Precision calculation of the bound-electron $g$ factor in molecular hydrogen ions

TL;DR

This paper presents highly precise calculations of the bound-electron $g$ factor in molecular hydrogen ions, incorporating relativistic and QED corrections, significantly improving accuracy for spectroscopic and fundamental physics applications.

Contribution

The study introduces a comprehensive calculation of the bound-electron $g$ factor in H$_2^+$ and HD$^+$, achieving unprecedented accuracy and including relativistic and QED effects up to order $ ext{α}^5$.

Findings

Achieved a relative accuracy of 4-5×10^{-11} for the scalar $g$ factor.

Improved previous calculations by over three orders of magnitude.

Provided results useful for molecular spectroscopy and tests of QED.

Abstract

We calculate the bound-electron $g$ factor for a wide range of rovibrational states of the molecular hydrogen ions H$_2^+$ and HD$^+$. Relativistic and QED corrections of orders up to $\alpha^5$ are taken into account. All contributions are calculated in a nonrelativistic QED framework, except for relativistic corrections of order $(Z\alpha)^4$ and above, which are obtained by calculating the relativistic $g$ factor using a precise minmax finite element solution of the two-center Dirac equation. A relative accuracy of $4-5 \times 10^{-11}$ is achieved for the scalar $g$ factor component, which represents an improvement by more than three orders of magnitude over previous calculations. These results are useful for internal state identification and rovibraional spectroscopy of single molecular hydrogen ions in Penning traps, and open a new avenue towards precision tests of QED.