Journal ArticleIrreducible three-loop vacuum-polarization correction in muonic bound systemsAdkins, Gregory S; Jentschura, Ulrich D<p>Three-loop electronic vacuum-polarization corrections due to irreducible diagrams are evaluated for two-body muonic ions with nuclear charge numbers<math display='inline'><mn>1</mn><mo>≤</mo><mi>Z</mi><mo>≤</mo><mn>6</mn></math>. The corrections are of order<math display='inline'><msup><mi>α</mi><mn>3</mn></msup><mo stretchy='false'>(</mo><mi>Z</mi><mi>α</mi><msup><mo stretchy='false'>)</mo><mn>2</mn></msup><msub><mi>m</mi><mi>r</mi></msub></math>, where<math display='inline'><mi>α</mi></math>is the fine-structure constant and<math display='inline'><msub><mi>m</mi><mi>r</mi></msub></math>is the reduced mass. Numerically, the energy corrections are found to be of the same order of magnitude as the largest of the order<math display='inline'><msup><mi>α</mi><mn>2</mn></msup><mo stretchy='false'>(</mo><mi>Z</mi><mi>α</mi><msup><mo stretchy='false'>)</mo><mn>6</mn></msup><msub><mi>m</mi><mi>r</mi></msub></math>corrections, and are thus phenomenologically interesting. Our method of calculation eliminates numerical uncertainty encountered in other approaches.</p> <sec><supplementary-material><permissions><copyright-statement>Published by the American Physical Society</copyright-statement><copyright-year>2025</copyright-year></permissions></supplementary-material></sec>American Physical Society2025-03-0110613830Physical Review D11150560162470-0010https://doi.org/10.1103/PhysRevD.111.0560162308792; 2110294QEDvacuum polarizationatomic physicsenergy levelsmuonic ionsmuonic hydrogenmuonic deuteriumNational Science Foundation