Quark model calculation ofη→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 $\eta \to l^{+}{l}^{-}$ ($l=e, \mu$). 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 $B_P\equiv \frac{\Gamma (\eta \to l^{+}{l}^{-})}{\Gamma (\eta \to \gamma \gamma )}$ 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 Bergström. 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 $\eta \to {\mu }^{+}{\mu }^{-}$ (unitary bound). However we find substantial deviations from the unitary bound in other kinematical regions, such as $\eta , {\pi }^0\to e^{+}{e}^{-}$. Using the experimental branching ratio for $\eta \to \gamma \gamma$ as input, these quark models yield $B(\eta \to {\mu }^{+}{\mu }^{-})\approx 4.3\times {10}^{-6\mathrm{}}$, within errors of the recent SATURNE measurement of 5.1±0.8×${10}^{-6\mathrm{}}$, and $B(\eta \to e^{+}{e}^{-})\approx 6.3\times {10}^{-9\mathrm{}}$. While an application of constituent quark models to the pion should be viewed with particular caution, the branching ratio $B({\pi }^0\to e^{+}{e}^{-})\approx 1.0\times {10}^{-7\mathrm{}}$ is independent of the details of the above quark model wave functions to within a few percent.