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Gross CP violation in the strong cubic intermediate vector Boson model
7~ Nuclear P h y s i c s B l l (1969) 253-258. North-tlolland Puhl. Comp.. A m s t e r d a m
~;ROSS C P V I O L A T I O N IN THE S T R O N G CUBIC INTERMEDIATE VECTOR BOSON MODEL R. E. M A R S H A K , R. N. M O H A P A T R A , S. OKUBO a n d J. SUBBA RAO
Unic<'r.,;ity of Roch('.s'tcr. l?~ochc.','h'r. N. Y. Received 14 M'Lv 1969 A b s t r a c t : The gross CP (or equivalently, t i m e - r e v e r s a l ) violation effects predicted by the strong cubic IVB model a r e discussed for weak decays involving induced neutral c u r r e n t s and for the IVB production p r o c e s s .
It h a s b e e n f a i r l y w e l l e s t a b l i s h e d t h a t , if p e r t u r b a t i v e c a l c u l a t i o n s in w e a k i n t e r a c t i o n s a r e c o r r e c t - a n d t h e y a g r e e with e x p e r i m e n t to w i t h i n a few p e r c e n t - t h e w e a k i n t e r a c t i o n c u t o f f A m u s t n e c e s s a r i l y b e s m a l l . In f a c t , t h e c a l c u l a t i o n of t h e ( i ~ - K~) m a s s d i f f e r e n c e [1, 2] l e a d s to a v a l u e of A ~ 5 G e V , i n d e p e n d e n t of t h e h a d r o n i c s t r o n g i n t e r a c t i o n s . T h e o r i g i n of s u c h a l o w v a l u e of A b e c o m e s a p r o b l e m of g r e a t i n t e r e s t a n d m o s t s p e c u l a t i o n s c o n c e r n i n g t h i s q u e s t i o n h a v e b e e n c a r r i e d out w i t h i n t h e f r a m e w o r k of t h e i n t e r m e d i a t e b o s o n m o d e l [3]. In one a p p r o a c h , t h e l o w v a l u e of A i s a t t r i b u t e d to s o m e f o r m of s t r o n g i n t e r a c t i o n of an i n t e r m e d i a t e v e c t o r b o s o n (IVB) w h i c h p r o v i d e s a ' n a t u r a l ' c u t o f f at A ~ m W ( m a s s of IVB). In p r i n c i p l e , t h e s t r o n g i n t e r a c t i o n of t h e IVB c a n h a v e a h a d r o n i c o r i g i n [4] o r r e s u l t f r o m a s t r o n g s e l f - i n t e r a c t i o n [5]. In t h i s n o t e we e x p l o r e f u r t h e r t h e c o n s e q u e n c e s of t h e l a t t e r t y p e of m o d e l ( c a l l e d t h e s t r o n g c u b i c IVB m o d e l ) w h i c h d i r e c t l y i n c o r p o r a t e s t h e p h e n o m e n o n of C P v i o l a tion. In t h e s t r o n g c u b i c IVB m o d e l , one a s s i g n s C P = - 1 to t h e t o t a l s e m i w e a k i n t e r a c t i o n a n d at t h e s a m e t i m e d e m a n d s t h a t a l l ' v i r t u a l ' f i r s t - o r d e r e f f e c t s in Z W ( c o u p l i n g c o n s t a n t of IVB) a r e f o r b i d d e n . T h i s is a c h i e v e d [5] b y i n t r o d u c i n g a W t r i p l e t w h i c h i n t e r a c t s e m i - w e a k l y w i t h h a d r o n s and l e p t o n s b u t h a v e a s t r o n g cubic__ i n t e r a c t i o n with e a c h o t h e r . A m o d e l L a g r a n g i a n f o r t h e W t r i p l e t [llApa);. a = 1, 2, 3] i s :
-
m 2 w ( a ) ( x ) w ( a ) ( x ) - if 0
Iz
I~
: 00[)C
[ W( a) ( v~ W( h ) ( v~ _~ w(C)lv ~ t
tz
x" ~
V
v ~Cl L
b'
~" )
- t¢((4 09 W(~)(x) a/j w(f )09] /~ v .
(1) '
254
R . E . MA[KSHAK et al.
