Bethe-Salpeter basis for quark-pair-creation model: understanding of VPP, VPγ, and Pγγ couplings

Abstract

A Bethe-Salpeter harmonic-oscillator framework, developed recently for qq̅ mesons in the instantaneous (null-plane) approximation, is employed for the construction of 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 qq̅ 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, ρ→ππ and ω→ρπ, as well as through the two electromagnetic processes ω→γπ⁰ and π⁰→γγ. The ρ→ππ and ω→γπ⁰ widths (via direct γ coupling) come out as 142.7 MeV and 888 keV, respectively, with no adjustable parameters. The ω→γπ⁰ matrix element also agrees to within 1%, with that obtained from the corresponding ωρπ coupling through vector-meson-dominance (VMD) substitution. The π⁰→γγ width (7.7 eV) agrees excellently with experiment when it is deduced from ωρπ coupling via double VMD substitution, but the corresponding amplitude obtained from direct γ coupling is only about half the experimental value. Theoretical reasons are given for this apparent paradox.