Subatomic Physics

350 The Weak Interaction
The notation V ≡ (V0, V ) is a reminder that the “sandwich” ψ ∗ Vψ transforms
like a four-vector. With Eqs. (11.37) and (11.38), the explicit form of V for a
nonrelativistic electron is
V≡(V0, V ), V0 =1, V = p
. (11.40)
mc
There are a number of differences between the electromagnetic and weak currents.
Whereas the electromagnetic current is always a neutral one that conserves charge,
the weak current has a charge-changing part, J (−)
w , in addition to the neutral one,
J (0)
w . For electrons, the corresponding weak current densities are written in analogy
to the electromagnetic ones as
J e(−)
w
= cψ ∗ eVψνe,
J e(0)
w = cψ∗ e Vψe, J ν(0)
w = cψ∗ νe Vψνe. (11.41)
The weak current is more complicated than the electromagnetic one in other ways.
We have seen in chapter 9 and earlier in this chapter that the weak interaction does
not respect parity. The operator V =(V0, V ) behaves under the parity operation
as
V0
P
−→ V0 V P
−→ −V . (11.42)
The fact that the vector part changes sign follows from Eq. (9.1). V0, on the other
hand, is a probability density, and it remains unchanged under the parity operation.
According to the golden rule, the transition rate for a reaction from a polarized or
unpolarized source is proportional to the square of a matrix element, or
��
�
wµ ∝ �
� d 3 xψ ∗ e Vψνe · ψ ∗ νµ Vψµ
�
�2
�
� .
The vector product V·V = V0V0 −V ·V remains unchanged under P ;ifw P µ denotes
the transition rate after the parity operation, it is equal to wµ:
w P µ
= wµ.
This result disagrees with the electron asymmetry observed in beta and muon decays.
How can the expression for the weak current be generalized in such a way
that the analogy to the electromagnetic current is not completely destroyed but that
parity nonconservation is included? A hint to the answer comes from comparing
linear and angular momentum. Under ordinary rotations, both behave in the same
way. We have not demonstrated this fact explicitly, but the proof is straightforward
if the arguments given in Section 8.2 are used. Under the parity operation, the polar
vector p and the axial vector J reveal their difference: p changes sign, whereas J
does not. These properties remain true for general operators V and A: V and A