Several hadronic observables are nowadays computed in lattice QCD with a sub-percent precision which requires the inclusion of strong isospin-breaking and electromagnetic effects. Most of the methods that implement the photon propagator in finite-volume lead to power-law suppressed finite-size effects and do not allow for a straightforward crosscheck against phenomenology and other calculations. Both issues can be avoided by working with a Pauli-Villars regulated photon propagator defined directly in the continuum and infinite volume. This methodology can be profitably exploited to improve the determination of leading-order electromagnetic corrections to several observables such as the HVP or nucleon masses. In this work we apply the strategy to the charged/neutral pion mass difference using CLS ensembles.

We present a lattice QCD calculation of the charged-neutral pion mass splitting M_π^+ - M_π^0 at 𝒪(α_em) using a recently proposed framework based on a Pauli-Villars (PV) regulated photon propagator defined in the continuum and infinite-volume limit, with Λ acting as an additional UV cutoff scale. The use of this propagator avoids power-law finite-volume effects, allowing for a straightforward treatment of the infinite-volume limit. We perform the calculation using CLS ensembles, studying finite-volume effects, the continuum limit and the extrapolation to the physical point for several values of the scale Λ. By means of the Cottingham formula, we further decompose the result into elastic and inelastic contributions at fixed Λ. Our final result, after removing the cutoff scale Λ, is M_π^+ - M_π^0 = 4.56(22) MeV, in good agreement with the experimental measurement. This calculation serves as a validation of the formalism in a well-controlled setting and offers useful insights into the application of electromagnetic corrections to other observables.

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.

We present an updated determination of the hadronic vacuum polarization contribution to the running of the electromagnetic coupling Δα_had^(5)(-Q^2), and of the electroweak mixing angle in the space-like momentum range up to 12 GeV^2. Using N_f=2+1 CLS ensembles at five values of the lattice spacing and several pion masses, including the physical point, we achieve a significantly enhanced precision over our previous result. A refined analysis strategy based on telescopic series and a new family of kernel functions enables a clean separation of distinct Euclidean regions, disentangling strong cutoff effects at short distances from the pronounced chiral dependence at larger ones. Employing the Euclidean split technique, we convert our lattice results into an ab initio estimate of Δα_had^(5)(M_Z^2). A comparison with results from other lattice calculations and phenomenology is performed. We also analyze improvement scenarios required to match the projected precision of future electroweak measurements at next-generation colliders.

In previous work, we determined the improvement coefficients cV and cV required for the massless O(a)-improvement of the local and point-split discretizations of the nonsinglet vector current for Nf = 3 nonperturbatively O(a)-improved Wilson fermions and the L & uuml;scher-Weisz gauge action, using ensembles of large-volume configurations generated by the coordinated lattice simulations (CLS) initiative. A new estimate for the mass-dependent improvement coefficient b & strns;effA has recently become available, differing from the one used in our earlier study, and on which our implementation via a massive axial Ward identity relied. Here, we update our analysis of the mass-independent vector improvement coefficients based on the new axial current improvement coefficient, and analyze additional ensembles with a different chiral trajectory in order to validate our results at two values of the bare coupling. We find that using the new estimate of b & strns;eff improves the consistency between the two chiral trajectories, as well as with a previous determination of the improvement coefficients directly in the massless limit on small volumes.

We present an update of our lattice QCD determination of the hadronic contribution to the running of the electromagnetic coupling, Δα_had^(5)(-Q^2), and of the electroweak mixing angle in the space-like momentum region up to Q^2=12 GeV^2. The calculation is based on CLS ensembles with N_f=2+1 flavours of O(a)-improved Wilson fermions, covering five lattice spacings between 0.039 and 0.085 fm and a range of pion masses, including the physical point. A refined analysis employing a telescopic window strategy allows for a clean separation of systematic effects across Euclidean distance scales. Statistical precision is further enhanced through low-mode averaging, combined with a spectral reconstruction of the vector-vector correlator at long distances on the most chiral ensembles. We confirm significant tensions of up to 7σ at space-like virtualities around Q^2= 1 GeV^2 between our lattice results for Δα_had^(5)(-Q^2) and the corresponding data-driven estimates based on e^+e^- cross section data. Combining our lattice data with perturbative QCD via the Euclidean split technique, we obtain at the Z-pole Δα_had^(5)(M_Z^2) = 0.027813(33)_lat(35)_pQCD, which is more than two times more precise than recent data-driven estimates. Our result deviates slightly, by 1-2σ, from the value produced by global electroweak fits. For the electroweak mixing angle, we present the hadronic contribution to its running and provide a precise determination of the octet-singlet mixing component ^(0,8), in good agreement with phenomenological models but with significantly higher precision.

