Numerical Study of $ K^0$--$\bar{K}^0$ Mixing and $ B_K $

We have computed $B_K$ with staggered fermions, using two different methods: both in the one spin trace form and two spin trace form. Renormalized results in both forms are in good agreement. The numerical simulations were performed on a $16^3 \times 40$ lattice in full QCD with $\beta = 5.7$. We also tried an improved wall source method in order to select only the pseudo-Goldstone bosons and compare the numerical results obtained with those from the conventional wall source method. We have studied $B_K$ with a series of non-degenerate quark anti-quark pairs and saw no effect on $B_K$, although dramatic effects in the chiral limit were seen on the individual terms making up $B_K$.