  We investigate the scaling behavior of the $B$ meson decay constant $f_B$ and
$f_{B_{s}}$ at $\beta$$=$$5.7, 5.9, 6.1$, employing the NRQCD heavy quark
action and the clover light quark action. Mixing effect from dimension-4
operator in the heavy-light axial-vector current is studied, and we find that
the $a$ dependence of $f_B$ is significantly reduced. Our preliminary result
for the decay constants in the quenched approximation is
$f_{B}$$=$$162(^{+35}_{-18})$ MeV, $f_{B_{s}}$$=$$190(^{+40}_{-19})$ MeV, and
$f_{B_{s}}/f_{B}$$=$$1.18(^{+6}_{-6})$.
