- Email: [email protected]
Spectra of Kμ4 decays
-~
Nuclear Physics B5 (1968) 85-92. North-Holland Publ. Comp.. Amsterdam
SPECTRA
OF
Kp4
DECAYS
K. SCHILCHER Arnerican University of Beirut, Beirut, Libanon Received 15 January 1968
Abstract: The K~4 decay spectrum and decay rate are calculated using currentalgebra results for the form factors.
1. INTRODUCTION In the past a l a r g e n u m b e r of t h e o r e t i c a l and e x p e r i m e n t a l investigations have been p e r f o r m e d [1-6] c o n c e r n i n g the decay of a kaon into two pions and a lepton pair. A r e l a t i v e l y l a r g e n u m b e r of events for the c a s e of the lepton being an e l e c t r o n h a s been o b s e r v e d , while the K~4 decay mode is r a t h e r r a r e due to the limited available p h a s e space. The weak i n t e r a c t i o n causing the K/4 decay is of c o u r s e i n t e r e s t i n g in itself but the special attention paid to this decay s t e m s f r o m the fact that one hopes to e x t r a c t information on the l o w - e n e r g y ~-n interaction since the a n a l y s i s is facilitated by the a b s e n c e of other strongly interacting p a r t i c l e s in the final state. One way of taking the n-n final state interaction into account c o n s i s t s in relating the K/4 f o r m f a c t o r s to the KK ~ nn amplitude by m e a n s of the PCAC h y p o t h e s i s for the kaon field; we shall d i s c u s s this a p p r o a c h briefly in sect. 2. This a s s u m p t i o n is equivalent to the one made in ref. [4], w h e r e decay r a t e and di pion s p e c t r u m w e r e calculated for the c a s e rnl=O. F o r the c a s e of the K~4 decay we have to take rn l ¢ 0 , however the extension of the above calculation b e c o m e s to c o m p l i c a t e d to be c a r r i e d out in p r a c t i c e . F o r t u n a t e l y e x p e r i m e n t s s e e m to suggest strongly that the n-n final state i n t e r a c t i o n is c o m p l e t e l y negligible. This conclusion is supp o r t e d by W e i n b e r g ' s [6] f a m o u s p r e d i c t i o n of the Ke4 decay r a t e which one finds in excellent a g r e e m e n t with p r e s e n t l y available e x p e r i m e n t a l data. W e i n b e r g ' s soft-pion method c o n s i s t s in exploiting PCAC and c u r r e n t c o m mutation r e l a t i o n s in a s y s t e m a t i c way by taking both pions off the m a s s shell, expanding in p o w e r s of t h e i r m o m e n t a and keeping only the l o w e s t o r d e r t e r m s . This p r o c e d u r e , applied to the K#4 decay, is justified at l e a s t a p o s t e r i o r i since one notices f r o m the pion s p e c t r u m , fig. 2, d e t e r mined with the help of W e i n b e r g ' s r e s u l t s that the pions a r e p r e d o m i n a n t l y emitted at the lower end of the s p e c t r u m . * Partially supported by Research Corporation grant.
86
K. SCHILCHER
In this note we shall c a l c u l a t e the decay r a t e f o r the K p 4 p r o c e s s and the di pion effective m a s s s p e c t r u m making u s e of the c u r r e n t a l g e b r a r e s u l t s f o r the f o r m f a c t o r s .
