Dimensional regularization and two-loop vacuum polarization operator: Master integrals, analytic results, and energy shifts

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>D</a:mi><a:mo>=</a:mo><a:mn>4</a:mn><a:mo>−</a:mo><a:mn>2</a:mn><a:mi>ε</a:mi></a:mrow></a:math> dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ε</c:mi></c:math>, to be combined with the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>1</e:mn><e:mo>/</e:mo><e:mi>ε</e:mi></e:math> divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>1</g:mn><g:mi>S</g:mi></g:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>2</i:mn><i:mi>S</i:mi></i:math>, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mn>2</k:mn><k:mi>P</k:mi></k:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>Z</m:mi><m:mi>α</m:mi></m:math>-expansion. We compare the results obtained to the existing literature. Published by the American Physical Society 2024