  There is a long-standing puzzle concerning the Coulomb solutions of the Dirac
equation, i.e., what is the physics governing the weakly divergent
near-the-origin behavior of the Dirac wave functions of the $nS_{1/2}$
hydrogen? As a sequel of our preceding work that aim to demystifying the
universal near-the-origin behavior of the atomic Schr\"{o}dinger and
Klein-Gordon wave functions, the goal of this work is to demonstrate that,
within the nonrelativistic effective field theory (NREFT) tailored for
Coulombic atoms, the universal logarithmic divergence of the Dirac wave
functions can be accounted by the perturbatively calculable Wilson coefficient
emerging from the operator product expansion (OPE) of the electron and the
nucleus fields. The cause is due to the relativistic kinetic correction and
Darwin (zitterbewegung) term in the NREFT. With the aid of renormalization
group equation, one can resum the leading logarithms to all orders in $Z\alpha$
and recover the $r^{-Z^2\alpha^2/2}$ anomalous scaling behavior exhibited by
the Dirac wave function for the $nS_{1/2}$ hydrogen. It appears somewhat
counterintuitive that these universal logarithmic divergences can not be
accounted by the OPE set up in the relativistic QED. We are thereby enforced to
conclude that the Dirac wave function must cease to be meaningful when $r$ is
shorter than the electron's Compton wavelength.
