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Calculation of the μ-e ratio for semi-leptonic hyperon decays considering q2 dependent form factors
~
Nuclear Physics B15 (1970)125-130. North-Holland Publ. Comp., Amsterdam
CALCULATION OF THE ~-e RATIO FOR SEMI-LEPTONIC HYPERON DECAYS CONSIDERING q2 DEPENDENT FORM FACTORS U. E. SCHRODER * Institut fiir Theoretische Physik der Universitttt Frankfurt~Main, Frankfurt~Main, Germany
Received 27 May 1969 Abstract: The calculation of the ]l-e ratio for semi-leptonic baryon decays with change of strangeness is reported, taking into account the q2 dependence of all form factors induced by regular currents. The calculated values are about 1-2% higher than those obtained by assuming constant form factors FI(0 ) and GI(0 ) . It is found that ff i r r e g u l a r currents were to be included the main contribution would come from the axial magnetism t e r m g2.
1. I N T R O D U C T I O N In a p r e v i o u s p a p e r [1] the r e s u l t s of the c a l c u l a t i o n of the ~ - e r a t i o of s t r a n g e n e s s c h a n g i n g b a r y o n d e c a y s w e r e r e p o r t e d t a k i n g into c o n s i d e r a t i o n t h e m o m e n t u m t r a n s f e r d e p e n d e n c e of t h e w e a k f o r m f a c t o r s . F o r the t h e o r e t i c a l a n d 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 s e d e c a y p r o c e s s e s it m a y b e u s e f u l to p r e s e n t a m o r e d e t a i l e d a c c o u n t of the c a l c u l a t i o n s . T h e t r e a t m e n t to b e d e s c r i b e d i n t h i s p a p e r l e a d s to f o r m u l a e f o r t h e t o t a l r a t e s of the ~ - and e - d e c a y s of b a r y o n s w h i c h e x h i b i t d e p e n d e n c e on e x p e r i m e n t a l p a r a m e t e r s s u c h a s m a s s e s , G A / G V , etc. T h e r e f o r e v a r i o u s a p p l i c a t i o n s a r e p o s s i b l e , s i n c e a n y one of t h e s e p a r a m e t e r s m a y b e e x p r e s s e d in t e r m s of t h e o t h e r s **. C l e a r l y , t h i s a p p r o a c h h a s c e r t a i n a d v a n t a g e s o v e r a numerical integration method. Our calculations are performed using a m i n i m u m n u m b e r of a p p r o x i m a t i o n s . T h e a s s u m p t i o n s c o n c e r n i n g t h e f o r m f a c t o r s a r e m e n t i o n e d only b r i e f l y , s i n c e a m o r e d e t a i l e d d i s c u s s i o n h a s b e e n g i v e n in r e f . [ 1].
2. CALCULATIONS AND RESULTS In o r d e r to s e t up the f o r m a l i s m f o r the d e s c r i p t i o n of a g e n e r a l s e m i l e p t o n i c b a r y o n d e c a y A ~ B + l+ Ul, (l = e, p) we s t a r t with the c u r r e n t x * P r e s e n t address: Institut f[ir Theoretische Physik der Universit~t Karlsruhe, Karlsruhe, Germany. ** For example, see ref. [1].
126
U.E. SCHR(~DER
× c u r r e n t f o r m of the L o r e n t z i n v a r i a n t weak i n t e r a c t i o n . A s s u m i n g f u r t h e r the V-A n a t u r e and local action of the lepton c u r r e n t , the total r a t e of the d e c a y A -~ B + l- + Pl (fig. 1) is given by 1 1 d3qld3q2d3p2 FA, B , I - (2~)5 2m A f 2q oI 2q 2o 2 P ~2 G
Tfi
6(4)(pl-P2
'
ql-
q2) ~
~
II'Tfi'2
spins
,
1
= --~[16mAmBm l m~,]~ ~/(q2)T~(1 +75)v~(ql)UB(P2)F p(Pl,P2)UA(Pl ) •
T h e v e r t e x function F # is d e t e r m i n e d by L o r e n t z i n v a r i a n c e in its m o s t general form Fu(pl,P2 )
= Fl(q2)y p + F2(q2)i(~pv qU + F3(q2)qt. L + [Gl(q2)?~+G2(q2)i(~pvqU+G3(q2)qp ]75 '
qu = ( P B - P A ) ~
and d e s c r i b e s the influence of the s t r o n g i n t e r a c t i o n of the b a r y o n s . The e l e c t r o m a g n e t i c c o r r e c t i o n s a r e neglected. A f t e r p e r f o r m i n g the s u m m a t i o n o v e r the lepton spins 1 G2 d3 q l d3q2 d 3p2 FA, B , l - (2~)5 2m A f o o 6 ( 4 ) ( p l - P 2 - q l - q2) 2q 1 2q 2 2p~
rXp Xp X NXU {qlq2 + q 2 q l - g~U(q 1 "q2 ) - ieXaUfiqlaq2fi } ,
NAp = ~
spins
~B(P2)rx U A ( P l ) [ ~ B ( P 2 ) F p U A ( P l ) ] * 4 m A m B ,
the integration over the lepton momenta yields FA, B, l 1 G2 - (2n)5 m A 12
2
d3p2 ( 1 -
f - 2P2
ml - - ~ ) N x p {2q~q~
(q
2 + 2 m / ) -2
gX~(2q2 + m~)}.
