A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation

Abstract

A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound spin- 1 2 quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons. Numerical solutions to the Bethe-Salpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic. Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The λ quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula. The decay amplitudes for π, K → μν are calculated, and it is found, independent of parameters, that ƒ π ≈ ƒ K for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for ϱ o , ω, φ → e + e −, μ + μ − are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment. The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as ω → π o γ. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model. The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters. The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with J PC = 0 +−) with masses of about 2 BeV.