o m p a r i n g t h i s e x p r e s s i o n w i t h the l o w e r b o u n d , eq. (1), we find i32 >/ 3.72 × 10 -5 .

(7)

(We m a y r e m a r k t h a t if we had u s e d the s t r o n g e r i n e q u a l i t i e s o b t a i n e d by L e e and C h r i s t 3 f o r F ( K ° ~ /a/Y), then we w o u l d have o b t a i n e d a s t r o n g e r bound 2.34 × 10!3]>~ ~2 >~ lS.16 × 10-4.) H o w e v e r , t h i s e s t i m a t e i s p e r h a p s c o n s e r v a t i v e s i n c e it d o e s not t a k e into a c c o u n t the s p e c i a l f e a t u r e s of o u r m o d e l . I n d e e d , W o l f e n s t e i n [4] h a s a n a l y z e d the p r o b l e m in d e t a i l on the b a s i s of a p u r e l y a x i a l v e c t o r H a m i l tonian H' = iG' sin 0 A i 7 )(x ) p ( x ) ;¢Xy5P(x),

and h a s o b t a i n e d the f o l l o w i n g s t r o n g e r bound 2.7 × 10 -2 > ~

1

( G ' / G ) > 0.8 × 10 -2 .

A s i m i l a r r e s u l t h a s a l s o b e e n o b t a i n e d by D a s s [4]. In t e r m s of o u r p a r a m e t e r ~, the a b o v e bound i m p l i e s 2.8 × 10 . 3 > ~2 > 2.3 × 10 -4 .

(7,)

T h i s e s t i m a t e i s a l s o a n a l o g o u s to t h o s e o b t a i n e d by C h r i s t and L e e [3]. A s we s h a l l s e e in the n e x t s e c t i o n , o u r t h e o r y of C P v i o l a t i o n can g i v e a v a l u e of fl c o n s i s t e n t w i t h eq. (7'). The c a l c u l a t i o n of K ~ -~ ~t7 d e c a y is a v e r y c o m p l i c a t e d m a t t e r . Now, the H a m i l t o n i a n of eq. (2) d o e s not a l l o w K~ ~ /a~ and we have to t a k e into a c c o u n t not only K ~ - K ~ m i x i n g [2, 3] but a l s o the e l e c t r o m a g n e t i c e f f e c t s [11, 12] a s w e l l p e r h a p s a s the s e c o n d o r d e r C P c o n s e r v i n g w e a k c o n t r i b u tion [13]. H o w e v e r , it i s o b v i o u s t h a t i t s d e c a y r a t e i s m u c h s m a l l e r than F ( K ~ s ~ p f i ) w h i c h i s p e r m i t t e d by H2(x ). We m a y a l s o r e m a r k t h a t

544

S. Okubo, M. Bac~e, CP ~,iolating ~nodel

_(me.2 [1 4( -e-]21½ 1-4(m]21-½ -~ 2.11 x 10 - 5 .

(8)

A s i m i l a r r a t e f o r F ( K ~ - e~)/F(K~ -. tJ-iJ~)is also e x p e c t e d b e c a u s e both e l e c t r o m a g n e t i c and weak c u r r e n t s a r e i n v a r i a n t u n d e r the c h i r a l t r a n s f o r m a t i o n e ( x ) - ' Y5e(x) and p(x) • 75t~(x) so that, by the s t a n d a r d a r g u m e n t [81, the d e c a y m a t r i x e l e m e n t s m u s t be p r o p o r t i o n a l to the lepton m a s s e s m e and m ~ , r e s p e c t i v e l y . This p r e d i c t i o n is quite d i f f e r e n t f r o m that of S e h g a l ' s t h e o r y [14] in which

r(K~-~ e~) ~ r(K~- ~g). Next, we notice that the c h a r g e i n d e p e n d e n c e and c h a r g e conjugation inv a r i a n c e g i v e s us ,

,

1

,V(4-iS)(x)IK+>,

< _ o iV {X7)(x) K o1 = 0 . T h e r e f o r e , we find that o u r t h e o r y gives us the following r e l a t i o n s (by n e g l e c t i n g s m a l l t e r m s p r o p o r t i o n a l to ( m e / i n K ) 2 ) ,

F(-)(K~

~ Oe~ ) = ½F(I~L_,

o~)

= ½F(K + ~ +~p-) 2

o

= F ( - ) ( K + ~ ~+ee} = ~7 F(K L : ½ 7 2 F ( K + --

Jv+e~)

~°e+~) '

r(-)(K~ _ O e e ) = r(-)(K~ _. o

~) = r(K~ -- o ~ )

= 0,

(9)

where F(-)(K~ -- ~°ed) and F(-)(K + ~ ,v+e~) represent the C P violating partial decay rates due to our C P = -I Hamiltonian, eq. (2), and we replaced O O 142 by K L since the correction is expected to be extremely stoat1. In reality, these s a m e decays can occur by the C P conserving Dalitz pair ~orocesses 0 + 0 + 0 + • . . KL(K ) - - 77 ( , v ) y - - ~ ( 7 ; ) e e . W r i t i n g t h e s e p a r n a l r a t e s as F~+)(K-- 7;e~), the t r u e d e c a y r a t e s a r e given by F(K~

7;°e~) ~ F(+)(K~ - ~°e~) + F ( - ) ( K °L ~ 7T°e~)

F(K + ~ ~+e~) = F(+)(K +-- ~+e~) + F(-)(K + -

~+e~).

(I0)

S. Okubo, M. Bac~e, CP violating model

Experimentally

545

[15], we know F ( K + ~ 7T+~P)£ 1.2 × 10 . 6 F ( K + -~ all).

T h e n , eq. (9) p r e d i c t s 2< ~' ~

2 . 4 7 × 10-

5

A l s o , f r o m the u p p e r l i m i t [15] F ( K + -- ~+e~) ~ 0.4 x 10 . 6 F ( K + ~ a l l ) t o g e t h e r w i t h e q s . (9) and (10), we find a s t r o n g e r bound 2 L 1.65 × 10 - 5 .

(11)

C o m p a r i n g t h i s with eq. (7'), it a p p e a r s that t h e m o s t i n t e r e s t i n g c h o i c e fi2 = ~,2 c o r r e s p o n d i n g to the e x a c t V - A c u r r e n t f o r J(p.7) is u n f o r t u n a t e l y i n c o n s i s t e n t w i t h the p r e s e n t e x p e r i m e n t a l d a t a by a f a c t o r of 14. We a l s o r e m a r k t h a t the s m a l l n e s s of F ( K + ~ v+e~) p r e s e n t s a s l i g h t d i f f i c u l t y to m a n y t h e o r e t i c a l e s t i m a t e s [16] of the C P c o n s e r v i n g D a l i t z p a i r d e c a y r a t e F(+)(K + ~ v+ee-). F o r e x a m p l e , P a k v a s a and S i m o n [16] c a l c u l a t e F(+)(K + - ~+ee) ~ 1.0 × 10 -6 F ( K + ~ a l l ) . A t any r a t e , it i s p o s s i b l e t h a t the C P v i o l a t i n g c o n t r i b u t i o n f o r K + -- ~+e~ ( o r ~ + p p ) i s of the s a m e o r d e r [7] a s t h a t of the C P c o n s e r v i n g one. Then we e x p e c t to have a g r o s s C P v i o l a t i o n in K + ~ ~+e~ ( o r 7r+/J~) d e c a y , w h i c h can m a n i f e s t i t s e l f in p o l a r i z a t i o n c o r r e l a t i o n m e a s u r e m e n t s . H o w e v e r , s i n c e t h i s p o i n t h a s b e e n f u r t h e r d i s c u s s e d by M a r s h a k et al. [7], we w i l l not go into d e t a i l s . O O -N e x t , l e t us c o n s i d e r the KT ~ 7r ee d e c a y . Since the C P c o n s e r v i n g d e c a y m a t r i x e l e m e n t f o r t h i s p r o c e s s i s p r o p o r t i o n a l t o g 2 02 = (e 2 /4~) 2~ 2 , we e x p e c t [6, 7] t h a t we can n e g l e c t i t , i . e . , F ( + ) ( K ~ - ~°e~) << F ( - ) ( K L ~ u ° e ~ ) .