(1)
In o r d e r to p e r f o r m the i n t e g r a t i o n o v e r the b a r y o n m o m e n t u m P 2 it is convenient to u s e the new i n t e g r a t i o n v a r i a b l e qZ -= s which is r e l a t e d to the kinetic e n e r g y T of the b a r y o n (B) by q2_--s = A 2 - 2 m A T ,
A = m A-m B .
$ The definitions of spinors, 7 - m a t r i c e s , etc. are those of Schweber, An introduction to relativistic quantum field theory (Row, Peterson and Co., Evanston, 1961); except that we use 75 = iT0Yl~2~3. The general formulae useful here are given for instance in ref. [2].
127
~-e RATIO
By m e a n s of the r e l a t i o n d3p2
A2
d3p2
1
ds[(y,, 2 - s)(A 2 -
2
s)]~f(s),
2
ml
ml
where E = m A + m B , the decay r a t e (1) then b e c o m e s [3] G2 FA, B, 1 - 384n 3
1 A2 3 f mA m~
(s
ds
2 2
- ml) s2
[ ( Z - s)(A 2 - s)] ½
s + 2m~
× {IFl(S)[2[A2(4s +
ml)2 + 2~,2A 2 - -
]F2(s) 12(A2 - s)(2~, 2 +s)(2s
(~2 + 2s)(2s +m2)]
2 + m l ) + tF3(s) '~123m/s(1:2- s)
+ 2ReFI(S)F2(s)3~(A2 - s)(2s +ml)2 _ 2ReFl(S)F3(s)3m2A(E2_s ) 2 s+2m~ + IGl(S) 12[E2(4s-ml )+2~2A2 s " (A2+2s)(2s+m2)] + IG2(slI2(E 2 - sl(2A 2 + s l ( 2 s +m 2l)+ IG3(s) ]2 3m~s(A 2 - s)
- 2ReGl(S)G2(s)3A(2s+m~)(~,2-s)+
(2)
2ReGl(S)G3(s)3m~Z(A2-s)}.
It should be noted that so far no approximations have been made in the ca1culations. In the following we shall neglect the f o r m f a c t o r s F 3 and G2 which a r e induced by i r r e g u l a r c u r r e n t s $ and we shall d e s c r i b e the q2 dependence of the r e m a i n i n g f o r m f a c t o r s by the l i n e a r r e l a t i o n s
Fi(q2) = Fi(O )-{ l + ~1a v2q 2 }, 1221
Gk(q2 ) =Gk(O ) ~l+~aAq ~ ,
i = 1,2, k = 1,3,(a VCaA) .
It is convenient to introduce the following d i m e n s i o n l e s s quantities
(~ = s / A 2, x l= m l / A , 5 = A / E , f i = E . F i , gk = E.Gk, a n d k = G A / G V , (G V = GFI(0), GA = GGI(0)). Now the integration of (2) can be p e r f o r m e d neglecting t e r m s O(54), and a f t e r some s t r a i g h t f o r w a r d (but lengthy) calculation one obtains $ Since time-reversal invariance is assumed to hold within the required accuracy, we shall write the remaining form factors as real functions. It is convenient to use (~2 _s)½~ E(1 -½5209, which leads to integrals of the type 1 f;2 d(~'(~n" 41 -0% -2 ~< n < 3, which have analytic solutions [4].
