$B_K$ using HYP-smeared staggered fermions in $N_f=2+1$ unquenched QCD

TL;DR

This paper calculates the kaon mixing parameter $B_K$ using HYP-smeared staggered fermions on MILC lattices, employing chiral perturbation theory for extrapolation, and provides a precise result consistent with other fermion formulations.

Contribution

The study introduces a detailed calculation of $B_K$ with HYP-smeared staggered fermions across multiple lattice spacings and quark masses, including new chiral perturbation theory fits and continuum matching techniques.

Findings

$B_K(2 ext{GeV})=0.529 ext{ with errors}$

$ ext{RGI } ext{value}=0.724 ext{ with errors}$

Results are consistent with other fermion discretizations.

Abstract

We present results for kaon mixing parameter $B_K$ calculated using HYP-smeared improved staggered fermions on the MILC asqtad lattices. 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 (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 $B_K(\text{NDR}, \mu = 2 \text{GeV}) = 0.529 \pm 0.009 \pm 0.032$ and $\hat{B}_K =B_K(\text{RGI})= 0.724 \pm 0.012 \pm 0.043$, 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.