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Relativistic and recoil corrections to light-fermion vacuum polarization for bound systems of spin-0, spin-1/2, and spin-1 particles

Gregory S. Adkins1 and Ulrich D. Jentschura2

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Phys. Rev. D 113, 056029 – Published 30 March, 2026

DOI: https://doi.org/10.1103/lll9-9wxz

Abstract

In bound systems whose constituent particles are heavier than the electron, the dominant radiative correction to energy levels is given by light-fermion (electronic) vacuum polarization. In consequence, relativistic and recoil corrections to the one-loop vacuum-polarization correction are phenomenologically relevant. Here, we generalize the treatment, previously accomplished for systems with orbiting muons, to bound systems of constituents with more general spins: spin-0, spin-1/2, and spin-1. We discuss the application of our more general expressions to various systems of interest, including spinless systems (pionium), muonic hydrogen and deuterium, and devote special attention to the excited non-S states of deuteronium, the bound system of a deuteron and its antiparticle. The obtained energy corrections are of order α5mr, where α is the fine-structure constant and mr is the reduced mass.

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Phys. Rev. D 113, 056029– Published 30 March, 2026
  • Received 1 January 2026
  • Accepted 4 February 2026
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