| The success of non-relativistic, Schrodinger equation (with relativistic corrections) fits of the charmonium and bottomonium spectra suggests the value of a relativistic extension to allow a treatment of the low-mass meson trajectories. After reviewing the various non-relativistic and semirelativistic treatments of quarkonia, I propose a covariant generalization of the two-particle Dirac equation, including a generalization of the famed "Coulomb-plus-linear" potential. After separating this equation and relating the eigenvalues of the separation to the observable mass of a bound state, I found that the combination of relativistic kinematics, a four-vector tensor form for the interaction, and a potential linear in the proper separation s(=(r('2) - t('2))(' 1/2)) of the two particles for large s implies asymptotically linear Regge trajectories. By solving the equation I was able to reproduce the (pi), (rho), K, and K* trajectories very accurately; my potential parameters were similar to those found in heavy quark studies. A consideration of the non-relativistic limit of my equation provides hints as to the source of the long-range weakening of the spin-spin part of the four-vector interaction which has been observed in connection with the hyperfine structure of the heavy quark states. |