w h e r e f o is a s t r o n g c o u p l i n g c o n s t a n t and the notation is o t h e r w i s e s t a n d a r d . F r o m the s t r u c t u r e of the i n t e r a c t i o n t e r m in eq. (1), it follows that the total c h a r g e of the t h r e e m e m b e r s of the W t r i p l e t m u s t be z e r o and a unique a s s i g n m e n t i s Q = 0, -1 +1 ( a = 1 , 2 , 3 ) if we take IQ I v 1. An imp o r t a n t p r o p e r t y of eq. (1) is its i n v a r i a n c e u n d e r the t r i a l i t y t r a n s f o r m a tion [6] given by
w(a)(x) -.xw(al(x) , IZ p
w(al(x) -'X*l~(a)(x) , /z 11
(2)
w h e r e ~ is a c o n s t a n t s a t i s f y i n g the cubic e q u a t i o n X3 = 1. F i n a l l y , a suitable CP = - 1 s e m i - w e a k i n t e r a c t i o n is (0 is the Cabibbo angle): 3 ~-~(,1) 3 HSW = igw{W(,1)[4z3 cosO+Jp3 sin n] - WIj i " [Jp2 +J~/J.3 sin 0]
+ W(2)[nljc°sO+~u]-W-~pz 12)[J~z~,l
c°s0+~U]
r (3)rr3 Tx1(3)rr 1 + ,,/j. t¢,#l s i n 0 + f g ] - , , p t~,p3 sin~9+ f p ] } ,
(3)
i . w h e r e J~j xs the o c t e t h a d r o n c u r r e n t (in t e n s o r notation) and ~p is the lepton c u r r e n t . It is now p o s s i b l e to show that ,all ' v i r t u a l ' f i r s t - o r d e r e f f e c t s in the s t r o n g cubic IVB m o d e l defined by eqs. (1)- (3) a r e f o r b i d d e n by t r i a l i t y c o n s e r v a t i o n ~. T h e f i r s t n o n - v a n i s h i n g e f f e c t s o c c u r in s e c o n d o r d e r of g~,, s i n c e a t e r m like (Wtj.(x) Wv(Y))o is c o n s i s t e n t with t r i a l i t y c o n s e r v a t i o n . In this way, one m a y d e d u c e the C P - c o n s e r v i n g (CP = +1) w e a k p r o c e s s e s of the u s u a l IVB m o d e l . T h e d i f f e r e n c e s which,,distinguish the s t r o n g cubic f r o m the u s u a l IVB m o d e l f i r s t a r i s e in o r d e r g~xr w h e r e one e n c o u n t e r s t e r m s of the t y p e (W~(x) • Wv(Y). Wx(z)) o which do not v a n i s h b e c a u s e of the cubic s e l f - c o u p l i n g a m o n g the W b o s o n s (and which c o n s e r v e t r i a l i t y a s a r e s u l t of condition (2)). M o r e o v e r , the t e r m s o c c u r r i n g in o r d e r g3W p o s s e s s CP = - 1 and a r e t h e r e f o r e C P - v i o l a t i n g when they i n t e r f e r e with t e r m s o f o r d e r g ~ v . Such t e r m s , of c o u r s e , cxplain the s m a l l C P - v i o l a t i n g e f f e c t s (of o r d e r g w ) o b s e r v e d in K~ -- 2n d e c a y s . But of p a r t i c u l a r i m p o r t a n c e f o r t e s t i n g the s t r o n g cubic IVB m o d e l is the p o s s i b i l i t y that the i n t e r f e r e n c e b e t w e e n the s t r o n g cubic and e l e c t r o m a g n e t i c c o n t r i b u t i o n s to the m a t r i x e l e m e n t f o r c e r t a i n weak p r o c e s s e s will r e s u l t ingross C P - v i o l a t i o n (i.e. w h e r e the CP = -1 and CP = +1 a m p l i t u d e s a r e c o m p a r a b l e ) . B e f o r e c o n s i d e r i n g s e v e r a l e x a m p l e s of g r o s s C P - v i o l a t i o n (or e q u i v a lently t i m e - r e v e r s ' a ] violation) p r e d i c t e d by this t h e o r y , we s k e t c h the c a l c u l a t i o n f o r K~. - . 2n d e c a y to i n d i c a t e the n a t u r e of the a p p r o x i m a t i o n s involved. T h e d i a g r a m c o n t r i b u t i n g to the d e c a y K~ ~ 2~ is given in fig. 1 and the m a t r i x e l e m e n t is: :~ The typical term which appears is (Wtt(x)) o (the expectation value is understood with respect to the wlcuum state of the W bosons with the Lagrangian defined by eq. (1)) and this must vanish by triality conservation.
GROSS C P VIOLA'FION
255
w + ( P - q-
;e2)
~ b 2 )
W°(p-q)
N - . ~ - (k 2 ) Fig. ]
K. oL
._,
2/r d i a g r a m .
3
'~w cos2 o sin 0 Aw(~_ q),~wck2) w M=- ~/8P0 kl0 k20 -- f i~4 / %8 ~ - q- I'2) × rafis(p , q, k2)A~(q)(K°(p) ! V/~ ]v-(q)) @-(q) I Vy 7r°(kl)} <~°(k2)]A u !0},
(4)
where AW~. is the IVB p r o p a g a t o r , Fairy is the t r i p l e W v e r t e x and Vp and
A v a r e the v e c t o r and axial v e c t o r hadron c u r r e n t s r e s p e c t i v e l y . F r o m symmetry considerations
F~f35 = f ( q 2 , q. p, q. k2)[ 5a~3(p _ q + 1..2)5 - 5as(2P - 2q - le2)fl + 5fi 5(/5 - q)a] • We a s s u m e that f(q2, q ' [ , contributions, we get:
q" k2) ~ fo and, retaining the most divergent
3 2 3 A4 gW . . . . . cos 2 0 sin 0 f o f ~ ~zK , 256~ 2 m 6
M
(5)
(6)
where f~ is the pion decay amplitude. Hence:
'M(K L
I I
-~ 27r)
(}
M(K~ -* 2~)
A 4 "~ 10-3 ]o (,,/'-) ~ w " k
(7)
W /
Another r e l a t i o n between f o and A is d e r i v e d by calculating the s e l f - m a s s of the W boson with cubic interactions; using the s a m e a p p r o x i m a t i o n s for the t r i p l e W v e r t e x , one gets: 5m2
or
2 9 f o A4 ~ m 2 , 32~ 2 ~n2
(8)
256
R.E. MARSIIAK et al. f o A2 m 2 ~ 2~ .
