Empirical Verification of Recently Proposed Hadron Mass Formulas

Abstract

For the nucleon resonances as well as for the heavy vector mesons different mass formulas are compared. Those corresponding to a band of rotational excitations of the particle seem to agree rather well with the existing data.It has become an increasing challenge to relate the overwhelming flow of data [1] on newly discovered particles to theoretical models. As a rather humble step towards this formidable goal it seems worthwhile to point out any regularities in the particle spectrum. Since this is particularly necessary for the corresponding energy levels we shall deal with the following issues:A) The angular momentum dependence of several deviating mass formulas will be compared with the data on the rotational bands of nucleon resonances. B) Three mass formulae for the recently discovered "photon family" of isospin singlet mesons are proposed and then related to the experiments. As a result, the mass of the "toponium" may be predicted.C) As a concluding remark some speculative observations on the origin of Barut's lepton mass formula [2] will be presented. A) Nucleon ResonancesStimulated by a recent w r ork of MacGregor [3] the band of nucleon and A -resonances will be considered. For these systems, the observed correlation between higher spin values and increasing energies suggests a dynamics corresponding to a rotational, "internal" motion of the particle.Such a phenomenon has been demonstrated in an exactly solvable "toy" model of non-linear and nonlocal interacting Klein-Gordon fields [4]. For a certain non-trivial stationary ansatz cp Q (see Eq. (6) where L denotes the quantum number of the angular momentum of the soliton and mo and a are arbitrary constants with the dimension of a mass. Although resembling the Gürsey-Radicati formula [6] common to non-relativistic theories with internal SU (6) symmetry, it should be noted that our result (1) of the Gell-Mann-Okubo type. Generalizing Wheeler's "no hair" conjecture, the mass of the black soliton depends only on the quantum numbers I of the isospin, Y of the hypercharge and J of the total angular momentum but not on other details of the internal structure (a = gr 2 /Ac is the dimensionless gauge coupling constant and ß a phenomenological constant resulting from a symmetry breaking). In the case of nucleon resonances we may simply probe the black soliton mass formula (2) in the simplified formMass formulas of this kind have already a long history. If a "relativistic rotator" is considered in the context of dynamical groups the relation (3) results from the eigenvalues of the first Casimir operator of an irreducible continuous representation of the de Sitter group SO (1,4) [10]. This