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 $B_K(NDR,2GeV)=0.62\pm 0.02(stat)\pm 0.02(syst)$. 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 $\mathrm{\Delta }I=3\mathrm{/}2$ electromagnetic penguin operators we find $B_7^{(3\mathrm{/}2)}=0.62\pm 0.03\pm 0.06$ and $B_8^{(3\mathrm{/}2)}=0.77\pm 0.04\pm 0.04$. We also use the ratio of unsmeared to smeared operators to make a partially non-perturbative 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 ${\alpha }_{\text{MSbar}}(q^{\ast }=1\mathrm{/}a)$.