Resonance on top of thresholds: the $\Lambda_c(2595)^+$ as an extremely fine-tuned state

A dedicated study of the $\pi\Sigma_c$ scattering around its threshold is carried out in this work to probe the nature of $\Lambda_c(2595)^+$. We first demonstrate that the effective range expansion approach fails to work near the $\Lambda_c(2595)^+$ pole position, due to the presence of a nearby CDD pole around the $\pi\Sigma_c$ thresholds. We then develop a general framework to properly handle the situation with a CDD pole accompanied by nearby thresholds, which is first elaborated for the single-channel case and then generalized to the coupled-channel study. The isospin breaking effects of the three $\pi\Sigma_c$ channels with different thresholds are specially taken into account in our study. The finite-width effects from the $\Sigma_c$ baryons are considered and found to be relevant to give the $\Lambda_c(2595)^+$ width fully compatible to its experimental value. Through the compositeness analysis, our robust conclusion is that the $\pi^0\Sigma_c^+$ component is subdominant inside the $\Lambda_c(2595)^+$.