We present results for the kaon mixing parameter $B_K$ calculated using HYP (hypercubic fat link)-smeared improved staggered fermions on the asqtad lattices generated by the MILC collaboration. We use three lattice spacings ($a\approx 0.12$, 0.09 and 0.06 fm), ten different valence-quark masses ($m\approx m_s/10-m_s$), and several light sea-quark masses in order to control the continuum and chiral extrapolations. We derive the next-to-leading order staggered chiral perturbation theory (SChPT) results necessary to fit our data, and use these results to do extrapolations based both on SU(2) and SU(3) SChPT. The SU(2) fitting is particularly straightforward because parameters related to taste breaking and matching errors appear only at next-to-next-to-leading order. We match to the continuum renormalization scheme [naïve dimensional regularization (NDR)] using one-loop perturbation theory. Our final result is from the SU(2) analysis, with the SU(3) result providing a (less accurate) cross check. We find $B_K(\mathrm{NDR},\mu =2\mathrm{GeV})=0.529\pm 0.009\pm 0.032$ and ${\widehat{B}}_K=B_K(\mathrm{RGI})=0.724\pm 0.012\pm 0.043$, where the first error is statistical and the second systematic. The error is dominated by the truncation error in the matching factor. Our results are consistent with those obtained using valence domain-wall fermions on lattices generated with asqtad or domain-wall sea quarks.

• Received 10 September 2010

DOI:https://doi.org/10.1103/PhysRevD.82.114509