# Numerical Study of $ K^0$--$\bar{K}^0$ Mixing and $ B_K $

## TL;DR

This study computes the kaon B-parameter ($ B_K $) using staggered fermions in full QCD, comparing different methods and sources, and finds consistent results with no effect from non-degenerate quark pairs.

## Contribution

It introduces a comparison of two computational methods and source techniques for calculating $ B_K $ in lattice QCD, demonstrating their agreement and robustness.

## Key findings

- Renormalized $ B_K $ results agree across methods
- Improved wall source effectively isolates pseudo-Goldstone bosons
- No effect of non-degenerate quark pairs on $ B_K $

## Abstract

We have computed $ B_K $ with staggered fermions, using two different methods: both in the one spin trace form and two spin trace form. Renormalized results in both forms are in good agreement. The numerical simulations were performed on a $ 16^3 \times 40 $ lattice in full QCD with $ \beta = 5.7 $. We also tried an improved wall source method in order to select only the pseudo-Goldstone bosons and compare the numerical results obtained with those from the conventional wall source method. We have studied $ B_K $ with a series of non-degenerate quark anti-quark pairs and saw no effect on $ B_K $, although dramatic effects in the chiral limit were seen on the individual terms making up $ B_K $.