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Systematic Uncertainties in $B_K$ with Improved Staggered Fermions

We study three sources of error in our calculation of $B_K$ using HYP-smeared staggered fermions on the MILC asqtad lattices. These are (1) dependence on the light sea quark mass; (2) finite volume effects; and (3) the impact of an order of magnitude increase in the number of measurements. Our main results are (1) the dependence on the light sea-quark mass is weaker than expected by naive dimensional analysis, (2) including finite volume effects in SU(2) staggered chiral perturbation theory fits leads to a very small change in $B_K$, of size $\approx 0.1%$, and (3) increasing the statistics on one of the coarse MILC lattices resolves a potential discrepancy with other coarse results.