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 D = 4 - 2e dimensions, and on -mass shell renormalization is employed. The one -loop effect is given to order e, to be combined with the 1 / e divergence of the two -loop amplitude. Master integrals are given. Final evaluations of two -loop energy shifts for 1 S, 2 S, and 2 P 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.