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 $\ensuremath{\eta}\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}}$ ($l=e, \ensuremath{\mu}$). 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 $\frac{\mathrm{p}}{{m}_{q}}$, where ${m}_{q}$ is the quark mass. We compute ${B}_{P}\ensuremath{\equiv}\frac{\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}})}{\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})}$ 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 consider a quark model calculation in the limit of extreme NR binding due to Bergstr\"om. Numerical calculations of ${B}_{P}$ 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 $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ (unitary bound). However we find substantial deviations from the unitary bound in other kinematical regions, such as $\ensuremath{\eta}, {\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$. Using the experimental branching ratio for $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ as input, these quark models yield $B(\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})\ensuremath{\approx}4.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, within errors of the recent SATURNE measurement of 5.1\ifmmode\pm\else\textpm\fi{}0.8\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$, and $B(\ensuremath{\eta}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})\ensuremath{\approx}6.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$. While an application of constituent quark models to the pion should be viewed with particular caution, the branching ratio $B({\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})\ensuremath{\approx}1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ is independent of the details of the above quark model wave functions to within a few percent.