(9)
T h e s e a p p r o x i m a t e c a l c u l a t i o n s ( t a k i n g s 2W ~ 10-2) show that the c o r r e c t o r d e r of m a g n i t u d e of the C P - v i o l a t i n g K~ -. 2u a m p l i t u d e is p r e d i c t e d by the s t r o n g cubic IVB m o d e l . One is t h e r e f o r e e n c o u r a g e d to look at s o m e o t h e r p r o c e s s e s w h e r e the C P - v i o l a t i n g a m p l i t u d e p r e d i c t e d by this m o d e l is s u b s t a n t i a l l y l a r g e r . O o n s i d e r f i r s t K~ --+ /x/x d e c a y ( in the s t r o n g cubic IVB m o d e l I ~L ~ #xix is f o r b i d d e n in the SU 3 limit). T h i s d e c a y i n v o l v e s the C P - v i o l a t i n g (1S0) ' s t r o n g c u b i c ' a m p l i t u d e of o r d e r A2 w and the C P - c o n s e r v i n g (3P0) ' e l e c t r o m a g n e t i c ' a m p l i t u d e of o r d e r / ~ 2 e 4 . T h e i n t e r f e r e n c e b e t w e e n t h e s e two c o m p a r a b l e a m p l i t u d e s ( s i n c e £'W ~ e4) should p r o d u c e g r o s s CP (or 7") violation e f f e c t s (e.g. p o l a r i z a t i o n c o r r e l a t i o n of the two final m u o n s [7], etc.). M o r e q u a n t i t a t i v e l y , the a m p l i t u d e f o r K~ ~ tJ,~ d e c a y can be w r i t t e n :
/L~(K~ ~ t~.i~.) = i<(0[a+ ;t'~'5]v(b ,
(10)
w h e r e a(b) is the C P - c o n s e r v i n g (-violating) p a r t and r e c e i v e s a c o n t r i b u tion f r o m fig. 2a(b). T h e d i a g r a m in fig. 2b is c a l c u l a t e d in the s a m e app r o x i m a t i o n s a s a b o v e and one g e t s finally, 3 m Is. ,,
-
.
.
.
4~2 .
.
.
J,
4"2
w h e r e f K is the k a o n d e c a y a m p l i t u d e . T o c a l c u l a t e the d i a g r a m shown in fig. 2a, we a s s u m e the following f o r m f o r the K~ - 2 ~ / v e r t e x (B is a c o n stant) :
~Os. 2"t'x = (;'~'2q2-B(e 1 " e2
q l " q2 - q l " e2 q2 • el) .
(12)
T h i s is g a u g e i n v a r i a n t and the o n e - p h o t o n i n t e r m e d i a t e s t a t e is f o r b i d d e n by g a u g e i n v a r i a n c e . T h e l o w e s t o r d e r c a l c u l a t i o n of fig. 2a y i e l d s : c///~
+
w +
Fig. 2a. K ~ . txtT: electromagnetic diagr'tm.
i~4"-Fig. 2b. K~ ~ ~ : strong cubic diagram.
:l: From the relation (;,/V:2 - sW ,2 //,.2 "W, one gets 2 g\V 10_ 5 (~zW,2 _ ~ .... ~ ,--~-N) ~ 1 . 5 × i 0 - 5 ( f o r mW ~: 5 m N ) o r g w ~ 4~ 4~v'2 \ "
1.5×10-2
"
GROSS C I ) VIOI~ATION
Ge 4B
a = -m~
257
;~ 1
1=
4~ 2
hi ( ~ K )
(13)
,
w h e r e A 1 is the e l e c t r o m a g n e t i c cutoff e n e r g y . A v e r y c r u d e e s t i m , t t e of B suggests: b - - ~ 50 , (i
(14)
i.e. g r o s s C P (or T) violation in K~ -/J./5 decay.~ It should be e m p h a s i z e d that in the u s u a l IVB t h e o r y , b is of o r d e r A~V so that b / a <..:. 1. S i m i l a r C P (or T) violation e f f e c t s a r e e x p e c t e d in o t h e r weak d e c a y s involving induced n e u t r a l c u r r e n t s (and h e n c e a c o m p e t i t i o n b e t w e e n the ' s t r o n g c u b i c ' and ' e l e c t r o m a g n e t i c ' c o n t r i b u t i o n s to the m a t r i x e l e m e n t ) . In K~ - - ~ ° e ~ d e c a y , the g r o s s C P (or 7") violation effect is s i m i l a r to that f o r I ~ - . e~ and if the s t r o n g cubic c o n t r i b u t i o n d o m i n a t e s , as expected, the d i f f e r e n t i a l t r a n s i t i o n r a t e should go [7] as sin 2 0 ((i is the angle b e t w e e n the pion and lepton in the r e s t f r a m e of the lepton pair). S i m i l a r l y , we expect to have s i z e a b l e (but not n e a r l y such l a r g e ) C P (or T) violation e f f e c t s in d e c a y s like E + -. pe~ and K + - ' 7r+pfi. As f o r K~ - ' 1~17 d e c a y , while it is f o r b i d d e n in the SU 3 limit, it will o c c u r if we allow f o r m a s s d i f f e r e n c e s in the W t r i p l e t ; thus a 10% m a s s d i f f e r e n c e (the s a m e as f o r h a d r o n s ) y i e l d s a b r a n c h i n g r a t i o w 10 -7 f o r K~ - ' tl~ and an e x p e