Bethe-Salpeter basis for quark-pair-creation model: Understanding of $\mathrm{VPP}$, $\mathrm{VP}\gamma$, and $P\gamma \gamma$ couplings

A Bethe-Salpeter harmonic-oscillator framework, developed recently for $q\overline{q}$ mesons in the instantaneous (null-plane) approximation, is employed for the construction of $\mathrm{MMM}$ couplings ($M=P or V$) through quark-loop diagrams. The resulting structure of the matrix elements exhibits the central feature of the quark-pair-creation model, that is, a multiplicative structure for the orbital and spin wave functions of all three $q\overline{q}$ states involved, together with a couple of essential kinematical factors (but only in the numerator) as part of the field-theoretic result. The model is illustrated through two typical meson couplings involving equal-mass quarks, $\rho \to \pi \pi$ and $\omega \to \rho \pi$, as well as through the two electromagnetic processes $\omega \to \gamma {\pi }^0$ and ${\pi }^0\to \gamma \gamma$. The $\rho \to \pi \pi$ and $\omega \to \gamma {\pi }^0$ widths (via direct $\gamma$ coupling) come out as 142.7 MeV and 888 keV, respectively, with no adjustable parameters. The $\omega \to \gamma {\pi }^0$ matrix element also agrees to within 1%, with that obtained from the corresponding $\omega \rho \pi$ coupling through vector-meson-dominance (VMD) substitution. The ${\pi }^0\to \gamma \gamma$ width (7.7 eV) agrees excellently with experiment when it is deduced from $\omega \rho \pi$ coupling via double VMD substitution, but the corresponding amplitude obtained from direct $\gamma$ coupling is only about half the experimental value. Theoretical reasons are given for this apparent paradox.