128
U. E. SCHR~DER
A
=
Fig. 1. General semi-leptonic baryon decay.
222., FA, B,1 -
5~ 3
~2
+ 222,
+ ~ \F22]
+
1
[52A2+~avA
A2]+F-- ~
÷
-
_-2W
'
F1
(3)
w h e r e the v a r i o u s f u n c t i o n s r(xl) , A i ( x l ) , Bi(Xl) , etc. a r e given in the appendix. So f a r the q u a n t i t i e s f 2 ( O ) / F l ( O ) , ~, g3(O)/Fl(O), a v , a A a r e still unknown. In o r d e r to get m o r e i n f o r m a t i o n on the f o r m f a c t o r s we a s s u m e t h a t the v e c t o r p a r t of the h a d r o n i c c u r r e n t is in the s a m e o c t e t of SU(3) a s is the e l e c t r o m a g n e t i c c u r r e n t . F r o m t h i s a s s u m p t i o n and SU(3) s y m m e t r y one g e t s the known r e l a t i o n s of F l ( q 2 ) and F2(q2 ) ~ (weak m a g n e t i s m ) to the c o r r e s p o n d i n g e l e c t r o m a g n e t i c q u a n t i t i e s , that m e a n s e.g. F I ( 0 ) = 1, and a V = J(r2~ in (3) is now the r m s r a d i u s as d e t e r m i n e d f r o m e l e c t r o n s c a t t e r i n g e x p e r i m e n t s . It is f u r t h e r a s s u m e d that the induced p s e u d o s c a l a r f o r m f a c t o r s G3(0 ) a r e r e l a t e d to GI(0) by the g e n e r a l i s e d G o l d b e r g e r T r e i m a n r e l a t i o n s [5] 2
c3(0) = c1(0) ~ (1- mK 2 mK
m
),
KA
w h e r e KA is the 1320 MeV r e s o n a n c e [6]. If e i t h e r of the not y e t fixed q u a n t i t i e s a A and ~ is known, the o t h e r can now in p r i n c i p l e be d e t e r m i n e d f r o m the pJ-e r a t i o . U s i n g the e x p e r i m e n t a l i n f o r m a t i o n f o r the v a r i o u s d e c a y s , the n u m e r i c a l c a l c u l a t i o n l e a d s to e q u a t i o n s of the f o r m F p
a + 3~2b
Fe
a' +3~2b '
and $ The expressions for F2(0 ) are given for some decays of interest in ref. [1], table 1.
]2-e RATIO
r~
129
2A2 ~+fia A
Fe - a' +fi,a2A 2 w h e r e t h e n u m b e r s a , e t c . and a, e t c . a r e d e t e r m i n e d by m e a n s of t h e f u n c t i o n s g i v e n in t h e a p p e n d i x . A m o r e d e t a i l e d d i s c u s s i o n of t h e s e r e s u l t s i s to b e found in r e f . [ 1].
3. DISCUSSION F r o m eq. (3) one c a n o b t a i n an e s t i m a t e of t h e e r r o r s i n v o l v e d by a s s u m i n g c o n s t a n t f o r m f a c t o r s E l ( 0 ) a n d GI(0) a n d n e g l e c t i n g a l l t e r m s p r o p o r t i o n a l to 5 2. S i n c e r(xe) ~ 1 in t h i s a p p r o x i m a t i o n F#
i . e . t h e r e i s no l o n g e r a d e p e n d e n c e on ~2. T h e v a l u e s F p / F e c a l c u l a t e d f r o m r(x(~)) a r e in g o o d a g r e e m e n t w i t h p r e s e n t e x p e r i m e n t a l d a t a and c h a n g e to h i g h e r v a l u e s a b o u t 1-2% if t h e m o r e c o m p l e t e r e s u l t (3) i s u s e d . In o r d e r t o c h e c k t h e s e p r e d i c t i o n s it w o u l d b e i n t e r e s t i n g to h a v e m o r e e x perimental information. F u r t h e r it i s n a t u r a l to u s e eq. (2) f o r an e s t i m a t e of t h e p o s s i b l e c o n t r i b u t i o n s of t h e n e g l e c t e d f o r m f a c t o r s f 3 ( 0 ) a n d g2(0). T h e r e s u l t i s s u m m a r i z e d f o r t h e d e c a y e . g . ~ - -~ n + l + ~l in t a b l e 1, w h e r e t h e n u m e r i c a l f a c t o r s f o r t h e c o r r e s p o n d i n g f o r m f a c t o r s a r e l i s t e d . One r e a l i s e s t h a t t h e m a i n c o n t r i b u t i o n s c o u l d c o m e f r o m g 2 . T h e r e f o r e it w o u l d b e i n t e r e s t i n g t o l o o k f o r p o s s i b l e d e v i a t i o n s f r o m t h e p u r e o c t e t r e s u l t s $ t a k i n g into a c c o u n t at l e a s t g2(q2). A t p r e s e n t no i n f o r m a t i o n on t h e p o s s i b l e m a g n i t u d e of g2(0) i s a v a i l a b l e . One c o u l d t r y to e s t i m a t e t h e v a l u e of g2(0) f o r i n s t a n c e f r o m a f i t of t h e s e m i - l e p t o n i c b a r y o n d e c a y s in t h e f r a m e w o r k of Cabibbo' s theory. T h e a u t h o r w o u l d l i k e to t h a n k P r o f e s s o r H. P i e t s c h m a n n f o r a c r i t i c a l r e a d i n g of t h e m a n u s c r i p t and D r . M. S a n g s t e r f o r c o r r e c t i n g t h e E n g l i s h .
