Three-pion effects in $K^0-\bar{K}^0$ mixing

The rate of mixing between a neutral kaon and an anti-kaon ( 𝐾 0 − ¯ 𝐾 0 ) is given, in part, by a long-range matrix element, deﬁned with two insertions of the weak Hamiltonian separated by physical, Minkowski time evolution. For physical quark masses, the kaon mass lies above the two-and three-pion thresholds and, as a result, this long-range matrix element receives contributions from intermediate on-shell 2 𝜋 and 3 𝜋 states. These contributions cannot easily be captured in a ﬁnite Euclidean spacetime, meaning that such matrix elements are not directly accessible via lattice QCD. In this talk, we present a strategy for combining quantities that can be extracted in numerical lattice QCD calculations in order to reproduce the physical, inﬁnite-volume long-range amplitude for 𝐾 0 − ¯ 𝐾 0 . The key novelty relative to published work is that we fully include the eﬀects of three-particle states that were previously neglected. The strategy is built on existing formalism for long-range matrix elements with two-particle intermediate states, together with the relativistic-ﬁeld-theory ﬁnite-volume formalism for extracting three-hadron weak decays.