2. MATRIX E L E M E N T FOR THE K/4 DECAY If we a s s u m e a local V - A coupling f o r the leptons then the K/4 decay m a t r i x e l e m e n t m a y be w r i t t e n to f i r s t o r d e r of p e r t u r b a t i o n t h e o r y in the weak i n t e r a c t i o n s , a s M(K/4) = (2~)46(PK
-P- q-Pl-Pv) ~2
( ~ t ( 1 +~5)/)
x (nqa npblA~ (0) [ K p K m ) ,
(1)
w h e r e PK, P, q, Pl, Pv a r e the m o m e n t a of the K - m e s o n , the ~ - m e s o n s , and the leptons r e s p e c t i v e l y , A~(0) is the axial v e c t o r , h i 3 = -n, AS = -1 s t r a n g e n e s s changing c u r r e n t . (It was shown by Shabalin [1] that the v e c t o r c u r r e n t c o n t r i b u t e s only v e r y little to the d e c a y s p e c t r a and m a y t h e r e f o r e be neglected), a and b a r e the pion i s o v e c t o r indices and m = +-~the 13 v a l u e of the K - m e s o n , G is the weak coupling constant G = 1.02 x 10-'5,2/rap. F r o m i n v a r i a n c e c o n s i d e r a t i o n s one can w r i t e the m o s t g e n e r a l f o r m of the axial vector current matrix element as
(Trqa TrpbIAgh(0) IKpKrr~ 1 {(p+q))tF1 + (p_q)xF2 + (pK_p_q)•F3}" mK
(2)
The f o r m f a c t o r s F i a r e in g e n e r a l functions of ( P K ' P ) , (PK" q), (P" q), a, b, rn and n. The PCAC h y p o t h e s i s f o r the s t r a n g e n e s s - c h a n g i n g c u r r e n t a s s e r t s a/l A p.(x) = k~bK(X),
(3)
w h e r e ~bK is the K - m e s o n field and ~ = F K . M 2 with F K being defined by
Using r e l a t i o n (3), the s t a n d a r d LSZ f o r m a l i s m and p e r f o r m i n g a p a r t i a l i n t e g r a t i o n one e s t a b l i s h e s the r e l a t i o n (2~)46(PK -
q - P - k) - ~1o
(" q~pl Au( O)[KPK) k~ i)t
k2_m2K (~q ~P ] s- I[KPKI~)"
(4)
One can t h e r e f o r e r e l a t e the f o r m f a c t o r s to the KI~ --* ~ s c a t t e r i n g a m p l i tude. The pole t e r m m a y be t r a c e d b a c k to the d i a g r a m in which the K m e s o n e m i t s two soft pions, continues a s a K which finally d e c a y s into l e p -
K]24 DECAY
87
ton plus neutrino. Since the K -~ l + v a m p l i t u d e is p r o p o r t i o n a l to k g = ( P K - P -q)tZ the pole t e r m c o n t r i b u t e s only to F 3. Our eq. 4 is identical to eq. (6) in ref. [4], obtained by d i s p e r s o n - t h e o r e t i c a l m e t h o d s neglecting the contribution of the continuum. F o r m l = 0 a p a r t i a l wave a n a l y s i s m a y be applied advantageously. T h i s m e t h o d s f a i l s h o w e v e r f o r m l ¢ 0 u n l e s s one m a k e s d r a s t i c a p p r o x i m a t i o n s . F o r this r e a s o n , and a l s o in o r d e r to obtain f e a s i b l e p h a s e s p a c e i n t e g r a l s , we m a k e u s e of the s i m p l e c u r r e n t a l g e b r a r e s u l t s f o r the f o r m f a c t o r s . F o r l a t e r convenience we e x t r a c t the pole t e r m f r o m the f o r m f a c t o r F 3 and write
F3 = F~1)
PK°(P-q) F~2)
+ PK" (-P+ q)
"
(5)
In t h i s notation W e i n b e r g ' s r e s u l t s [6] r e a d
F 1 =A 5ab 5urn ,
F 2 = -iA Cabc(~'c)nm,
F~1) = B6ab 5rim ,
F~2) = iBeabc(Tc)nm ,
(6)
w h e r e A and B a r e c o n s t a n t s r e l a t e d to the e x p e r i m e n t a l K/3 f o r m f a c t o r s f±(PK" q) by
mK
A = 2f+(0) G F---~-'
B = (f+(0)+f_(0))
GmK
F-~-
(7)
Using the e x p e r i m e n t a l v a l u e s [7] f o r F u and the K/3 f o r m f a c t o r s one obtains [A[ = 1 . 2 0 + 0.07,
1BI = 1.75 + 0.33.