Then,

from

(12)

eqs. (9) and (ii), we predict O

F ( K ~ ~ ~ ee-)

0.10 y 2 ~

1.65 × 10 . 6 .

(13)

F(K~ ~ all) F o r a d i s c u s s i o n of I ~L ~ v ° ~ y d e c a y , we note t h a t the r e s u l t d e p e n d s upon the ~ and ;~+ p a r a m e t e r s of the c o r r e s p o n d i n g K + ~ 7r°~g d e c a y . A s we s h a l l s h o w in a p p e n d i x 1, we find

S. Okubo, M. Bac~e, CP violating n~odel

546

F(K~-- o j . ~ ) F ( I ~L -- ~Oe~ )

F(-)(K~2 -~ ~Oe~ )

= 0.37 + 0.16~ + 0.03~ 2 + 1.51X+ + 0 . 0 7 8 ~ + ~ , where

we

(14)

have set

/+(l ) = 74_(0)[1 + X+l/m2] ,

x_ = 0,

~ :%_(o)//+(o).

The value of the X+ and ~ parameters is stilluncertain experimentally, if we use the value [171 ~ = -0.6 and X+ = 0.03, the right-hand side of eq. (14) will become 0.32. Also in appendix 1, we will give a formula for the energ~y spectrum of the pion in the decay. Neglecting mass differences between K' and K ° and between ~+ and r,°, we also predict

F(-)(K+ -- ~+lJ g) _ F(-)(K~ - ~°~7)

(15)

F~Y)(K+-, ,v+eN)F(-)(K~ -- ~°e~) As has been discussed by Marshak et al. [7], the Hamiltonian of eq. (2) may give us a large CPviolating effect for E + ~ pe~decay. Also we predict +

~ 2 > ~ F(E

~ pu/~Yp)>~ ~Y'2 ,

(16)

F(E -* neF e) and a s i m i l a r one f o r F(E + -* pUeYe). H o w e v e r , in view of the s m a l l v a l u e s f o r ¢72 and y2, this will give an e x t r e m e l y s m a l l r a t e f o r E + ~ p u F d e c a y . P e r h a p s a b e t t e r way to t e s t eq. (16) will be to m e a s u r e the r a r e n e u t r i n o

reaction ;~oPs ~e~ef~;c~:i:hoh;S ~;i; Ito1:itYn~Hal Imbl~:k g:ou n:q Other sib . (2), are decays such as O

+

K L-~

-

--

~ ee, ypp, ye@.

O

First, consider the Ke4 decay. The decay matrix element due to H2(x) is proportional to o

+

(17)

This matrix element is calculable in terms of the corresponding matrix element of the K +~ 77+7~-u6decay. It is not difficult to show that the vector part proportional to y gives a very small contribution while the axial vector term proportional to ~ is calculable from the p-wave 77+7r- scattering param

S. Oleubo, M. Bace, CP violating model

e t e r s [18] of K + ~ v + v - v ~ d e c a y . If we u s e the e x p e r i m e n t a l p a r a m e t e r s S c h w e i n b e r g e r e t al. [19], we find

r ( -)(K o2 _, ~ + ~ - e g ) F(K~

1.5~? 2 × 1 0 - 5 .

547

of

(18)

all)

H o w e v e r , the n u m e r i c a l r e s u l t i s s e n s i t i v e to the p - w a v e p a r a m e t e r . F o r e x a m p l e , if we had u s e d v a l u e s o b t a i n e d by E l y et al. [20] we would have h a d to m u l t i p l y the r i g h t - h a n d s i d e of eq. (18) by f a c t o r s of 1.3 and 0.93, r e s p e c t i v e l y , f o r t h e i r s o l u t i o n s l a b e l e d a s (A) and (B). U s i n g the l o w e r b o u n d , eq. ( 7 ' ) , f o r i~2, we find F (-)(K~

7r+ ~ - eg)

L 3.4 × 10 -9.

(19)

I ' ( K ~ ' all) H o w e v e r , t h e r e i s a n o t h e r c o n t r i b u t i o n too K °L +_. ~r+Tr-e~ w h i c h c o m e s f r o m the C P c o n s e r v i n g D a l i t z p a i r p r o c e s s K L ~ 7r u-~' ~ u+Tr-eg. M a j u m d a r and S m i t h [2111 e s t i m a t e it to be r ( + ) ( Z ~ ~Tr + ~ - e~) -

~ 1 . 7 × 10 - 7 .

(20)

F ( K ~ - , all) T h i s i s m u c h l a r g e r than eq. (19) and h e n c e the new c o n t r i b u t i o n s w i l l b e c o m e i n s i g n i f i c a n t , a l t h o u g h t h e r e i s no e x p e r i m e n t a l v a l u e a v a i l a b l e to c h e c k eq. (20) at the p r e s e n t t i m e . 0 ± 0 ~ In t h i s c o n n e c t i o n , we n o t i c e t h a t the d e c a y K L S -~ 7r 7r e v can p r o c e e a by the C P c o n s e r v i n g C a b i b b o H a m i l t o n t a n , eq. (4I. Its m a t r i x e l e m e n t i s a g a i n c a l c u l a b l e f r o m the c o r r e s p o n d i n g K + -~ 7r+Tr-vg d e c a y . U s i n g the p w a v e p a r a m e t e r of S c h w e i n b e r g e r e t al. [19], we c o m p u t e F ( K ~ - ' ~+~°e:~ v)

__

~ 1.23 × 10 -4,

F(K~-" all) F ( K ~ ~ 7r±Tr°eTv) 2.05 x 10 - 7 .

(21)

all) If we u s e the v a l u e of E l y e t al. [20], then we h a v e to m u l t i p l y t h e s e n u m b e r s by a f a c t o r of 1.3 f o r s o l u t i o n (A) and 0.93 f o r s o l u t i o n (B). T h e r e f o r e , the e x p e r i m e n t a l d e c a y r a t e f o r K ~ , S ~ ~+~°e~v m a y be u s e f u l to d i s c r i m i + n a t e b e t w e e n t h e s e v a r i o u s s o l u t i o n s of Ke4 d e c a y p a r a m e t e r s . So f a r , no e x p e r i m e n t a l n u m b e r f o r the d e c a y r a t e s i s a v a i l a b l e . S o m e d e t a i l s of the c a l c u l a t i o n w i l l be g i v e n in a p p e n d i x 2. F i n a l l y , l e t u s c o m m e n t upon K ~ ~ ye~ d e c a y . The H a m i l t o n i a n H2(x) w i l l g i v e a C P v i o l a t i n g t e r m p r o p o r t i o n a l to .em

~qf d4x e x p ( - iqx) {0 I[#H. (x), a ~ ) ( 0 ) ] + I K~}.