c t a t i o n of g r o s s C P (or 7") violation, i.e. b / a ~ 0.3. F i n a l l y , we note the p o s s i b i l i t y of s i z e a b l e C P (or T) violation e f f e c t s in W p r o d u c t i o n on the b a s i s of the s t r o n g cubic IVB model. In the usual IVB t h e o r y , W p r o d u c t i o n o c c u r s only t h r o u g h the e l e c t r o m a g n e t i c m e c h a n i s m shown in fig. 3a but in the p r e s e n t m o d e l , fig. 3b a l s o c o n t r i b u t e s to W p r o duction. Both c r o s s s e c t i o n s should be c o m p a r a b l e , with the c o h e r e n t photop r o d u c t i o n c o n t r i b u t i n g m o r e in the f o r w a r d d i r e c t i o n and c o n v e r s e l y for the s t r o n g cubic d i a g r a m at l a r g e a n g l e s . C a r e f u l e x a m i n a t i o n s h o w s that t h e r e a r e o t h e r d i s t i n g u i s h i n g f e a t u r e s in the d i f f e r e n t i a l c r o s s s e c t i o n if fig. 3b is a m e c h a n i s m in W p r o d u c t i o n . F i r s t of all, if the d e c a y of the p r o d u c e d W is o b s e r v e d in the lepton p a i r ( l ~ P p ) channel, one finds for the c o n t r i b u t i o n f r o m fig. 3b:
IM b 12 •
~y_ [[. . . . . . 1 ~ . . . . . . . . . . 1 ( - _ _ ' 2 E-m(
2)+~ic(2)
E-m(
3)+2iF(3)[
•
p
'
b'
p
i.'ig. 3a. Wproduction: electromagnetic diagram.
N
(15) ,
N
Fig. 3b. Wproduction: strong cubic diagram.
258
R.E. MARStIAK et al.
(a) (a) w h e r e E is the e n e r g y of the o u t g o i n ~ W in its r e s t s y s t e m a n d m and F a r e the m a s s a n d d e c a y width of w(a). In the e x a c t SUa s y m m e t r y l i m i t f o r the W b o s o n s , m(2) = m(3) and 1"(2) = F(3) a n d h e n c e i~M I 2 ~ 0. H o w e v e r , we e x p e c t SU 3 s y m m e t r y to b e b r o k e n (i.e. m(2) - m(3) ¢ 0) a n d m o r e o v e r Ira(2) - m ( 3 ) ~; >> ½F; a d o u b l e p e a k s t r u c t u r e is t h e n p r e d i c t e d in the t o t a l c r o s s s e c t i o n n e a r the W m a s s . Such a s t r u c t u r e in the W p r o d u c t i o n c r o s s s e c t i o n would be p o s i t i v e e v i d e n c e in f a v o r of t h i s m o d e l . Note that in t h i s c a s e , we m a y have a l a r g e CP (or T) v i o l a t i o n which m a y be d e t e c t e d by l o o k i n g for the t r a n s v e r s e p o l a r i z a t i o n of the e m e r g e n t m u o n . It s h o u l d be p o i n t e d out that, e v e n in the SU 3 s y m m e t r y l i m i t , no c a n c e l l a t i o n o c c u r s in the lAY! = 1, W p r o d u c t i o n r e a c t i o n ; t h u s , for a p r o c e s s l i k e
v p + N ~ I~-+W+ +A I_~p+ + 1)~ , we h a v e IMI 2 ~
]'E
1 2 _ m(3) +½iF(3 )
(16)
T h e r e is, h o w e v e r , no p o s s i b i l i t y now of d e t e c t i n g CP (or T) v i o l a t i o n eff e c t s s i n c e the e l e c t r o m a g n e t i c m e c h a n i s m is i n o p e r a t i v e in a ! AY i = 1 r e a c t i o n . N e v e r t h e l e s s , it would b e v e r y i n t e r e s t i n g to s e a r c h f o r t h e ! AY i = 1 p r o d u c t i o n r e a c t i o n s i n c e it c o u l d h a v e a s u b s t a n t i a l l y l a r g e r c r o s s s e c t i o n in the s t r o n g c u b i c t h a n in the u s u a l IVB m o d e l . T h i s w o r k w a s s u p p o r t e d in p a r t by the U.S. A t o m i c E n e r g y C o m m i s s i o n .
REFERENCES t11 B. L. Ioffe and E. P. Shabalin, Sov. J. Nucl. Phys. 6 (1967) 828. [2] R. N./vlohapatra, J. S. Rao and R. E. Marshak, Phys. Rev. Letters 20 (1968) t08; Phys. Rev. 171 (1968) 1502. [3] B. L. loffe, Proc. Int. Conf. on particles and fields at ltochester (John Wiley, t967); M.Gell-Mann, M.Goldberger, [,'.Low and N.Kroll, Phys.ltcv. 179 (1969) 1518. [41 S. V. Pepper, C. Ryan, S. Okubo and R. E. Marshak, Phys. Rev. 137 (t965) Bl259. [51 S. Okubo, Nuovo Cirnento 54A (1908) 491; 57A (1968) 794; Ann. of Phys. 49 (1968) 2t9. [6] S. Okubo, C. Ryan and R. E. Marshak, Nuovo Cimento 34 (1954) 753; 34 (1964) 759. [7] A. Pais and S.B. T r e i m a n , Phys. Rev. 176 (1968) 1974.