Numerical
factors of the contribution
Table 1 from (f3/FI)2,
etc. to the total decay
rate for
~ - -* n + / - +P/. Mode
e
(f3/F1)2
f3/F1
(g2/F1)2
Glg2/F2
7 x 10 -4 5 x 10 -7
-2.7 x 10 -2
1.2 x 10 -2
-2.4 x 10 -1
< 10 -6
2.4 x 10 -2
-4.8 x 10-1
$ According to a result by Cabibbo [7] time-reversal violation occurs only by irregular currents if the hadron current are members of the same SU(3) octet. However, first-order breaking effects could be important, since no counterpart to the Ademollo-Gatto theorem exists for the axial-vector current.
130
U.E. SCHR(JDER
APPENDIX
The functions introduced in r e l a t i o n (3) a r e x=
m l mA - mB
;
l=e,p,
L ( x ) = log
2 ~/1 - x 2
r(x) : (1-x2)~(1-~x 2)+~x 4L,
AI(X) : (1 -x2)~ (1 - ~ x 2 - 6x 4) + 1~--~5x4 L , A2(x ) = (1-x2)~ ( 1 - ~ x 2 + ~ x 4)-~-~x 6 L , BI(X) = ( 1 - x 2 ) ~ ( 1 - ~ x 2 + ~ x ) - 22 4" ~x 4 L , B2(x ) = x 2 ( 1 - x 2 ) ~ ( l + ~ x A'I(X ) = (1 - x2) ~
2 ) - 5 ( x 4 + ¼ x 6) L ,
(1 + ~ 2 +~ ~4) _ ,~_~(~4 + ~ 6 ) L ,
~ -~6)-~6L, A ~ ( x ) = ( 1 - x 2 ) ~ (1 - T39~, 2 ~'95,4
A,2(x) = ( l _ x 2 ) ~ ( l _
~ x 2 _ ~51x 4 - ~ x 6 ) + ~ 2 5 x 6 L
,
B ' l ( X ) = ( 1 - x 2 ) ~ ( 1 - ~ x 2 + ~ x 4 ) - ~ (x4 + ½ x 6) L ,
B'~(~) = (1-~2){ (1 +~ ~2_ ~ 4
+~6) +~ ~6 L ,
B ' 2 ( x ) = x 2 ( 1 - x 2 ) ~ (1- ~ x 2 - ~x 4) + ~ x 6 L
.
REFERENCES [1] U.E.SchrSder, Nucl. Phys. Bll (1969) 595. [2] H.Pietschmann, Acta Phys. Austriaca, Suppl. V (1968) 88. [3] J. Nilsson and H. Pietschmann, An introduction to weak interaction physics (McGraw Hill Publ. Comp., New York) (in preparation). [4] I. S. Gradshteyn and I.M.Ryzhik, Table of integrals, series and products (Academic Press, New York, 1965). [5] H.T.Nieh, Phys. Rev. 164 (1967) 1780. [6] N. Barash-Schmidt, A. Barbaro-Galtieri, L.R. Price, Marts Roos, A.H. Rosen. feld, P. SSding and C.H.Wohl, Review of particle properties (UCRL-8030, August, 1968). [7] N. Cabibbo, Phys. Letters 12 (1964) 137.
