Near-the-origin divergence of Klein-Gordon wave functions for hydrogen-like atoms and operator product expansion

There have been some long-standing puzzles related to the Coulomb solutions of the Klein-Gordon and Dirac equations, namely how to understand the physics underlying the weakly divergent near-the-origin behavior of the $S$-wave wave functions for the hydrogen-like atoms. Taking the Klein-Gordon wave function as a simpler example, in this work we demonstrate that, with the aid of the renormalization group equation, this universal short-distance behavior can be successfully taken into account by the operator product expansion (OPE) formulated in the nonrelativistic effective field theory (EFT), which is tailored for Coulombic atoms. The key is to include the relativistic kinetic correction in the EFT. Somewhat counterintuitively, these universal near-the-origin logarithmic divergences can not be addressed by the OPE set up in the relativistic scalar QED. We conclude that the Klein-Gordon wave function at a length scale shorter than the electron Compton wavelength may cease to make physical significance.