(8)
3. DECAY R A T E AND D I - P I O N E F F E C T I V E - M A S S SPECTRUM To f a c i l i t a t e the p h a s e s p a c e i n t e g r a t i o n s we introduce the following variables
Q=p+q,
R =p - q ,
K=Pl+Pv,
L=Pl-Pv"
(9)
If we s q u a r e the m a t r i x e l e m e n t eq. (1), s u m o v e r the lepton p o l a r i z a t i o n s we m a y obtain the total decay r a t e in the K r e s t s y s t e m
F _
f d3q d3p dapl d3pv 54(Q+K_PK)
G2
16(2
3
% × ~1{ ( . WK )
2
-
( V ' L ) 2 - V 2 ( K 2 m~)},
(10)
where
Vg = F 1 Qtz + F 2 R g + F3Kg."
(11)
88
K. SCHILCHER
The phase space integrations are lengthy but conventional [8] we state only the result after the L, R and K integrations: /
4m2~
-
16(2~-)6m3K
7n2] 2
-Li - ~J 2
2
Qz/L
+ K2 /
K2/J
2
+ 2F~I)Flm~(Q.K)
w h e r e K = q - Q. The effective mass of the two ~ mesons is fQ2. Expression (13) is now integrated over dQo in the limits Q --< Qo "< (Q2 + m2 _ m2)/2mK and over d~ to yield the effective mass spectrum
16(2~)5 36
+ F(1)2 (1) ~.2, ~(2)2a(2)i~2~ 3
A33 ( ~ ) + - 3
"'33~ i
(13) where Z2 = Q2/m2. The functions Aik(Z 2) were determined to be
L'~
b~
ba
b~
v
v
v
II
II
+ <.. b~
II
v
II
+
%
+ I
I
II
+
+ <...
+
"i" +
%
,'7
+"
+
b~ !
I
+
v
b~ I
+
ba
i,--i
+ I
I
b~
b~
b~
E ~
~ + I
t
I
t I'--4
o r/l
N 1:::
+
g~"
+
+ ÷
+ -
4-
b~
+
b~ +
i ~
"1= +
I I
1=
+
>
I
+
I
+ 1::
~
I
+
b~
'
b~
+
b~ v
L'~ I
"7"
"i" t~
b~
I
b~ I
b~
v
b~ + b~
+
+
+
~
-~
+ +
+ "1::: + "1::
~
a
"1::
L _ _ J
'I:::
t~ I
N !
+ I
I
+
1:::
+ N
t~
1::
i,,,-q
90
K. S C H I L C H E R
2
3
(2) _2, , 4mu .~ A33(~ ) =~(1 - m--~] {-616Z2(Z2+l-2~)+
- 692 (1 +Z2) 2 in (II) + 48
~2(l+Z2)]in(I)
1+ Z2) 2
arcos
2Z Z2+l -
+ ~ ( Z 2 + 1 - ~ ) 2 _ 4 Z 2 [ Z 4 + Z 2 ( 5 8 _ l l g ) + 1 - l l g + 10~ 2
+
6g (Z4+l-2Z 2-2gZ 2-2g)]} I+Z 2
w h e r e ~ = (m l / i n K )2 and in (I) = in
Z2+ 1 - ~z+~/(Z2+l-~)2-4Z2 2Z
in (II) = in (I+Z2)(I+Z 2-~)
2Z~ - 4Z 2 + ( 1 - Z 2 ) ~ / ( Z 2 + l _ ~ ) 2 _ 4 Z 2 "
F o r m l = 0 o u r r e s u l t a g r e e s with that of Cabibbo and M a k s y m o w i c z except that t h e i r r e s u l t should be m u l t i p l i e d by a f a c t o r 4 a s pointed out by W e i n b e r g [6]. P a r t of the e f f e c t i v e m a s s s p e c t r u m m a y a l s o be found in ref. [1], one should h o w e v e r r e p l a c e in (I) by -ln (I) in that p a p e r . The functions A i k a r e plotted in fig. 1. If one i n t e g r a t e s n u m e r i c a l l y o v e r Z 2 within the l i m i t s ( 2 m ~ / i n K ) 2 --< Z 2 --< (Ira K - m ~ ] / m K ) 2 one obtains the d e c a y r a t e : F-
(2~) 5" 16" 36
We now c o n s i d e r the s p e c i f i c d e c a y K + --* ~+ + n- + ~+ + v which is m o s t i m p o r t a n t f r o m the e x p e r i m e n t a l standpoint of view. T h e f o r m f a c t o r s f o r this process are
F1-- F2-- A,
B,
(15)
u s i n g the e x p e r i m e n t a l v a l u e s , eq. (8), one obtains an effective m a s s s p e c t r u m plotted in fig. 2. T h e p r e d i c t e d d e c a y r a t e f o r K + -~ ~+ + ~- + IZ+ - v is: F = 5.5 × 102 sec -1 ,
(16)
K~4 DECAY
9[
I
/
I
91
I
\
6-
?:J
\
4-
,:
',,
2- /i/...... :.t'/
\,\ 4rV . , , .. ~ r ~) . . .....
