The operator is referred to as the chirality operator which transforms as:

With the momentum along the z-axis, the helicity operator becomes:

Since

we have:

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:

where

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:

where

It follows that:

and

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]:

- Only left-handed neutrinos and right-handed antineutrinos are coupled to charged leptons by the weak current.
- Parity violation: if
**P**is the parity operator, then leads to a neutrino with positive helicity which is not allowed. - Charge conjugation violation: if
**C**is the charge conjugation operator, then (where describes the neutrino) leads to an antineutrino still with negative helicity: this is not allowed. - Time reversal is conserved. Since
**T**reverses both spin and momentum, the original state is unchanged. -
**CP**is conserved because such an operation transforms a left-handed neutrino into a right-handed antineutrino. However, from the studies of the weak decays of the neutral kaon, small**CP**violation has been established.

Naturally, one would hope that all weak processes are described the
V**-**A interactions with a universal coupling constant.

Mon Jan 15 14:57:06 MET 1996