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 μH) and hydrogen-like helium-3 ions (^3He^+ and μ^3He^+), 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^5α^6) hVP-finite-size correction, which is by no means negligible in μ^3He^+. 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) μeV in μ^3He^+, as well as 0.0860(4)kHz and -0.476(17)kHz in ordinary H and ^3He^+, respectively. Notably, our result for μH differs from previous evaluations by roughly ten times the experimental precision anticipated by the upcoming CREMA and FAMU measurements.

The evaluation of the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment requires precise knowledge of the pion transition form factor (TFF). In this work, we present a feasibility study for a combined analysis of lattice QCD (LQCD) and experimental data. Our methodology is driven by the goal of combining complementary datasets to leverage their respective kinematic advantages: while LQCD provides robust predictions for the doubly-virtual TFF, e^+e^- scattering experiments offer high-precision singly-virtual measurements up to large momentum transfers. To ensure a statistically rigorous combination, we implement a global one-stage fitting approach based on the modified z-expansion, utilizing synthetic jackknife replicate sampling and a normalized χ^2 weighting scheme. We demonstrate that the inclusion of experimental data substantially tightens the constraints on the pion TFF, yielding up to a factor of three reduction in uncertainty in the singly-virtual limit. In contrast, the uncertainty of the resulting pion-pole contribution to the muon g-2 improves by a factor of 1.5. This more modest improvement reflects the fact that the g-2 integral is heavily dominated by the low-Q^2 region, which is already well constrained by physical normalization constraints.

We attempt to rehabilitate a sum rule (proposed long ago by Bernab\'eu and Tarrach) which relates the electric polarizability of a particle to the total photoabsorption of quasi-real longitudinally polarized photons by that particle. We discuss its perturbative verification in QED, which is largely responsible for the scepticism about its validity. The failure of the QED test can be understood via the Sugawara-Kanazawa theorem and is due to the non-vanishing contour contribution in the pertinent dispersion relation. We show another example where this contribution is absent and the perturbative test works exactly. On the empirical side, we show that the sum rule gives a reasonable estimate of the $\pi N$-channel contribution to the proton electric polarizability. If this sum rule is valid indeed, there should be a sum rule for the so-called ``subtraction function'' entering the data-driven calculations of the polarizability effects in the Lamb shift. We have written down a possible sum rule for the subtraction function and verified it in a perturbative calculation.

A comprehensive theory of the Lamb shift in light muonic atoms such as mu H, mu D, mu 3He+, and mu 4He+ is presented, with all quantum electrodynamic corrections included at the precision level constrained by the uncertainty of nuclear structure effects. This analysis can be used in the global adjustment of fundamental constants and in the determination of nuclear charge radii. Further improvements in the understanding of electromagnetic interactions of light nuclei will allow for a promising test of fundamental interactions by comparison with "normal" atomic spectroscopy, in particular, with H -D and 3He-4He isotope shifts.

Two groups, ours (Mainz) and Bochum, have recently been re-evaluating the spin polarizabilities and spin structure functions at low $Q$, using the baryon chiral perturbation theory (B$\chi$PT), the manifestly-covariant counterpart of the heavy-baryon chiral perturbation theory (HB$\chi$PT). Whilst the two groups agree that the B$\chi$PT framework works better than HB$\chi$PT in this sector, their quantitative results disagree in some of the quantities; most notably, the proton spin polarizabilities $\gamma_0$ and $\delta_{LT}$. These discrepancies are especially intriguing in light of new experimental data coming from the Jefferson Lab "Spin Physics Program". The preliminary data on the proton are reported by Karl Slifer in a plenary session of this workshop. Another theoretical discrepancy is emerging in the proton-polarizability contribution to the hyperfine splitting (hfs) in hydrogen and muonic hydrogen. Our B$\chi$PT calculation shows a significantly smaller effect than the state-of-the-art data-driven evaluations based on empirical spin structure functions. The smaller polarizability contribution leads to a smaller Zemach radius of the proton. This discrepancy could be relevant for the planned first-ever measurement of the ground-state hfs in muonic hydrogen.