548

S. Okubo, M. Bac'e, CP violating ~odel

This matrix element is the analogue of the so-called structure-dependent form factor of the corresponding decay K+ - e~,y. However, our new matrix element is easily shown to vanish in the exact SU(3) limit. We have made an estimate of this t e r m , following the f a m i l i a r method of refs. [22,23], and O found that it gives an extremely small contribution for g L ~ yee in comparison to the CP conserving Dalitz pair mechanism and hence we will not consider it further. As we shall show in appendix 3, the Dalitz pair p r o cess K ~ ~ y ~ - ' y ~ - p r e d i c t s

F ( K ~ ~ yeT) 8.8 × 10 . 6 ,

r(~:~-~

all)

F(K~-~ ~,~) 2.3 × 10 . 7

(22)

r(z~-' all) if we neglect again a small effect due to KS - K L mixing. Also, we will calculate the energy spectrum of the photon as well as a correction due to the non-zero form factor in the appendix, since this process will give a background for the more interesting decays K ~ - - ~°ee and o /7. In ending this section the following comment is in order. To explain a O large KS ~ ~ decay rate, we have introduced the effective neutral axialvector leptonic current. This is essentially due to the fact that our theory of the CP violation predicts its existence, as we shall see in the next s e c tion. Another reason for the p r e f e r e n c e of the axial vector current is as follows. Suppose that instead of eq. (2) we assume a p s e u d o s c a l a r form i H3(x) = ,~2 G s i n 0 {~P(7)(x)[~(x

)(a+ib 75) ~.(x) + ~(x)(a +ib 7,5) e(x)] ,

(23) where P(a)(x) (a = i . . . . 8) is an octet of pseudoscalar densities and a,b and O fi are real constants. This Hamiltonian allows the decay K1 ~ ~ as before. O -Hence, we could also explain a large KS ~ H-~-by this Hamlltonian. However, O it predicts now a large KS ~ ee rate in addition. In fact, we can easily prove

F(K~ ~

e6)

F(K~ ~

~1

>/ ( m

-4m

)~(rn

-4m

)-5 > 1

(24)

if we a s s u m e eq. (23). (The e q u a l i t y in eq. (24) o c c u r s for a = 0.) Note the d i f f e r e n c e b e t w e e n this e x p r e s s i o n and eq. I8). The r e a s o n why we no l o n g e r have the s m a l l m u l t i p l y i n g f a c t o r (me/rnp)2 h e r e is due to the fact that the H a m i l t o n i a n eq. (23) is not i n v a r i a n t u n d e r the c h i r a l i t y t r a n s f o r m a t i o n ~(x) ~ y5~(x), (~ = e o r p). H o w e v e r , the e l e c t r o m a g n e t i c i n t e r a c t i o n is i n v a r i a n t [8] u n d e r the c h i r a l i t y t r a n s f o r m a t i o n , so that the d e c a y m a t r i x O O e l e m e n t for K 1 ~ e~ and K 2 ~ e~ due to e l e c t r o m a g n e t i c i n t e r m e d i a t e s t a t e s is s u p p r e s s e d by a v e r y s m a l l f a c t o r rne/m p. (See, f o r e x a m p l e , e x p l i c i t c a l c u l a t i o n s by M a r t i n et al. [11] and by Sehgal ~ 2 ] . ) T h e r e f o r e , t h e r e is .~ _ O no m e c h a n i s m of c a n c e l l i n g a r e l a t i v e l y l a r g e K L ~ ee r a t e due to K S - K ~

m i x i n g in c o n t r a s t

S. Okubo, M. Bac~e, CP violating model o to the K L ~ tz/~ c a s e . I n d e e d , we e x p e c t F(K~ 0

e~) ~ I ~ 1 2 F ( K ~

549 to have now

e~-),

0

w h e r e ~ is the s t a n d a r d K L - K S m i x i n g p a r a m e t e r [8] with 16 I - 10-3- T h i s g i v e s , in view of eqs. (1) and (24), F ( K ~ ~ ee) > 4 . 5 × 1 0 -8 '

F(K~ ~ all) which contradicts the present experimental upper limit [15] of 1.9 × 10 -9 . Thus, this can be used as another argument against the pure pseudoscalar interaction eq. (23). Of course, the dilemma can be avoided if we give up -e universality by discarding the electron current in eq. (23). But in view of the universal validity of p - e symmetry, this would be somewhat a r b i trary.

2. STRONG CUBIC I N T E R M E D I A T E V E C T O R BOSON THEORY In the p r e v i o u s s e c t i o n , we have i n v e s t i g a t e d v a r i o u s c o n s e q u e n c e s of the e x i s t e n c e of the new CP v i o l a t i n g s e m i - l e p t o n i c i n t e r a c t i o n , eq. (2). As we have n o t e d e l s e w h e r e [6, 7], i t s e x i s t e n c e can follow f r o m a u n i f y i n g t h e o r y of the w e a k i n t e r a c t i o n b a s e d upon a t r i p l e t o r o c t e t of i n t e r m e d i a t e v e c t o r b o s o n s w h i c h i n t e r a c t s t r o n g l y a m o n g t h e m s e l v e s . H e r e , we s h a l l d e m o n s t r a t e m o r e o v e r that in the l o c a l l i m i t , the e f f e c t i v e i n t e r a c t i o n can be i n d e e d w r i t t e n in the f o r m of eq. (2). We c o n s i d e r the c a s e of the o c t e t of W m e s o n s whose f i e l d o p e r a t o r w i l l be d e s c r i b e d by (Wt~)~(x), (a, b = 1, 2 , 3 ) , a l t h o u g h we could a c h i e v e the s a m e r e s u l t with a t r i p l e t of W m e s o n s . It is s o m e t i m e s c o n v e n i e n t to r e g a r d ( W p ) ~ ( x ) as a 3 × 3 m a t r i x ~Vt~(x) with (~Tt~)ba =- (WH.) ~. T h e n , the Y a n g - M i l l s L a g r a n g i a n for the W m e s o n i s w r i t t e n as

1 o2 Tr(I~p(x) I~p(x)), L o = _ ~Tr(/~/~(x)-~/~u(x)) - ~rn

(25)

w h e r e the t r a c e is with r e s p e c t to the SU(3) i n d i c e s and

: Note that

W~(x)

a

%(x/-

(26)

i s s e l f - c o n j u g a t e , i.e.

Vd~(x): ~ ( x ) ,

or

[(Wp)~(x)] j"

=(w,)b(x).

