Researchers have developed a finite-volume formalism that enables the study of meson interaction amplitudes, such as the $K \overline K \to \pi^+ \pi^0 \pi^-$ decay, using lattice quantum chromodynamics (Lattice QCD) calculations. This advancement is crucial for understanding the Wess-Zumino-Witten (WZW) term in chiral perturbation theory, a manifestation of chiral anomalies that describes fundamental hadronic interactions.

The proposed formalism allows access to these amplitudes from the two- and three-particle spectra in finite volumes, which can be directly obtained from Lattice QCD simulations. Specifically, the formalism has been presented for the $K \overline K + 3 \pi$ systems with isospin $I=0$ and for $ \pi K + \pi \pi K$ with isospin $I=1/2$ and $I=3/2$. This approach bridges the gap between theoretical predictions from chiral theory and numerical results from Lattice QCD, providing a robust tool for low-energy particle physics.

Furthermore, the study has determined the threshold expansions for the $2 \leftrightarrow 3$ and $3 \leftrightarrow 3$ particle K matrices that are essential to the formalism. Leading-order predictions from chiral perturbation theory for the coefficients entering these expansions have also been calculated. This not only validates the formalism with known theoretical predictions but also establishes a framework for future detailed comparisons with Lattice QCD data, opening new avenues for investigating meson properties and the strong force.