Higher-order corrections to spin-spin scalar interactions in HD$^+$ and H$_2^+$

TL;DR

This paper calculates high-precision hyperfine interaction coefficients in HD$^+$ and H$_2^+$, including relativistic and QED corrections, and compares results with experiments, revealing a significant discrepancy in HD$^+$ hyperfine splitting.

Contribution

The paper provides the most accurate calculations to date of hyperfine coefficients in HD$^+$ and H$_2^+$, incorporating higher-order relativistic and QED corrections.

Findings

Calculated hyperfine coefficients with uncertainties below 1 ppm.

Excellent agreement with experiments for H$_2^+$.

Detected a 4$\sigma$ discrepancy in HD$^+$ hyperfine splitting.

Abstract

The largest hyperfine interaction coefficients in the hydrogen molecular ion HD$^+$, i.e. the electron-proton and electron-deuteron spin-spin scalar interactions, are calculated with estimated uncertainties slightly below 1~ppm. The $(Z\alpha)^2 E_F$ relativistic correction, for which a detailed derivation is presented, QED corrections up to the order $\alpha^3 \ln^2 (\alpha)$ along with an estimate of higher-order terms, and nuclear structure corrections are taken into account. Improved results are also given for the electron-proton interaction coefficient in H$_2^+$, in excellent agreement with RF spectroscopy experiments. In HD$^+$, a 4$\sigma$ difference is found in the hyperfine splitting of the $(v,L)=(0,3) \to (9,3)$ two-photon transition that was recently measured with high precision. The origin of this discrepancy is unknown.