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Franziska Hagelstein

Publications and source records attributed to Franziska Hagelstein.

At least 19 recordsLinked to original sources
arXiv2026

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 $μ$H) and hydrogen-like helium-3 ions ($^3$He$^+$ and $μ^3$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 $O(Z^5α^6)$ hVP-finite-size correction, which is by no means negligible in $μ^3$He$^+$. Our results for the hVP contribution to the HFS deviate significantly from all previous evaluations. For the ground-state HFS, we obtain $2.153(11)~μ$eV in $μ$H and $-15.19(57)~μ$eV in $μ^3$He$^+$, as well as $0.0860(4)~$kHz and $-0.476(17)~$kHz in ordinary H and $^3$He$^+$, respectively. Notably, our result for $μ$H differs from previous evaluations by roughly ten times the experimental precision anticipated by the upcoming CREMA and FAMU measurements.

physics.atom-ph↗
arXiv2026

Combined Analysis of Lattice QCD and Experimental Data on the Pion Transition Form Factor

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.

hep-lat↗
arXiv2024

New Spin Structure Constraints on Hyperfine Splitting and Proton Size

The 1S hyperfine splitting in hydrogen is measured to an impressive ppt precision and will soon be measured to ppm precision in muonic hydrogen. The latter measurement will rely on theoretical predictions, which are limited by knowledge of the proton polarizability effect $Δ_\text{pol}$. Data-driven evaluations of $Δ_\text{pol}$ have long been in significant tension with baryon chiral perturbation theory. Here we present improved results for $Δ_\text{pol}$ driven by new spin structure data, reducing the long-standing tension between theory and experiment and halving the dominating uncertainty in hyperfine splitting calculations.

nucl-ex↗
arXiv2024

Chiral perturbation theory of the hyperfine splitting in (muonic) hydrogen

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 $R_\mathrm{Z}$, if one uses the well-known experimental $1S$ hfs in H or the $2S$ hfs in $μ$H. We, respectively, obtain $R_\mathrm{Z}(\mathrm{H}) = 1.010(9)$ fm, $R_\mathrm{Z}(μ\mathrm{H}) = 1.040(33)$ 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 $1S$ hfs in $μ\text{H}$ is $182.640(18)\,\mathrm{meV}.$

nucl-th↗
arXiv2023

The QED of Bernabéu-Tarrach sumrule for electric polarizability and its implication for the Lamb shift

We attempt to rehabilitate a sum rule (proposed long ago by Bernabéu 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 $πN$-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.

hep-ph↗
arXiv2023

Radiative Corrections: From Medium to High Energy Experiments

Radiative corrections are crucial for modern high-precision physics experiments, and are an area of active research in the experimental and theoretical community. Here we provide an overview of the state of the field of radiative corrections with a focus on several topics: lepton-proton scattering, QED corrections in deep-inelastic scattering, and in radiative light-hadron decays. Particular emphasis is placed on the two-photon exchange, believed to be responsible for the proton form-factor discrepancy, and associated Monte-Carlo codes. We encourage the community to continue developing theoretical techniques to treat radiative corrections, and perform experimental tests of these corrections.

hep-ph↗
arXiv2023

Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

Lattice QCD calculations of the hadronic vacuum polarization (HVP) have reached a precision where the electromagnetic (e.m.) correction can no longer be neglected. This correction is both computationally challenging and hard to validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared (IR) effects associated with the massless photon. While we precisely determine the UV divergence using the operator-product expansion, we propose to introduce a separation scale $Λ\sim400\;$MeV into the internal photon propagator, whereby the calculation splits into a short-distance part, regulated in the UV by the lattice and in the IR by the scale $Λ$, and a UV-finite long-distance part to be treated with coordinate-space methods, thereby avoiding power-law finite-size effects altogether. In order to predict the long-distance part, we express the UV-regulated e.m. correction to the HVP via the forward hadronic light-by-light (HLbL) scattering amplitude and relate the latter via a dispersive sum rule to $γ^*γ^*$ fusion cross-sections. Having tested the relation by reproducing the two-loop QED vacuum polarization (VP) from the tree-level $γ^*γ^*\to e^+e^-$ cross-section, we predict the expected lattice-QCD integrand resulting from the $γ^*γ^*\toπ^0$ process.

hep-lat↗
arXiv2022

Two-photon exchange in (muonic) deuterium at N3LO in pionless effective field theory

We present a study of the two-photon-exchange ($2γ$-exchange) corrections to the $S$-levels in muonic ($μ$D) and ordinary (D) deuterium within the pionless effective field theory (pionless EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the pionless 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 $2γ$-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 $2γ$-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 $2γ$-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 pionless 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 $2γ$-exchange effects evaluated in a unified pionless EFT approach, are in perfect agreement.

nucl-th↗
arXiv2022

Muonic-Atom Spectroscopy and Impact on Nuclear Structure and Precision QED Theory