(27)

The Y a n g - M i l l s L a g r a n g i a n L o c o n t a i n s a s t r o n g c u b i c i n t e r a c t i o n with the s t r o n g c o u p l i n g c o n s t a n t f o a m o n g W f i e l d s . The m o t i v a t i o n s for i n t r o d u c i n g s u c h a s t r o n g i n t e r a c t i o n a r e t h r e e f o l d . F i r s t , i t s e x i s t e n c e could g e n e r a t e a l a r g e W m e s o n m a s s [24] of m o r e than 5 GeV as a s e l f - e n e r g y e f f e c t , if the W m e s o n s e x i s t a t all. S e c o n d , it could a l s o e x p l a i n a r e l a t i v e l y

S. Ok~bo, M. Ba~e, CP t,iolating model

550

low weak i n t e r a c t i o n c u t - o f f of 5 ~ 10 GeV, w h i c h is n e c e s s a r y to e x p l a i n [13] m a s s d i f f e r e n c e s of the K ~ = - K ° c o m p l e x a s w e l l a s a p o r t i o n of the K ~ : " /./= d e c a y r a t e [25] r e s u l t i n g f r o m the s e c o n d = o r d e r weak e f f e c t of the o r d e r G 2. I n d e e d , the p r e s e n c e of a s t r o n g i n t e r a c t i o n a m o n g W m e s o n s would p r o d u c e the w e a k i n t e r a c t i o n c u t - o f f of the o r d e r of the W m e s o n m a s s . T h i r d , the c u b i c s t r o n g i n t e r a c t i o n d e t e r m i n e s the r e l a t i v e p a r i t y and r e l a t i v e c h a r g e c o n j u g a t i o n p a r i t y of W m e s o n s a m o n g t h e m s e l v e s . Es= p e c i a l l y , the p r o d u c t of p a r i t i e s (or c h a r g e c o n j u g a t i o n p a r i t i e s ) of t h r e e W m e s o n s i s a b s o l u t e l y fixed to be u n i t y when total e l e c t r i c and h y p e r = c h a r g e s of the t h r e e m e s o n s a r e s u m m e d up to be z e r o . A l t h o u g h this l a s t r e q u i r e m e n t is s u f f i c i e n t for o u r p u r p o s e s , it is c o n v e n i e n t to a s s i g n d e f i n i t e P ( p a r i t y ) and C ( c h a r g e c o n j u g a t i o n ) p a r i t i e s of the W m e s o n s to be e x a c t l y the s a m e as those of the u s u a l v e c t o r o c t e t p, K*, K* and w8With this c o n v e n t i o n , the s e m i = w e a k i n t e r a c t i o n L a g r a n g i a n Ll(X) is as= s u m e d to be m a x i m a l l y CP v i o l a t i n g , i.e. CP = =1, of the f o r m L l(x) =

-@{wl(x)(j~(x)+cosOf (x))+W~(x)(jl3(X)+ sin(+~(x)) + W23(X)(j2(X)+ cJ~(x))} + h e r m i t i a n c o n j u g a t e ,

(28)

w h e r e we have s u p p r e s s e d a l l L o r e n t z v e c t o r i n d i c e s for s i m p l i c i t y and

Qx(x) =i[g(x) Tx(1 +75) u p ( x ) + e(x) 7X(1 + 7,5) ~e(X)] i s the s t a n d a r d c h a r g e d l e p t o n i c c u r r e n t . s u m e d to have the u s u a l V-A f o r m •

(29)

The h a d r o n i c c u r r e n t j b ( x ) is a s -

(I

(jX)b(X) = (Vk)~(x) + (Ax)~(x) ,

(30)

w h i l e (JX)~(x), e n t e r i n g into the n e u t r a l h a d r o n i c c u r r e n t in eq. (28), is t a k e n to be an a r b i t r a r y l i n e a r c o m b i n a t i o n of V and A

with r e a l c o n s t a n t s , teresting possibility p u r e V-A c u r r e n t s . mental results seem

/3', /~", V' and 7" of the o r d e r of u n i t y . The m o s t i n is o b v i o u s l y to s e t F~' = Y' = ;~" = 7" = 1 w h i c h g i v e s the H o w e v e r , as we s h a l l see s h o r t l y , the p r e s e n t e x p e r i to r e q u i r e i~' ¢ V'. F o r the q u a r k m o d e l we have

(Vk)~(x) :iqa(X)TX qb(x),

(Ax)~(x) =i~a(X)7xy 5 qb(x).

B e f o r e going into d e t a i l s , we m a y r e m a r k that the i m a g i n a r y f a c t o r m u l t i p l y i n g L 1 is n e c e s s a r y to e n s u r e the CP = =1 p r o p e r t y of L 1. As we have e m p h a s i z e d e l s e w h e r e [26], we c a n n o t a b s o r b this f a c t o r i by i n c o r -

S. Okubo, M. Bac~, CP violating model

551

p o r a t i n g it into a r e d e f i n i t i o n of the W f i e l d s i n c e it a f f e c t s the s t r o n g c u b i c i n t e r a c t i o n a m o n g IV f i e l d s . T h e r e f o r e , the C P v i o l a t i o n is g e n u i n e . F o r any r e a c t i o n w i t h o u t e m i s s i o n o r a b s o r p t i o n of r e a l W m e s o n s , we o b tain no e f f e c t in the o r d e r g e n (n = 0, 1,+2 . . . . ) w h e r e e is the e l e c t r o m a g n e t i c c o u p l i n g c o n s t a n t . T h i s is due to the c o n s e r v a t i o n of the w e a k h y p e r c h a r g e and w e a k e l e c t r i c c h a r g e s of W m e s o n s , as h as b e e n e x p l a i n e d e l s e w h e r e [6]. In the o r d e r g2, we r e p r o d u c e the s t a n d a r d C P c o n s e r v i n g Cab i b b o t h e o r y f o r l e p t o n i e and s e m i - I e p t o n i c i n t e r a c t i o n s in the l o c a l l i m i t ,

H l ( x ) = 1 G { j l ( x ) c o s ~ + j l ( x ) s i n 0 } ~(x)

1

{j21(X) cos8

j~(x) s i n O } ~ ( x ) +

1

(32)

if we i d e n t i f y

G/~-2 = g2/m 2 ,

(33)

w h e r e m is the c o m m o n m a s s of the W m e s o n s . H o w e v e r , the o r i g i n of C P c o n s e r v i n g n o n - l e p t o n i c i n t e r a c t i o n in the o r d e r g2 is e n t i r e l y d i f f e r e n t . It r e s u l t s in o u r t h e o r y f r o m the i n t e r f e r e n c e of n e u t r a l h a d r o n i c c u r r e n t s p r o p o r t i o n a l to W 3 w i t h t h e i r h e r m i t i a n c o n j u g a t e c o u n t e r - p a r t in eq. (28). The r e s u l t i n g i n t e~r a c t i o n a u t o m a t i c a l l y s a t i s f i e s the AI = ½ r u l e f o r AS = +1 and A I = 1 , 0 f o r AS = 0 r e a c t i o n . We h a v e no IASI >/ 2 t e r m s in t h i s o r d e r . We note that o u r r e s u l t has p e r h a p s a d e s i r a b l e f e a t u r e . It h as b e e n 1 s t r e s s e d by S a k u r a i and o t h e r s [27] t h a t a A [ = ~, C P c o n s e r v i n g c u r r e n t times current non-leptonic transition involves a coupling constant remarkably c l o s e to F e r m i ' s c o n s t a n t , it s e e m s , t h e r e f o r e , t h a t the e x p e r i m e n t d o e s not a l l o w f o r the s m a l l cosO s i n 0 f a c t o r of the Cab i b b o t h e o r y . In o u r t h e o r y , no s u c h f a c t o r n e e d s to a p p e a r . Now, the o r d e r g 3 t e r m g i v e s a n o n - z e r o c o n t r i b u t i o n in o u r t h e o r y in c o n t r a s t to the u s u a l one. T h i s i s due to the p r e s e n c e of the s t r o n g c u b i c i n t e r a c t i o n a m o n g W m e s o n s . A l s o , t h i s t e r m has C P p a r i t y -1 f o r r e a c t i o n s w h i c h do not i n v o l v e any r e a l W m e s o n s , no m a t t e r how we a s s i g n d i f f e r e n t c h o i c e s of r e l a t i v e P and C p a r i t i e s a m o n g W m e s o n s . T h i s i s due to o u r e a r l i e r r e m a r k t h a t the C P p a r i t y of the v a c u u m e x p e c t a t i o n v a l u e of

i s u n i q u e l y s p e c i f i e d u n l e s s it is i d e n t i c a l l y z e r o . To v e r i f y t h i s s t a t e m e n t e x p l i c i t l y , l e t us c o m p u t e the s e m i - l e p t o n i c d e c a y of a h a d r o n A into a n o t h e r h a d r o n ( o r v a c u u m ) B by A-~B+~

