Complex 2D matrix model and geometrical map on the complex-Nc plane

Abstract

We study the parameter dependence of the internal structure of resonance states by formulating a complex two-dimensional (2D) matrix model, where the two dimensions represent two levels of resonances. We calculate a critical value of the parameter at which a “nature transition” with character exchange occurs between two resonance states, from the viewpoint of geometry on complex-parameter space. Such a critical value is useful for identifying the internal structure of resonance states with variation of the parameter in the system. We apply the model to analyze the internal structure of hadrons with variation of the color number |$N_c$| from |$\infty $| to a realistic value 3. By regarding |$1/N_c$| as the variable parameter in our model, we calculate a critical color number of the nature transition between hadronic states in terms of a quark–antiquark pair and a mesonic molecule as exotics from the geometry on the complex-|$N_c$| plane. For large-|$N_c$| effective theory, we employ the chiral Lagrangian induced by holographic QCD with a D4/D8/|$\overline {\mathrm {D8}}$| multi-D brane system in type IIA superstring theory.