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 Lüscher and Lellouch and Lüscher, 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 $\mathrm{\Delta }{M}_K$ and the $\mathrm{CP}$-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.

• Received 13 April 2015

DOI:https://doi.org/10.1103/PhysRevD.91.114510