+~,

w h e r e f now s t a n d s f o r any one of e , /~, v e o r v/j. The r e l e v a n t F e y n m a n d i a g r a m f o r t h i s p r o c e s s in the o r d e r g3 i s d e p i c t e d in fig. 1. In the l o w e s t o r d e r of f o , t h i s d i a g r a m g i v e s a q u a d r a t i c a l l y d i v e r g e n t r e s u l t . H o w e v e r , a s we h a v e r e m a r k e d , the p r e s e n c e of the s t r o n g c u b i c i n t e r a c t i o n a m o n g

S. Okz~bo, M. Bac~e, CP violati~zg model

552

B\

wY

7

Fig. 1. A l'~eynman diagram in the o r d e r g 3 responsible for CP :: -1 neutral semi-leptonic decays of hadrons A ~ B + f + ~. W m e s o n m a y g i v e r i s e to an e f f e c t i v e c u t - o f f of the s a m e o r d e r as the W m e s o n m a s s m, if we a v o i d the p e r t u r b a t i o n with r e s p e c t to fo" U n f o r t u n a t e l y , we do not know how to a c h i e v e this end at the p r e s e n t t i m e . It m a y be that the p r o g r a m of S a l a m et al. [28] on s u c h a p r o b l e m m a y e v e n t u a l l y s o l v e the d i f f i c u l t y . At any r a t e , h e r e we s i m p l y i n t r o d u c e a c u t - o f f p a r a m e t e r A in the c a l c u l a t i o n . Then as we s h a l l p r o v e in a p p e n d i x 4, the f i n a l r e s u l t in the l o c a l l i m i t is e q u i v a l e n t to the i n t r o d u c t i o n of an e f f e c t i v e l o c a l i n t e r a c t i o n of the f o r m H2(x) = \ ~1 i G sin O z

{(Jx)2(a ") (Jx)3(x)[ ~iO)(x),

(34a)

;~O)(x) = i {]i(x) yX(1 + y 5 ) p ( x ) + ~(x)yX(1 + ¥5 ) e(x)

where

the constant

~p(x) y x ( l + Y 5 ) ~ p ( x ) -

z is calculated

~-e(X)yx(l+Y5)~e(X)},

to be

3-2fogcosO (A22)

z~

16~

y = -2zy',

i~

to eq. (2) of the previous

section :

mesons

gives

essen-

(36)

= -2zi3',

if we

(Jx)2(x) (jx)23(x) 2ij(7)(x). -

(35)

m

for A -> co. The calculation based upon a triplet of W tially the same answer [7]. With the identification

eq. (34) reduces

(34b)

notice

(37)

B e f o r e g o i n g into d e t a i l s , we e m p h a s i z e that by the l o c a l l i m i t we m e a n n e g l e c t i n g t e r m s of the o r d e r of q 2 / m 2 , q 2 / A 2 , and m 2 / m 2 w h e r e q and rn~ a r e the m o m e n t u m t r a n s f e r of the h a d r o n s and the l e p t o n m a s s , r e s p e c t i v e ly. S i n c e we e x p e c t A ~ m ~ 5 GeV ( r e f . [24]), t h o s e t e r m s n e g l e c t e d a r e v e r y s m a l l . F o r i n s t a n c e , e v e n f o r K -~ pfi d e c a y , we h a v e q2 ~ m 2 so that -+ 10-2. Now, it i s e a s y to v e r i f y that o u r H a m i l t o n i a n , eq. (34a), has C P = -1. A l s o , we n o t i c e that in the l o c a l l i m i t , we h a v e no s e m i - l e p t o n i c n e u t r a l i n t e r a c t i o n with AS = 0. F u r t h e r m o r e , i n d e p e n d e n t of m a k i n g the l o c a l l i m i t ,

iq2/m2]

/

S° Okubo, M. Bace, CP ~,iolating model

553

t h e r e i~ no I ASI >~ 2 i n t e r a c t i o n up to the o r d e r g3. Since we have three unknown parameters g, f o and A, it is d i f f i c u l t to e s t i m a t e the p r e c i s e v a l u e of z. H o w e v e r , f o l l o w i n g M a r s h a k et al. [71, w e c a n m a k e a r o u g h o r d e r of m a g n i t u d e e s t i m a t e ,

i43o

1 ~ 10,

g=

10 -2 ~ 10 - 3 ,

= 1 ~ 10.

(38)

