Perturbative renormalization of the $\Delta B=2$ four-quark operators in   lattice NRQCD
  We present a perturbative calculation of the renormalization constants for the $\Delta B=2$ four-quark operators, which is needed to evaluate the bag parameters $B_B$ and $B_S$ in lattice QCD. The one-loop coefficients are calculated for the $O(1/M)$ NRQCD heavy quark action and the $O(a)$-improved Wilson action for light quark. Using these coefficients, which remove $O(\alpha_s/aM)$ error, we reanalyze our previous calculation of the bag parameters. 
S. Hashimoto, K-I. Ishikawa, T. Onogi, M. Sakamoto, N. Tsutsui, and N.   Yamada