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Two-loop vacuum polarization in a Coulomb field

S. A. Volkov, V. A. Yerokhin, Z. Harman, and C. H. Keitel

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Phys. Rev. A 112, 062814 – Published 15 December, 2025

DOI: https://doi.org/10.1103/bm5f-11hh

Abstract

The leading-order two-loop vacuum-polarization potential, linear in the Coulomb field of a nucleus, was first derived in the seminal 1955 work of Källén and Sabry [Dan. Mat. Fys. Medd. 29, 1 (1955)]. The higher-order two-loop vacuum-polarization corrections, however, have remained unknown until now. In this work, we compute Coulomb corrections to the Källén-Sabry potential, specifically those involving three, five, and seven Coulomb interactions inside the vacuum-polarization loop. The potentials are evaluated in momentum space and subsequently used to calculate one-electron energy shifts. Our results reduce the theoretical uncertainty of the two-loop vacuum-polarization contribution to transition energies, which is required for next-generation tests of bound-state QED in heavy one- and few-electron ions as well as for the determination of nuclear charge radii.

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Phys. Rev. A 112, 062814– Published 15 December, 2025
  • Received 1 October 2025
  • Accepted 14 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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