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 $(gJ_L+g'J_R)^2$, dimensional homogeneity of the Lagrangian gives
$$
\boxed{[g]=[g']=-1}.
$$
Equivalently, the expanded <four-fermion interaction> has coefficients $g^2,gg',g'^2$ of mass dimension minus two.

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

Solution

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

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A four-fermion interaction is a local operator containing four fermion fields. In four spacetime dimensions it has mass dimension six, so its coefficient has mass dimension minus two; writing it as the square of a current multiplied by a coefficient inside the square gives that coefficient mass dimension minus one.


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 Incoming links: Four-fermion interaction