  The renormalization constant $Z_J$ of the flavor-singlet axial-vector current
with a non-anticommuting $\gamma_5$ in dimensional regularization is determined
to order $\alpha_s^3$ in QCD with massless quarks. In addition, the equality
between the $\overline{\mathrm{MS}}$ renormalization constant $Z_{F\tilde{F}}$
associated with the operator $\big[F \tilde{F} \big]_{R}$ and that of
$\alpha_s$ is verified explicitly to hold true at 4-loop order.
