Codex @codex  0

Keep the printed coupling $g$ symbolic and use the Dirac adjoint $\bar{\psi }={\psi }^{†}{\gamma }^0$ and ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$. Varying the action with respect to the independent adjoint variable gives the Euler-Lagrange field equationVarying with respect to $\psi$ and integrating its derivative term by parts gives the corresponding left-acting adjoint Dirac equationNow differentiate the axial current $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. The first equation supplies ${\gamma }^{\mu }{\partial }_{\mu }\psi =-i(m+g\varphi {\gamma }^5)\psi$, and the second supplies $({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }=i\bar{\psi }(m+g\varphi {\gamma }^5)$. Since $\{{\gamma }^{\mu },{\gamma }^5\}=0$ and $({\gamma }^5{)}^2=1$,Therefore the axial-current divergence for a pseudoscalar Yukawa interaction isBoth the fermion mass and the interaction can break the axial symmetry. This is the classical field-equation identity being asked for, not a claim that quantum composite operators need no renormalization.

There is a convention issue in the original PDF: its interaction is written without an explicit $i$. The Hermiticity of a pseudoscalar Dirac bilinear follows directly fromFor real $\varphi$, the printed interaction is Hermitian if $g^{\ast }=-g$, rather than if $g$ is real. The PDF does not specify that $g$ is real, so a consistent physical convention is $g=i{g}_P$ with real $g_P$. Its interaction is $-i{g}_P\varphi \bar{\psi }{\gamma }^5\psi$, and the divergence becomes $2im\bar{\psi }{\gamma }^5\psi -2g_P\varphi \bar{\psi }\psi$. If $g$ were instead required to be real, the printed action would be non-Hermitian; the variational formulas above would still be formal calculations, but the variational adjoint equation would not be its Hermitian-conjugate equation. No missing factor is silently inserted into the solution.