T h e n , w e c a l c u l a t e r o u g h l y z = (10 .2 - 10 -3 ) w h i c h g i v e s the r i g h t o r d e r of m a g n i t u d e f o r c o n s t a n t s i3 and 7 r e q u i r e d in t h e p r e v i o u s s e c t i o n . N o t e t h a t s i n c e it a p p e a r s t h a t ~2 < ~g2 e x p e r i m e n t a l l y , we unfortunately cannot set a 7 ' = iT', i . e . the c u r r e n t (Jx)b(X) c a n n o t h a v e t h e i d e a l V - A s t r u c t u r e . A s w e n o t e d in t h e p r e v i o u s s e c t i o n , o u r e f f e c t i v e l o c a l H a m i l t o n i a n eq. (2) o r eq. (34) a l l o w s K o1 ~ lJ~ii b u t n o t K o2 t~tJ. H o w e v e r , K o2 -- lJ.//c a n o c c u r in o u r F e y n m a n d i a g r a m c a l c u l a t i o n a s a n o n - l o c a l c o r r e c t i o n . Its m a g n i t u d e t h e r e f o r e w i l l be s u p p r e s s e d by a f a c t o r (mK/rn) 2 - 10 -2 in o c o m p a r i s o n to the K 1 ~ t.~/i a m p l i t u d e . A l s o , in the e x a c t w e a k SU(3) l i m i t in w h i c h we n e g l e c t tile m a s s d i f f e r e n c e s a m o n g v a r i o u s W m a s s e s , we c a n s h o w t h a t K 7 ~ p ~ i s f o r b i d d e n i r r e s p e c t i v e of w h e t h e r we t a k e the l o c a l l i m i t o r not. T h u s , it i s r e a s o n a b l e to e x p e c t t h a t the m a t r i x e l e m e n t f o r K~2 -~ p.~ d u e to o u r m e c h a n i s m w i l l be s m a l l e r by a f a c t o r of 103 at l e a s t o in c o m p a r i s o n to K 1 -, V-/L A s we n o t e d a l r e a d y , t h i s i s e x a c t l y w h a t i s n e e d e d to e x p l a i n a p o s s i b l y l a r g e F ( K ~ - V / J . ) / F ( K ° L " t~tJ) r a t i o . In r e f . [71, t h i s e x t r a d a m p i n g f a c t o r d u e to the l o c a l l i m i t h a s n o t b e e n t a k e n into a c c o u n t . A s a r e s u l t , an i n c o r r e c t l y l a r g e F ( K ~ -- / ~ ) r a t e h a s b e e n p r e O d t c t e d t h e r e . A n o t h e r c o m p l i c a t i o n in t h e c a l c u l a t i o n of K L - tJ-lJ~ d e c a y i s t h e f a c t t h a t w e h a v e to t a k e i n t o a c c o u n t t h e C P e o n s e r v i n g g 4 ( o r G 2) e f f e c t [131 a s w e l l a s the e l e c t r o m a g n e t i c e f f e c t of the o r d e r g 2 e 4 s i n c e a l l t h e s e a r e c o n t r i b u t i n g to K ~ ~ /ly~ d e c a y . H o w e v e r , f o r K~ -* tJ~lJ- the e f f e c t of H 2 ( x ) i s d o m i n a n t o v e r a l l o t h e r s , s o t h a t w e c a n n e g l e c t t h e s e c o m p l i cations. N e x t , l e t us c o n s i d e r C P v i o l a t i n g c o r r e c t i o n s to the s t a n d a r d s e m i leptonicdecay A ~ B e F o r A ~ B / ~ . A t y p i c a l d i a g r a m f o r t h i s p r o c e s s in t h e o r d e r g3 i s d e p i c t e d in fig. 2. T h e c a l c u l a t i o n f o r t h i s d i a g r a m i s m o r e i n v o l v e d and l e s s t r u s t w o r t h y t h a n the p r e v i o u s c a s e , s i n c e we h a v e to t a k e i n t o a c c o u n t n o w e f f e c t s of s t r o n g i n t e r a c t i o n s f o r h a d r o n i c v e r t i c e s . In t h i s c a s e , the e f f e c t i v e c u t - o f f is of the s a m e o r d e r a s the n u c l e o n m a s s m N r a t h e r t h a n the v e c t o r m e s o n m a s s m. T h e r e a s o n f o r t h i s i s t h a t at the hadron vertex, we must calculate matrix elements such as - "

•

(B(P') IJtj(O) IB(P)). However, as we know from the electromagnetic form factor of the nucleon, these matrix elements involve another effective cut-off at the high-momentum transfer of the order of the p or A I masses. Taking this damping factor into consideration, we can make the following rough order of magnitude estimate for the contribution from the diagram of fig. 2 which is roughly smaller by a factor of 10 -3 - 10 -4 in comparison to the standard Cabibbo theory. Then, the CPviolating effect will be small by a similar factor and

S. Okubo, M. Bac/e, CP violating model

554 B

e (,u.)

.-B

g S ''/

g

fo

fo I

g

~,.

Fig. 2. A typical graph giving a CP violating c o r rection to the Cabibbo semi -leptonic decay A ~ B e f + 12.

Fig. 3. A more c o mp l i cated Feynman diagram. an additional contribution to the diagram of fig. 2.

Fig. 4. The simplest diagram responsible for the possible e l e c t r i c dipole mome~_t of the neutron.

the t o t a l e f f e c t i v e h a d r o n i c c u r r e n t w i l l be m o d i f i e d in the l o c a l l i m i t to

Gv{Vx +igvV~} + GA {Ax +iOAA~},

(39)

w i t h OV, OA ~ (10 -3 ~ 10-4). Of c o u r s e , f o r o u r p r o c e s s , t h e r e a r e m a n y a d d i t i o n a l F e y n m a n d i a g r a m s s u c h as fig. 3, w h o s e m a g n i t u d e w i l l be d i f f i c u l t to e s t i m a t e . At any r a t e , in the l o c a l l i m i t , a l l t h e s e t e r m s w i l l be s u m m e d up to the f o r m of eq. (39), due to the v e c t o r n a t u r e of the W m e s o n . So f a r , we h a v e d i s c u s s e d only s e m i - l e p t o n i c d e c a y s . I r i s e a s y to s e e t h a t the p u r e l e p t o n i c d e c a y s a r e not a f f e c t e d at al l by o u r t h e o r y in the o r d e r g3 so that t h e s e m u s t be p u r e l y CP c o n s e r v i n g up to t h i s o r d e r . W i t h r e s p e c t to n o n - l e p t o n i c d e c a y s in the o r d e r g 3, we g et a CP = -1 c o n t r i b u tion. We can c h e c k that we h a v e no t e r m s w i t h !AS 1 >~ 2 and that the 1 AS = +1 p a r t c o n t a i n s in g e n e r a l both AI = g a n d A I = ]. S i n c e the p r e s e n t e x p e r i m e n t on K ~ ~ 2~ d e c a y s d e m a n d s the d o m i n a n c e of the AI = ½ p a r t , we h a v e to a p p e a l to the i d e a of o c t e t e n h a n c e m e n t . O r it m a y be that the d o m i n a n c e of A I = ½ r e s u l t s f r o m a m e c h a n i s m p r o p o s e d by M a t h u r and O l e s s e n [29] who s h o w e d that the m o s t d i v e r g e n t p a r t in the o r d e r g2 w i l l in g e n e r a l obey the A! = ½ r u l e . H o w e v e r , if we i n s i s t upon o b t a i n i n g a p u r e AI =g,1 CP v i o l a t i n g i n t e r a c t i o n in the o r d e r g 3 , we w o u l d a c h i e v e it if we e n l a r g e the s y m m e t r , y a g r O u p f o r the W m e s o n f r o m the SU(3) to SU(4). T h e n , we have to u s e (lVtz)'b(X) f o r the W f i e l d w h e r e the i n d i c e s a and b now a s s u m e v a l u e s a,b = 1 , 2 , 3 , 4 . If we a s s u m e in a d d i t i o n to Ll(X) of eq. (28) the e x i s t e n c e of a n o t h e r p u r e l y n e u t r a l h a d r o n i c i n t e r a c t i o n L2(x) =

-ig[W4(x)j~(x)+

W 4 ( x ) j 3 ( x )] + h e r m i t i a n c o n j u g a t e ,

(40)

then the new c o n t r i b u t i o n c a n c e l s the AI = ~ p a r t , thus l e a v i n g a p u r e AI = ½ n o n - l e p t o n i c H a m t l t o n i a n in the o r d e r g 3. A l s o it d o e s not a f f e c t at a l l the l e p t o n i c o r s e m i - l e p t o n i c i n t e r a c t i o n s d i s c u s s e d b e f o r e , e x c e p t that in the orderg 2, it can g i v e a n o t h e r c o n t r i b u t i o n to AS = 0 n o n - l e p t o n i c i n t e r a c tion. A l t h o u g h we could now in p r i n c i p l e c o m p u t e the d e c a y m a t r i x e l e m e n t s f o r K L ~ 2,v, tt is too c o m p l i c a t e d to do any r e l i a b l e c a l c u l a t i o n . H o w e v e r , an o r d e r of m a g n i t u d e e s t i m a t e d o e s g i v e a q u a l i t a t i v e l y c o r r e c t r e s u l t [7]. S i m i l a r l y , the e l e c t r i c d i p o l e m o m e n t of the n e u t r o n is a l l o w e d in the o r d e r

