Meson and Baryon Families as Vibronic States in sl(2) Quantum Universal Enveloping Algebra

Abstract

A mass formula of the q·deformed modified harmonic oscillator type in the sl(2) quantum universal enveloping algebra is proposed for the meson and baryon families, by taking into account the known theories as a guide. Specifying the vibronic quantum number, the deformation parameter and associated ones of the theory are determined from available data for the scalar, pseudoscalar, vector meson and baryon families. The parameters determined from totally ten families not only predict many unobserved states, but also give restrictions on the observable number of- states. The method may admit taking into account no.n·perturbative effects. mation, specified by a parameter q, into the th~ory of angular momentum. 2 ),3) The original idea of these works has been started by Faddeev 4 ) and his schoolS) from the examination of an exact solvability of classical and quantum two-dimensional models in statistical mechanics. The present author6) has pointed out that the quadratic Casimir operator and its eigenvalue discovered by Witten have many applications in the angular momentum associated problems. The purpose of this paper is to analyze the vibronic spectra of hadrons by making use of the q-deformed harmonic oscillator and its modified form. In order to con­ struct the mass formula we have taken (i) the MIT bag modeF) and a scalar ),.¢4 theory8) in two space-time dimensions as a guide, and (ii) the q-deformed version of one-dimensional oscillator and its modified form. In the former the zero-point energy is omitted by taking a normal ordering. The application of the one-dimensional quantum oscillator in a universal enveloping (UE) algebra is quite common in the analysis of the molecular spectra 9 ) under the light of the Born-Oppenheimer approxi­ mation. In order to determine the terminating quantum number for oscillation we use the analogous idea which is operative in the estimate of the dissociation energies for molecules. lO ) We consider that the total spins of baryon (meson) families belong­ ing to the oscillatory states consist of intrinsic spins of quarks (quark and antiquark), viz., with no orbital excitation. There are totally ten families suitable to analyze with spin 0, 1/2 and 1 states, for the time being, if we include all members listed in the full listings of Tables of Particle Properties.ll) The observed decay modes of ha­ drons classified as members of one and the same family are quite similar in nature except for the difference arising from the threshold effect due to mass difference. This is counted as another indication of the usefulness of our assignment on the