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# Chiral Invariance and V-A

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:

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.

Next: Conserved Vector Current Up: Introduction Previous: Massless Fermion

Bernward Krause
Mon Jan 15 14:57:06 MET 1996