"*'o,'~"~:...,
1- //
\
\ '\",
X\\ \
\.. \~X~ x ""'- .:::.,:~
~.
~, ~,,,,
........~ :..~...:..~-~___~ .32
' .34
.~6
' 4 1.
.38
0
~
.50
.610'. 6 2
z2
Fig. 1. Functions entering into the general expression [eq. (13)] for the di-pion effective-mass spectrum. w h i c h ought to b e c o m p a r e d with the e x p e r i m e n t a l r e s u l t s [9] F ~ < 1 . 1 x 1 0 3 sec - 1 . I 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 b r i n g i n g t h e p r o b l e m to m y a t t e n t i o n a n d f o r h i s i n t e r e s t . I a m a l s o g r a t e f u l f o r a n u m b e r of c o n v e r s a t i o n s with P r o f e s s o r s R. A c h a r y a , H . H . A l y a n d H. A. M a v r o m a t i s . D. K e o s h e y a n a n d t h e o t h e r s t a f f of t h e A U B C o m p u t e r C e n t r e w e r e of g r e a t h e l p i n t h e c o u r s e of t h e n u m e r i c a l c a l c u l a t i o n s .
92
K. SCHILCHER
5
4
3
2
I
32 34 36 38 40
50
i
I
l
60 62
i
I
i
,E
Fig. 2. Di-pion e f f e c t i v e - m a s s spectrum for K+--~ 7[+ + ~ - + /l + + ~. A f t e r t h i s w o r k w a s d o n e it c a m e to t h e a u t h o r ' s a t t e n t i o n t h a t c a l c u l a t i o n s on r e l a t e d l i n e s a r e b e i n g c a r r i e d out by A. D o n n a c h i e , G, C. O a d e s a n d F. B e r e n d s . NOTE
ADDED
IN P R O O F
A m o r e r e c e n t e x p e r i m e n t a l v a l u e f o r the K + -~ ~+~-/x+v r a t e i s F = ( 1 . ] + 0 . 7 ) x 1 0 3 s e c -1 T h i s r e s u l t i s t a k e n f r o m V. B i s i , R. C e s t e r , A. M a r z a r i C h i e s a and M. V i g o n e , P h y s i c s L e t t e r s 25B (1967) 572. REFERENCES [1] E. P. Shabalin, ZhETF (USSR) 39 (1961) 345 [ JETP (Soy. Phys.) 10 (1960) 1252]. [2] N. Cabibbo and A. Maksymowicz, Phys. Rev. 137 (1965) B438, and r e f e r e n c e s therein. [3] C. Kacs er , P. Singer and T. N. Truong, Phys. Rev. 137 (1965) B1605. [4] N. V. Hieu, ZhETF 44 (1963) 162 [ J E T P (Sov. Phys.) 17 (1963) 113]. [5] R.W. Birge et al., Phys. Rev. 139 (1965) B1600. [6] S. Weinberg, Phys. Rev. L e t te r s 17 (1966) 336. [7] G. H. Trilling, Argonne National Laboratory Report No. ANL 7130, 1965. [8] J. D. Jackson, 1962 Brandeis Lectures. [9] A. H. Rosenfeld et al., UCRL-8030 (revised September 1967).
