Past exam of the particle physics course of the University of Oxford / 2021 / 4 / a / Solution

The chiral projectors $P_{L,R}=(1\mp {\gamma }^5)/2$ obey ${\gamma }^{\mu }{P}_L=P_R{\gamma }^{\mu }$, while the Dirac adjoints are ${\bar{\psi }}_L=\bar{\psi }{P}_R$ and ${\bar{\psi }}_R=\bar{\psi }{P}_L$. Consequently the cross terms vanish and the vector current isThe vector interaction preserves fermion chirality at a vertex and couples equally to left and right components. This statement about the vertex does not assert that a fermion mass term preserves chirality. Under parity, left and right components are exchanged; their equal coefficients allow a parity-invariant contraction with a true vector field.

An axial vector must couple to the axial current to obtain the analogous parity-invariant interaction. Since ${\gamma }^5{\psi }_L=-{\psi }_L$ and ${\gamma }^5{\psi }_R={\psi }_R$,The relative sign changes; the overall sign is conventional. A pure axial interaction can itself conserve parity when $A_{\mu }$ transforms as an axial vector. Simply contracting the original vector current with an axial-vector field would instead produce a parity-odd interaction.