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