Precision Spectroscopy of the Pionic Helium-4.

The transition frequency of (n,ℓ)=(17,16)→(16,15) in pionic helium-4 is calculated to an accuracy of 4 ppb (parts per billion), including relativistic and quantum electrodynamic corrections up to O(R_{∞}α^{5}). Our calculations significantly improve the recent theoretical values [Hori et al., Phys. Rev. A 89, 042515 (2014)PLRAAN1050-294710.1103/PhysRevA.89.042515]. In addition, collisional effects between pionic helium and target helium on transition frequencies are estimated. Once measurements reach the ppb level, our Letter will improve the value of the π^{-} mass by 2-3 orders of magnitude.

Nonadiabatic, Relativistic, and Leading-Order QED Corrections for Rovibrational Intervals of

The rovibrational intervals of the ^{4}He_{2}^{+} molecular ion in its X ^{2}Σ_{u}^{+} ground electronic state are computed by including the nonadiabatic, relativistic, and leading-order quantum-electrodynamics corrections. Good agreement of theory and experiment is observed for the rotational excitation series of the vibrational ground state and the fundamental vibration. The lowest-energy rotational interval is computed to be 70.937 69(10) cm^{-1} in agreement with the most recently reported experimental value, 70.937 589(23)(60)_{sys} cm^{-1} [L. Semeria et al., Phys. Rev. Lett. 124, 213001 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.213001].

Fundamental Transitions and Ionization Energies of the Hydrogen Molecular Ions with Few ppt Uncertainty.

We calculate ionization energies and fundamental vibrational transitions for H_{2}^{+}, D_{2}^{+}, and HD^{+} molecular ions. The nonrelativistic quantum electrodynamics expansion for the energy in terms of the fine structure constant α is used. Previous calculations of orders mα^{6} and mα^{7} are improved by including second-order contributions due to the vibrational motion of nuclei. Furthermore, we evaluate the largest corrections at the order mα^{8}. That allows us to reduce the fractional uncertainty to the level of 7.6×10^{-12} for fundamental transitions and to 4.5×10^{-12} for the ionization energies.

Theoretical Hyperfine Structure of the Molecular Hydrogen Ion at the 1 ppm Level.

We revisit the mα^{6}(m/M) order corrections to the hyperfine splitting in the H_{2}^{+} ion and find a hitherto unrecognized second-order relativistic contribution associated with the vibrational motion of the nuclei. Inclusion of this correction term produces theoretical predictions which are in excellent agreement with experimental data [K. B. Jefferts, Phys. Rev. Lett. 23, 1476 (1969)], thereby concluding a nearly 50-year-long theoretical quest to explain the experimental results within their 1-ppm error. The agreement between the theory and experiment corroborates the proton structural properties as derived from the hyperfine structure of atomic hydrogen. Our work furthermore indicates that, for future improvements, a full three-body evaluation of the mα^{6}(m/M) correction term will be mandatory.

mα(7)-order corrections in the hydrogen molecular ions and antiprotonic helium.

We present a calculation of the complete set of QED corrections of order mα7 for one-electron two-center systems. Leading corrections of order mα8 are also considered, which allows us to estimate the magnitude of yet uncalculated contributions. The theoretical uncertainty on the frequencies of rovibrational transitions in the hydrogen molecular ions H2+ and HD+, and of two-photon transition in antiprotonic helium is reduced by about 1 order of magnitude, down to (3-4)×10-11 and 10-10, respectively. These results open new perspectives for improved determination of the proton- and antiproton-to-electron mass ratios by precision spectroscopy experiments.

High-precision calculation of the hyperfine structure of the HD+ ion.

High-precision laser spectroscopy of ultracold hydrogen molecular ions has the potential of improving the precision of the electron-to-proton mass ratio. An accurate knowledge of the spin structure of the transition is required in order to permit precise comparison with experimental transition frequencies. We calculate with a relative accuracy of the order of O(alpha2) the hyperfine splitting of the rovibrational states of HD+ with orbital momentum L