Conformal symmetry algebra of the quark potential and degeneracies in the hadron spectra Conference Paper uri icon

abstract

  • The essence of the potential algebra concept [Y. Alhassid, F. Gürsey, F. Yachello. Phys. Rev. Lett. 50 (1983)] is that quantum mechanical free motions of scalar particles on curved surfaces of given isometry algebras can be mapped on 1D Schrödinger 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. In effect, the excitation modes considered are classified according to the irreducible representations of the algebra of interest and are characterized by typical degeneracies. In consequence, also the spectra of the equivalent Schrödinger operators are classified according to the same irreducible representations and carry the same typical degeneracies. A subtle point concerns the representation of the algebra elements which may or may not be unitarily equivalent to the standard one generating classical groups like SO(n), SO(p,q), etc. To be specific, any similarity transformations of an algebra that underlies, say, an orthogonal group, always conserve the commutators among the elements, but a non-unitarily transformed algebra must not generate same group. One can then consider the parameters of the non-unitary similarity transformation as group symmetry breaking scales and seek to identify them with physical observables. 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) ⊂ so(2,4) isometry algebra of the S3 ball to a particular non-unitary similarity transformation and obtain a deformed isometry copy to S3 such that free motion on the copy is equivalent to a cotangent perturbed motion on S3, and to the 1D Schrödinger 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 highlying mesons and obtain a value compatible with the light dilaton mass. We conclude that while the conformal group symmetry of QCD following from AdS5/CFT4 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. © 2012 American Institute of Physics.

publication date

  • 2012-01-01