~;ROSS C P V I O L A T I O N IN THE S T R O N G CUBIC INTERMEDIATE VECTOR BOSON MODEL R. E. M A R S H A K , R. N. M O H A P A T R A , S. OKUBO a n d J. SUBBA RAO
Unic<'r.,;ity of Roch('.s'tcr. l?~ochc.','h'r. N. Y. Received 14 M'Lv 1969 A b s t r a c t : The gross CP (or equivalently, t i m e - r e v e r s a l ) violation effects predicted by the strong cubic IVB model a r e discussed for weak decays involving induced neutral c u r r e n t s and for the IVB production p r o c e s s .
It h a s b e e n f a i r l y w e l l e s t a b l i s h e d t h a t , if p e r t u r b a t i v e c a l c u l a t i o n s in w e a k i n t e r a c t i o n s a r e c o r r e c t - a n d t h e y a g r e e with e x p e r i m e n t to w i t h i n a few p e r c e n t - t h e w e a k i n t e r a c t i o n c u t o f f A m u s t n e c e s s a r i l y b e s m a l l . In f a c t , t h e c a l c u l a t i o n of t h e ( i ~ - K~) m a s s d i f f e r e n c e [1, 2] l e a d s to a v a l u e of A ~ 5 G e V , i n d e p e n d e n t of t h e h a d r o n i c s t r o n g i n t e r a c t i o n s . T h e o r i g i n of s u c h a l o w v a l u e of A b e c o m e s a p r o b l e m of g r e a t i n t e r e s t a n d m o s t s p e c u l a t i o n s c o n c e r n i n g t h i s q u e s t i o n h a v e b e e n c a r r i e d out w i t h i n t h e f r a m e w o r k of t h e i n t e r m e d i a t e b o s o n m o d e l [3]. In one a p p r o a c h , t h e l o w v a l u e of A i s a t t r i b u t e d to s o m e f o r m of s t r o n g i n t e r a c t i o n of an i n t e r m e d i a t e v e c t o r b o s o n (IVB) w h i c h p r o v i d e s a ' n a t u r a l ' c u t o f f at A ~ m W ( m a s s of IVB). In p r i n c i p l e , t h e s t r o n g i n t e r a c t i o n of t h e IVB c a n h a v e a h a d r o n i c o r i g i n [4] o r r e s u l t f r o m a s t r o n g s e l f - i n t e r a c t i o n [5]. In t h i s n o t e we e x p l o r e f u r t h e r t h e c o n s e q u e n c e s of t h e l a t t e r t y p e of m o d e l ( c a l l e d t h e s t r o n g c u b i c IVB m o d e l ) w h i c h d i r e c t l y i n c o r p o r a t e s t h e p h e n o m e n o n of C P v i o l a tion. In t h e s t r o n g c u b i c IVB m o d e l , one a s s i g n s C P = - 1 to t h e t o t a l s e m i w e a k i n t e r a c t i o n a n d at t h e s a m e t i m e d e m a n d s t h a t a l l ' v i r t u a l ' f i r s t - o r d e r e f f e c t s in Z W ( c o u p l i n g c o n s t a n t of IVB) a r e f o r b i d d e n . T h i s is a c h i e v e d [5] b y i n t r o d u c i n g a W t r i p l e t w h i c h i n t e r a c t s e m i - w e a k l y w i t h h a d r o n s and l e p t o n s b u t h a v e a s t r o n g cubic__ i n t e r a c t i o n with e a c h o t h e r . A m o d e l L a g r a n g i a n f o r t h e W t r i p l e t [llApa);. a = 1, 2, 3] i s :
-
m 2 w ( a ) ( x ) w ( a ) ( x ) - if 0
Iz
I~
: 00[)C
[ W( a) ( v~ W( h ) ( v~ _~ w(C)lv ~ t
tz
x" ~
V
v ~Cl L
b'
~" )
- t¢((4 09 W(~)(x) a/j w(f )09] /~ v .
(1) '
254
R . E . MA[KSHAK et al.
