The operator is referred to as the chirality operator which transforms as:
With the momentum along the z-axis, the helicity operator becomes:
It follows that the solutions to the Dirac equation for a massless fermion are eigenstates of . This is not the case for a massive fermion. However, the wavefunction can be written as a linear combination of the right-handed and left-handed components:
Consider a 4-fermion interaction which is assumed to be invariant under the chiral transformation defined above. The interaction amplitude can be written as:
Chiral invariance leads to:
In order for the amplitude to be invariant under a Lorentz transformation, the operator O can be shown to be a linear combination of the following:
It follows that:
Of these five possible operators, only the vector and axial-vector operators V and A anticommute with . Therefore,
From equation (), one has a=-b since and anticommutes with . The operator O becomes:
where a is a constant. The amplitude M takes the form:
where the currents and are given below:
This is the modification of the current-current Fermi interaction --- at a point --- to accommodate the axial-vector contribution originally absent in Fermi's theory.
It should be pointed out that theoretically, one could arrive at a form of the weak charged current in which only S, T and P contribute. However, the experimental evidences are in favor of the (V,A) form.
The following conclusions follow from the V-A form of the weakly charged current [Che-79]:
Naturally, one would hope that all weak processes are described the
V-A interactions with a universal coupling constant.