Nuclear Physics B15 (1970)125-130. North-Holland Publ. Comp., Amsterdam
CALCULATION OF THE ~-e RATIO FOR SEMI-LEPTONIC HYPERON DECAYS CONSIDERING q2 DEPENDENT FORM FACTORS U. E. SCHRODER * Institut fiir Theoretische Physik der Universitttt Frankfurt~Main, Frankfurt~Main, Germany
Received 27 May 1969 Abstract: The calculation of the ]l-e ratio for semi-leptonic baryon decays with change of strangeness is reported, taking into account the q2 dependence of all form factors induced by regular currents. The calculated values are about 1-2% higher than those obtained by assuming constant form factors FI(0 ) and GI(0 ) . It is found that ff i r r e g u l a r currents were to be included the main contribution would come from the axial magnetism t e r m g2.
1. I N T R O D U C T I O N In a p r e v i o u s p a p e r [1] the r e s u l t s of the c a l c u l a t i o n of the ~ - e r a t i o of s t r a n g e n e s s c h a n g i n g b a r y o n d e c a y s w e r e r e p o r t e d t a k i n g into c o n s i d e r a t i o n t h e m o m e n t u m t r a n s f e r d e p e n d e n c e of t h e w e a k f o r m f a c t o r s . F o r the t h e o r e t i c a l a n d 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 s e d e c a y p r o c e s s e s it m a y b e u s e f u l to p r e s e n t a m o r e d e t a i l e d a c c o u n t of the c a l c u l a t i o n s . T h e t r e a t m e n t to b e d e s c r i b e d i n t h i s p a p e r l e a d s to f o r m u l a e f o r t h e t o t a l r a t e s of the ~ - and e - d e c a y s of b a r y o n s w h i c h e x h i b i t d e p e n d e n c e on e x p e r i m e n t a l p a r a m e t e r s s u c h a s m a s s e s , G A / G V , etc. T h e r e f o r e v a r i o u s a p p l i c a t i o n s a r e p o s s i b l e , s i n c e a n y one of t h e s e p a r a m e t e r s m a y b e e x p r e s s e d in t e r m s of t h e o t h e r s **. C l e a r l y , t h i s a p p r o a c h h a s c e r t a i n a d v a n t a g e s o v e r a numerical integration method. Our calculations are performed using a m i n i m u m n u m b e r of a p p r o x i m a t i o n s . T h e a s s u m p t i o n s c o n c e r n i n g t h e f o r m f a c t o r s a r e m e n t i o n e d only b r i e f l y , s i n c e a m o r e d e t a i l e d d i s c u s s i o n h a s b e e n g i v e n in r e f . [ 1].
2. CALCULATIONS AND RESULTS In o r d e r to s e t up the f o r m a l i s m f o r the d e s c r i p t i o n of a g e n e r a l s e m i l e p t o n i c b a r y o n d e c a y A ~ B + l+ Ul, (l = e, p) we s t a r t with the c u r r e n t x * P r e s e n t address: Institut f[ir Theoretische Physik der Universit~t Karlsruhe, Karlsruhe, Germany. ** For example, see ref. [1].
126
U.E. SCHR(~DER
× c u r r e n t f o r m of the L o r e n t z i n v a r i a n t weak i n t e r a c t i o n . A s s u m i n g f u r t h e r the V-A n a t u r e and local action of the lepton c u r r e n t , the total r a t e of the d e c a y A -~ B + l- + Pl (fig. 1) is given by 1 1 d3qld3q2d3p2 FA, B , I - (2~)5 2m A f 2q oI 2q 2o 2 P ~2 G
Tfi
6(4)(pl-P2
'
ql-
q2) ~
~
II'Tfi'2
spins
,
1
= --~[16mAmBm l m~,]~ ~/(q2)T~(1 +75)v~(ql)UB(P2)F p(Pl,P2)UA(Pl ) •
T h e v e r t e x function F # is d e t e r m i n e d by L o r e n t z i n v a r i a n c e in its m o s t general form Fu(pl,P2 )
= Fl(q2)y p + F2(q2)i(~pv qU + F3(q2)qt. L + [Gl(q2)?~+G2(q2)i(~pvqU+G3(q2)qp ]75 '
qu = ( P B - P A ) ~
and d e s c r i b e s the influence of the s t r o n g i n t e r a c t i o n of the b a r y o n s . The e l e c t r o m a g n e t i c c o r r e c t i o n s a r e neglected. A f t e r p e r f o r m i n g the s u m m a t i o n o v e r the lepton spins 1 G2 d3 q l d3q2 d 3p2 FA, B , l - (2~)5 2m A f o o 6 ( 4 ) ( p l - P 2 - q l - q2) 2q 1 2q 2 2p~
rXp Xp X NXU {qlq2 + q 2 q l - g~U(q 1 "q2 ) - ieXaUfiqlaq2fi } ,
NAp = ~
spins
~B(P2)rx U A ( P l ) [ ~ B ( P 2 ) F p U A ( P l ) ] * 4 m A m B ,
the integration over the lepton momenta yields FA, B, l 1 G2 - (2n)5 m A 12
2
d3p2 ( 1 -
f - 2P2
ml - - ~ ) N x p {2q~q~
(q
2 + 2 m / ) -2
gX~(2q2 + m~)}.
