Effects of finite volume on the $K_L$-$K_S$ mass difference

TL;DR

This paper extends finite-volume correction methods in lattice QCD to accurately compute the $K_L$-$K_S$ mass difference and CP violation parameters, accounting for near-degenerate states and second-order mixing effects.

Contribution

It generalizes existing finite-volume correction techniques to second-order mixing of neutral kaons, providing a unified approach for precise lattice QCD calculations of kaon mass differences.

Findings

Finite-volume effects on $ riangle M_K$ and $ ext{Re}( ext{epsilon}_K)$ are quantified.

Large finite-volume corrections can occur near degeneracy of kaon and two-pion energies.

The methods improve the accuracy of lattice QCD computations of kaon mixing parameters.

Abstract

Phenomena that involve two or more on-shell particles are particularly sensitive to the effects of finite volume and require special treatment when computed using lattice QCD. In this paper we generalize the results of L\"uscher, and Lellouch and L\"uscher, which determine the leading order effects of finite volume on the two-particle spectrum and two-particle decay amplitudes to determine the finite-volume effects in the second order mixing of the $K^0$ and $\overline{K^0}$ states. We extend the methods of Kim, Sachrajda and Sharpe to provide a direct, uniform treatment of these three, related, finite-volume corrections. In particular, the leading, finite-volume corrections to the $K_L$-$K_S$ mass difference $\Delta M_K$ and the CP violating parameter $\epsilon_K$ are determined, including the potentially large effects which can arise from the near degeneracy of the kaon mass and the energy of a finite-volume, two-pion state.