Theoretical study of the $D^0 \to K^- \pi^+ \eta$ reaction

We develop a model to study the $D^0 \to K^- \pi^+ \eta$ weak decay, starting with the color favored external emission and Cabibbo favored mode at the quark level. A less favored internal emission decay mode is also studied as a source of small corrections. Some pairs of quarks are allowed to hadronize producing two pseudoscalar mesons, which posteriorly are allowed to interact to finally provide the $K^- \pi^+ \eta$ state. The chiral unitary approach is used to take into account the final state interaction of pairs of mesons, which has as a consequence the production of the $\kappa$ ($K^*_0(700)$) and the $a_0(980)$ resonances, very well visible in the invariant mass distributions. We also introduce the $\bar{K}^{*0} \eta$ production in a phenomenological way and show that the $s$-wave pseudoscalar interaction together with this vector excitation mode are sufficient to provide a fair reproduction of the experimental data. The agreement with the data, in particular the relative weight of the $a_0(980)$ to the $\kappa$ excitation, provides extra support to the picture used, in which these two resonances are a consequence of the interaction of pseudoscalar mesons and not ordinary $q \bar{q}$ mesons.