Tests of Two-Body Dirac Equation Wave Functions in the Decays of Quarkonium and Positronium into Two Photons

Two-Body Dirac equations of constraint dynamics provide a covariant framework to investigate the problem of highly relativistic quarks in meson bound states. This formalism eliminates automatically the problems of relative time and energy, leading to a covariant three dimensional formalism with the same number of degrees of freedom as appears in the corresponding nonrelativistic problem. It provides bound state wave equations with the simplicity of the nonrelativistic Schroedinger equation. Here we begin important tests of the relativistic sixteen component wave function solutions obtained in a recent work on meson spectroscopy, extending a method developed previously for positronium decay into two photons. Preliminary to this we examine the positronium decay in the 3P_{0,2} states as well as the 1S_0. For the 3P_{0,2} positronium states we find decay amplitudes distinctly different from earlier results obtained with other formalisms. The two-gamma quarkonium decays that we investigate are for the eta_c, eta_c', chi_c0, chi_c2, pi^0, pi_2, a_2, and f_2' mesons. Our results for the four charmonium states compare well with those from other quark models and show the particular importance of including all components of the wave function. The results for the pi^0, although well off the experimental rate, is much closer than the usual expectations from a potential model. We conclude that the Two-Body Dirac equations lead to wave functions which provide good descriptions of the two-gamma decay amplitude and can be used with some confidence for other purposes