Experiments showed strange asymmetries in weak decays, beta decays, but no one related this to parity violation. Tsung-Dao Lee and Chen-Ning Yang carefully reviewed all experiments. They concluded: There was no experimental proof that weak interactions conserve parity. They proposed: Do an experiment in beta decay with polarized nuclei.

If parity is conserved → electron emission should be symmetric.
If parity is violated → electrons will prefer one direction.

This was radical. Most physicists thought it would obviously conserve parity. Then Chien-Shiung Wu performed the test and verified their bold conclusion.

Theoretical physicists have to come up with an explanation, The solution that emerged was the V–A theory. The weak current has the structure:

The part of 1-gamma 5 is the key, it projects out only left-handed components. Then what Is γ⁵?

To understand how this part shows up, we need to derive from Lagrange equation of the field.