Continuum extrapolation of $B_K$ with staggered fermions

We report on recent progress in the calculation of $B_K$ using HYP-smeared staggered fermions on the MILC asqtad lattices. Our main focus is on the continuum extrapolation, which is done using (up to) four different lattice spacings—$a\approx$ 0.12, 0.09, 0.06 and 0.045 fm. Since Lattice 2010, we have reduced the statistical errors on the $a\approx 0.09;$fmlatticesbyafactorof$\sim 3$,androughlydoubledthesizeofthe$a\approx0.045;$fm ensemble. We find that these improvements have a very significant impact on the continuum extrapolation, with the $a\approx 0.12$fm data lying outside the range of applicability of simple functional forms. Hence we use only the three smallest lattice spacings to perform the extrapolation, finding $\hat{B}_K = B_K($\text{RGI}$) = 0.725 \pm 0.004($\text{stat}$) \pm 0.038($\text{sys}$) $. This value is consistent with our published value from 2010 (based the three coarsest lattice spacings), but has smaller errors.