QED corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ for HD$^+$ rovibrational transitions beyond Born-Oppenheimer approximation

TL;DR

This paper derives and computes high-order QED corrections for HD+ rovibrational transitions, improving accuracy beyond the Born-Oppenheimer approximation by using effective operators and regularization techniques.

Contribution

It introduces a finite-value effective operator formulation for $m ext{--} ext{alpha}^6$ corrections and combines recent second-order calculations to enhance precision.

Findings

Reduced uncertainty in $m ext{--} ext{alpha}^6$ corrections by a factor of three.

Implemented regularization scheme for divergent recoil operators.

Achieved more accurate theoretical predictions for HD$^+$ rovibrational transitions.

Abstract

The effective Hamiltonian of $m\alpha^6$ and $m\alpha^6(m/M)$ order corrections for hydrogen molecular ions has been derived in [ Z.-X. Zhong, \emph{et al.}, Phys. Rev. A {\bf98}, 032502(2018).], in this work we express the energy correction in the form of finite-value effective operators. The cut-off regularization scheme is used to determine finite part of divergent operators of the leading-order recoil corrections. Numerical calculations of first-order contributions are performed in the Hylleraas basis set. Combining the second-order terms calculated in recent work [V. I. Korobov, \emph{et al.}, Mol. Phys. e2563023 (2025).], the $m\alpha^6$-order corrections for the fundamental rovibrational transition are obtained with an uncertainty three times smaller than in previous calculations.