Bound states at lowest order in ~

Bound states poles in scattering amplitudes are generated by the divergence of the perturbative series due to enhanced Coulomb scattering near thresholds. This suggests to organize bound state calculations according to an expansion in ~, i.e., in the number of loops. I study QED and QCD bound states at lowest order in ~, which are analogous to usual Born amplitudes. The absence of loops allows the use of retarded boundary conditions where particles only propagate forward in time, facilitating a hamiltonian approach. The instantaneous A0 field is determined by the equations of motion separately for each Fock component of the bound state. The field equations allow also a linear A0 potential as a homogeneous, non-perturbative solution. Stationarity of the action sets the direction of the ensuing constant electric field to be along the fermion pair separation in each Fock state. Applying this approach to relativistic quark-antiquark states in QCD results in a bound state equation which was previously proposed without derivation and shown to provide a reasonable description of the meson spectrum, including linear Regge trajectories. The equal-time wave functions have unique Lorentz transformation properties, which ensure the correct dependence of the bound state energy on the center-of-mass momentum. ar X iv :0 90 9. 30 45 v1 [ he pph ] 1 6 Se p 20 09