Quarkonia with a variational method

Abstract

A variational method, based in independent minimization of energy levels, is applied to quantum mechanical quarkonium systems. A discussion is presented of the method, with emphasis on quark mass dependence of energy levels (Feynman-Hellman theorem), on the behaviour of wave functions at the origin (Martin's theorem) and on the ordering of energy levels. The potential used in the applications is the Coulomb (4-vector) + Linear (4-scalar) potentials, the fine and hyperfine splittings being included by means of the Fermi-Breit hamiltonian. An attempt is made for an overall treatment of the known splittings (radials-wave, hyperfine and fine splittings) for all quarkonium systems, using an approximately flavour independent potential. Thep-wave splittings, as well as the hyperfine splittings, are shown to be particularly sensitive to the nature and mass dependence of the potential. In particular, asymptotic freedom can be more easily tested there. On the other hand, radial excitations provide the place where the set up of short range potential effects should be first detected. Some results concerning the ordinary mesons are presented, and it is specifically pointed out that strong restrictions exist for the masses of the ρ′ and theA1. Predictions for the toponium family are also presented.