Schematic model of mesons based on analytic propagators

Abstract

Formation of mesons is studied in a model where the confinement phenomenon is parametrized through entire-analytic and asymptotically free propagators for "would-be interpolating" fields with quark and gluon quantum numbers. It is pointed out that with such propagators the corresponding proper vertices necessarily have principal-value infrared singularities in external ${(\mathrm{mass})}^{2}$ variables. With the simplest $\mathrm{qqg}$ vertex model embodying those singularities, the quark-antiquark Bethe-Salpeter equation driven by dressed one-gluon exchange possesses an exact analytic solution. The solution is studied in a simplified scalar model omitting spinor structure. It exhibits a meson pole in the color-singlet channel only and no cuts, the meson mass depending nonanalytically on the coupling constant. The model illustrates, on a phenomenological level, an unconventional binding mechanism: Field configurations having no particle character---while looking like particles at large momentum transfers---can nevertheless interact to form stable hadrons, through a mechanism that invokes neither a confining potential nor the notion of a constituent mass.