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Light two-body systems such as hydrogen, muonic hydrogen, or muonium are the most viable candidates for tests of the Standard Model of fundamental interactions at low energies. It is because these simple systems allow for highly accurate theoretical predictions. Therefore, from the comparison of theory with precise experimental data we are able to search for new physics and to determine values of fundamental constants such as the electron mass, the nuclear magnetic moments, and nuclear charge radii. In the framework of Nonrelativistic QED (NRQED), the nonrelativistic energy of these systems can be obtained by analytically solving the Schrodinger equation, and relativistic and QED corrections can be obtained perturbatively in powers of the fine structure constant $\alpha$ and the nuclear charge $Z$ with exact dependence on the ratio of masses of the orbiting particle to the nucleus.
Two-body antiprotonic atoms are exotic systems where the nucleus is orbited by an antiproton. These systems are interesting candidates for QED tests [1] and for exploring possible new physics, such as a long-range interaction between hadrons. In general, for heavy antiprotonic atoms one cannot use the NRQED since $Z\alpha$ is no longer a small parameter and the expansion of energy levels does not converge well. On the other hand, one also cannot use Dirac equation because the mass ratio of the orbiting particle and nucleus is not a small parameter. However, in the case of highly excited circular states, it can be shown that the NRQED expansion parameter is $Z\alpha/n$ where $n$ is the principal quantum number, and the NRQED series thus converges rapidly. In our work [2] we demonstrated the usability of this approach by calculating the energy levels for various two-body antiprotonic atoms, which are being considered for measurements by the PAX collaboration [1]. By including the vacuum polarization contribution in the Schrodinger equation and solving it numerically, we obtained the most accurate theoretical predictions for these systems which are valid for an arbitrary ratio of masses of the orbiting particle and the nucleus. Moreover, we show that the comparison of theory and experimental data can potentially be used for very accurate determination of nuclear charge radii. Our code can be further extended to muonic atoms [3] and can account for hyperfine interaction.
[1] G. Baptista, S. Rathi, M. Roosa, Q. Senetaire, J. Sommerfeldt, T. Azuma, D. Becker, F. Butin, O. Eizenberg, J. Fowler et al., PoS EXA-LEAP2024, 085 (2025).
[2] V. Patkóš and K. Pachucki, Phys. Rev. A 112, 052808 (2025).
[3] B. Ohayon, A. Abeln, S. Bara, T. E. Cocolios, O. Eizenberg, A. Fleischmann, L. Gastaldo, C. Godinho, M. Heines, D. Hengstler et al., Physics 6, 206 (2024)