Non-degenerate quark mass effect on $B_K$ with a mixed action

Abstract

We investigate the effect of non-degenerate quarks on $B_K$. This effect is noticeably large for $B_K$ (significantly larger than statistical uncertainty). Hence, it is important to include this effect in order to determine $B_K$ with higher precision. We also observe that the quality of fitting for $B_K$ gets better when we include non-degenerate combinations to fit to the prediction by Van de Water and Sharpe. However, this effect on $B_7^{(3/2)}$ and $B_8^{(3/2)}$ turns out to be relatively small.