Quark model calculation of $\eta \to l^{+}{l}^{-}$ to all orders in the bound state relative momentum

Abstract

The electromagnetic box diagram for the leptonic decays of pseudoscalar mesons in the quark model is evaluated to all orders in $\mathbf{p}/m_q$, where $\mathbf{p}$ is the relative three-momentum of the quark-antiquark pair and $m_q$ is the quark mass. We compute $B_P\equiv \Gamma (\eta \to l^{+}{l}^{-})/\Gamma (\eta \to \gamma \gamma )$ using a popular nonrelativistic (NR) harmonic oscillator wave function, and with a relativistic momentum space wave function that we derive from the MIT bag model. We also compare with a calculation in the limit of extreme NR binding due to Bergström. Numerical calculations of $B_P$ using these three parameterizations of the wave function agree to within a few percent over a wide kinematical range. We find 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 $\eta ,{\pi }^0\to e^{+}{e}^{-}$. These quark models yield $B(\eta \to {\mu }^{+}{\mu }^{-})\approx 4.3\times 10^{-6}$, within errors of the recent SATURNE measurement of $5.1\pm 0.8\times 10^{-6}$, $B(\eta \to e^{+}{e}^{-})\approx 6.3\times 10^{-9}$, and $B({\pi }^0\to e^{+}{e}^{-})\approx 1.0\times 10^{-7}$.

Comments: 17 pages REVTEX. Postscript figures appended