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 ($R_\infty$) and the fine-structure ($α$) constants.

nucl-th↗
arXiv2022

A reassessment of nuclear effects in muonic deuterium using pionless effective field theory at N3LO

We provide a systematic assessment of the order-$α^5$ nuclear contributions to the Lamb shift of muonic deuterium, including the accompanying radiative corrections due to vacuum polarization, up to next-to-next-to-next-to-leading order (N3LO) within the pionless effective field theory (EFT). 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, $r_d(μ\text{D})=2.12763(13)_\text{exp}(77)_\text{theory}$~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 pionless EFT series.

nucl-th↗
arXiv2022

Theoretical discrepancies in the nucleon spin structure and the hyperfine splitting of muonic hydrogen

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$χ$PT), the manifestly-covariant counterpart of the heavy-baryon chiral perturbation theory (HB$χ$PT). Whilst the two groups agree that the B$χ$PT framework works better than HB$χ$PT in this sector, their quantitative results disagree in some of the quantities; most notably, the proton spin polarizabilities $γ_0$ and $δ_{LT}$. 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$χ$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.

hep-ph↗
arXiv2022

The proton structure in and out of muonic hydrogen

Laser spectroscopy of muonic atoms has been recently used to probe properties of light nuclei with unprecedented precision. We introduce nuclear effects in hydrogen-like atoms, nucleon structure quantities (form factors, structure functions, polarizabilities) and their effects in the Lamb shift and hyperfine splitting (HFS) of muonic hydrogen ($μ$H). Updated theory predictions for the Lamb shift and HFS in $μ$H are presented. We review the challenges of the ongoing effort to measure the ground-state HFS in $μ$H and its impact on our understanding of the nucleon spin structure. To narrow down this search, we present a novel theory prediction obtained by scaling the measured HFS in hydrogen leveraging radiative corrections. We also summarize recent developments in the spectroscopy of simple atomic and molecular systems and emphasize how they allow for precise determinations of fundamental constants, bound-state QED tests and New Physics searches.

nucl-th↗
arXiv2022

Deuteron VVCS and nuclear structure effects in muonic deuterium at N3LO in pionless EFT

We present our studies of the forward unpolarised doubly-virtual Compton scattering (VVCS) off the deuteron and the closely related two-photon-exchange ($2γ$-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 $2γ$-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 $2γ$-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 $2γ$-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 $2γ$-exchange correction in (muonic) deuterium.

nucl-th↗
arXiv2021

The subtraction contribution to the muonic-hydrogen Lamb shift: a point for lattice QCD calculations of the polarizability effect

The proton-polarizability contribution to the muonic-hydrogen Lamb shift is a major source of theoretical uncertainty in the extraction of the proton charge radius. An empirical evaluation of this effect, based on the proton structure functions, requires a systematically improvable calculation of the "subtraction function", possibly using lattice QCD. We consider a different subtraction point, with the aim of accessing the subtraction function directly in lattice calculations. A useful feature of this subtraction point is that the corresponding contribution of the structure functions to the Lamb shift is suppressed. The whole effect is dominated by the subtraction contribution, calculable on the lattice.

hep-ph↗
arXiv2021

Short-distance constraints for the longitudinal component of the hadronic light-by-light amplitude: an update

We reassess the impact of short-distance constraints for the longitudinal component of the hadronic light-by-light amplitude on the anomalous magnetic moment of the muon, $a_μ=(g-2)_μ/2$, by comparing different solutions that have recently appeared in the literature. In particular, we analyze the relevance of the exact axial anomaly and its impact on $a_μ$ and conclude that it remains rather limited. We show that all recently proposed solutions agree well within uncertainties on the numerical estimate of the impact of short-distance constraints on $a_μ$, despite differences in the concrete implementation. We also take into account the recently calculated perturbative corrections to the massless quark loop to update our estimate and outline the path towards future improvements.

hep-ph↗
arXiv2021

Forward doubly-virtual Compton scattering off the nucleon in chiral perturbation theory: II. Spin polarizabilities and moments of polarized structure functions

We examine the polarized doubly-virtual Compton scattering (VVCS) off the nucleon using chiral perturbation theory ($χ$PT). The polarized VVCS contains a wealth of information on the spin structure of the nucleon which is 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 the polarized VVCS amplitudes $S_1(ν, Q^2)$ and $S_2(ν, Q^2)$, and the corresponding polarized spin structure functions $g_1(x, Q^2)$ and $g_2(x,Q^2)$. Our results for the moments of polarized structure functions, partially related to different spin polarizabilities, are compared to other theoretical predictions and "data-driven" evaluations, as well as to the recent Jefferson Lab measurements. 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 baryon $χ$PT (B$χ$PT) and heavy-baryon (HB$χ$PT) frameworks.

hep-ph↗