(1)
In o r d e r to p e r f o r m the i n t e g r a t i o n o v e r the b a r y o n m o m e n t u m P 2 it is convenient to u s e the new i n t e g r a t i o n v a r i a b l e qZ -= s which is r e l a t e d to the kinetic e n e r g y T of the b a r y o n (B) by q2_--s = A 2 - 2 m A T ,
A = m A-m B .
$ The definitions of spinors, 7 - m a t r i c e s , etc. are those of Schweber, An introduction to relativistic quantum field theory (Row, Peterson and Co., Evanston, 1961); except that we use 75 = iT0Yl~2~3. The general formulae useful here are given for instance in ref. [2].
127
~-e RATIO
By m e a n s of the r e l a t i o n d3p2
A2
d3p2
1
ds[(y,, 2 - s)(A 2 -
2
s)]~f(s),
2
ml
ml
where E = m A + m B , the decay r a t e (1) then b e c o m e s [3] G2 FA, B, 1 - 384n 3
1 A2 3 f mA m~
(s
ds
2 2
- ml) s2
[ ( Z - s)(A 2 - s)] ½
s + 2m~
× {IFl(S)[2[A2(4s +
ml)2 + 2~,2A 2 - -
]F2(s) 12(A2 - s)(2~, 2 +s)(2s
(~2 + 2s)(2s +m2)]
2 + m l ) + tF3(s) '~123m/s(1:2- s)
+ 2ReFI(S)F2(s)3~(A2 - s)(2s +ml)2 _ 2ReFl(S)F3(s)3m2A(E2_s ) 2 s+2m~ + IGl(S) 12[E2(4s-ml )+2~2A2 s " (A2+2s)(2s+m2)] + IG2(slI2(E 2 - sl(2A 2 + s l ( 2 s +m 2l)+ IG3(s) ]2 3m~s(A 2 - s)
- 2ReGl(S)G2(s)3A(2s+m~)(~,2-s)+
(2)
2ReGl(S)G3(s)3m~Z(A2-s)}.
It should be noted that so far no approximations have been made in the ca1culations. In the following we shall neglect the f o r m f a c t o r s F 3 and G2 which a r e induced by i r r e g u l a r c u r r e n t s $ and we shall d e s c r i b e the q2 dependence of the r e m a i n i n g f o r m f a c t o r s by the l i n e a r r e l a t i o n s
Fi(q2) = Fi(O )-{ l + ~1a v2q 2 }, 1221
Gk(q2 ) =Gk(O ) ~l+~aAq ~ ,
i = 1,2, k = 1,3,(a VCaA) .
It is convenient to introduce the following d i m e n s i o n l e s s quantities
(~ = s / A 2, x l= m l / A , 5 = A / E , f i = E . F i , gk = E.Gk, a n d k = G A / G V , (G V = GFI(0), GA = GGI(0)). Now the integration of (2) can be p e r f o r m e d neglecting t e r m s O(54), and a f t e r some s t r a i g h t f o r w a r d (but lengthy) calculation one obtains $ Since time-reversal invariance is assumed to hold within the required accuracy, we shall write the remaining form factors as real functions. It is convenient to use (~2 _s)½~ E(1 -½5209, which leads to integrals of the type 1 f;2 d(~'(~n" 41 -0% -2 ~< n < 3, which have analytic solutions [4].
128
U. E. SCHR~DER
A
=
Fig. 1. General semi-leptonic baryon decay.