S. Okubo , M. Bac~e , CP violating model

555

g 3 e . A t y p i c a l d i a g r a m is shown in fig. 4. A g a i n , it is too c o m p l i c a t e d to m a k e any r e l i a b l e c o m p u t a t i o n s . F i n a l l y , we s h a l l b r i e f l y m e n t i o n that the C P v i o l a t i n g e f f e c t s f o r Ix'° L ~ y y (Ix"° S ~ ),7,) and K + ~ n+Tr°y a r e e x p e c t e d to o c c u r in the o r d e r g 3 e 2 a n d g 3 e r e s p e c t i v e l y f o r o u r t h e o r y . H e n c e , the C P v i o l a t i o n f o r these d e c a y s a r e e x p e c t e d to be s m a l l e r by a f a c t o r of 10 -3 than the C P c o n s e r v i n g m a t r i x e l e m e n t . This will be quite d i f f e r e n t f r o m r e s u l t s of s o m e o t h e r t h e o r i e s of C P violation [301 . P r e s u m a b l y a study [311 of photon p o l a r i z a tions in t h e s e d e c a y m o d e s m a y d i s t i n g u i s h o u r t h e o r y f r o m o t h e r s . L a s t , we r e m a r k that f o r o u r t h e o r y , the n o n - l e p t o n i c i n t e r a c t i o n will not have the s t a n d a r d c u r r e n t t i m e s c u r r e n t f o r m with e x a c t V - A s t r u c t u r e , s i n c e {~ ¢ y is e x p e r i m e n t a l l y r e q u i r e d . Then, the a n a l y s i s of n o n - l e p t o n i c d e c a y p r o b l e m s on the b a s i s of the a l g e b r a of c u r r e n t s will have to be r e i n v e s t i g a t e d in o u r model. The a u t h o r s would like to e x p r e s s t h e i r g r a t i t u d e f o r i l l u m i n a t i n g c o n v e r s a t i o n s with P r o f . V. S. M a t h u r , and a l s o to P r o f . L. W o l f e n s t e i n f o r v a r i ous v a l u a b l e s u g g e s t i o n s .

APPENDIX

1

Here, we shall prove eq. (14). First, by the charge independence and charge conjugation we notice ( O ( p , ) iViT)(0)! K~(p)) = (4PoPoV2)-½

× ½i{(P+P')xf+(t) + w

ere

aro

standard

,or

(P-P')x f_(t)},

actors

de

with t = _ ( p _ p , ) 2 .

T h e n , o u r H a m i l t o n i a n , eq. (2), g i v e s us

F(-)(K~ ~ o ~ )

1 1 G2 sin20 7,2 (2,7) 3 24rn K

_

w° × f

_ dw(w2

2 ! rnn)z(1-

(A.1)

2

ay

1

4m~,~ I - / {A(w)[f+(t)]2

m..z

+ B ( o ) ) f + ( l ) f (t) + C(w)[f_(l)]2}, where

w is the pion

energy

in the rest

system

of the kaon,

(A.2) and

I and

w o are

t = m 2 + m 2 - 2mKW , w o = (m2K+ m 2~.- 4 r n 2 ) / ( 2 m K ).

(A.3)

/

S. Okubo, M. Bace, CP violating model

556

Also A(w), B(w) and C(co) are given by

2 2K - r n 2 ) A(w) =4m2K(CO2 - r n 2 )+ rn~[3(m

2 m2)]/t,

2 2K - rn2v) , B(w) = 6m~(rn C(a~) = 3rn~ t.

(A.4)

If we approximate

/+(t) ~/+(o)[I

+ x+Um2],

f_(t) ~ f (o),

and setting

= f_(0)//+(0), we integrate the integral numerically neglecting t e r m s of o r d e r ~ ~ and ~ 4 2 to obtain eq. (14). Eq. (A.2) gives the energy spectrum of the o meson orequivalently the distribution of the square of the invariant lepton m a s s , which is equal to t. APPENDIX 2 0

+

-

--

0

+

0

_

Here we shall compute K L~ v v ee and K L-~ v v eudecay rate. We shall follow the notations of Treiman and Pals [18] and of Ely et al. [20] for the K + ~ v+v-e+u decay. First, consider the decay K~ -~ ~+~-ee by our Hamiltonian, eq. (2). Its decay matrix element is proportional to

By CP invariance of the theory, it is not difficult to see that the final two pion systems must be in odd angular momentum states in their rest frame. Following refs. [18,20], we restrict ourselves only to the discussion of S and P waves. Then it is not difficult to show that the vector part does not give any contribution at all in the approximation used by the authors of refs. [18,20]. Thus, we compute F(-)(K~ ~ ~+~-e~)

-

1

1

G2 sin20(½/3)2

(2v) 5 72(rnK)5

× f f d s ~ d s ~ ( -s~ s~ where

we have

•

X{(-PlP)2+2(-F2P )2}'

(A. 5)

S. Okubo, M. Bac~e, CP t,iolating ~nodel s f)2

x--

Fp1~

= / sr, - 4 m 2 . ½ ~ [ ,v "% r , . . z ' ~ ,

F2p =[sf(s

1,

- 4s:f}

/

557

Z

I

2

-4m2)]½~,

(A.6)

in the notation of ref. [20]. I n t e g r a t i n g this n u m e r i c a l l y , we find F(-)(K~ ~ v+n-e~) F(K + ~

~+~-ev)

= ½d2C,

(A.7)

w h e r e the value of the c o n s t a n t C is c o m p u t e d to be C = 0.29 and 0.20 f o r s o l u t i o n s A and B o f Ely et al. [20] r e s p e c t i v e l y and C = 0.22 f o r the s o l u tion given by S c h w e i n b e r g et al. [19]. We r e m a r k that c o m p u t i n g K ~ ~ ~ ° n ° e ~ s i m i l a r l y we obtain r ( - ) ( g ~ ~ 7r°Tr°e~)

= ½fi2C'

(A.7')

F(K + ~ ~+7;-e v) w h e r e C ' = 0 . 2 9 , 0.27 and 0.31 f o r the t h r e e S wave s o l u t i o n s . S i m i l a r l y , we could c o m p u t e the g~ ~ ~+~°e~ d e c a y . Now, this p r o c e s s can p r o c e e d by the s t a n d a r d Cabibbo H a m i l t o n i a n eq. (4). Again the v e c t o r c o n t r i b u t i o n is found to give an e x t r e m e l y s m a l l c o n t r i b u t i o n and the d e c a y r a t e can be corn puted f r o m eq. (A.5) by r e p l a c i n g the p a r a m e t e r /3 t h e r e by 2. Then, i n s e r t ing the known d e c a y r a t e for K + ~ n+~-6v, we obtain eq. (21). A l s o , we have

r(K~ ~

T;+~7°e~) = F ( K ~ ~ 77+~°e~)

b e c a u s e of the AS = AQ rule.

APPENDIX

3 0

We s h a l l c o m p u t e h e r e the D a l i t z p a i r d e c a y r a t e K 2 ~ ~,>, ~ ~,~f (f = e o r ~). Let us s e t I

.era

(4koPoV2) ~ ~,(k) y p

( 0 " )IK~(p)} = 2 i f 6 u a ~ p u k a 6 / ~ ( k ) F(t) ,

(A.8)

w h e r e the c o u p l i n g p a r a m e t e r f c o r r e s p o n d s to that of the e f f e c t i v e H a m i l tonian

H ' ( x ) = i f ¢pvafi~ p A v ( x ) ~ a A f i ( x ) K ~ ( x ) , and the f o r m f a c t o r F(t) is n o r m a l i z e d to s a t i s f y

(A.9)

/

558

S. Okubo, M. B a c e , C P violating model

/7(0) = 1, First,

the c o u p l i n g p a r a m e t e r

t = -(p-/e) 2 .

