We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in $D=4-2\epsilon$ dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order $\epsilon$, to be combined with the $1/\epsilon$ divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for $1S$, $2S$, and $2P$ states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the $Z\alpha$-expansion. We compare the results obtained to the existing literature.

中文翻译：

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维数正则化和二环真空极化算子：主积分、分析结果和能量转移

我们对量子电动力学中光子传播器的不可约双环真空偏振校正进行了完整的重新评估，即在费米子传播器中使用电子-正电子对。积分是通过将积分简化为一组有限的主积分来进行的，这些主积分是使用按部分积分恒等式计算的。维数正则化用于$D=4-2\epsilon$尺寸，并采用质量壳重整化。单环效果给定$\epsilon$，与$1/\epsilon$两环振幅的散度。给出了主积分。双环能量转移的最终评估$1S$,$2S$， 和$2磷$状态通过分析完成，并给出结果，重点是μ子氢。对于相对论狄拉克-库仑参考态，获得的高阶系数为$Z\alpha$-扩张。我们将获得的结果与现有文献进行比较。