Abstract
We report on progress in calculation of the spin–orbit interaction for the HD\(^+\) molecular ion. This interaction is currently the largest source of theoretical uncertainty in determination of the hyperfine structure of ro-vibrational transition lines. The corrections of order \(m\alpha ^7\ln (\alpha )\) are derived and numerically calculated. Theoretical hyperfine intervals are compared with experimental data, and the observed discrepancies are discussed.
Graphical abstract
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study].
References
S. Alighanbari, G.S. Giri, F.L. Constantin, V.I. Korobov, S. Schiller, Precise test of quantum electrodynamics and determination of fundamental constants with HD\(^+\) ions. Nature 581, 152 (2020)
S. Patra, M. Germann, J.-P. Karr, M. Haidar, L. Hilico, V.I. Korobov, F.M.J. Cozijn, K.S.E. Eikema, W. Ubachs, J.C.J. Koelemeij, Proton–electron mass ratio from laser spectroscopy of HD\(^+\) at the part-per-trillion level. Science 369, 1238 (2020)
I. Kortunov, S. Alighanbari, M.G. Hansen, G.S. Giri, S. Schiller, V.I. Korobov, Proton–electron mass ratio by high-resolution optical spectroscopy of ion ensemble in the resolved-carrier regime. Nat. Phys. 17, 569 (2021)
V.I. Korobov, L. Hilico, J.-Ph. Karr, Fundamental transitions and ionization energies of the hydrogen molecular ions with few ppt uncertainty. Phys. Rev. Lett. 118, 233001 (2017)
V.I. Korobov, J.-Ph. Karr, Rovibrational spin-averaged transitions in the hydrogen molecular ions. Phys. Rev. A 104, 032806 (2021)
V.I. Korobov, Precision spectroscopy of the hydrogen molecular ions: present status of theory and experiment. Phys. Part. Nucl. 53, 787–789 (2022)
M. Haidar, Z.-X. Zhong, V.I. Korobov, J.-Ph. Karr, NRQED approach to the fine and hyperfine structure corrections of order \(m\alpha ^6\) and \(m\alpha ^6(m/M)\): application to the hydrogen atom. Phys. Rev. A 101, 022501 (2020)
V.I. Korobov, J.-P. Karr, M. Haidar, Z.-X. Zhong, Hyperfine structure in the H\(_2^+\) and HD\(^+\) molecular ions at \(m\alpha ^6\) order. Phys. Rev. A 102, 022804 (2020)
J.-P. Karr, M. Haidar, L. Hilico, Z.-X. Zhong, V.I. Korobov, Higher-order corrections to spin–spin scalar interactions in HD\(^+\) and H\(_2^+\). Phys. Rev. A 102, 052827 (2020)
W.E. Caswell, G.P. Lepage, Effective Lagrangians for bound state problems in QED, QCD, and other field theories. Phys. Lett. B 167, 437 (1986)
T. Kinoshita, M. Nio, Radiative corrections to the muonium hyperfine structure: the \(\alpha ^2(Z\alpha )\) correction. Phys. Rev. D 53, 4909 (1996)
G. Paz, An introduction to NRQED. Mod. Phys. Lett. A 80, 1550128 (2015)
A.V. Manohar, Heavy quark effective theory and nonrelativistic QCD Lagrangian to order \(a_s/m^3\). Phys. Rev. D 56, 230 (1997)
R.J. Hill, G. Lee, G. Paz, M.P. Solon, NRQED Lagrangian at order \(1/M^4\). Phys. Rev. D 87, 053017 (2013)
D. Bakalov, V.I. Korobov, S. Schiller, High-precision calculation of the hyperfine structure of the \(\text{ HD}^+\) ion. Phys. Rev. Lett. 97, 243001 (2006)
N. Ramsey, Electron coupled interaction between nuclear spins in molecule. Phys. Rev. 91, 303 (1953)
Acknowledgements
This work was done in collaboration with Laurent Hilico, Mohammad Haidar (LKB), and Zhen-Xiang Zhong (Wuhan, WIPM CAS) that is gratefully acknowledged.
Author information
Authors and Affiliations
Contributions
Both authors equally contributed to the derivation of formulas and numerical calculations presented in the text.
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Korobov, V.I., Karr, JP. Spin–orbit interaction in the HD\(^+\) ion. Eur. Phys. J. D 76, 197 (2022). https://doi.org/10.1140/epjd/s10053-022-00522-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjd/s10053-022-00522-3