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Use the momentum-space Feynman rules with all momenta conserved at each vertex. The free scalar propagator, free Dirac propagator and pseudoscalar Yukawa interaction vertex are respectivelyThe Dirac propagator follows by multiplying the inverse kinetic matrix $/p-m$ by $/p+m$, using the Clifford algebra to obtain $(/p-m)(/p+m)=(p^2-m^2)I_4$. The vertex follows by expanding $e^{i{S}_{\mathrm{int}}}$ once: there is no factorial because it has one scalar and two distinct spinor field factors. An internal scalar has no spinor indices; an internal fermion line carries the displayed matrix with its momentum directed along fermion flow.

These rules use the coefficient exactly as printed. With the standard Hermitian ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$, the Hermiticity of a pseudoscalar Dirac bilinear gives $(\bar{\psi }{\gamma }^5\psi {)}^{†}=-\bar{\psi }{\gamma }^5\psi$. A Hermitian physical interaction therefore requires ${\lambda }^{\ast }=-\lambda$; writing $\lambda =ig$ with real $g$ changes the vertex to $g{\gamma }^5$. This convention issue does not alter the formal rules above, and no unstated real-coupling assumption is needed.

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.