We present our studies of the forward unpolarised doubly-virtual Compton scattering (VVCS) off the deuteron and the closely related two-photon-exchange ($2\gamma$-exchange) corrections to the Lamb shift of muonic deuterium. The deuteron VVCS amplitude is calculated in the framework of pionless effective field theory, up to next-to-next-to-next-to-leading order (N3LO) for the longitudinal and next-to-leading order (NLO) for the transverse amplitude. The charge elastic form factor of the deuteron, obtained from the residue of the longitudinal VVCS amplitude, is used to extract the value of the single unknown two-nucleon one-photon contact coupling that enters the longitudinal amplitude at N3LO. The obtained deuteron VVCS amplitude serves as a high-precision model-independent input to examine the $2\gamma$-exchange corrections. Substantial differences with the recent dispersive evaluations are identified, namely, the elastic contribution appears to be larger by several standard deviations, thus ameliorating the current discrepancy between theory and experiment on the size of $2\gamma$-exchange effects. A correlation between the values of the deuteron charge and Friar radii is found that can be used to judge on the quality of a parametrisation of the deuteron charge elastic form factor. The discrepancy between the theory and the empirical result for the $2\gamma$-exchange correction in muonic deuterium appears to be completely eliminated. To further confirm this, we revisit the hydrogen-deuterium isotope shift in the same framework. Our work provides an alternative self-consistent and high-precision evaluation of the $2\gamma$-exchange correction in (muonic) deuterium.

The ongoing experimental efforts to measure the hyperfine transition in muonic hydrogen prompt an accurate evaluation of the proton-structure effects. At the leading order in α , which is O(α ^5) in the hyperfine splitting (hfs), these effects are usually evaluated in a data-driven fashion, using the empirical information on the proton electromagnetic form factors and spin structure functions. Here we perform a first calculation based on the baryon chiral perturbation theory (B χ PT). At leading orders it provides a prediction for the proton polarizability effects in hydrogen (H) and muonic hydrogen ( μ H). We find large cancellations among the various contributions leading to, within the uncertainties, a zero polarizability effect at leading order in the B χ PT expansion. This result is in significant disagreement with the current data-driven evaluations. The small polarizability effect implies a smaller Zemach radius R_Z , if one uses the well-known experimental 1 S hfs in H or the 2 S hfs in μ H. We, respectively, obtain R_Z(H) = 1.010(9) fm, R_Z(μH) = 1.040(33) fm. The total proton-structure effect to the hfs at O(α ^5) is then consistent with previous evaluations; the discrepancy in the polarizability is compensated by the smaller Zemach radius. Our recommended value for the 1 S hfs in μH is 182.640(18) meV.

We present a study of the two-photon-exchange (2 -exchange) corrections to the S -levels in muonic ( μ D) and ordinary (D) deuterium within the pionless effective field theory ( EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the EFT expansion. The only unknown low-energy constant entering the calculation at this order corresponds to the coupling of a longitudinal photon to the nucleon–nucleon system. To minimise its correlation with the deuteron charge radius, it is extracted using the information about the hydrogen–deuterium isotope shift. We find the elastic 2 -exchange contribution in μ D larger by several standard deviations than obtained in other recent calculations. This discrepancy ameliorates the mismatch between theory and experiment on the size of 2 -exchange effects, and is attributed to the properties of the deuteron elastic charge form factor parametrisation used to evaluate the elastic contribution. We identify a correlation between the deuteron charge and Friar radii, which can help one to judge how well a form factor parametrisation describes the low-virtuality properties of the deuteron. We also evaluate the higher-order 2 -exchange contributions in μ D, generated by the single-nucleon structure and expected to be the most important terms beyond N3LO. The uncertainty of the theoretical result is dominated by the truncation of the EFT series and is quantified using a Bayesian approach. The resulting extractions of the deuteron charge radius from the μ D Lamb shift, the 2S-1S transition in D, and the 2S-1S hydrogen–deuterium isotope shift, with the respective 2 -exchange effects evaluated in a unified EFT approach, are in perfect agreement.

We provide a systematic assessment of the order-α5 nuclear contributions to the Lamb shift of muonic deuterium, including the accompanying radiative corrections due to vacuum polarisation, up to next-to-next-to-next-to-leading order (N3LO) within the pionless effective field theory (π̸EFT). We also evaluate higher-order corrections due to the single-nucleon structure, which are expected to be the most important corrections beyond N3LO. We find a correlation between the deuteron charge and Friar radii, which can be useful to judge the quality of charge form factor parametrisations. We refine the theoretical description of the 2γ-exchange contribution, especially in the elastic contribution and the radiative corrections, ameliorating the original discrepancy between theory and experiment in the size of 2γ-exchange effects. Based on the experimental Lamb shift of muonic deuterium, we obtain the deuteron charge radius, rd(μD)=2.12763(13)exp(77)theory fm, which is consistent with (but less precise than) the value obtained by combining the H-D isotope shift with the muonic-hydrogen Lamb shift. The theory uncertainty is evaluated using a Bayesian procedure and is dominated by the truncation of the π̸EFT series.

