The continuum limit of $f_B$ from the lattice in the static approximation

We present an analysis of the continuum extrapolation of $f_B$ in the static approximation from lattice data. The method described here aims to uncover the systematic effects which enter in this extrapolation and has not been described before. Our conclusions are that we see statistical evidence for scaling of $f_B^{stat}$ for inverse lattice spacings $\text{gtap}2$ GeV but not for $\text{ltap}2$ GeV. We observe a lack of {\em asymptotic} scaling for a variety of quantities, including $f_B^{stat}$, at all energy scales considered. This can be associated with finite lattice spacing systematics. Once these effects are taken into account, we obtain a value of 230(35) MeV for $f_B^{stat}$ in the continuum where the error represents uncertainties due to both the statistics and the continuum extrapolation. In this method there is no error due to uncertainties in the renormalization constant connecting the lattice and continuum effective theories.