Understanding the success of nonrelativistic potential models for relativistic quark-antiquark bound states

We investigate the connection between relativistic potential models for quark-antiquark bound states and the nonrelativistic models that have been used successfully to fit and predict the spectra of relativistic systems, as in the work of Martin. We use Martin’s operator inequality (Formula presented) to motivate the approximation of the relativistic kinetic energy terms in the spinless Salpeter equation by expressions of the nonrelativistic form (Formula presented) for each quark. To investigate the validity of the resulting approximation numerically, we generate energy spectra for (Formula presented) mesons composed of two light or two heavy quarks using the spinless Salpeter equation with the linear-plus-Coulomb potential typical of phenomenological fits to (Formula presented) data, and then fit the lowest few states of each type using the effective Schrödinger description with the same potential. We find good fits to the lowest four calculated (Formula presented) and the lowest three (Formula presented) states either taking M fixed at the value (Formula presented) that minimizes the Martin bound, or allowing (Formula presented) to vary in the fit. The energies of the lowest few (Formula presented) states are then predicted with similar accuracy. The reasons for the success of the nonrelativistic approximation are identified, and explain the success of Martin’s nonrelativistic predictions for the spectra of relativistic light-heavy mesons. However, we note that the agreement between the nonrelativistic and relativistic wave functions is not good, a point of potential concern for the calculation of transition matrix elements.