Near-the-origin divergence of Dirac wave functions of hydrogen and operator product expansion

TL;DR

This paper explains the near-origin divergence of hydrogen's Dirac wave functions using nonrelativistic effective field theory and operator product expansion, revealing the role of relativistic corrections and resummation of logarithms.

Contribution

It demonstrates that the universal divergence of Dirac wave functions can be explained by Wilson coefficients in NREFT, linking it to relativistic effects and operator product expansion.

Findings

Logarithmic divergence explained by Wilson coefficients in NREFT

Resummation of leading logarithms reproduces anomalous scaling

Dirac wave function ceases to be meaningful below electron Compton wavelength

Abstract

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.