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

Abstract

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. We use Martin's operator inequality \sqrt{p^2+m^2} <= (p^2+M^2+m^2)/2M to motivate the approximation of the relativistic kinetic energy terms in the spinless Salpeter equation by expressions of the nonrelativistic form M+\epsilon+p^2/2M for each quark. To investigate the validity of the resulting approximation numerically, we generate energy spectra for q\bar{q} mesons composed of two light or two heavy quarks using the spinless Salpeter equation with the linear-plus-Coulomb potential, and then fit the lowest few states of each type using the effective Schr\"{o}dinger description with the same potential. We find good fits to the lowest four calculated c\bar{c} and the lowest three s\bar{s} states either taking M fixed at the value M_q=\sqrt{<p^2> +m_q^2}, or allowing M_q to vary in the fit. The energies of the lowest few c\bar{s} 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.