Nuclear Physics B5 (1968) 85-92. North-Holland Publ. Comp.. Amsterdam
SPECTRA
OF
Kp4
DECAYS
K. SCHILCHER Arnerican University of Beirut, Beirut, Libanon Received 15 January 1968
Abstract: The K~4 decay spectrum and decay rate are calculated using currentalgebra results for the form factors.
1. INTRODUCTION In the past a l a r g e n u m b e r of t h e o r e t i c a l and e x p e r i m e n t a l investigations have been p e r f o r m e d [1-6] c o n c e r n i n g the decay of a kaon into two pions and a lepton pair. A r e l a t i v e l y l a r g e n u m b e r of events for the c a s e of the lepton being an e l e c t r o n h a s been o b s e r v e d , while the K~4 decay mode is r a t h e r r a r e due to the limited available p h a s e space. The weak i n t e r a c t i o n causing the K/4 decay is of c o u r s e i n t e r e s t i n g in itself but the special attention paid to this decay s t e m s f r o m the fact that one hopes to e x t r a c t information on the l o w - e n e r g y ~-n interaction since the a n a l y s i s is facilitated by the a b s e n c e of other strongly interacting p a r t i c l e s in the final state. One way of taking the n-n final state interaction into account c o n s i s t s in relating the K/4 f o r m f a c t o r s to the KK ~ nn amplitude by m e a n s of the PCAC h y p o t h e s i s for the kaon field; we shall d i s c u s s this a p p r o a c h briefly in sect. 2. This a s s u m p t i o n is equivalent to the one made in ref. [4], w h e r e decay r a t e and di pion s p e c t r u m w e r e calculated for the c a s e rnl=O. F o r the c a s e of the K~4 decay we have to take rn l ¢ 0 , however the extension of the above calculation b e c o m e s to c o m p l i c a t e d to be c a r r i e d out in p r a c t i c e . F o r t u n a t e l y e x p e r i m e n t s s e e m to suggest strongly that the n-n final state i n t e r a c t i o n is c o m p l e t e l y negligible. This conclusion is supp o r t e d by W e i n b e r g ' s [6] f a m o u s p r e d i c t i o n of the Ke4 decay r a t e which one finds in excellent a g r e e m e n t with p r e s e n t l y available e x p e r i m e n t a l data. W e i n b e r g ' s soft-pion method c o n s i s t s in exploiting PCAC and c u r r e n t c o m mutation r e l a t i o n s in a s y s t e m a t i c way by taking both pions off the m a s s shell, expanding in p o w e r s of t h e i r m o m e n t a and keeping only the l o w e s t o r d e r t e r m s . This p r o c e d u r e , applied to the K#4 decay, is justified at l e a s t a p o s t e r i o r i since one notices f r o m the pion s p e c t r u m , fig. 2, d e t e r mined with the help of W e i n b e r g ' s r e s u l t s that the pions a r e p r e d o m i n a n t l y emitted at the lower end of the s p e c t r u m . * Partially supported by Research Corporation grant.
86
K. SCHILCHER
In this note we shall c a l c u l a t e the decay r a t e f o r the K p 4 p r o c e s s and the di pion effective m a s s s p e c t r u m making u s e of the c u r r e n t a l g e b r a r e s u l t s f o r the f o r m f a c t o r s .
