$O(a\alpha_s)$ matching coefficients for the $\Delta B$=2 operators in the lattice static theory

Abstract

We present the perturbative matching coefficient to $O(a\alpha_s)$ which relates the $\Delta B$=2 operator in the continuum to that of the lattice static theory, which is important in the accurate extraction of the continuum value of the $B_B$ from lattice simulations. The coefficients are obtained by the one-loop calculations in both of the continuum and lattice theory. We find that two new dimension seven operators appear at the $O(a\alpha_s)$ with the O(1) coefficients. We also discuss the possible cancellation of $O(a\alpha_s)$ correction in the ratio $B_B=<\bar{B}|\Op_L|B>/ ((8/3)(f_{B}M_{B})^2)$ qualitatively.