(A.10)

f can be c o m p u t e d f r o m the f o r m u l a

f2

F(K~ - ' yy) = 4rr ~(mK)3"

(a. ll)

O

T h e n , the d e c a y r a t e f o r the D a l i t z p r o c e s s K 2 -~ ),1[~ i s now g i v e n by [32] 1

F(K~-' y~)_e

F(K~2~ rV)

5

2 2 f dr(1-y)3(y+½YolY-~(y-yo)~(F(t))2 4rr 3v Yo

(A.121

w h e r e we have s e t (2m~, 2 Yo = . ~ ) .

y =t/m~,

(A.13t

N o t i c e t h a t if co i s the p h o t o n e n e r g y in the r e s t f r a m e of the k a o n , then we have y = 1 -

2 --

co.

(A.14)

m K 0

0

If we a p p r o x i m a t e F ( I ) = F ( 0 ) = 1, then we c o m p u t e K L ~ ~°e~ and v ~.~ a s in eq. (22). Note t h a t f o r the K% - ~°e~ c a s e , the d o m i n a n t c o n t r i b u t i o n c o m e s f r o m the s m a l l y r e g i o n , i . e . , s m a l l e l e c t r o n e n e r g y . I n d e e d , in t h a t c a s e , Yo i s e x t r e m e l y s m a l l , and we can i n t e g r a t e eq. (A.12) to o b t a i n F ( K ~ - - ye~)

e2 4 ~

F(K~ ~ y,y)

4g 3~

mK

~rn e , 2

by setting F(t) = i, again. This formula reproduces of course the well known result of Dalitz [32]. The last approximation may not be so good for large t or for a value ofy near one. If the p and co dominance for the electromagnetic form factorF(t) is reasonable, then we expect to have

mp2 F(I) ~

2 _ --. t mp

(A.16)

If we u s e eq. (A.16), then the n u m e r i c a l v a l u e of eq. (22) w i l l be m o d i f i e d to F(K~ ~ ye~) . . . .

9.06 × 10 -6 ,

F(K~ ~ a l l ) F(K~ ~ ),g/~) = 3.1 × 10 -7 ,

(A.17)

r ( ~ 2 ~ all) w h e r e we u s e d [15] F(K~ -" ),~,) = 5.6 x 10 -4 F(K~ -* all). We m a y r e m a r k t h a t the r a d i a t i v e c o r r e c t i o n s [331 m a y a c t to c a n c e l the

/

S. Okubo, M.Bace, CP violating model

559

e f f e c t of the f o r m f a c t o r F(t). Since we can n e g l e c t a s m a l l c o n t r i b u t i o n f r o m the K ~ - I ~L m i x i n g p a r a m e t e r E, we can r e p l a c e K~ in o u r d i s c u s s i o n O by K L without m a k i n g any s e r i o u s e r r o r .

APPENDIX 4 We shall c o m p u t e the F e y n m a n d i a g r a m , =

~'fo Tr ~Wta(x), Wv(x)]

fig. 1. L e t us s e t

~ Wv (x) - ~xv

which is a p a r t of the Y a n g - M i l l s L a g r a n g i a n , have to c o m p u t e the i n t e g r a l

eqs. (25) and (26). Then we

I = f f f f d4wd4xd4yd 4 z exp {iklx + ik2Y + ik3z ~ c W e L (w)]+I0} . x
(A.19)

F o r s i m p l i c i t y , we h e r e a f t e r a s s u m e that all i n t e r m e d i a t e v e c t o r m e s o n s have the s a m e c o m m o n m a s s rn (the w e a k SU(3) limit). Then we c o m p u t e

a c be - 6_Csesa., b d f]

I = -(27r) 4 5(4)(kl+k2+k3)folH.vk(kl,k2,k3)[6dSf5

(A.20)

w h e r e Ipv~. h a s m o r e o v e r the following s t r u c t u r e

I~vx(k 1, k2, k3) = Fal~y(kl, k2, k3) Dt~c~(kl) Dv~(k2) D ~ ' ( k 3 ) ' F a ~ y ( k l , k2, k3) = (k I - k2) Y 6a3 + (k 2 - k3) a 53y + (k 3 - kl) ~ 6./a,

D ~a(k) = -i[6 ~a + )ln~ kp k a](k2 + m 2 ) - 1 . Then, the F e y n m a n d i a g r a m ,

(A.21)

fig. 1, is now c o m p u t e d to be

s (A(k) ~ B(k ') + ~(k 1) ÷ ~(-k2)}

= (2u) 4 i6(4)(k - k '

sinO c o s 0

x {(B(k')[ (Jx)23(0)1A(k)} - (B(k ')I(Jx)2(0)!A(k)}}

x g f ( k 1 ) ½(1 - r5) Qx ½( 1 + r5)u ~(k2),

(A. 22)

S. Ok~¢bo, M° Bac/e, CP violating nzodel

560 w h e r e Qx i s g i v e n by

Qx=-i

l w D ~ a ( q ) f d 4 p F at3-/( q, P - k l , k2 - [) Dp fl(p - le1 )

(27r)-~

--

× Dz,),(k 2 -

q= k-

p)(1 - ),5 ) )'t~ p 2 + (m ~)2 "Zu(1 + V5 ) ,

(A.23)

k' = k 1 - k 2.

The i n t e g r a l i s f o r m a l l y q u a r t i c a l l y d i v e r g e n t . H o w e v e r , it i s not d i f f i c u l t to s e e that the m o s t d i v e r g e n t p a r t c a n c e l e a c h o t h e r , l e a v i n g a q u a d r a t i c a l ly d i v e r g e n t i n t e g r a l . To e v a l u a t e it e x p l i c i t l y , we i n s e r t the F e y n m a n c u t off f a c t o r A 4 ( p 2 + A 2 ) -2 i n s i d e the i n t e g r a l . Since we e x p e c t A 2 ~ rn 2, we n e g l e c t t e r m s p r o p o r t i o n a l to q 2 / m 2 , q 2 / A 2 , and m 2 / m 2 T h e n , a f t e r s o m e c a l c u l a t i o n s , it i s not d i f f i c u l t to find _ QX

1 4 (27T)4 rn 2 ~ ' x ( l + ~ 5 ) J ,

1

J=-i

A4

3

p2}.

f d 4 p ( p 2 + m 2 ) 2 ( p 2 + A 2 ) 2 1~- + 4 ~m

(A.24)

The integral J is easily evaluated to be

J = ½,72

f o r A 2 = m 2.

(A.25)

T h e n , the e x p r e s s i o n l e a d s to the e x p l i c i t f o r m eq. (35) f o r z of the text. A l t h o u g h we h a v e a s s u m e d t h a t a l l v e c t o r m e s o n s have the s a m e m a s s , t h i s i s r e a l l y not n e c e s s a r y . In the l o c a l l i m i t , the f i n a l r e s u l t e s s e n t i a l l y remains unchanged.

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