Irreducible three-loop vacuum-polarization correction in muonic bound systems

Three-loop electronic vacuum-polarization corrections due to irreducible diagrams are evaluated for two-body muonic ions with nuclear charge numbers 1 <= Z <= 6. The corrections are of order alpha 3 & eth;Z alpha & THORN;2mr, where alpha is the fine-structure constant and mr is the reduced mass. Numerically, the energy corrections are found to be of the same order of magnitude as the largest of the order alpha 2 & eth;Z alpha & THORN;6mr corrections, and are thus phenomenologically interesting. Our method of calculation eliminates numerical uncertainty encountered in other approaches.