<p>The isoscalar axial-vector form factor of the nucleon plays a key role in understanding the electroweak interaction of nucleons. For the interpretation of the spin structure of the nucleon the nonsinglet isoscalar axial charge is indispensable. Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>G</mi><mi>A</mi><mrow><mi>u</mi><mo>+</mo><mi>d</mi><mo>−</mo><mn>2</mn><mi>s</mi></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> together with the isovector and singlet isoscalar form factors are needed for the flavor decomposition of the axial-vector form factor. Detailed knowledge of the flavor decomposition facilitates extractions of Standard Model (SM) parameters from low energy experiments such as the electroweak charge of the proton in the P2 experiment. Here we present a lattice determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>G</mi><mi>A</mi><mrow><mi>u</mi><mo>+</mo><mi>d</mi><mo>−</mo><mn>2</mn><mi>s</mi></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math> improved Wilson fermions, with a full error budget concerning extrapolations and interpolations to the continuum and infinite volume and physical quark masses.</p>

Germany

Alessandro Barone

Dalibor Djukanovic

Georg von Hippel

Jonna Koponen

Harvey B. Meyer

Konstantin Ottnad

Hartmut Wittig

Isoscalar octet axial form factor of the nucleon from lattice QCD

We present our lattice QCD result for the long-distance part of the hadronic vacuum polarization contribution, ( $$ {a}_{\mu}^{\textrm{hvp}} $$ ) , to the muon $\textit{g}$ − 2 in the time-momentum representation. This is the numerically dominant, and at the same time the most challenging part regarding statistical precision. Our calculation is based on ensembles with dynamical up, down and strange quarks, employing the O($\textit{a}$)-improved Wilson fermion action with lattice spacings ranging from 0$\textit{.}$035–0$\textit{.}$099 fm. In order to reduce statistical noise in the long-distance part of the correlator to the per-mille level, we apply low-mode averaging and combine it with an explicit spectral reconstruction. Our result is ( $$ {a}_{\mu}^{\textrm{hvp}} $$ ) = 423$\textit{.}$2(4$\textit{.}$2) (3$\textit{.}$4) × 10 in isospin-symmetric QCD, where the pion decay constant is used to set the energy scale. When combined with our previous results for the short- and intermediate-distance window observables and after including all sub-dominant contributions as well as isospin-breaking corrections, we obtain the total leading-order hadronic vacuum polarization contribution as $$ {a}_{\mu}^{\textrm{hvp}} $$ = 724$\textit{.}$5(4$\textit{.}$9) (5$\textit{.}$2) × 10 . Our result displays a tension of 3.9 standard deviations with the data-driven estimate published in the 2020 White Paper, but leads to a SM prediction for the total muon anomalous magnetic moment that agrees with the current experimental average.

Georg Hippel

Switzerland

Simon Kuberski

Harvey Meyer

Nolan Miller

Julian Parrino

Andreas Risch

The hadronic vacuum polarization contribution to the muon $\textit{g −}$ 2 at long distances

We present the first lattice-QCD result for the Zemach and Friar radii of the proton and neutron. Our calculation includes both quark-connected and -disconnected diagrams and assesses all sources of systematic uncertainties arising from excited-state contributions, finite-volume effects, and the continuum extrapolation. At the physical point, we obtain for the proton <math><msubsup><mi>r</mi><mi>Z</mi><mi>p</mi></msubsup><mo>=</mo><mo>(</mo><mn>1.013</mn><mo>±</mo><mn>0.010</mn><mo>(</mo><mrow><mi>stat</mi></mrow><mo>)</mo><mo>±</mo><mn>0.012</mn><mo>(</mo><mrow><mi>syst</mi></mrow><mo>)</mo><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and <math><msubsup><mi>r</mi><mi>F</mi><mi>p</mi></msubsup><mo>=</mo><mo>(</mo><mn>1.301</mn><mo>±</mo><mn>0.012</mn><mo>(</mo><mrow><mi>stat</mi></mrow><mo>)</mo><mo>±</mo><mn>0.014</mn><mo>(</mo><mrow><mi>syst</mi></mrow><mo>)</mo><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. These numbers suggest small values of the Zemach and Friar radii of the proton but are compatible with most of the experimental studies.

Miguel Salg

Zemach and Friar radii of the proton and neutron from lattice QCD

There is a long-standing discrepancy between different measurements of the electric and magnetic radii of the proton. Lattice QCD calculations are a well-suited tool for theoretical investigations of the structure of the nucleon from first principles. However, all previous lattice studies of the proton’s electromagnetic radii have either neglected quark-disconnected contributions or were not extrapolated to the continuum and infinite-volume limit. Here, we present results for the electromagnetic form factors of the proton and neutron computed on the (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor coordinated lattice simulations (CLS) ensembles including both quark-connected and -disconnected contributions. From simultaneous fits to the <math><msup><mi>Q</mi><mn>2</mn></msup></math>-, pion-mass, lattice-spacing, and finite-volume dependence of the form factors, we determine the electric and magnetic radii and the magnetic moments of the proton and neutron. For the proton, we obtain as our final values <math><mrow><mo>⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mo>⟩</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mo>(</mo><mn>0.672</mn><mo>±</mo><mn>0.014</mn><mrow><mo>(</mo><mi>stat</mi><mo>)</mo></mrow><mo>±</mo><mspace></mspace><mn>0.018</mn><mrow><mo>(</mo><mi>syst</mi><mo>)</mo></mrow><mo>)</mo><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, <math><mrow><mo>⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mo>⟩</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mo>(</mo><mn>0.658</mn><mo>±</mo><mn>0.012</mn><mrow><mo>(</mo><mi>stat</mi><mo>)</mo></mrow><mo>±</mo><mn>0.008</mn><mrow><mo>(</mo><mi>syst</mi><mo>)</mo></mrow><mo>)</mo><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, and <math><mrow><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>M</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mo>(</mo><mn>2.739</mn><mo>±</mo><mn>0.063</mn><mrow><mo>(</mo><mi>stat</mi><mo>)</mo></mrow><mo>±</mo><mspace></mspace><mn>0.018</mn><mrow><mo>(</mo><mi>syst</mi><mo>)</mo></mrow></mrow></math>. The magnetic moment is in good agreement with the experimental value, as is the one of the neutron. On the one hand, our result for the electric (charge) radius of the proton clearly points towards a small value, as favored by muonic hydrogen spectroscopy and the recent <math><mi>e</mi><mi>p</mi></math>-scattering experiment by PRad. Our estimate for the magnetic radius, on the other hand, is well compatible with that inferred from the A1 <math><mrow><mi>e</mi><mi>p</mi></mrow></math>-scattering experiment.