2. MATRIX E L E M E N T FOR THE K/4 DECAY If we a s s u m e a local V - A coupling f o r the leptons then the K/4 decay m a t r i x e l e m e n t m a y be w r i t t e n to f i r s t o r d e r of p e r t u r b a t i o n t h e o r y in the weak i n t e r a c t i o n s , a s M(K/4) = (2~)46(PK
-P- q-Pl-Pv) ~2
( ~ t ( 1 +~5)/)
x (nqa npblA~ (0) [ K p K m ) ,
(1)
w h e r e PK, P, q, Pl, Pv a r e the m o m e n t a of the K - m e s o n , the ~ - m e s o n s , and the leptons r e s p e c t i v e l y , A~(0) is the axial v e c t o r , h i 3 = -n, AS = -1 s t r a n g e n e s s changing c u r r e n t . (It was shown by Shabalin [1] that the v e c t o r c u r r e n t c o n t r i b u t e s only v e r y little to the d e c a y s p e c t r a and m a y t h e r e f o r e be neglected), a and b a r e the pion i s o v e c t o r indices and m = +-~the 13 v a l u e of the K - m e s o n , G is the weak coupling constant G = 1.02 x 10-'5,2/rap. F r o m i n v a r i a n c e c o n s i d e r a t i o n s one can w r i t e the m o s t g e n e r a l f o r m of the axial vector current matrix element as
(Trqa TrpbIAgh(0) IKpKrr~ 1 {(p+q))tF1 + (p_q)xF2 + (pK_p_q)•F3}" mK
(2)
The f o r m f a c t o r s F i a r e in g e n e r a l functions of ( P K ' P ) , (PK" q), (P" q), a, b, rn and n. The PCAC h y p o t h e s i s f o r the s t r a n g e n e s s - c h a n g i n g c u r r e n t a s s e r t s a/l A p.(x) = k~bK(X),
(3)
w h e r e ~bK is the K - m e s o n field and ~ = F K . M 2 with F K being defined by
Using r e l a t i o n (3), the s t a n d a r d LSZ f o r m a l i s m and p e r f o r m i n g a p a r t i a l i n t e g r a t i o n one e s t a b l i s h e s the r e l a t i o n (2~)46(PK -
q - P - k) - ~1o
(" q~pl Au( O)[KPK) k~ i)t
k2_m2K (~q ~P ] s- I[KPKI~)"
(4)
One can t h e r e f o r e r e l a t e the f o r m f a c t o r s to the KI~ --* ~ s c a t t e r i n g a m p l i tude. The pole t e r m m a y be t r a c e d b a c k to the d i a g r a m in which the K m e s o n e m i t s two soft pions, continues a s a K which finally d e c a y s into l e p -
K]24 DECAY
87
ton plus neutrino. Since the K -~ l + v a m p l i t u d e is p r o p o r t i o n a l to k g = ( P K - P -q)tZ the pole t e r m c o n t r i b u t e s only to F 3. Our eq. 4 is identical to eq. (6) in ref. [4], obtained by d i s p e r s o n - t h e o r e t i c a l m e t h o d s neglecting the contribution of the continuum. F o r m l = 0 a p a r t i a l wave a n a l y s i s m a y be applied advantageously. T h i s m e t h o d s f a i l s h o w e v e r f o r m l ¢ 0 u n l e s s one m a k e s d r a s t i c a p p r o x i m a t i o n s . F o r this r e a s o n , and a l s o in o r d e r to obtain f e a s i b l e p h a s e s p a c e i n t e g r a l s , we m a k e u s e of the s i m p l e c u r r e n t a l g e b r a r e s u l t s f o r the f o r m f a c t o r s . F o r l a t e r convenience we e x t r a c t the pole t e r m f r o m the f o r m f a c t o r F 3 and write
F3 = F~1)
PK°(P-q) F~2)
+ PK" (-P+ q)
"
(5)
In t h i s notation W e i n b e r g ' s r e s u l t s [6] r e a d
F 1 =A 5ab 5urn ,
F 2 = -iA Cabc(~'c)nm,
F~1) = B6ab 5rim ,
F~2) = iBeabc(Tc)nm ,
(6)
w h e r e A and B a r e c o n s t a n t s r e l a t e d to the e x p e r i m e n t a l K/3 f o r m f a c t o r s f±(PK" q) by
mK
A = 2f+(0) G F---~-'
B = (f+(0)+f_(0))
GmK
F-~-
(7)
Using the e x p e r i m e n t a l v a l u e s [7] f o r F u and the K/3 f o r m f a c t o r s one obtains [A[ = 1 . 2 0 + 0.07,
1BI = 1.75 + 0.33.