Recent progress in laser and x-ray spectroscopy of muonic atoms offers promising long-term possibilities at the intersection of atomic, nuclear and particle physics. In muonic hydrogen, laser spectroscopy measurements will determine the ground-state hyperfine splitting (HFS) and additionally improve the Lamb shift by a factor of 5. Precision spectroscopy with cryogenic microcalorimeters has the potential to significantly improve the charge radii of the light nuclei in the Z=3-8 range. Complementary progress in precision should be achieved on the theory of nucleon- and nuclear-structure effects. The impact of this muonic-atom spectroscopy program will be amplified by the upcoming results from H and He^+ spectroscopy, simple molecules such as HD^+ and Penning trap measurements. In this broader context, one can test ab-initio nuclear theories, bound-state QED for two- or three-body systems, and determine fundamental constants, such as the Rydberg (R_∞) and the fine-structure (α) constants.

We merge the dispersive relation approach and the ab initio method to compute nuclear structure corrections to the Lamb shift in muonic deuterium. We calculate the deuteron response functions and corresponding uncertainties up to next-to-next-to-next-to-leading order in chiral effective field theory and compare our results to selected electromagnetic data to test the validity of the theory. We then feed response functions calculated over a wide range of kinematics to the dispersion-theory formalism and show that an improved accuracy is obtained compared to that with the use of available experimental data in the dispersive analysis. This opens up the possibility of applying this hybrid method to other light muonic atoms and supplementing experimental data with ab initio theory for kinematics where data are scarce or difficult to measure with the goal of reducing uncertainties in estimates of nuclear structure effects in atomic spectroscopy.

We calculate the forward unpolarised doubly-virtual Compton scattering (VVCS) off the deuteron in the framework of pionless effective field theory, up to next-to-next-to-next-to-leading order (N3LO) for the longitudinal and next-to-leading order (NLO) for the transverse amplitude. The charge elastic form factor of the deuteron, obtained from the residue of the longitudinal VVCS amplitude, is used to extract the value of the single unknown two-nucleon one-photon contact coupling that enters the longitudinal amplitude at N3LO. We also study the lowest spin-independent generalised polarisabilities of the deuteron. The calculated unpolarised VVCS amplitude provides a high-precision model-independent input for a future calculation of the two-photon-exchange correction to the Lamb shift of muonic deuterium.

The forward doubly-virtual Compton scattering (VVCS) off the nucleon contains a wealth of information on nucleon structure, relevant to the calculation of the two-photon-exchange effects in atomic spectroscopy and electron scattering. We report on a complete next-to-leading order (NLO) calculation of low-energy VVCS in chiral perturbation theory ($\chi$PT). Here we focus on the unpolarized VVCS amplitudes $T_1(\nu, Q^2)$ and $T_2(\nu, Q^2)$, and the corresponding structure functions $F_1(x, Q^2)$ and $F_2(x,Q^2)$. Our results are confronted, where possible, with "data-driven" dispersive evaluations of low-energy structure quantities, such as nucleon polarizabilities. We find significant disagreements with dispersive evaluations at very low momentum-transfer $Q$; for example, in the slope of polarizabilities at zero momentum-transfer. By expanding the results in powers of the inverse nucleon mass, we reproduce the known "heavy-baryon" expressions. This serves as a check of our calculation, as well as demonstrates the differences between the manifestly Lorentz-invariant (B$\chi$PT) and heavy-baryon (HB$\chi$PT) frameworks.

The first experimental investigation of the near-threshold cross section for incoherent pi(-) photoproduction on the deuteron gamma d -> pi(-) pp is presented. The experimental technique involved detection of the approximate to 131 MeV gamma ray resulting from the radiative capture of photoproduced pi(-) in the target. The total cross section was measured using an unpolarized tagged-photon beam, a liquid-deuterium target, and three very large NaI(Tl) spectrometers. The data are compared to theoretical models that give insight into the elementary reaction gamma n -> pi(-)p and pion-nucleon and nucleon-nucleon final-state interactions.

