= Axial-current divergence for a pseudoscalar Yukawa interaction
{title2=$\partial_\mu j_5^\mu=2im\bar\psi\gamma^5\psi+2ig\phi\bar\psi\psi$}
For $\mathcal L_{\rm int}=-g\phi\bar\psi\gamma^5\psi$, the variational <Dirac equation> and <adjoint Dirac equation> are $(i\not\partial-m-g\phi\gamma^5)\psi=0$ and $i(\partial_\mu\bar\psi)\gamma^\mu+m\bar\psi+g\phi\bar\psi\gamma^5=0$. Substitute these into the <derivative> of $j_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi$. Anticommuting the chirality matrix past $\gamma^\mu$ gives two identical mass terms and two identical interaction terms, proving the displayed identity. These are classical field equations; the formula does not assert an unrenormalized quantum composite-operator identity. With Hermitian coupling $g=i g_P$, the second term is $-2g_P\phi\bar\psi\psi$.
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