(8)
3. DECAY R A T E AND D I - P I O N E F F E C T I V E - M A S S SPECTRUM To f a c i l i t a t e the p h a s e s p a c e i n t e g r a t i o n s we introduce the following variables
Q=p+q,
R =p - q ,
K=Pl+Pv,
L=Pl-Pv"
(9)
If we s q u a r e the m a t r i x e l e m e n t eq. (1), s u m o v e r the lepton p o l a r i z a t i o n s we m a y obtain the total decay r a t e in the K r e s t s y s t e m
F _
f d3q d3p dapl d3pv 54(Q+K_PK)
G2
16(2
3
% × ~1{ ( . WK )
2
-
( V ' L ) 2 - V 2 ( K 2 m~)},
(10)
where
Vg = F 1 Qtz + F 2 R g + F3Kg."
(11)
88
K. SCHILCHER
The phase space integrations are lengthy but conventional [8] we state only the result after the L, R and K integrations: /
4m2~
-
16(2~-)6m3K
7n2] 2
-Li - ~J 2
2
Qz/L
+ K2 /
K2/J
2
+ 2F~I)Flm~(Q.K)
w h e r e K = q - Q. The effective mass of the two ~ mesons is fQ2. Expression (13) is now integrated over dQo in the limits Q --< Qo "< (Q2 + m2 _ m2)/2mK and over d~ to yield the effective mass spectrum
16(2~)5 36
+ F(1)2 (1) ~.2, ~(2)2a(2)i~2~ 3
A33 ( ~ ) + - 3
"'33~ i
(13) where Z2 = Q2/m2. The functions Aik(Z 2) were determined to be
L'~
b~
ba
b~
v
v
v
II
II
+ <.. b~
II
v
II
+
%
+ I
I
II
+
+ <...
+
"i" +
%
,'7
+"
+
b~ !
I
+
v
b~ I
+
ba
i,--i
+ I
I
b~
b~
b~
E ~
~ + I
t
I
t I'--4
o r/l
N 1:::
+
g~"
+
+ ÷
+ -
4-
b~
+
b~ +
i ~
"1= +
I I
1=
+
>
I
+
I
+ 1::
~
I
+
b~
'
b~
+
b~ v
L'~ I
"7"
"i" t~
b~
I
b~ I
b~
v
b~ + b~
+
+
+
~
-~
+ +
+ "1::: + "1::
~
a
"1::
L _ _ J
'I:::
t~ I
N !
+ I
I
+
1:::
+ N
t~
1::
i,,,-q
90
K. S C H I L C H E R
2
3
(2) _2, , 4mu .~ A33(~ ) =~(1 - m--~] {-616Z2(Z2+l-2~)+
- 692 (1 +Z2) 2 in (II) + 48
~2(l+Z2)]in(I)
1+ Z2) 2
arcos
2Z Z2+l -
+ ~ ( Z 2 + 1 - ~ ) 2 _ 4 Z 2 [ Z 4 + Z 2 ( 5 8 _ l l g ) + 1 - l l g + 10~ 2
+
6g (Z4+l-2Z 2-2gZ 2-2g)]} I+Z 2
w h e r e ~ = (m l / i n K )2 and in (I) = in
Z2+ 1 - ~z+~/(Z2+l-~)2-4Z2 2Z
in (II) = in (I+Z2)(I+Z 2-~)
2Z~ - 4Z 2 + ( 1 - Z 2 ) ~ / ( Z 2 + l _ ~ ) 2 _ 4 Z 2 "
F o r m l = 0 o u r r e s u l t a g r e e s with that of Cabibbo and M a k s y m o w i c z except that t h e i r r e s u l t should be m u l t i p l i e d by a f a c t o r 4 a s pointed out by W e i n b e r g [6]. P a r t of the e f f e c t i v e m a s s s p e c t r u m m a y a l s o be found in ref. [1], one should h o w e v e r r e p l a c e in (I) by -ln (I) in that p a p e r . The functions A i k a r e plotted in fig. 1. If one i n t e g r a t e s n u m e r i c a l l y o v e r Z 2 within the l i m i t s ( 2 m ~ / i n K ) 2 --< Z 2 --< (Ira K - m ~ ] / m K ) 2 one obtains the d e c a y r a t e : F-
(2~) 5" 16" 36
We now c o n s i d e r the s p e c i f i c d e c a y K + --* ~+ + n- + ~+ + v which is m o s t i m p o r t a n t f r o m the e x p e r i m e n t a l standpoint of view. T h e f o r m f a c t o r s f o r this process are
F1-- F2-- A,
B,
(15)
u s i n g the e x p e r i m e n t a l v a l u e s , eq. (8), one obtains an effective m a s s s p e c t r u m plotted in fig. 2. T h e p r e d i c t e d d e c a y r a t e f o r K + -~ ~+ + ~- + IZ+ - v is: F = 5.5 × 102 sec -1 ,
(16)
K~4 DECAY
9[
I
/
I
91
I
\
6-
?:J
\
4-
,:
',,
2- /i/...... :.t'/
\,\ 4rV . , , .. ~ r ~) . . .....
