Staggered fermion matrix elements using smeared operators

We investigate the use of two kinds of staggered fermion operators, smeared and unsmeared. The smeared operators extend over a $4^4$ hypercube, and tend to have smaller perturbative corrections than the corresponding unsmeared operators. We use these operators to calculate kaon weak matrix elements on quenched ensembles at $\beta =6.0,$ 6.2, and 6.4. Extrapolating to the continuum limit, we find $B_K(\mathrm{N}\mathrm{D}\mathrm{R},2\mathrm{}\mathrm{}\mathrm{GeV})=0.62\mathrm{}\pm 0.02\left(\mathrm{stat}\right)\pm 0.02\left(\mathrm{syst}\right).\mathrm{}$ The systematic error is dominated by the uncertainty in the matching between lattice and continuum operators due to the truncation of perturbation theory at one loop. We do not include any estimate of the errors due to quenching or to the use of degenerate $s$ and $d$ quarks. For the $\Delta I=3/2\mathrm{}$ electromagnetic penguin operators we find $B_7^{\mathrm{}(3/2)\mathrm{}}=0.62\mathrm{}\pm 0.03\pm 0.06$ and $B_8^{\mathrm{}(3/2)\mathrm{}}=0.77\mathrm{}\pm 0.04\pm \mathrm{0.04.}$ We also use the ratio of unsmeared to smeared operators to make a partially nonperturbative estimate of the renormalization of the quark mass for staggered fermions. We find that tadpole improved perturbation theory works well if the coupling is chosen to be ${\alpha }_{\overline{\mathrm{MS}}}{(q}^{*}=1/a).\mathrm{}$