222., FA, B,1 -
5~ 3
~2
+ 222,
+ ~ \F22]
+
1
[52A2+~avA
A2]+F-- ~
÷
-
_-2W
'
F1
(3)
w h e r e the v a r i o u s f u n c t i o n s r(xl) , A i ( x l ) , Bi(Xl) , etc. a r e given in the appendix. So f a r the q u a n t i t i e s f 2 ( O ) / F l ( O ) , ~, g3(O)/Fl(O), a v , a A a r e still unknown. In o r d e r to get m o r e i n f o r m a t i o n on the f o r m f a c t o r s we a s s u m e t h a t the v e c t o r p a r t of the h a d r o n i c c u r r e n t is in the s a m e o c t e t of SU(3) a s is the e l e c t r o m a g n e t i c c u r r e n t . F r o m t h i s a s s u m p t i o n and SU(3) s y m m e t r y one g e t s the known r e l a t i o n s of F l ( q 2 ) and F2(q2 ) ~ (weak m a g n e t i s m ) to the c o r r e s p o n d i n g e l e c t r o m a g n e t i c q u a n t i t i e s , that m e a n s e.g. F I ( 0 ) = 1, and a V = J(r2~ in (3) is now the r m s r a d i u s as d e t e r m i n e d f r o m e l e c t r o n s c a t t e r i n g e x p e r i m e n t s . It is f u r t h e r a s s u m e d that the induced p s e u d o s c a l a r f o r m f a c t o r s G3(0 ) a r e r e l a t e d to GI(0) by the g e n e r a l i s e d G o l d b e r g e r T r e i m a n r e l a t i o n s [5] 2
c3(0) = c1(0) ~ (1- mK 2 mK
m
),
KA
w h e r e KA is the 1320 MeV r e s o n a n c e [6]. If e i t h e r of the not y e t fixed q u a n t i t i e s a A and ~ is known, the o t h e r can now in p r i n c i p l e be d e t e r m i n e d f r o m the pJ-e r a t i o . U s i n g the e x p e r i m e n t a l i n f o r m a t i o n f o r the v a r i o u s d e c a y s , the n u m e r i c a l c a l c u l a t i o n l e a d s to e q u a t i o n s of the f o r m F p
a + 3~2b
Fe
a' +3~2b '
and $ The expressions for F2(0 ) are given for some decays of interest in ref. [1], table 1.
]2-e RATIO
r~
129
2A2 ~+fia A
Fe - a' +fi,a2A 2 w h e r e t h e n u m b e r s a , e t c . and a, e t c . a r e d e t e r m i n e d by m e a n s of t h e f u n c t i o n s g i v e n in t h e a p p e n d i x . A m o r e d e t a i l e d d i s c u s s i o n of t h e s e r e s u l t s i s to b e found in r e f . [ 1].
3. DISCUSSION F r o m eq. (3) one c a n o b t a i n an e s t i m a t e of t h e e r r o r s i n v o l v e d by a s s u m i n g c o n s t a n t f o r m f a c t o r s E l ( 0 ) a n d GI(0) a n d n e g l e c t i n g a l l t e r m s p r o p o r t i o n a l to 5 2. S i n c e r(xe) ~ 1 in t h i s a p p r o x i m a t i o n F#
i . e . t h e r e i s no l o n g e r a d e p e n d e n c e on ~2. T h e v a l u e s F p / F e c a l c u l a t e d f r o m r(x(~)) a r e in g o o d a g r e e m e n t w i t h p r e s e n t e x p e r i m e n t a l d a t a and c h a n g e to h i g h e r v a l u e s a b o u t 1-2% if t h e m o r e c o m p l e t e r e s u l t (3) i s u s e d . In o r d e r t o c h e c k t h e s e p r e d i c t i o n s it w o u l d b e i n t e r e s t i n g to h a v e m o r e e x perimental information. F u r t h e r it i s n a t u r a l to u s e eq. (2) f o r an e s t i m a t e of t h e p o s s i b l e c o n t r i b u t i o n s of t h e n e g l e c t e d f o r m f a c t o r s f 3 ( 0 ) a n d g2(0). T h e r e s u l t i s s u m m a r i z e d f o r t h e d e c a y e . g . ~ - -~ n + l + ~l in t a b l e 1, w h e r e t h e n u m e r i c a l f a c t o r s f o r t h e c o r r e s p o n d i n g f o r m f a c t o r s a r e l i s t e d . One r e a l i s e s t h a t t h e m a i n c o n t r i b u t i o n s c o u l d c o m e f r o m g 2 . T h e r e f o r e it w o u l d b e i n t e r e s t i n g t o l o o k f o r p o s s i b l e d e v i a t i o n s f r o m t h e p u r e o c t e t r e s u l t s $ t a k i n g into a c c o u n t at l e a s t g2(q2). A t p r e s e n t no i n f o r m a t i o n on t h e p o s s i b l e m a g n i t u d e of g2(0) i s a v a i l a b l e . One c o u l d t r y to e s t i m a t e t h e v a l u e of g2(0) f o r i n s t a n c e f r o m a f i t of t h e s e m i - l e p t o n i c b a r y o n d e c a y s in t h e f r a m e w o r k of Cabibbo' s theory. T h e a u t h o r w o u l d l i k e to t h a n k P r o f e s s o r H. P i e t s c h m a n n f o r a c r i t i c a l r e a d i n g of t h e m a n u s c r i p t and D r . M. S a n g s t e r f o r c o r r e c t i n g t h e E n g l i s h .