w h e r e f o is a s t r o n g c o u p l i n g c o n s t a n t and the notation is o t h e r w i s e s t a n d a r d . F r o m the s t r u c t u r e of the i n t e r a c t i o n t e r m in eq. (1), it follows that the total c h a r g e of the t h r e e m e m b e r s of the W t r i p l e t m u s t be z e r o and a unique a s s i g n m e n t i s Q = 0, -1 +1 ( a = 1 , 2 , 3 ) if we take IQ I v 1. An imp o r t a n t p r o p e r t y of eq. (1) is its i n v a r i a n c e u n d e r the t r i a l i t y t r a n s f o r m a tion [6] given by
w(a)(x) -.xw(al(x) , IZ p
w(al(x) -'X*l~(a)(x) , /z 11
(2)
w h e r e ~ is a c o n s t a n t s a t i s f y i n g the cubic e q u a t i o n X3 = 1. F i n a l l y , a suitable CP = - 1 s e m i - w e a k i n t e r a c t i o n is (0 is the Cabibbo angle): 3 ~-~(,1) 3 HSW = igw{W(,1)[4z3 cosO+Jp3 sin n] - WIj i " [Jp2 +J~/J.3 sin 0]
+ W(2)[nljc°sO+~u]-W-~pz 12)[J~z~,l
c°s0+~U]
r (3)rr3 Tx1(3)rr 1 + ,,/j. t¢,#l s i n 0 + f g ] - , , p t~,p3 sin~9+ f p ] } ,
(3)
i . w h e r e J~j xs the o c t e t h a d r o n c u r r e n t (in t e n s o r notation) and ~p is the lepton c u r r e n t . It is now p o s s i b l e to show that ,all ' v i r t u a l ' f i r s t - o r d e r e f f e c t s in the s t r o n g cubic IVB m o d e l defined by eqs. (1)- (3) a r e f o r b i d d e n by t r i a l i t y c o n s e r v a t i o n ~. T h e f i r s t n o n - v a n i s h i n g e f f e c t s o c c u r in s e c o n d o r d e r of g~,, s i n c e a t e r m like (Wtj.(x) Wv(Y))o is c o n s i s t e n t with t r i a l i t y c o n s e r v a t i o n . In this way, one m a y d e d u c e the C P - c o n s e r v i n g (CP = +1) w e a k p r o c e s s e s of the u s u a l IVB m o d e l . T h e d i f f e r e n c e s which,,distinguish the s t r o n g cubic f r o m the u s u a l IVB m o d e l f i r s t a r i s e in o r d e r g~xr w h e r e one e n c o u n t e r s t e r m s of the t y p e (W~(x) • Wv(Y). Wx(z)) o which do not v a n i s h b e c a u s e of the cubic s e l f - c o u p l i n g a m o n g the W b o s o n s (and which c o n s e r v e t r i a l i t y a s a r e s u l t of condition (2)). M o r e o v e r , the t e r m s o c c u r r i n g in o r d e r g3W p o s s e s s CP = - 1 and a r e t h e r e f o r e C P - v i o l a t i n g when they i n t e r f e r e with t e r m s o f o r d e r g ~ v . Such t e r m s , of c o u r s e , cxplain the s m a l l C P - v i o l a t i n g e f f e c t s (of o r d e r g w ) o b s e r v e d in K~ -- 2n d e c a y s . But of p a r t i c u l a r i m p o r t a n c e f o r t e s t i n g the s t r o n g cubic IVB m o d e l is the p o s s i b i l i t y that the i n t e r f e r e n c e b e t w e e n the s t r o n g cubic and e l e c t r o m a g n e t i c c o n t r i b u t i o n s to the m a t r i x e l e m e n t f o r c e r t a i n weak p r o c e s s e s will r e s u l t ingross C P - v i o l a t i o n (i.e. w h e r e the CP = -1 and CP = +1 a m p l i t u d e s a r e c o m p a r a b l e ) . B e f o r e c o n s i d e r i n g s e v e r a l e x a m p l e s of g r o s s C P - v i o l a t i o n (or e q u i v a lently t i m e - r e v e r s ' a ] violation) p r e d i c t e d by this t h e o r y , we s k e t c h the c a l c u l a t i o n f o r K~. - . 2n d e c a y to i n d i c a t e the n a t u r e of the a p p r o x i m a t i o n s involved. T h e d i a g r a m c o n t r i b u t i n g to the d e c a y K~ ~ 2~ is given in fig. 1 and the m a t r i x e l e m e n t is: :~ The typical term which appears is (Wtt(x)) o (the expectation value is understood with respect to the wlcuum state of the W bosons with the Lagrangian defined by eq. (1)) and this must vanish by triality conservation.
GROSS C P VIOLA'FION
255
w + ( P - q-
;e2)
~ b 2 )
W°(p-q)
N - . ~ - (k 2 ) Fig. ]
K. oL
._,
2/r d i a g r a m .
3
'~w cos2 o sin 0 Aw(~_ q),~wck2) w M=- ~/8P0 kl0 k20 -- f i~4 / %8 ~ - q- I'2) × rafis(p , q, k2)A~(q)(K°(p) ! V/~ ]v-(q)) @-(q) I Vy 7r°(kl)} <~°(k2)]A u !0},
(4)
where AW~. is the IVB p r o p a g a t o r , Fairy is the t r i p l e W v e r t e x and Vp and
A v a r e the v e c t o r and axial v e c t o r hadron c u r r e n t s r e s p e c t i v e l y . F r o m symmetry considerations
F~f35 = f ( q 2 , q. p, q. k2)[ 5a~3(p _ q + 1..2)5 - 5as(2P - 2q - le2)fl + 5fi 5(/5 - q)a] • We a s s u m e that f(q2, q ' [ , contributions, we get:
q" k2) ~ fo and, retaining the most divergent
3 2 3 A4 gW . . . . . cos 2 0 sin 0 f o f ~ ~zK , 256~ 2 m 6
M
(5)
(6)
where f~ is the pion decay amplitude. Hence:
'M(K L
I I
-~ 27r)
(}
M(K~ -* 2~)
A 4 "~ 10-3 ]o (,,/'-) ~ w " k
(7)
W /
Another r e l a t i o n between f o and A is d e r i v e d by calculating the s e l f - m a s s of the W boson with cubic interactions; using the s a m e a p p r o x i m a t i o n s for the t r i p l e W v e r t e x , one gets: 5m2
or
2 9 f o A4 ~ m 2 , 32~ 2 ~n2
(8)
256
R.E. MARSIIAK et al. f o A2 m 2 ~ 2~ .
