Bethe-Salpeter basis for quark-pair-creation model: Understanding ofVPP,VPγ, andPγγcouplings

Abstract

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, $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}$, as well as through the two electromagnetic processes $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$. The $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\pi}}^{0}$ widths (via direct $\ensuremath{\gamma}$ coupling) come out as 142.7 MeV and 888 keV, respectively, with no adjustable parameters. The $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\pi}}^{0}$ matrix element also agrees to within 1%, with that obtained from the corresponding $\ensuremath{\omega}\ensuremath{\rho}\ensuremath{\pi}$ coupling through vector-meson-dominance (VMD) substitution. The ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ width (7.7 eV) agrees excellently with experiment when it is deduced from $\ensuremath{\omega}\ensuremath{\rho}\ensuremath{\pi}$ coupling via double VMD substitution, but the corresponding amplitude obtained from direct $\ensuremath{\gamma}$ coupling is only about half the experimental value. Theoretical reasons are given for this apparent paradox.