One Spin Trace Formalism for $ B_K $

TL;DR

This paper introduces a new one spin trace formalism for calculating $ B_K $ with staggered fermions, detailing its implementation, renormalization, and how it compares to the traditional two spin trace method.

Contribution

It presents the first detailed explanation and renormalization procedure for the one spin trace formalism in $ B_K $ calculations with staggered fermions, expanding the methodological toolkit.

Findings

The one spin trace operators require additional chiral partner operators.

The renormalization of the one spin trace formalism is described and compared.

The formalism maintains continuum chiral behavior with added operators.

Abstract

It has been known for some time that there are two methods to calculate $ B_K $ with staggered fermions: one is the two spin trace formalism and the other is the one spin trace formalism. Until now, the two spin trace formalism has been exclusively used for weak matrix element calculations with staggered fermions. Here, the one spin trace formalism to calculate $ B_K $ with staggered fermions is explained. It is shown that the one spin trace operators require additional chiral partner operators in order to keep the continuum chiral behavior. The renormalization of the one spin trace operators is described and compared with the two spin trace formalism.