introduction-weak-interaction-volume-one

and then applying the matrix element (3 .5 .1) separately to each
term of the decomposition . We denote the nonrelativistic Schrddinger wav e
function of the initial nucleon by 14 1 a
and that of the final one b
(X) = 2I (2) e J E x
y and we Fourier decompose each of these according to the standard formulae :
(3 .5 .2a )
T(2) = 5 e -j P x y ( l ) .
(3 .5 .2b)
Thus the matrix element between the nuclear states 4a> and lb>, each o f
which contain several nucleons labelled by the index r, becomes :
r sp Irb (r) (E.) 1C (r) cap ) < 9p' a., gv 3 En>
Dropping the phase factor (exp(—j x0 ( En — Ep —
(3 .5 .3 )
Ee — Ev) ) ), we may evaluate
this matrix element explicitly, using nonrelativistic approximations (38) :
< b IHIla> _ 2i J Yb`r)(.) of Va(r)(x) ex p( —j x(3 e + S~) X
X (1/d2) ue(+)(ae) of (Ci + Ci y5) uv (-) (-av )
(3 .5 .4 )
Because of the comparative smallness of the nuclear radius, we find (38 )
that we may replace the exponential factor in this formula by unity, an d
hence we may writ e
(b i Hl la> = 2E . %i ue (+) (ye ) F. uv (-) (- 3v ) , (3 .5 .5a )
F . /= ( 1/f) 0i ( ci + Ci Y5 ) , (3 .5 .5b )
Mi
=
r vo (r)( a) o f ' a(r)(x)
. (3 .5 .5c )
Using standard nonrelativistic approximations for the 0 i
, we find tha t
Mi = < 1> , i = s, V , (3 .5 .6a )
Mi
± , i = T, A , (3 .5 .6b )
N. = 0 , i = p . (3 .5 .6c)
In order to compare our predictions with experiment, we us e
the usual formula for the intensity of particles with a given momentum :
1 (4) _ (1/27N) 5 q2 (E max - E)2 X d1Z e dRv , (3 .5 .7a )