The use of improved staggered actions (HYP, Asqtad) has been proved to reduce the scaling corrections that affected previous calculations of $B_K$ with unimproved (standard) staggered fermions in the quenched approximation. This improved behavior allows us to perform a reliable calculation of $B_K$ including quark vacuum polarization effects, using the MILC configurations with $n_f=2+1$ flavors of sea fermions. We perform such a calculation for a single lattice spacing, $a=0.125\mathrm{fm}$, and with kaons made up of degenerate quarks with $m_s/2$. The valence strange quark mass $m_s$ is fixed to its physical value and we use two different values of the light sea quark masses. After a chiral extrapolation of the results to the physical value of the sea quark masses, we find ${\widehat{B}}_K=0.83\pm 0.18$, where the error is dominated by the uncertainty in the lattice to continuum matching at $\mathcal{O}({\alpha }_s^2)$. The matching will need to be improved to get the precision needed to make full use of the experimental data on ${\epsilon }_K$ to constrain the unitarity triangle.

• Received 28 March 2006

DOI:https://doi.org/10.1103/PhysRevD.73.114502