Hadronic vacuum polarization effect in muonic atoms

The hadronic vacuum polarization (HVP) corrections to energy levels in muonic atoms are studied systematically across the periodic table. Two nuclear charge distribution models have been considered, with a partly analytical solution for a homogeneously charged sphere model and direct numerical integration for a Fermi nuclear charge distribution. Our comparison reveals a good agreement between both approaches with relative differences typically below a few percent. The magnitude of HVP corrections increases with nuclear charge, displaying irregularities that correlate directly with the nonmonotonic behavior of nuclear radii. Despite strong relativistic effects at high Z , the ratio of hadronic to muonic vacuum polarization remains remarkably stable, deviating by less than 10 % from the established 67 % nonrelativistic benchmark value.