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

This is the authors' abstract. We don't add key points for this paper.

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.