The singlet axial form factor of the nucleon provides essential input for a complete understanding of the nucleon axial structure. Together with the isovector and isoscalar octet channels, in the forward limit it forms the basis for a full flavor decomposition of the proton spin. In this work we present a lattice QCD determination of the singlet axial form factor $G^{u+d+s}_A(Q^2)$ and related strange contribution $G^{s}_A(Q^2)$ using a set of $N_f = 2 + 1$ CLS gauge ensembles with $O(a)$-improved Wilson fermions, with a full error budget for the extrapolation to the chiral, continuum and infinite-volume limits. Particular focus is placed on the treatment of the disconnected contributions, which constitute the crucial element for the extraction of the strange component. Together with determinations of the isovector and isoscalar octet axial form factors, this work provides a comprehensive lattice QCD determination of the nucleon axial structure across different flavor channels.

We present the first lattice QCD calculation of the next-to-leading order (NLO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment with sub-percent precision. We employ the time-momentum representation combined with the spatially summed vector correlator computed on CLS ensembles with N_f=2+1 flavors of O(a)-improved Wilson fermions, spanning six lattice spacings (0.039-0.097fm) and a range of pion masses including the physical value. After accounting for finite-size corrections and isospin-breaking effects, we obtain in the continuum limit a_μ^hvp, nlo = (-101.57 ± 0.26_ stat± 0.54_ syst) × 10^-11, corresponding to a total relative error of 0.6%. Our result lies 1.4σ below the estimate of the 2025 White Paper update and is two times more precise. It also shows a tension of 4.6σ with data-driven evaluations based on hadronic cross section measurements prior to the CMD-3 result.

Abstract Electromagnetic corrections to the n-point functions of lattice QCD can be evaluated using a position-space photon propagator defined in infinite volume. Here we address the computational challenge arising from the volume-squared sum over the endpoints of the photon propagator. We consider a class of integral representations of the photon propagator that lead to a factorization of the two volume-sums, the Fourier representation being one instance thereof. An alternative choice is based on expressing the free scalar propagator as the autoconvolution of the corresponding five-dimensional propagator. We compare the performance of three different choices in the context of electromagnetic corrections to the hadronic vacuum polarization, on a gauge ensemble of size 483 × 128 with a pion mass of 286 MeV. As an outlook, we discuss more generally the factorization of sums over internal vertices, taking as an example the hadronic light-by-light contribution to the muon (g − 2).

We present the first lattice QCD calculation of the next-to-leading order hadronic vacuum polarization contribution to the muon anomalous magnetic moment with sub-percent precision. We employ the time-momentum representation for the space-like kernel, which is combined with the spatially summed vector correlator computed on CLS ensembles with Nf = 2 + 1 flavors of O(a)-improved Wilson fermions, covering six lattice spacings between 0.039 and 0.097 fm and a range of pion masses including the physical value. After accounting for finite-size corrections and isospin-breaking effects, we obtain as our final, continuum-extrapolated result a_μ^hvp,0.30em nlo = −101.57(26)stat(54)syst × 10−11. It lies below the estimate provided by the 2025 White Paper of the Muon (g – 2) Theory Initiative by 1.4σ but is two times more precise. It also exhibits a strong tension of 4.6σ with data-driven evaluations based on hadronic cross section measurements excluding the recent result by CMD-3.

We present a high-precision calculation of the hadronic running of electroweak gauge couplings from first principles. Employing lattice QCD in the low-energy regime, we achieve permille precision for virtualities Q^{2}≲12  GeV^{2}. At Q^{2}≃1  GeV^{2}, our determination deviates by up to 7σ from estimates based on e^{+}e^{-} measurements. Combining lattice QCD with perturbative QCD via the Euclidean split technique, we obtain for the electromagnetic coupling Δα_{had}^{(5)}(M_{Z}^{2})=0.027821(34)_{lat}(35)_{pQCD}, which is more than twice as precise as recent phenomenological determinations. We assess improvement scenarios by which the precision target for next-generation electroweak measurements could be reached.