Numerical
factors of the contribution
Table 1 from (f3/FI)2,
etc. to the total decay
rate for
~ - -* n + / - +P/. Mode
e
(f3/F1)2
f3/F1
(g2/F1)2
Glg2/F2
7 x 10 -4 5 x 10 -7
-2.7 x 10 -2
1.2 x 10 -2
-2.4 x 10 -1
< 10 -6
2.4 x 10 -2
-4.8 x 10-1
$ According to a result by Cabibbo [7] time-reversal violation occurs only by irregular currents if the hadron current are members of the same SU(3) octet. However, first-order breaking effects could be important, since no counterpart to the Ademollo-Gatto theorem exists for the axial-vector current.
130
U.E. SCHR(JDER
APPENDIX
The functions introduced in r e l a t i o n (3) a r e x=
m l mA - mB
;
l=e,p,
L ( x ) = log
2 ~/1 - x 2
r(x) : (1-x2)~(1-~x 2)+~x 4L,
AI(X) : (1 -x2)~ (1 - ~ x 2 - 6x 4) + 1~--~5x4 L , A2(x ) = (1-x2)~ ( 1 - ~ x 2 + ~ x 4)-~-~x 6 L , BI(X) = ( 1 - x 2 ) ~ ( 1 - ~ x 2 + ~ x ) - 22 4" ~x 4 L , B2(x ) = x 2 ( 1 - x 2 ) ~ ( l + ~ x A'I(X ) = (1 - x2) ~
2 ) - 5 ( x 4 + ¼ x 6) L ,
(1 + ~ 2 +~ ~4) _ ,~_~(~4 + ~ 6 ) L ,
~ -~6)-~6L, A ~ ( x ) = ( 1 - x 2 ) ~ (1 - T39~, 2 ~'95,4
A,2(x) = ( l _ x 2 ) ~ ( l _
~ x 2 _ ~51x 4 - ~ x 6 ) + ~ 2 5 x 6 L
,
B ' l ( X ) = ( 1 - x 2 ) ~ ( 1 - ~ x 2 + ~ x 4 ) - ~ (x4 + ½ x 6) L ,
B'~(~) = (1-~2){ (1 +~ ~2_ ~ 4
+~6) +~ ~6 L ,
B ' 2 ( x ) = x 2 ( 1 - x 2 ) ~ (1- ~ x 2 - ~x 4) + ~ x 6 L
.
REFERENCES [1] U.E.SchrSder, Nucl. Phys. Bll (1969) 595. [2] H.Pietschmann, Acta Phys. Austriaca, Suppl. V (1968) 88. [3] J. Nilsson and H. Pietschmann, An introduction to weak interaction physics (McGraw Hill Publ. Comp., New York) (in preparation). [4] I. S. Gradshteyn and I.M.Ryzhik, Table of integrals, series and products (Academic Press, New York, 1965). [5] H.T.Nieh, Phys. Rev. 164 (1967) 1780. [6] N. Barash-Schmidt, A. Barbaro-Galtieri, L.R. Price, Marts Roos, A.H. Rosen. feld, P. SSding and C.H.Wohl, Review of particle properties (UCRL-8030, August, 1968). [7] N. Cabibbo, Phys. Letters 12 (1964) 137.