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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

G. S. Adkins, U. D. Jentschura

TL;DR

This work develops a general NRQED-based framework to compute relativistic and recoil corrections to the one-loop electronic vacuum polarization (eVP) for bound states formed from spin-0, spin-1/2, and spin-1 constituents. By employing an optimized Coulomb gauge and integrating the VP spectral function, the authors derive a unified $H^{(5)}_{\mathrm{eVP}}$ operator and explicit spin-dependent expressions valid to order $\alpha^5 m_r$, applicable to pionium, muonic hydrogen, muonic deuterium, and deuteronium. They provide detailed analyses for fine- and hyperfine-structure effects and finite-size/quadrupole contributions, together with numerical results for deuteronium, illustrating the framework's predictive power and uncertainties. The results enable high-precision tests of low-energy Standard Model physics and offer a robust tool for interpreting future spectroscopy of heavy bound systems and potential New Physics couplings.

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 alpha^5*m_r, where alpha is the fine-structure constant and m_r is the reduced mass.

Relativistic and Recoil Corrections to Light-Fermion Vacuum Polarization for Bound Systems of Spin-0, Spin-1/2, and Spin-1 Particles

TL;DR

This work develops a general NRQED-based framework to compute relativistic and recoil corrections to the one-loop electronic vacuum polarization (eVP) for bound states formed from spin-0, spin-1/2, and spin-1 constituents. By employing an optimized Coulomb gauge and integrating the VP spectral function, the authors derive a unified operator and explicit spin-dependent expressions valid to order , applicable to pionium, muonic hydrogen, muonic deuterium, and deuteronium. They provide detailed analyses for fine- and hyperfine-structure effects and finite-size/quadrupole contributions, together with numerical results for deuteronium, illustrating the framework's predictive power and uncertainties. The results enable high-precision tests of low-energy Standard Model physics and offer a robust tool for interpreting future spectroscopy of heavy bound systems and potential New Physics couplings.

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 alpha^5*m_r, where alpha is the fine-structure constant and m_r is the reduced mass.
Paper Structure (13 sections, 123 equations, 1 figure, 4 tables)

This paper contains 13 sections, 123 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. NRQED Lagrangian and Feynman Rules
  3. Spin-1 Breit Hamiltonian
  4. Relativistic and Recoil Corrections to eVP
  5. Applications
  6. Spinless Bound Systems: Pionium
  7. Fine--Structure of Muonic Hydrogen
  8. Hyperfine Structure of Muonic Hydrogen
  9. Fine--Structure of Muonic Deuterium
  10. Hyperfine Structure of Muonic Deuterium
  11. Numerical Values for Deuteronium
  12. Conclusions
  13. Fourier Transforms

Figures (1)

  • Figure 1: Graphs contributing to the spin-1 Breit Hamiltonian. Graph (a) represents the basic Coulomb interaction. Graphs (b)-(d) also involve the Coulomb photon, and represent the Darwin, spin-orbit, and quadrupole contributions. Graphs (e)-(g) contain transverse photons, and represent the magnetic contribution (e). which involves convection currents (conv), the Fermi spin-orbit contribution (f), and the Fermi spin-spin contribution (g). Additional contributions with the vertices on the two spin-1 particles interchanged must also be included for graphs (b), (c), and (f). For the notation, we refer to Table \ref{['table1']}.