One-loop vacuum polarization at $m\alpha^7$ order for the two center problem

TL;DR

This paper calculates the one-loop vacuum polarization effects at the $m ext{alpha}^7$ order for two-center systems, providing numerical results for hydrogen molecular ions and antiprotonic helium, advancing precision in quantum electrodynamics calculations.

Contribution

It introduces a detailed calculation of the vacuum polarization contribution at high order for two-center problems within the NRQED framework, including specific numerical results for fundamental molecular transitions.

Findings

Numerical vacuum polarization contributions for H$_2^+$ and HD$^+$ at $m ext{alpha}^7$ order.

Results for fundamental transitions in two-center systems.

Enhanced precision in quantum electrodynamics for molecular ions.

Abstract

We present calculations of the one-loop vacuum polarization contribution (Uehling potential) for the two-center problem in the NRQED formalism. The cases of hydrogen molecular ions ($Z_1=Z_2=1$) as well as antiprotonic helium ($Z_1=2$, $Z_2=-1$) are considered. Numerical results of the vacuum polarization contribution at $m\alpha^7$ order for the fundamental transitions $(v=0,L=0)\to(v'=1,L'=0)$ in H$_2^+$ and HD$^+$ are presented.