Abstract We develop elements of coordinate-space perturbation theory for massive quantum field theories in general d-dimensional Euclidean space. Using the expansion in Gegenbauer polynomials, we provide analytic expressions for several correlation functions in theories with one massive and one massless field. To this end, a class of antiderivatives of products of two Bessel functions multiplied by a power of their common argument are studied systematically. We expect these results to be useful in perturbative calculations at finite temperature and/or in finite volume, in quantum-field theories with high-degree vertices, as well as in auxiliary perturbative computations for a lattice QCD treatment of hadronic effects in precision observables. As an illustration, we compute the one-loop coordinate-space propagator of a massive particle coupled to a massless one, both in the vacuum and at finite temperature. We also provide a master formula for computing the electroweak triangle-diagram contribution to the muon (g − 2) in lattice QCD.

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.

The size of the proton is of lasting and high interest in the subatomic physics community. The most well-known example is the electric radius which has been the subject of the proton radius puzzle for more than a decade. While tremendous progress in $ep$-scattering, atomic spectroscopy, and lattice QCD has brought this puzzle closer to its resolution, one also finds discrepant results for the magnetic radius. In light of the upcoming high-precision measurements of the hyperfine splitting (HFS) in muonic hydrogen, other definitions of radii gain relevance as well. On the one hand, to infer the electric radius from the observed Lamb shift in muonic hydrogen, higher-order nuclear structure corrections need to be subtracted, which depend on the Friar radius of the proton. The magnetic properties of the proton, on the other hand, only enter the HFS via the proton's Zemach radius. Based on our previous calculation of the electromagnetic form factors of the proton and neutron, which includes both quark-connected and -disconnected contributions and assesses all sources of systematic uncertainties, we now present results for the Zemach and Friar radii. For the proton, we obtain $r_Z^p = ( 1.013 \pm 0.010\ (\mathrm{stat}) \pm 0.012\ (\mathrm{syst}) )~\mathrm{fm}$ and $r_F^p = ( 1.301 \pm 0.012\ (\mathrm{stat}) \pm 0.014\ (\mathrm{syst}) )~\mathrm{fm}$. These numbers suggest small values of the Zemach and Friar radii of the proton, and have a precision which is sufficient to make a meaningful comparison to data-driven evaluations.

We present the status of the Mainz group's lattice QCD calculation of the pion transition form factor $\mathcal{F}_{\pi^0\gamma^\ast\gamma^\ast}$, which describes the interaction of an on-shell pion with two off-shell photons. This form factor is the main ingredient in the calculation of the pion-pole contribution to hadronic light-by-light scattering in the muon $g-2$. We use the $N_f = 2 + 1$ CLS gauge ensembles, and we update our previous work by including a physical pion mass ensemble (E250). We compute the transition form factor in the pion rest frame as well as in a moving frame in order to have access to a wider range of photon virtualities. In addition to the quark-line connected correlator we also compute the quark-line disconnected diagrams that contribute to the form factor. In this final stage of the analysis, we combine the result on E250 with the previous work published in 2019 to extrapolate the form factor to the continuum and to physical quark masses. Testing different ans\"atze for the fit, we explore the systematic uncertainties of the extrapolation. The contribution from the disconnected diagrams is also scrutinized.

Determining the spectrum of photons emitted by the quark-gluon plasma nonperturbatively remains an open computational challenge. In this paper, we calculate two moments of that spectrum at a tem-perature T 254 MeV, employing lattice QCD with two flavors of O(a)-improved Wilson fermions, without facing an inverse problem. Our central value for the difference of these two moments, which is sensitive to photon energies w greater than or similar to irT, is lower than, but compatible with that obtained by integrating the leading-order weak-coupling photon spectrum. This study informs the "direct photon puzzle" in heavy-ion collision phenomenology, where it has proved difficult to simultaneously explain the yield and azimuthal anisotropy of photons not originating from final-state hadronic decays.

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.

We report on our update to [1] on the hadronic running of electroweak couplings from O(a)-improved Wilson fermions with N_f=2+1 flavours. The inclusion of additional ensembles at very fine lattice spacings together with a number of techniques to split the different contributions for a better control of cutoff effects allows us to substantially improve the precision. We employ two different discretizations of the vector current to compute the subtracted Hadronic Vacuum Polarization (HVP) functions Π̅^γγ and Π̅^Zγ for Euclidean time momenta up to Q^2≤ 9 GeV^2 . To reduce cutoff effects in the short distance region we apply a suitable subtraction to the TMR kernel function, which cancels the leading x_0^4 behaviour. The subtracted term is then computed in perturbative QCD using the Adler function and added back to compensate for the subtraction. Chiral-continuum extrapolations are performed with five values of the lattice spacing and several pion masses, including its physical value, and several fit ansätze are explored to estimate the systematics arising from model selection. Our results show excellent prospects for high-precision estimates of Δα_had^(5)(M_Z^2) at the Z-pole.