Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 1 / iv

In four spacetime dimensions a Weyl spinor has mass dimension $3/2$, so each chiral current $J_L^{\mu }={\bar{\psi }}_L{\bar{\sigma }}^{\mu }{\psi }_L$ or $J_R^{\mu }={\bar{\psi }}_R{\sigma }^{\mu }{\psi }_R$ has dimension three. Since the interaction is $(g{J}_L+g^{\prime }{J}_R{)}^2$, dimensional homogeneity of the Lagrangian givesEquivalently, the expanded four-fermion interaction has coefficients $g^2,g{g}^{\prime },g^{\prime 2}$ of mass dimension minus two.

Parity exchanges $J_L$ and $J_R$. The square is invariant precisely when the two same-chirality coefficients agree, $g^2=g^{\prime 2}$, while the mixed coefficient is already symmetric. Therefore