On the Glue Content in Heavy Quarkonia

Eur.Phys.J.C30:233-240,2003 Starting with two coupled Bethe-Salpeter equations for the quark-antiquark, and for the quark-glue-antiquark component of the quarkonium, we solve the bound state equations perturbatively. The resulting admixture of glue can be partially understood in a semiclassical way, one has, however, to take care of the different use of time ordered versus retarded Green functions. Subtle questions concerning the precise definition of the equal time wave function arise, because the wave function for the Coulomb gluon is discontinuous with respect to the relative time of the gluon. A striking feature is that a one loop non abelian graph contributes to the same order as tree graphs, because the couplings of transverse gluons in the tree graphs are suppressed in the non relativistic bound state, while the higher order loop graph can couple to quarks via non suppressed Coulomb gluons. We also calculate the amplitude for quark and antiquark at zero distance in the quark-glue-antiquark component of the P-state. This quantity is of importance for annihilation decays of P-states. It shows a remarkable compensation between the tree graph and the non abelian loop graph contribution. An extension of our results to include non perturbative effects is possible.