Continuum extrapolation of $B_K$ with staggered fermions

TL;DR

This paper advances the calculation of the kaon bag parameter $B_K$ by improving statistical precision and continuum extrapolation techniques using staggered fermions on multiple lattice spacings, resulting in a more accurate and reliable estimate.

Contribution

It introduces refined continuum extrapolation methods with additional lattice spacings and reduced errors, enhancing the precision of $B_K$ calculations with staggered fermions.

Findings

Continuum extrapolation using three smallest lattice spacings yields $B_K=0.725$ with reduced errors.

Statistical errors on $a extasciitilde 0.09$ fm lattices reduced by a factor of 3.

Improved data quality significantly impacts the continuum limit analysis.

Abstract

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 $a\approx 0.09\;$fm lattices by a factor of $\sim 3$, and roughly doubled the size of the $a\approx0.045\;$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.