(9)
T h e s e a p p r o x i m a t e c a l c u l a t i o n s ( t a k i n g s 2W ~ 10-2) show that the c o r r e c t o r d e r of m a g n i t u d e of the C P - v i o l a t i n g K~ -. 2u a m p l i t u d e is p r e d i c t e d by the s t r o n g cubic IVB m o d e l . One is t h e r e f o r e e n c o u r a g e d to look at s o m e o t h e r p r o c e s s e s w h e r e the C P - v i o l a t i n g a m p l i t u d e p r e d i c t e d by this m o d e l is s u b s t a n t i a l l y l a r g e r . O o n s i d e r f i r s t K~ --+ /x/x d e c a y ( in the s t r o n g cubic IVB m o d e l I ~L ~ #xix is f o r b i d d e n in the SU 3 limit). T h i s d e c a y i n v o l v e s the C P - v i o l a t i n g (1S0) ' s t r o n g c u b i c ' a m p l i t u d e of o r d e r A2 w and the C P - c o n s e r v i n g (3P0) ' e l e c t r o m a g n e t i c ' a m p l i t u d e of o r d e r / ~ 2 e 4 . T h e i n t e r f e r e n c e b e t w e e n t h e s e two c o m p a r a b l e a m p l i t u d e s ( s i n c e £'W ~ e4) should p r o d u c e g r o s s CP (or 7") violation e f f e c t s (e.g. p o l a r i z a t i o n c o r r e l a t i o n of the two final m u o n s [7], etc.). M o r e q u a n t i t a t i v e l y , the a m p l i t u d e f o r K~ ~ tJ,~ d e c a y can be w r i t t e n :
/L~(K~ ~ t~.i~.) = i<(0[a+ ;t'~'5]v(b ,
(10)
w h e r e a(b) is the C P - c o n s e r v i n g (-violating) p a r t and r e c e i v e s a c o n t r i b u tion f r o m fig. 2a(b). T h e d i a g r a m in fig. 2b is c a l c u l a t e d in the s a m e app r o x i m a t i o n s a s a b o v e and one g e t s finally, 3 m Is. ,,
-
.
.
.
4~2 .
.
.
J,
4"2
w h e r e f K is the k a o n d e c a y a m p l i t u d e . T o c a l c u l a t e the d i a g r a m shown in fig. 2a, we a s s u m e the following f o r m f o r the K~ - 2 ~ / v e r t e x (B is a c o n stant) :
~Os. 2"t'x = (;'~'2q2-B(e 1 " e2
q l " q2 - q l " e2 q2 • el) .
(12)
T h i s is g a u g e i n v a r i a n t and the o n e - p h o t o n i n t e r m e d i a t e s t a t e is f o r b i d d e n by g a u g e i n v a r i a n c e . T h e l o w e s t o r d e r c a l c u l a t i o n of fig. 2a y i e l d s : c///~
+
w +
Fig. 2a. K ~ . txtT: electromagnetic diagr'tm.
i~4"-Fig. 2b. K~ ~ ~ : strong cubic diagram.
:l: From the relation (;,/V:2 - sW ,2 //,.2 "W, one gets 2 g\V 10_ 5 (~zW,2 _ ~ .... ~ ,--~-N) ~ 1 . 5 × i 0 - 5 ( f o r mW ~: 5 m N ) o r g w ~ 4~ 4~v'2 \ "
1.5×10-2
"
GROSS C I ) VIOI~ATION
Ge 4B
a = -m~
257
;~ 1
1=
4~ 2
hi ( ~ K )
(13)
,
w h e r e A 1 is the e l e c t r o m a g n e t i c cutoff e n e r g y . A v e r y c r u d e e s t i m , t t e of B suggests: b - - ~ 50 , (i
(14)
i.e. g r o s s C P (or T) violation in K~ -/J./5 decay.~ It should be e m p h a s i z e d that in the u s u a l IVB t h e o r y , b is of o r d e r A~V so that b / a <..:. 1. S i m i l a r C P (or T) violation e f f e c t s a r e e x p e c t e d in o t h e r weak d e c a y s involving induced n e u t r a l c u r r e n t s (and h e n c e a c o m p e t i t i o n b e t w e e n the ' s t r o n g c u b i c ' and ' e l e c t r o m a g n e t i c ' c o n t r i b u t i o n s to the m a t r i x e l e m e n t ) . In K~ - - ~ ° e ~ d e c a y , the g r o s s C P (or 7") violation effect is s i m i l a r to that f o r I ~ - . e~ and if the s t r o n g cubic c o n t r i b u t i o n d o m i n a t e s , as expected, the d i f f e r e n t i a l t r a n s i t i o n r a t e should go [7] as sin 2 0 ((i is the angle b e t w e e n the pion and lepton in the r e s t f r a m e of the lepton pair). S i m i l a r l y , we expect to have s i z e a b l e (but not n e a r l y such l a r g e ) C P (or T) violation e f f e c t s in d e c a y s like E + -. pe~ and K + - ' 7r+pfi. As f o r K~ - ' 1~17 d e c a y , while it is f o r b i d d e n in the SU 3 limit, it will o c c u r if we allow f o r m a s s d i f f e r e n c e s in the W t r i p l e t ; thus a 10% m a s s d i f f e r e n c e (the s a m e as f o r h a d r o n s ) y i e l d s a b r a n c h i n g r a t i o w 10 -7 f o r K~ - ' tl~ and an e x p