Abstract The Standard Model prediction of the muon $$g-2$$ g - 2 increasingly depends on lattice QCD computations of the hadronic vacuum polarization (HVP), where the isospin-breaking (IB) effects remain a significant source of uncertainty. To complement the lattice QCD evaluations, the data-driven approach to HVP has been used to assess some of the electromagnetic IB effects, in particular from the channels with a photon in the final state, e.g., $$e^+e^-\rightarrow \pi ^0 \gamma $$ e + e - → π 0 γ . Here we argue that such contributions are largely canceled by virtual electromagnetic corrections to the purely hadronic channels: $$\pi ^+ \pi ^-$$ π + π - , $$\pi ^+ \pi ^- \pi ^0$$ π + π - π 0 , etc. We identify these leading corrections by performing a field-theoretic calculation in a vector-meson dominance model, thereby reconciling the timelike and spacelike approaches to electromagnetic effects. Although these virtual corrections are more difficult to extract in a systematic manner, addressing them is essential for the data-driven method to consistently complement the lattice QCD program.

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.

We propose the linear combination of the anomalous magnetic moments of the muon and electron, a_μ-e≡ a_μ- (m_μ/m_e)^2 a_e, as a natural low-energy window quantity, with enhanced sensitivity to the current discrepancy between the data-driven and lattice-QCD evaluations of hadronic contributions. The rescaling ensures an exact cancellations the short-range effects, thereby improving the UV behavior and bypassing a number of issues that arise in a_μ or a_e separately. The hadronic-vacuum-polarization effect in a^ HVP_μ-e, together with its uncertainty, is reduced as compared to a^ HVP_μ by ∼ 85%. This is promising for tests of New Physics, conditional to significant improvements in experimental measurements of a_e and α. One can foresee the improvements in tests of New Physics with some degree of flavor non-universality, as well as for the flavor-universal sub-GeV states.

The Standard Model prediction of the muon g-2 increasingly depends on lattice QCD computations of the hadronic vacuum polarization (HVP), where the isospin-breaking (IB) effects remain a significant source of uncertainty. To complement the lattice QCD evaluations, the data-driven approach to HVP has been used to assess some of the electromagnetic IB effects, in particular from the channels with a photon in the final state, e.g., e^+e^-→π ^0 γ . Here we argue that such contributions are largely canceled by virtual electromagnetic corrections to the purely hadronic channels: π ^+ π ^- , π ^+ π ^- π ^0 , etc. We identify these leading corrections by performing a field-theoretic calculation in a vector-meson dominance model, thereby reconciling the timelike and spacelike approaches to electromagnetic effects. Although these virtual corrections are more difficult to extract in a systematic manner, addressing them is essential for the data-driven method to consistently complement the lattice QCD program.

In order to reach a precision of 0.2

We present our calculation of the isospin-violating part of the hadronic vacuum polarisation (HVP) contribution to muon $(g-2)$ in lattice QCD at the $SU(3)_{\mathrm{f}}$ symmetric point. The computation of the contributing fully connected diagrams with one internal photon as well as the computation of the only (mass) counterterm are shown. The latter is determined from the charged-neutral kaon mass splitting. We employ coordinate-space methods and a photon propagator which is regulated \`a la Pauli-Villars with a cutoff scale $\Lambda$ well below the lattice cutoff. This regularization makes it possible for us to do crosschecks of individual contributions with calculations in the continuum. Our continuum extrapolated results show little to no dependence on $\Lambda$. This makes our final limit $\Lambda \rightarrow \infty$ straightforward.

The Burkhardt–Cottingham sum rule is an exact superconvergence relation for a spin-structure function, derived from general principles of light absorption and scattering, and valid at any momentum transfer Q2. I illustrate how a class of such relations emerges from the Siegert point, an unphysical kinematical point where both the probe and the target are at rest. From light-by-light scattering, new sum rules for γ*γ* fusion are emerging, valid for arbitrary photon virtualities. Regarding the convergence of these relations, there is a simple argument for the suppression of longitudinal photon polarizations at high energy. Among its consequences is the prediction σL/σT→0 at high energy, for the ratio of unpolarized nucleon photoabsorption cross sections.

We compute the isospin-violating part a_μ^HVP,38 of the hadronic-vacuum-polarization (HVP) contribution to the muon (g − 2) in lattice QCD at the SU(3)f-symmetric point where Mπ = MK ≃ 416 MeV. All diagrams involving internal photons are evaluated in coordinate space, employing a Pauli-Villars-regulated photon propagator with a cutoff scale Λ well below the lattice cutoff. The counterterm (mu – md), whose Λ dependence is consistent with the expected logarithmic behaviour, is calibrated using the experimental kaon mass splitting as input. The bare electromagnetic contribution at fixed Λ is compared to a phenomenological estimate based on the kaon-loop and pseudoscalar-pole contributions to the forward light-by-light amplitude. An extension of these calculations to physical pion and kaon masses appears promising.

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 Δpol. Data-driven evaluations of Δpol have long been in significant tension with baryon chiral perturbation theory. Here we present improved results for Δpol driven by new spin structure data, reducing the long-standing tension between theory and experiment and halving the dominating uncertainty in hyperfine splitting calculations.

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.

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.

Abstract 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 ~ 400 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∗→ e+e− cross-section, we predict the expected lattice-QCD integrand resulting from the γ∗γ∗→ π0 process.