Modelling duality between bound and resonant meson spectra by means of   free quantum motions on the de Sitter space time dS4
  We seek for a pair of a well and barrier potentials such that the real parts of the complex energies of the resonances transmitted through the barrier equal the energies of the states bound within the well and find the hyperbolic Poeschl-Teller barrier, ~sech^2\rho, and the trigonometric Scarf well, ~ \sec^2\chi. The potentials are shown to be conformally symmetric by the aid of the de Sitter space time, dS4, related to flat conformal space time by a conformal map. Namely, we transform the quantum mechanical wave equations with the above potentials to free quantum motions on the respective open time like hyperbolic and the closed space like hyper spherical, S3, geodesics of dS4, the former by itself is related to Minkowski space time by a conformal map.We formulate a conformal symmetry respecting classification scheme for mesons seen either as resonances in scattering, or as states bound within a potential, according to trajectories in which the total spin of the meson, l-depends linearly on the first power of the invariant mass, M, and not as in the canonical Regge formalism on the squared mass.We analyze 71 reported mesons in this scheme and in finding good agreement with data predict the masses of 12 missing mesons. We observe that on S3 the color quantum number of mesons is limited to neutral. To the amount physics is independent of the choice of the set of coordinates, this property remains valid in the flat geometry too. We conclude on the usefulness of conformal maps of flat to curved space times as tools for modelling and interpreting the color neutrality of hadrons required by the confinement phenomenon and on the relevance of trigonometric and hyperbolic potentials for constituent quark models. Finally, all involved potentials have been motivated by Wilson loops with cusps. 
M. Kirchbach and C. Compean
hep-ph