Scaling property of domain-wall QCD in perturbation theory

We estimate the lattice artifacts in loop corrections perturbatively for domain-wall QCD with an infinite number of extra flavors. We find that there appear no $\mathcal{O}\mathrm{}\left(a\right)\mathrm{}$ errors in renormalization factors of the quark wave function, mass, and quark bilinear operators at the one- and two-loop levels with off-shell quark momentum. Our proof is based on the even- or oddness of the quantum correction in terms of the quark external momentum and mass, and it can be extended to any loop level.