The Uehling Potential at Its Own Range: Why Muonic Atoms See the Vacuum's Shortest Reach

Vacuum polarisation modifies the Coulomb potential only within a short reach of the charge: the Uehling potential's range is 193 femtometres, set by the mass of the lightest charged particle. An electronic hydrogen atom, with a Bohr radius of fifty-three thousand femtometres, samples that reach in a tiny fraction of its volume. A muonic hydrogen atom has a Bohr radius of 285 femtometres, the same size as the reach itself, and samples it completely. This paper computes both radii from the reduced masses, forms the volume-overlap measure for electronic and muonic systems, and finds a gain of about six orders of magnitude, which accounts for the observed prominence of polarisation in muonic spectra. The argument is a scaling computation from stated quantities, not a measurement, and it identifies the general design rule for vacuum-sensitive atoms: match the atom's size to the range of the vacuum effect being studied.