Quark model calculation of η→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 Bergstroem. Numerical calculations of BP using these three parametrizations 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-)∼4.3x10-6, within errors of the recent SATURNE measurement of 5.1±0.8x10-6, and B(η→e+e-)∼6.3x10-9