A relativistic formulation of the quark model for mesons

Abstract

Mesons are treated as bound states in quark-antiquark scattering using the Bethe-Salpeter equation (this does not necessarily imply that the properties of mesons are dominated by the Q- Q state). It is shown that the existence of very tightly bound states with a nonrelativistic spinor structure (such as are considered in the usual norelativistic quark model) requires a Dirac scalar potential (or sum of irreducible graphs) as well as small internal three momentum (q 2 ⪡ M Q 2). The assumption that the pseudoscalar mesons have a particular simple relativistic spinor structure leads naturally to the correct value of Γ(π→μν) Γ(K→μν) (in contrast to the usual model) and gives the prediction γ ρ 2:γ φ 2:γ ω 2=M ρ 2: 3M φ 2 cos 2θ : 3M ω 2 sin 2θ (where γ V are the usual V − γ coupling constants and θ is the ω − φ mixing angle) which is compatible with the data. The assumption that the internal momentum is small and the potential is spin-independent can lead to approximate SU(6) symmetry, as in the usual model, although the spinor structure is relativistic. Using the value of the weak axial vector coupling constant for quarks given by the nonrelativistic quark model for baryons, these assumptions lead to the Kawarabayashi-Suzuki relation 2 γ ϱ f π = M ϱ which agrees with experiment. The states which have L = 1 in the usual model are discussed. The model leads naturally to quadratic mass formulae for mesons. The processes π 0→ γγ and η 0→ γγ are discussed using the relativistic formalism.