Test of the theoretical hyperfine structure of the molecular hydrogen ion at the 1-ppm level

TL;DR

This paper refines the theoretical calculation of hyperfine splitting in H₂⁺ by including a previously overlooked second-order relativistic correction, achieving excellent agreement with experimental data at the 1-ppm level.

Contribution

It introduces a new second-order relativistic correction related to nuclear vibrational motion, resolving a long-standing discrepancy between theory and experiment.

Findings

Theoretical predictions now match experimental data within 1-ppm accuracy.

A previously unrecognized correction significantly improves the hyperfine structure calculations.

The work supports the proton structure properties derived from atomic hydrogen hyperfine data.

Abstract

We revisit the $m \alpha^6 (m/M)$ order corrections to the hyperfine splitting in the H$_2^+$ ion, and find a hitherto unrecognized second-order relativistic contribution associated with the vibrational motion of the nuclei. Inclusion of this correction term produces theoretical predictions which are in excellent agreement with experimental data [K. B. Jefferts, Phys.\ Rev.\ Lett.\ \textbf{23}, 1476 (1969)], thereby concluding a nearly fifty years long theoretical quest to explain the experimental results within their 1-ppm error. The agreement between theory and experiment corroborates the proton structural properties as derived from the hyperfine structure of atomic hydrogen. Our work furthermore indicates that for future improvements, a full three-body evaluation of the $m \alpha^6 (m/M)$ correction term will be mandatory.