Determination of $B_K$ using improved staggered fermions (III) Finite volume effects

TL;DR

This paper investigates finite-volume effects on the calculation of $B_K$ using improved staggered fermions, comparing theoretical predictions with direct lattice results to quantify volume-related errors.

Contribution

It provides a detailed comparison of SU(3) and SU(2) chiral perturbation theory predictions with direct lattice calculations for finite-volume effects in $B_K$.

Findings

Finite volume effects are approximately 2% for SU(3) and 0.9% for SU(2).

Finite volume effects are consistent with chiral perturbation theory predictions.

The study examines how statistical errors and autocorrelations depend on measurement strategies.

Abstract

We study the finite-volume effects in our calculation of $B_K$ using HYP-smeared improved staggered valence fermions. We calculate the predictions of both SU(3) and SU(2) staggered chiral perturbation theory at one-loop order. We compare these to the results of a direct calculation, using MILC coarse lattices with two different volumes: $20^3$ and $28^3$. From the direct calculation, we find that the finite volume effect is $\approx 2%$ for the SU(3) analysis and $\approx 0.9%$ for the SU(2) analysis. We also show how the statistical error depends on the number of measurements made per configuration, and make a first study of autocorrelations.