e c t a t i o n of g r o s s C P (or 7") violation, i.e. b / a ~ 0.3. F i n a l l y , we note the p o s s i b i l i t y of s i z e a b l e C P (or T) violation e f f e c t s in W p r o d u c t i o n on the b a s i s of the s t r o n g cubic IVB model. In the usual IVB t h e o r y , W p r o d u c t i o n o c c u r s only t h r o u g h the e l e c t r o m a g n e t i c m e c h a n i s m shown in fig. 3a but in the p r e s e n t m o d e l , fig. 3b a l s o c o n t r i b u t e s to W p r o duction. Both c r o s s s e c t i o n s should be c o m p a r a b l e , with the c o h e r e n t photop r o d u c t i o n c o n t r i b u t i n g m o r e in the f o r w a r d d i r e c t i o n and c o n v e r s e l y for the s t r o n g cubic d i a g r a m at l a r g e a n g l e s . C a r e f u l e x a m i n a t i o n s h o w s that t h e r e a r e o t h e r d i s t i n g u i s h i n g f e a t u r e s in the d i f f e r e n t i a l c r o s s s e c t i o n if fig. 3b is a m e c h a n i s m in W p r o d u c t i o n . F i r s t of all, if the d e c a y of the p r o d u c e d W is o b s e r v e d in the lepton p a i r ( l ~ P p ) channel, one finds for the c o n t r i b u t i o n f r o m fig. 3b:
IM b 12 •
~y_ [[. . . . . . 1 ~ . . . . . . . . . . 1 ( - _ _ ' 2 E-m(
2)+~ic(2)
E-m(
3)+2iF(3)[
•
p
'
b'
p
i.'ig. 3a. Wproduction: electromagnetic diagram.
N
(15) ,
N
Fig. 3b. Wproduction: strong cubic diagram.
258
R.E. MARStIAK et al.
(a) (a) w h e r e E is the e n e r g y of the o u t g o i n ~ W in its r e s t s y s t e m a n d m and F a r e the m a s s a n d d e c a y width of w(a). In the e x a c t SUa s y m m e t r y l i m i t f o r the W b o s o n s , m(2) = m(3) and 1"(2) = F(3) a n d h e n c e i~M I 2 ~ 0. H o w e v e r , we e x p e c t SU 3 s y m m e t r y to b e b r o k e n (i.e. m(2) - m(3) ¢ 0) a n d m o r e o v e r Ira(2) - m ( 3 ) ~; >> ½F; a d o u b l e p e a k s t r u c t u r e is t h e n p r e d i c t e d in the t o t a l c r o s s s e c t i o n n e a r the W m a s s . Such a s t r u c t u r e in the W p r o d u c t i o n c r o s s s e c t i o n would be p o s i t i v e e v i d e n c e in f a v o r of t h i s m o d e l . Note that in t h i s c a s e , we m a y have a l a r g e CP (or T) v i o l a t i o n which m a y be d e t e c t e d by l o o k i n g for the t r a n s v e r s e p o l a r i z a t i o n of the e m e r g e n t m u o n . It s h o u l d be p o i n t e d out that, e v e n in the SU 3 s y m m e t r y l i m i t , no c a n c e l l a t i o n o c c u r s in the lAY! = 1, W p r o d u c t i o n r e a c t i o n ; t h u s , for a p r o c e s s l i k e
v p + N ~ I~-+W+ +A I_~p+ + 1)~ , we h a v e IMI 2 ~
]'E
1 2 _ m(3) +½iF(3 )
(16)
T h e r e is, h o w e v e r , no p o s s i b i l i t y now of d e t e c t i n g CP (or T) v i o l a t i o n eff e c t s s i n c e the e l e c t r o m a g n e t i c m e c h a n i s m is i n o p e r a t i v e in a ! AY i = 1 r e a c t i o n . N e v e r t h e l e s s , it would b e v e r y i n t e r e s t i n g to s e a r c h f o r t h e ! AY i = 1 p r o d u c t i o n r e a c t i o n s i n c e it c o u l d h a v e a s u b s t a n t i a l l y l a r g e r c r o s s s e c t i o n in the s t r o n g c u b i c t h a n in the u s u a l IVB m o d e l . T h i s w o r k w a s s u p p o r t e d in p a r t by the U.S. A t o m i c E n e r g y C o m m i s s i o n .
REFERENCES t11 B. L. Ioffe and E. P. Shabalin, Sov. J. Nucl. Phys. 6 (1967) 828. [2] R. N./vlohapatra, J. S. Rao and R. E. Marshak, Phys. Rev. Letters 20 (1968) t08; Phys. Rev. 171 (1968) 1502. [3] B. L. loffe, Proc. Int. Conf. on particles and fields at ltochester (John Wiley, t967); M.Gell-Mann, M.Goldberger, [,'.Low and N.Kroll, Phys.ltcv. 179 (1969) 1518. [41 S. V. Pepper, C. Ryan, S. Okubo and R. E. Marshak, Phys. Rev. 137 (t965) Bl259. [51 S. Okubo, Nuovo Cirnento 54A (1908) 491; 57A (1968) 794; Ann. of Phys. 49 (1968) 2t9. [6] S. Okubo, C. Ryan and R. E. Marshak, Nuovo Cimento 34 (1954) 753; 34 (1964) 759. [7] A. Pais and S.B. T r e i m a n , Phys. Rev. 176 (1968) 1974.