"*'o,'~"~:...,
1- //
\
\ '\",
X\\ \
\.. \~X~ x ""'- .:::.,:~
~.
~, ~,,,,
........~ :..~...:..~-~___~ .32
' .34
.~6
' 4 1.
.38
0
~
.50
.610'. 6 2
z2
Fig. 1. Functions entering into the general expression [eq. (13)] for the di-pion effective-mass spectrum. w h i c h ought to b e c o m p a r e d with the e x p e r i m e n t a l r e s u l t s [9] F ~ < 1 . 1 x 1 0 3 sec - 1 . I 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 b r i n g i n g t h e p r o b l e m to m y a t t e n t i o n a n d f o r h i s i n t e r e s t . I a m a l s o g r a t e f u l f o r a n u m b e r of c o n v e r s a t i o n s with P r o f e s s o r s R. A c h a r y a , H . H . A l y a n d H. A. M a v r o m a t i s . D. K e o s h e y a n a n d t h e o t h e r s t a f f of t h e A U B C o m p u t e r C e n t r e w e r e of g r e a t h e l p i n t h e c o u r s e of t h e n u m e r i c a l c a l c u l a t i o n s .
92
K. SCHILCHER
5
4
3
2
I
32 34 36 38 40
50
i
I
l
60 62
i
I
i
,E
Fig. 2. Di-pion e f f e c t i v e - m a s s spectrum for K+--~ 7[+ + ~ - + /l + + ~. A f t e r t h i s w o r k w a s d o n e it c a m e to t h e a u t h o r ' s a t t e n t i o n t h a t c a l c u l a t i o n s on r e l a t e d l i n e s a r e b e i n g c a r r i e d out by A. D o n n a c h i e , G, C. O a d e s a n d F. B e r e n d s . NOTE
ADDED
IN P R O O F
A m o r e r e c e n t e x p e r i m e n t a l v a l u e f o r the K + -~ ~+~-/x+v r a t e i s F = ( 1 . ] + 0 . 7 ) x 1 0 3 s e c -1 T h i s r e s u l t i s t a k e n f r o m V. B i s i , R. C e s t e r , A. M a r z a r i C h i e s a and M. V i g o n e , P h y s i c s L e t t e r s 25B (1967) 572. REFERENCES [1] E. P. Shabalin, ZhETF (USSR) 39 (1961) 345 [ JETP (Soy. Phys.) 10 (1960) 1252]. [2] N. Cabibbo and A. Maksymowicz, Phys. Rev. 137 (1965) B438, and r e f e r e n c e s therein. [3] C. Kacs er , P. Singer and T. N. Truong, Phys. Rev. 137 (1965) B1605. [4] N. V. Hieu, ZhETF 44 (1963) 162 [ J E T P (Sov. Phys.) 17 (1963) 113]. [5] R.W. Birge et al., Phys. Rev. 139 (1965) B1600. [6] S. Weinberg, Phys. Rev. L e t te r s 17 (1966) 336. [7] G. H. Trilling, Argonne National Laboratory Report No. ANL 7130, 1965. [8] J. D. Jackson, 1962 Brandeis Lectures. [9] A. H. Rosenfeld et al., UCRL-8030 (revised September 1967).