Lattice Determination of Heavy-Light Decay Constants

We report on the MILC collaboration's calculation of $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$ and their ratios. We work on both quenched lattices and $N_F=2$ dynamical staggered lattices, and use Wilson light valence quarks and Wilson and static heavy quarks. All sources of systematic error, both within the quenched approximation and from quenching itself, are estimated. We find, for example, $f_B=159 \pm 11 {}^{+22}_{-9} {}^{+21}_{-0} \MeV$, $f_{B_s}/f_B = 1.11 \pm 0.02 {}^{+0.04}_{-0.03} \pm 0.03$, $f_{D_s} = 213 \pm 9 {}^{+23}_{-9} {}^{+17}_{-0} \MeV$ and $f_{B}/f_{D_s} = 0.75 \pm 0.03 {}^{+0.04}_{-0.02} {}^{+0.07}_{-0.00} $, where the errors are statistical, systematic (within the quenched approximation), and systematic (of quenching), respectively.