A dispersive study of final-state interactions in $K\to\pi\pi\pi$ amplitudes

A system of dispersive representations of the Omn\`es-Khuri-Treiman-Sawyer-Wali type for the final-state interactions in the amplitudes of the $K\to\pi\pi\pi$ weak transitions is constructed, under the assumptions that CP and isospin symmetries are conserved. Both the $\Delta I = 1/2$ and $\Delta I = 3/2$ transition channels are considered. The set of single-variable functions involved in these representations is identified, and the polynomial ambiguities in their definitions are discussed. Numerical solutions for the system of coupled integral relations satisfied by these functions are provided, and the determination of the subtraction constants in terms of the experimental information on the amplitudes in the decay region is addressed.