The $��_c(2595)$ resonance as a dynamically generated state: the compositeness condition and the large $N_c$ evolution

Recent studies have shown that the well established $��_c(2595)$ resonance contains a large meson-baryon component, which can vary depending on the specific formalism. In this work, we examine such a picture by utilizing the compositeness condition and the large number of colors ($N_c$) expansion. We examine three different models fulfilling two body unitarity in coupled-channels, and adopting renormalization schemes where the mass of the $��_c(2595)$ resonance is well described, but not necessarily its width, since we do not consider three body channels and work at the isospin symmetric limit. Both approximations might have an effect larger on the width than on the mass. In this context, our studies show that the compositeness of the $��_c(2595)$ depends on the number of considered coupled channels, and on the particular regularization scheme adopted in the unitary approaches and, therefore, is model dependent. In addition, we perform an exploratory study of the $��_c(2595)$ in the large $N_c$ expansion, within a scheme involving only the $����_c$ and $K��'_c$ channels, whose dynamics is mostly fixed by chiral symmetry. In this context and formulating the leading-order interaction as a function of $N_c$, we show that for moderate $N_c> 3$ values, the mass and width of the $��_c(2595)$ deviate from those of a genuine $qqq$ baryon, implying the relevance of meson-baryon components in its wave function. Furthermore, we study the properties of the $��_c(2595)$, in the strict $N_c \to \infty $ limit, using an extension of the chiral Weinberg-Tomozawa interaction to an arbitrary number of flavors and colors. This latter study hints at the possible existence of a (perhaps) sub-dominant $qqq$ component in the $��_c(2595)$ resonance wave function, which would become dominant when the number of colors gets sufficiently large.
33 pages, 2 figures