Matrix elements of four-fermion operators with quenched Wilson fermions

We present results for the matrix elements of a variety of four-fermion operators calculated using quenched Wilson fermions. Our simulations are done on 170 lattices of size ${32}^3\times 64$ at $\beta =6.0\mathrm{}$. We find $B_K=0.74\mathrm{}\pm 0.04\pm 0.05$, $B_D=0.78\mathrm{}\pm 0.01$, $B_7^{3/2}=0.58\mathrm{}\pm {0.02}_{-0.03}^{+0.07\mathrm{}}$, $B_8^{3/2}=0.81\mathrm{}\pm {0.03}_{-0.02}^{+0.03\mathrm{}}$, with all results being in the NDR scheme at $\mu =2\mathrm{}$ GeV. We also calculate the $B$ parameter for the operator ${\mathcal{Q}}_s,$ which is needed in the study of the difference of $B$-meson lifetimes. Our best estimate is $B_S(\mathrm{NDR},\mu =1/a=2.33\mathrm{}\hspace{0.5em}\mathrm{GeV})$ $=0.80\mathrm{}\pm 0.01$. This is given at the lattice scale since the required two-loop anomalous dimension matrix is not known. In all these estimates, the first error is statistical, while the second is due to the use of truncated perturbation theory to match continuum and lattice operators. Errors due to quenching and lattice discretization are not included. We also present new results for the perturbative matching coefficients, extending the calculation to all Lorentz scalar four-fermion operators, and using NDR as the continuum scheme.