K→3πDecays in Dual Models

The dual representation of $K\to 3\pi$ decay amplitudes is studied. It is found that $K\to 3\pi$ decay amplitudes in a generalized Veneziano model are incompatible with current-algebra relations. For example, pion poles and kaon poles are not dual to other poles. An example of realistic $K\to 3\pi$ decay amplitudes which contain pion and kaon poles, which contain both $\left|\Delta \mathbf{I}\right|=\frac{1}{2} \mathrm{and} \frac{3}{2}$ parts, and which are compatible with current-algebra relations is obtained. Our results seem to suggest that we should prefer the charged-current × charged-current nonleptonic weak Hamiltonian to Hamiltonians with pure $\left|\Delta \mathbf{I}\right|=\frac{1}{2}$.