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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>1</a:mn><a:mo>≤</a:mo><a:mi>Z</a:mi><a:mo>≤</a:mo><a:mn>6</a:mn></a:math>. The corrections are of order <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>α</c:mi><c:mn>3</c:mn></c:msup><c:mo stretchy="false">(</c:mo><c:mi>Z</c:mi><c:mi>α</c:mi><c:msup><c:mo stretchy="false">)</c:mo><c:mn>2</c:mn></c:msup><c:msub><c:mi>m</c:mi><c:mi>r</c:mi></c:msub></c:math>, where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>α</g:mi></g:math> is the fine-structure constant and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>m</i:mi><i:mi>r</i:mi></i:msub></i: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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>α</k:mi><k:mn>2</k:mn></k:msup><k:mo stretchy="false">(</k:mo><k:mi>Z</k:mi><k:mi>α</k:mi><k:msup><k:mo stretchy="false">)</k:mo><k:mn>6</k:mn></k:msup><k:msub><k:mi>m</k:mi><k:mi>r</k:mi></k:msub></k:math> corrections, and are thus phenomenologically interesting. Our method of calculation eliminates numerical uncertainty encountered in other approaches.