α(Zα)2Vacuum-Polarization Correction in Muonic Atoms

We use the Laplace transform obtained by Wichmann and Kroll for the vacuum-polarization charge density of order ${\left(Z\alpha \right)}^3$ induced by a point nucleus to determine the first few terms in the expansion of the charge density for small distances from the nucleus. The result is used to estimate the effect of the polarization charge near the nucleus on the muonic x rays studied experimentally by Dixit et al. The effect is found to be somewhat smaller than an earlier estimate, and the discrepancy between theory and experiment is silightly reduced, but the discrepancy remains large for the high-$Z$ muonic atoms, as much as three or four standard devitations in lead and barium.