We review the recent baryon chiral perturbation theory results for the nucleon polarisabilities that describe the different regimes of nucleon Compton scattering --- real, virtual, and doubly virtual. We stress the importance of the empirical verification of the theory in the context of the calculation of the inelastic nucleon structure corrections, such as the two-photon exchange contributions. We also discuss the recently obtained constraints that relate the different regimes of nucleon Compton scattering and can provide additional information on the nucleon structure.

Low-energy Compton scattering off the proton is used as an experimental tool for determination of the proton polarizabilities. However, the present empirical determinations rely heavily on the theoretical description(s) of the experimental cross sections in terms of polarizabilities. The state-of-art determinations are based on either the fixed-t dispersion relations (DR) or chiral perturbation theory in the single-baryon sector (χPT). The two approaches obtain rather different results for proton polarizabilities, most notably for βM1 (magnetic dipole polarizability). We attempt to resolve this discrepancy by performing a partial-wave analysis of the world data on proton Compton scattering below threshold. We find a large sensitivity of the extraction to a few “outliers”, leading us to conclude that the difference between DR and χPT extraction is a problem of the experimental database rather than of “model-dependence”. We have specific suggestions for new experiments needed for an efficient improvement of the database. With the present database, the difference of proton scalar polarizabilities is constrained to a rather broad interval: αE1−βM1=(6.8…10.9)×10−4fm3, with their sum fixed much more precisely [to 14.0(2)] by the Baldin sum rule.

We argue that the recently published CLAS results on the deuteron spin polarizability γ_0 [Adhikari et al., Phys. Rev. Lett. 120, 062501 (2018)], as well as their comparisons with chiral perturbation theory (χPT), are misleading. In reality, the deuteron polarizability is larger by 4 orders of magnitude, as demonstrated here by a novel calculation in pionless EFT. The CLAS paper, on the other hand, presents only a tiny correction to it, based on a partial evaluation of a sum rule for γ_0. The sum rule is assumed to have the same form as for the nucleon; we argue it does not. Moreover, their "test of χPT" tacitly involves assumptions which, as we demonstrate, may not be valid at the claimed accuracy.

We derive two new sum rules for the unpolarized doubly virtual Compton scattering process on a nucleon, which establish novel low-Q(2) relations involving the nucleon's generalized polarizabilities and moments of the nucleon's unpolarized structure functions F-1 (x, Q(2)) and F-2 (x, Q(2)). These relations facilitate the determination of some structure constants which can only be accessed in off-forward doubly virtual Compton scattering, not experimentally accessible at present. We perform an empirical determination for the proton and compare our results with a next-to-leading-order chiral perturbation theory prediction. We also show how these relations may be useful for a model-independent determination of the low-Q(2) subtraction function in the Compton amplitude, which enters the two-photon-exchange contribution to the Lamb shift of (muonic) hydrogen. An explicit calculation of the Delta(1232)-resonance contribution to the muonic-hydrogen 2P - 2S Lamb shift yields -1 +/- 1 mu eV, confirming the previously conjectured smallness of this effect.

The nucleon generalized polarizabilities (GPs), probed in virtual Compton scattering (VCS), describe the spatial distribution of the polarization density in a nucleon. They are accessed experimentally via the process of electron–proton bremsstrahlung (\(ep\rightarrow ep\gamma \)) at electron-beam facilities, such as MIT-Bates, CEBAF (Jefferson Lab), and MAMI (Mainz). We present the calculation of the nucleon GPs and VCS observables at next-to-leading order in baryon chiral perturbation theory (B\(\chi \)PT), and confront the results with the empirical information. At this order our results are predictions, in the sense that all the parameters are well known from elsewhere. Within the relatively large uncertainties of our calculation we find good agreement with the experimental observations of VCS and the empirical extractions of the GPs. We find large discrepancies with previous chiral calculations – all done in heavy-baryon \(\chi \)PT (HB\(\chi \)PT) – and discuss the differences between B\(\chi \)PT and HB\(\chi \)PT responsible for these discrepancies.

The scalar dipole polarizabilities, α_E1 and β_M1, are fundamental properties related to the internal dynamics of the nucleon. The currently accepted values of the proton polarizabilities were determined by fitting to unpolarized proton Compton scattering cross section data. The measurement of the beam asymmetry Σ_3 in a certain kinematical range provides an alternative approach to the extraction of the scalar polarizabilities. At the Mainz Microtron (MAMI) the beam asymmetry was measured for Compton scattering below pion photoproduction threshold for the first time. The results are compared with model calculations and the influence of the experimental data on the extraction of the scalar polarizabilities is determined.