introduction-weak-interaction-volume-one

(~ w/ St) (d3Pe)/(2n )5 J J d 3p v d3
v (p~ – pe – p v – pv )
P
1 2 1 2
X 1 ue(-F) ()Ol u ~(+) uv(+)(~ ) Fl uv( – )(_, ) I 2
2,1
I>
2 2
(4 .2 .4 )
Since the two neutrinos in the decay (4.1 .4) are not observed, we sum over thei r
polarization directions, obtaining
> ry v
1 2
where
(l/(4Ev1Ev2)
~ . 1 u e (+) (Pe ) Ol u(+)(Pp) v(-')(~1) Fl v2 –)(_~ 2 ) 2
i .j
u e(+) e ) of u r(+) (P) a,,' ) (27) 0. ue ({) ( ) x
(P
X Tr (j r v Pi j Y pv2
rJ+ )
1
0 j = Y4 0~ y4 = y4 0 j y 4 . (4 .2 .6 )
Using the fact that 0 i and 0it differ only by a sign, and substituting wit h
(4 .2 .5) in (4 .2 .4) we have
(Ew/fit) = (d 0)/(2) 5
X 0-) (
(C
i ,J e(+)(Pe) 0 i u` (}) ( u ) u (+) ( P
r) 0 X
dp v 2 dp v1 ~(Pv2) (2, 2 ) e(2 ) e(Pv )
2
1 2_
X S( p1. – pe – pv – pv ) T r (J Ypv oi ( C i + Ci Y5 ) j X
1 2 1
X pv2(C . – C 'j Y 5 ) 0j ) (4 .2 .7 )
0
We now take the special case of (4 .2 .7) for unpolarized decaying muons, and ,
using the techniques outlined in :Callen : Elementary Particle Physics, Addison–
'Wesley 1964, pp . 380-386, we have
( T,9/ .t) _ ( d3p e)/(3 8 4 '-v 4 E e Et ) (3( ICS I2+ I O SI2+ I CPI 2+ I CPI 2) X
X Q2(pe p1) + 2( (c I2t IC',)2 + ! CA I 2 +ICAI 2 )(Q2 (pe Pr) 1'
t 2( PeQ)(PpQ) )+ 2( I Cp 12+ ) C1 I 2 )(4(PeQ)(p,„Q) – Q 2( P epr, )) )
(4 .2 .8 )
where Q is the difference between the muon and the electron momenta . We define
the intensity of outgoing electrons, I(x), b y
( w/ 3-t) = 1(x) dx, (4 .2 .9 )
where
(4 .2 .5)
X