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

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

Abstract

The effective Hamiltonian of $mα^6$ and $mα^6(m/M)$ 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 $mα^6$-order corrections for the fundamental rovibrational transition are obtained with an uncertainty three times smaller than in previous calculations.

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

Abstract

The effective Hamiltonian of and 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 -order corrections for the fundamental rovibrational transition are obtained with an uncertainty three times smaller than in previous calculations.
Paper Structure (22 sections, 106 equations, 4 tables)

This paper contains 22 sections, 106 equations, 4 tables.

Table of Contents

  1. introduction
  2. the $m\alpha^6$ order effective Hamiltonian
  3. Elimination of Singularities and Regularization
  4. Singularity in nonrecoil correction
  5. Singularity in recoil correction
  6. the regularized effective Hamiltonian
  7. Numerical Calculations and Conclusion
  8. Coordinate cutoff regularization
  9. Divergent matrix elements
  10. Matrix elements $\langle V_a^2\vb{p}_e^2\rangle$, $\langle V_a^2\vb{p}_a^2\rangle$, $\langle V_a\vb{p}_e^2V_a\rangle$, $\langle V_a\vb{p}_a^2V_a\rangle$ and $\langle{V}\vb{p}_a^2V\rangle$
  11. Matrix elements $\langle\{\vb{p}_e^2,\rho_a\}\rangle$, $\langle\{\vb{p}_e^4,V_a\}\rangle$, $\langle\{\vb{p}_e^4,V_{12}\}\rangle$ and $\langle \vb{p}_e^2V_a\vb{p}_e^2\rangle$
  12. Matrix element $\langle U(E_0-H_0)U\rangle$
  13. Derivation of modified effective Hamiltonian
  14. expansion of $\delta H_1$
  15. expansion of $\delta H_2$
  16. ...and 7 more sections