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

We determine the behavior of the solution of the relativistic wave equation [(-${\nabla }^2$+$m_1$ ${}^2$ ${)}^{1/2}$+(- ${\nabla }^2$+$m_2$ ${}^2$ ${)}^{1/2}$ +V(r)-M]ψ(r)=0 for r→0 for the QCD-inspired running Coulomb potential V(r)∼-${\alpha }_0$/r ln($r_0$/r), r≪$r_0$. 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 $R_l$(r)∼$r^l$[ln($r_0$/r${\left)\right]}_l^{\lambda }$ for r→0 with ${\lambda }_l$>0 a known constant. The severity of the (power-law) divergence of $r^{\mathrm{-}l}$ $R_l$ at r=0 noted previously for a Coulomb potential is therefore reduced (but not eliminated) for the running potential.