Conformal symmetry algebra of the quark potential and degeneracies in the hadron spectra

Abstract

The essence of the potential algebra concept [3] is that quantum mechanical free motions of scalar particles on curved surfaces of given isometry algebras can be mapped on 1D Schroedinger equations with particular potentials. As long as the Laplace-Beltrami operator on a curved surface is proportional to one of the Casimir invariants of the isometry algebra, free motion on the surface is described by means of the eigenvalue problem of that very Casimir operator and the excitation modes are classified according to the irreps of the algebra of interest. In consequence, also the spectra of the equivalent Schroedinger operators are classified according to the same irreps. We here use the potential algebra concept as a guidance in the search for an interaction describing conformal degeneracies. For this purpose we subject the so(4) isometry algebra of the S^3 ball to a particular non-unitary similarity transformation and obtain a deformed isometry copy to S^3 such that free motion on the copy is equivalent to a cotangent perturbed motion on S^3, and to the 1D Schroedinger operator with the trigonometric Rosen-Morse potential as well. The latter presents itself especially well suited for quark-system studies insofar as its Taylor series decomposition begins with a Cornell-type potential and in accord with lattice QCD predictions. We fit the strength of the cotangent potential to the spectra of the unflavored high-lying mesons and obtain a value compatible with the light dilaton mass. We conclude that while the conformal group symmetry of QCD following from AdS_5/CFT_4 may be broken by the dilaton mass, it still may be preserved as a symmetry algebra of the potential, thus explaining the observed conformal degeneracies in the unflavored hadron spectra, both baryons and mesons.