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

Hadronic vacuum polarization (hVP) enters simple atomic systems at a level that is small yet decisive for the precision spectroscopy now underway.

We evaluate the hVP contributions to the Lamb shift and the hyperfine splitting (HFS) in ordinary and muonic hydrogen (H and $\mu$H) and hydrogen-like helium-3 ions ($^3$He$^+$ and $\mu^3$He$^+$), using the dispersive data-driven approach and state-of-the-art empirical parametrizations of the $R$ ratio.

At the centre of the analysis is the interplay of recoil and finite-size effects: the recoil corrections that dominate the HFS in muonium (Mu), where both constituents are pointlike, are shown to be suppressed by the nuclear elastic form factors (FFs).

Our results for the leading hVP contribution to the Lamb shift agree with the literature within uncertainties.

Furthermore, we present a first evaluation of the subleading $O(Z^5\alpha^6)$ hVP-finite-size correction, which is by no means negligible in $\mu^3$He$^+$.

Our results for the hVP contribution to the HFS deviate significantly from all previous evaluations.

For the ground-state HFS, we obtain $2.153(11)~\mu$eV in $\mu$H and $-15.19(57)~\mu$eV in $\mu^3$He$^+$, as well as $0.0860(4)~$kHz and $-0.476(17)~$kHz in ordinary H and $^3$He$^+$, respectively.

Notably, our result for $\mu$H differs from previous evaluations by roughly ten times the experimental precision anticipated by the upcoming CREMA and FAMU measurements.