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

TL;DR

This paper develops a method to accurately compute the long-range contributions to $K^0-ar{K}^0$ mixing, including previously neglected three-pion effects, by combining lattice QCD data with finite-volume formalism.

Contribution

It introduces a novel approach that fully incorporates three-particle states into the calculation of long-range matrix elements in kaon mixing.

Findings

Includes three-pion effects in the analysis of $K^0-ar{K}^0$ mixing.

Provides a strategy to extract infinite-volume amplitudes from lattice QCD.

Enhances the accuracy of theoretical predictions for kaon mixing phenomena.

Abstract

The rate of mixing between a neutral kaon and an anti-kaon ($K^0-\bar{K}^0$) 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 $2\pi$ and $3\pi$ 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 $K^0-\bar{K}^0$. 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.