Sixth order QED radiative corrections to lepton anomalies due to the fourth order vacuum polarization insertions within the Mellin-Barnes representation

The explicit form of the sixth order radiative corrections to the lepton <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>L</a:mi> </a:math> ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mi>L</c:mi> <c:mo>=</c:mo> <c:mi>e</c:mi> </c:mrow> </c:math> , <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:mi>μ</e:mi> </e:mrow> </e:math> , and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>τ</g:mi> </g:math> ) anomalous magnetic moment from QED Feynman diagrams with insertion of the fourth order polarization operators consisting of either two closed lepton loops or one lepton loop crossed by a photon line is discussed in detail. The approach is based on the consistent application of dispersion relations for vacuum polarization operators and the Mellin-Barnes transform for massive photon propagators. Explicit analytical expressions for corrections to the lepton anomaly are obtained for the first time in the whole interval <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mn>0</i:mn> <i:mo>&lt;</i:mo> <i:mi>r</i:mi> <i:mo>&lt;</i:mo> <i:mi>∞</i:mi> </i:math> of the ratio <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>r</k:mi> </k:math> of lepton masses <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>m</m:mi> <m:mo>ℓ</m:mo> </m:msub> <m:mo>/</m:mo> <m:msub> <m:mi>m</m:mi> <m:mi>L</m:mi> </m:msub> </m:math> . Asymptotic expansions in the limit of both small <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>r</o:mi> <o:mo>≪</o:mo> <o:mn>1</o:mn> </o:math> and large <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>r</q:mi> <q:mo>≫</q:mo> <q:mn>1</q:mn> </q:math> computed from the exact expressions are found to be in perfect agreement with the ones earlier reported in the literature. We argue that in the region where the physical lepton mass ratios are located, the asymptotic expansions hold with an accuracy higher than the experimentally measured anomalies. The two loop diagrams with all three leptons different from each other are computed numerically and compared with the corresponding corrections from the pure two-bubble and one-bubble mixed diagrams. It is shown that there are regions of ratios <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>r</s:mi> </s:math> where all three types of the fourth order polarization operator contribute equally to the anomaly.