= Solution
Keep the printed coupling $g$ symbolic and use the <Dirac adjoint> $\bar\psi=\psi^\dagger\gamma^0$ 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 equation>
$$
\boxed{(i\gamma^\mu\partial_\mu-m-g\phi\gamma^5)\psi=0.}
$$
Varying with respect to $\psi$ and integrating its derivative term by parts gives the corresponding left-acting <adjoint Dirac equation>
$$
i(\partial_\mu\bar\psi)\gamma^\mu+m\bar\psi+g\phi\bar\psi\gamma^5=0.
$$
Now differentiate the <axial current> $j_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi$. The first equation supplies $\gamma^\mu\partial_\mu\psi=-i(m+g\phi\gamma^5)\psi$, and the second supplies $(\partial_\mu\bar\psi)\gamma^\mu=i\bar\psi(m+g\phi\gamma^5)$. Since $\{\gamma^\mu,\gamma^5\}=0$ and $(\gamma^5)^2=1$,
$$
\begin{aligned}
\partial_\mu j_5^\mu
&=(\partial_\mu\bar\psi)\gamma^\mu\gamma^5\psi
+\bar\psi\gamma^\mu\gamma^5\partial_\mu\psi\\
&=i\bar\psi(m+g\phi\gamma^5)\gamma^5\psi
+i\bar\psi\gamma^5(m+g\phi\gamma^5)\psi.
\end{aligned}
$$
Therefore the <axial-current divergence for a pseudoscalar Yukawa interaction> is
$$
\boxed{\partial_\mu j_5^\mu=2im\bar\psi\gamma^5\psi+2ig\phi\bar\psi\psi.}
$$
Both 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 from
$$
(\bar\psi\gamma^5\psi)^\dagger=\psi^\dagger\gamma^5\gamma^0\psi
=-\psi^\dagger\gamma^0\gamma^5\psi=-\bar\psi\gamma^5\psi.
$$
For real $\phi$, the printed interaction is Hermitian if $g^*=-g$, 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 $-i g_P\phi\bar\psi\gamma^5\psi$, and the divergence becomes $2im\bar\psi\gamma^5\psi-2g_P\phi\bar\psi\psi$. 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.
Back to article page
