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Signatures of the two $K_1(1270)$ poles in $D^+\to νe^+ V P$ decay

We analyze theoretically the $D^+\to νe^+ ρ\bar K$ and $D^+\to νe^+ \bar K^* π$ decays to see the feasibility to check the double pole nature of the axial-vector resonance $K_1(1270)$ predicted by the unitary extensions of chiral perturbation theory (UChPT). Indeed, within UChPT the $K_1(1270)$ is dynamically generated from the interaction of a vector and a pseudoscalar meson, and two poles are obtained for the quantum numbers of this resonance. The lower mass pole couples dominantly to $K^*π$ and the higher mass pole to $ρK$, therefore we can expect that different reactions weighing differently these channels in the production mechanisms enhance one or the other pole. We show that the different final $VP$ channels in $D^+\to νe^+ V P$ weigh differently both poles, and this is reflected in the shape of the final vector-pseudoscalar invariant mass distributions. Therefore, we conclude that these decays are suitable to distinguish experimentally the predicted double pole of the $K_1(1270)$ resonance.