Quark model calculation of eta -->l+l- to all orders in the bound-state relative momentum.

We analyze the electromagnetic amplitude for the leptonic decays of pseudoscalar mesons in the quark model, with particular emphasis on η → l+l− (l = e, μ). We evaluate the electromagnetic box diagram for a quark-antiquark pair with an arbitrary distribution of relative three-momentum p: the amplitude is obtained to all orders in p/mq, where mq is the quark mass. We compute BP ≡ Γ(η → l+l−)/Γ(η → γγ) using a harmonic oscillator wave function that is widely used in nonrelativistic (NR) quark model calculations, and with a relativistic momentum space wave function that we derive from the MIT bag model. We also compare with a quark model calculation in the limit of extreme NR binding due to Bergström. Numerical calculations of BP using these three parameterizations of the wave function agree to within a few percent over a wide kinematical range. Our results show that the quark model leads in a natural way to a negligible value for the ratio of dispersive to absorptive parts of the electromagnetic amplitude for η → μ+μ− (unitary bound). However we find substantial deviations from the unitary bound in other kinematical regions, such as η, π0 → e+e−. Using the experimental branching ratio for η → γγ as input, these quark models yield B(η → μ+μ−) ≈ 4.3 × 10−6, within errors of the recent SATURNE measurement of 5.1± 0.8 × 10−6, and B(η → e+e−) ≈ 6.3 × 10−9. While an application of constituent quark models to the pion should be viewed with particular caution, the branching ratio B(π0 → e+e−) ≈ 1.0×10−7 is independent of the details of the above quark model wave functions to within a few percent.