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This paper presents a data-driven evaluation of the hadronic vacuum polarization (hVP) contributions to the Lamb shift and hyperfine splitting (HFS) in hydrogen-like atoms and ions, specifically muonium (Mu), hydrogen (H), muonic hydrogen (µH), helium-3 ion (³He+), and muonic helium-3 ion (µ³He+). The study emphasizes the interplay between recoil and finite-size effects, finding that nuclear elastic form factors significantly suppress recoil corrections in systems with composite nuclei, unlike in pointlike muonium. The paper reports new results for the hVP contribution to the HFS that deviate significantly from previous evaluations.

The paper addresses the following challenges:

The need for precise theoretical predictions of hadronic vacuum polarization (hVP) to match the increasing precision of atomic spectroscopy experiments.
The non-perturbative nature of QCD, which makes direct calculation of hVP difficult.
The complex interplay of recoil and finite-size effects in atomic systems, which can lead to significant cancellations and require careful treatment.
Discrepancies in previous evaluations of hVP contributions, particularly for the hyperfine splitting, which the authors attribute to issues with rescaling muonic vacuum polarization and calculational errors.

A consistent evaluation of hVP contributions to the Lamb shift and hyperfine splitting for Mu, H, µH, ³He+, and µ³He+.
A detailed analysis of the interplay between recoil and finite-size effects, demonstrating how nuclear elastic form factors suppress recoil corrections in composite systems.
A first evaluation of the subleading O(Z⁵α) hVP-finite-size correction, found to be non-negligible in µ³He+.
New results for the hVP contribution to the HFS that differ significantly from previous evaluations, with improved precision.

The study employs a data-driven dispersive approach to evaluate the hVP contributions. This involves using empirical parametrizations of the R ratio, which represents the cross-section of electron-positron annihilation into hadrons. The method incorporates nuclear recoil and finite-size effects on an equal footing with the hVP.

Key Equations: The paper utilizes the Lorentz- and gauge-invariant VP tensor: Πµν (q) = (gµνq² – q“q″) Π(q²) where Π(q²) is a scalar function satisfying a once-subtracted dispersion relation. The R ratio is related to the imaginary part of the hVP tensor by: Im IhVP (s) = -α/3 * R(s) The formalism also involves weighting functions, such as W(t) for HFS and U(t) for Lamb shift, which are used to combine the VP contributions with recoil and finite-size effects.

Data Source: The analysis uses the DHMZ parametrization of the R ratio [20, 31].
Nuclear Structure: Elastic electromagnetic form factors (FFs) of the nucleus are used to account for finite-size effects. For protons, FFs from [26] are used, and for helium, FFs from [28] are employed.
Recoil and Finite-Size Interplay: The method explicitly considers the interplay of recoil and finite-size effects, particularly how elastic FFs cut off the loop integral at a softer scale, suppressing recoil corrections in composite nuclei.
Calculational Approach: The paper calculates contributions to the Lamb shift and HFS by integrating the VP spectral function weighted by specific functions that incorporate recoil and finite-size effects.
Uncertainty Estimation: Uncertainties are propagated from the R ratio parametrization and from the elastic FFs. For proton FFs, a 4% model uncertainty is assigned but not included in the main tables.

Datasets: The study uses the DHMZ parametrization of the R ratio. Nuclear elastic form factors from [26] for protons and [28] for helium are used.
Metrics: Results are reported for the Lamb shift and hyperfine splitting in units of kHz and µeV.
Baselines: The paper compares its results with previous calculations from literature, including those by Sapirstein et al. [25], Faustov et al. [53], Czarnecki et al. [54], Nomura and Teubner [55], Keshavarzi et al. [56], Karshenboim and Shelyuto [1], and Borie [66].
Numeric Results:
- For the ground-state HFS in µH, the paper obtains 2.153(11) µeV.
- For the ground-state HFS in µ³He+, the paper obtains -15.19(57) µeV.
- For the ground-state HFS in ordinary H, the paper obtains 0.0860(4) kHz.
- For the ground-state HFS in ³He+, the paper obtains -0.476(17) kHz.
- The results for the hVP contribution to the Lamb shift agree with the literature within uncertainties.
- A first evaluation of the subleading O(Z⁵α) hVP-finite-size correction is presented, which is significant for µ³He+.
Runtime/Training Cost: Not stated in the paper.

The paper assigns a 4% model uncertainty to calculations involving proton FFs, which is not included in the presented tables.
The DHMZ evaluation of the hVP faces tensions among different experimental datasets for the R ratio, particularly in the π⁺π⁻ channel. While the DHMZ framework is designed to address these, the paper notes that to reconcile experimental and theoretical values of αμ, the uncertainty of ahVP would need to be inflated by a factor of 2.44.

Method evidence: "We evaluate the hVP contributions to the Lamb shift and the hyperfine splitting (HFS) in ordinary and muonic hydrogen (H and µH) and hydrogen-like helium-3 ions (³He+ and µ³He+), using the dispersive data-driven approach and state-of-the-art empirical parametrizations of the R ratio." (page: Abstract)
Result evidence: "Our results for the hVP contribution to the HFS deviate significantly from all previous evaluations. For the ground-state HFS, we obtain 2.153(11) µeV in µH and −15.19(57) μεV in µ³He⁺, as well as 0.0860(4) kHz and -0.476(17) kHz in ordinary H and He+, respectively." (page: Abstract)
Limitation evidence: "The DHMZ approach enables combining cross-section data with arbitrary point spacing or binning, employing spline-based interpolations to redistribute them in a fine common binning. The average weights, derived through a x² minimisation, also account for the different bin sizes and point-spacings of measurements. In addition, the x² evaluation enables a local test of the level of agreement among the input measurements." (page: 9)

Hadronic vacuum polarization in hydrogen-like atoms and ions amid the interplay of recoil and finite-size effects

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