introduction-weak-interaction-volume-one

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Jr(x) — g 't1p(x) Ir 7rn(x) . (5 .2 .1 )
Using isospin, (5 .2 .1) become s
— (€/ 2) ( ( x ) Yr. T *~J(x) ) (5 .2 .2 )
or
( g/2
-
,1r2 ) [l1'(x),
( rT + l(x)] . (5 .2 .3 )
The electromagnetic current (4 .4 .3) of the nucleon system i s
Jel ( x ) = J e ( ' p ( x ) Yr p ( x ) ) , (5 .2 .4 )
or, in isospin formalism :
(je/4) DTI(x), Y r (1 + T3 ) V ( x )] . (5 .2 .5 )
From (5 .2 .5) we see that (5 .2 .4) may be decomposed into an isospin scalar an d
an isospin vector (an isoscalar and isovector) :
J S
Jel(x)
(x) + J (x) , (5 .2 .6 )
where
T
r
Jr (x) _ (J e/4) [(x), Yr W(x)]
(5 .2 .7 )
and
JJr ( x ) = (j e /4) [ r(x), Y r T 3 1p (x)7 . (5 .2 .8)
Obviously the current (5 .2 .1) may be considered as another component of (5 .2 .8) ,
so tha t
J (x) = J 2 g/e (+) (x) . (5 .2 .9 )
Unfortunately
r
the electromagnetic nucleon current (5 .2 .4) does not obey a
continuity equation of the form (1 .4 .12) . Howeve r
a Jel
' D
0 (5 .2 .10 )
t
J being define d
J r
(x)
= je Caa(X) ~(x) —
t(x)2'ah))
(5 .2 .11 )
where cp is the compler x pion field in (5 .1 .22) and (5 .1 .23) . By decomposing
the latter field into real and imaginary components, we may write
( O J (x)/2xr ) + ( 2/d xr )(JJ (x) + , (x) ) = 0 . (5 .2 .12)
Thus reach term in (5 .2 .12) is conserved . We assume that the interaction which w e
are considering is invariant under rotations in isospin space, and hence othe r
components of the second term in (5 .2 .12) will also be conserved :
J
( a fix) (JRV (+) ( x ) + ,2j (4) (x) ) = 0 .
(5 .2 .13 )