Isospin symmetry and analyticity in $D\to \bar{K}\pi \pi$ decays

We perform a detailed study of the consequences of isospin symmetry in the Cabibbo favoured $D\to \text{Kbar}\pi \pi$ decays. These processes are important for precision testing of the Standard Model and for hadronic physics. Combining isospin symmetry with a dispersive reconstruction theorem we derive a representation in terms of one-variable functions which allows one to predict all the $D\to \text{Kbar}\pi \pi$ amplitudes given inputs from one $D^{+}$ mode and one $D^0$ mode. From this, using dispersion relations and unitarity, we derive a set of 6+6 Khuri-Treiman type integral equations which enable to take three-body rescattering effects into account. A first test of this approach is presented using experimental results on the $D^{+}\to K_S\text{piz}\text{pip}$ and the $D^0\to K_S\text{pim}\text{pip}$ modes.