QED corrections of orders <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><msup><mi>α</mi><mn>6</mn></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><msup><mi>α</mi><mn>6</mn></msup><mrow><mo>(</mo><mi>m</mi><mo>/</mo><mi>M</mi><mo>)</mo></mrow></mrow></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>HD</mi></mrow><mo>+</mo></msup></math> spin-independent rovibrational transitions beyond Born-Oppenheimer approximation

Zhen-Xiang Zhong; Ping Yang; Vladimir I. Korobov; Chun Li; Ting-Yun Shi

doi:10.1103/x5q9-pdcj

QED corrections of orders $m α^{6}$ and $m α^{6} (m / M)$ for ${HD}^{+}$ spin-independent rovibrational transitions beyond Born-Oppenheimer approximation

Center for Theoretical Physics, School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China

Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China and University of Chinese Academy of Sciences, Beijing 100049, China

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia

School of Mathematics, Nanjing University, Nanjing Jiangsu 210093, China

Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China

^*Contact author: zxzhong@hainanu.edu.cn

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Phys. Rev. A 114, 032814 – Published 10 September, 2026

DOI: https://doi.org/10.1103/x5q9-pdcj

Abstract

The effective Hamiltonian of $m α^{6}$ - and $m α^{6} (m / M)$ -order corrections for hydrogen molecular ions has been derived by Zhong et al. [Z.-X. Zhong, W.-P. Zhou, and X.-S. Mei, Phys. Rev. A 98, 032502 (2018)]; in this work we express the energy correction in the form of finite-value effective operators. The cutoff regularization scheme is used to determine the 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, J.-P. Karr, and Z.-X. Zhong, Mol. Phys. e2563023 (2025)], the $m α^{6}$ -order corrections for the fundamental rovibrational transition are obtained with an uncertainty three times smaller than in previous calculations.

Physical Review A

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QED corrections of orders $m α^{6}$ and $m α^{6} (m / M)$ for ${HD}^{+}$ spin-independent rovibrational transitions beyond Born-Oppenheimer approximation

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QED corrections of orders mα6 and mα6(m/M) for HD+ spin-independent rovibrational transitions beyond Born-Oppenheimer approximation

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QED corrections of orders $m α^{6}$ and $m α^{6} (m / M)$ for ${HD}^{+}$ spin-independent rovibrational transitions beyond Born-Oppenheimer approximation