Long-distance contribution to $\epsilon_K$ from lattice QCD

TL;DR

This paper proposes a lattice QCD method to calculate the long-distance contribution to the CP violation parameter 5_K, which is about 5% of the total and difficult to determine, demonstrating feasibility with an exploratory calculation.

Contribution

It introduces a novel lattice QCD approach to estimate the long-distance part of 5_K, providing a pathway for first-principles calculations with controlled errors.

Findings

Demonstrated the feasibility of lattice QCD to estimate long-distance 5_K contributions.

Performed an exploratory calculation with unphysical quark masses.

Projected future work to achieve 10% accuracy in the calculation.

Abstract

A lattice QCD approach to the calculation of the long-distance contributions to $\epsilon_K$ is presented. This parameter describes indirect CP violation in $K\to\pi\pi$ decay. While the short-distance contribution to $\epsilon_K$ can be accurately calculated in terms of standard model parameters and a single hadronic matrix element, $B_K$, there is a long-distance part which is estimated to be approximately $5\%$ of the total and is more difficult to determine. A method for determining this small but phenomenologically important contribution to $\epsilon_K$ using lattice QCD is proposed and a complete exploratory calculation of the contribution is presented. This exploratory calculation uses an unphysical light quark mass corresponding to a 339 MeV pion mass and an unphysical charm quark mass of 968 MeV, expressed in the $\overline{\mathrm{MS}}$ scheme at 2 GeV. This calculation demonstrates that future work should be able to determine this long-distance contribution from first principles with a controlled error of 10\% or less.