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 Castillejo-Dalitz-Dyson (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 with 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)(+).