Behavior of relativistic wave functions near the origin for a QCD potential

We determine the behavior of the solution of the relativistic wave equation [(-∇²+m₁{}²)1/2+(- ∇²+m₂{}²)1/2 +V(r)-M]{ψ}(r)=0 for r{→}0 for the QCD-inspired running Coulomb potential V(r){~}-α₀/r ln(r₀/r), r{}r₀. This equation appears in the theory of relativistic quark-antiquark bound states in a spinless, instantaneous approximation. We find that the radial wave function for angular momentum l behaves as Rₗ(r){~}rˡ[ln(r₀/r)]ₗ^λ for r{→}0 with λₗ>0 a known constant. The severity of the (power-law) divergence of r^-l{R}ₗ$ at r=0 noted previously for a Coulomb potential is therefore reduced (but not eliminated) for the running potential.