Perturbative $O(\alpha_s a)$ matching in static heavy and domain-wall light quark system

We discuss the perturbative $O(\alpha_s a)$ matching in the static heavy and domain-wall light quark system. The gluon action is the Iwasaki action and the link smearing is performed in the static heavy action. The chiral symmetry of the light quark realized by using the domain-wall fermion formulation does not prohibit the mixing of the operators at $O(a)$. The application of $O(a)$ improvement to the actual data shows that the B meson decay constant $f_B$, the matrix elements ${\cal M}_B$ and the B parameter $B_B$ have non-negligible effects, while the effect on the SU(3) breaking ratio $\xi$ is small.