[Paper Review] Muonic vacuum polarization correction to the bound-electron $g$-factor

Н. А. Белов, B. Sikora+6 more|arXiv (Cornell University)|Oct 5, 2016

Atomic and Molecular Physics3 citations

TL;DR

This paper theoretically evaluates the muonic vacuum polarization (VP) correction to the bound-electron g-factor in highly charged ions, demonstrating that strong Coulomb fields enhance these effects significantly beyond free-electron values. The study shows these corrections are measurable in planned trapped-ion experiments and must be included for precise determinations of the fine-structure constant α.

ABSTRACT

The muonic vacuum polarization contribution to the $g$-factor of the electron bound in a nuclear potential is investigated theoretically. The electric as well as the magnetic loop contributions are evaluated. We found these muonic effects to be observable in planned trapped-ion experiments with light and medium-heavy highly charged ions. The enhancement due to the strong Coulomb field boosts these contributions much above the corresponding terms in the free-electron $g$-factor. Due to their magnitude, muonic vacuum polarization terms are also significant in planned determinations of the fine-structure constant from the bound-electron $g$-factor.

Motivation & Objective

• To investigate the contribution of muonic vacuum polarization to the g-factor of electrons bound in highly charged ions.
• To assess the detectability of these corrections in upcoming trapped-ion experiments with medium- and high-Z ions.
• To evaluate the impact of muonic VP effects on the precision determination of the fine-structure constant α.
• To compare the magnitude of muonic VP contributions with other QED and nuclear effects in the g-factor shift.
• To determine whether weighted g-factor differences can suppress nuclear uncertainties and isolate the muonic VP signal.

Proposed method

• Calculated electric and magnetic loop contributions of muonic vacuum polarization using relativistic quantum electrodynamics (QED) in the presence of a strong Coulomb potential.
• Evaluated Uehling and Wichmann-Kroll potentials for muon loops, incorporating all-order Coulomb-Dirac propagators for both the bound electron and virtual muons.
• Used the Dirac equation with nuclear potential to solve for electron wave functions and compute matrix elements of the VP corrections.
• Applied weighted differences of g-factors in H-like and Li-like ions to cancel nuclear size and finite-size effects.
• Estimated hadronic VP contributions via experimental e⁺e⁻ annihilation data, scaling them relative to the muonic VP effect.
• Assessed theoretical uncertainties and identified dominant missing contributions, particularly two-loop QED effects beyond (Zα)⁴ order.

Experimental results

Research questions

• RQ1 How do muonic vacuum polarization corrections affect the g-factor of electrons bound in high-Z ions?
• RQ2 Can these muonic VP effects be experimentally resolved in trapped-ion setups with intermediate- and high-Z ions?
• RQ3 What is the relative magnitude of muonic VP corrections compared to other QED and nuclear effects in the g-factor shift?
• RQ4 To what extent can weighted g-factor differences suppress nuclear uncertainties and isolate the muonic VP signal?
• RQ5 How do hadronic vacuum polarization contributions compare to muonic ones in the context of bound-electron g-factor measurements?

Key findings

• The muonic vacuum polarization correction to the bound-electron g-factor is significantly enhanced in strong Coulomb fields, exceeding the free-electron value by orders of magnitude in high-Z ions.
• For Z > 14, the muonic VP contribution becomes large enough to be observable in planned trapped-ion experiments such as ALPHATRAP.
• The muonic VP effect is approximately 5.4 × 10⁻¹² in magnitude, comparable to the leading-order QED corrections in free-electron g-factor measurements.
• Hadronic vacuum polarization contributions are estimated to increase the total light-by-light scattering effect by 20–30% of the muonic contribution, making them potentially observable via weighted g-factor differences.
• Weighted differences of g-factors in H-like and Li-like ions can suppress nuclear size uncertainties, enabling isolation of the muonic VP signal even when nuclear parameters are imperfectly known.
• The muonic VP correction must be included in future high-precision determinations of the fine-structure constant α from bound-electron g-factors to avoid systematic errors.