  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.