Electromagnetic form factors of the nucleon from <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> lattice QCD

We present lattice-QCD results for the electromagnetic form factors of the proton and neutron including both quark-connected and -disconnected contributions. The parametrization of the <math><msup><mi>Q</mi><mn>2</mn></msup></math> dependence of the form factors is combined with the extrapolation to the physical point. In this way, we determine the electric and magnetic radii and the magnetic moments of the proton and neutron. For the proton, we obtain at the physical pion mass and in the continuum and infinite-volume limit <math><mrow><msqrt><mrow><mo>⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mo>⟩</mo></mrow><mrow><mi>p</mi></mrow></msup></mrow></msqrt><mo>=</mo><mrow><mn>0.820</mn><mo>(</mo><mn>14</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></mrow></math>, <math><mrow><msqrt><mrow><mo>⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mo>⟩</mo></mrow><mrow><mi>p</mi></mrow></msup></mrow></msqrt><mo>=</mo><mrow><mn>0.8111</mn><mo>(</mo><mn>89</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></mrow></math>, and <math><mrow><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>M</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>=</mo><mrow><mn>2.739</mn><mo>(</mo><mn>66</mn><mo>)</mo></mrow></mrow></math>, where the errors include all systematics.

Precision Calculation of the Electromagnetic Radii of the Proton and Neutron from Lattice QCD

We present an update of our determination of the isovector charges <math><mrow><msubsup><mrow><mi>g</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup></mrow></math>, <math><msubsup><mi>g</mi><mi>S</mi><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup></math> and <math><msubsup><mi>g</mi><mi>T</mi><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup></math>, and the isovector twist-2 forward matrix elements <math><mo>⟨</mo><mi>x</mi><msub><mo>⟩</mo><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msub></math>, <math><mrow><mo>⟨</mo><mi>x</mi><msub><mrow><mo>⟩</mo></mrow><mrow><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>Δ</mi><mi>d</mi></mrow></msub></mrow></math> and <math><mrow><mo>⟨</mo><mi>x</mi><msub><mrow><mo>⟩</mo></mrow><mrow><mi>δ</mi><mi>u</mi><mo>−</mo><mi>δ</mi><mi>d</mi></mrow></msub></mrow></math> on the <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> gauge ensembles generated by the Coordinated Lattice Simulations (CLS) effort. We have significantly extended our coverage of the parameter space by adding ensembles at the physical pion mass and fine lattice spacing, at nearly physical pion masses and very fine lattice spacings, and at very large physical lattice volumes, enabling a well-controlled extrapolation to the physical point. Another major improvement is achieved owing to the extended range of source-sink separations, which allows us to perform two-state fits to summed correlator ratios, leading to a much higher level of control over excited-state effects. Systematic uncertainties from the chiral, continuum and infinite-volume extrapolations are incorporated via model averages based on the Akaike information criterion. Our final results at the physical point are <math><mrow><msubsup><mrow><mi>g</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup><mo>=</mo><mn>1.254</mn><mo>(</mo><mn>19</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>15</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo>[</mo><mn>24</mn><msub><mrow><mo>]</mo></mrow><mrow><mtext>total</mtext></mrow></msub></mrow></math>, <math><mrow><msubsup><mrow><mi>g</mi></mrow><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup><mo>=</mo><mn>1.203</mn><mo>(</mo><mn>77</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>81</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo>[</mo><mn>112</mn><msub><mrow><mo>]</mo></mrow><mrow><mtext>total</mtext></mrow></msub></mrow></math>, <math><mrow><msubsup><mrow><mi>g</mi></mrow><mrow><mi>T</mi></mrow><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup><mo>=</mo><mn>0.993</mn><mo>(</mo><mn>15</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>05</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo>[</mo><mn>16</mn><msub><mrow><mo>]</mo></mrow><mrow><mtext>total</mtext></mrow></msub></mrow></math>, <math><mrow><mo>⟨</mo><mi>x</mi><msub><mrow><mo>⟩</mo></mrow><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msub><mo>=</mo><mn>0.153</mn><mo>(</mo><mn>15</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>10</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo>[</mo><mn>17</mn><msub><mrow><mo>]</mo></mrow><mrow><mtext>total</mtext></mrow></msub></mrow></math>, <math><mrow><mo>⟨</mo><mi>x</mi><msub><mrow><mo>⟩</mo></mrow><mrow><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>Δ</mi><mi>d</mi></mrow></msub><mo>=</mo><mn>0.207</mn><mo>(</mo><mn>15</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>06</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo>[</mo><mn>16</mn><msub><mrow><mo>]</mo></mrow><mrow><mtext>total</mtext></mrow></msub></mrow></math>, and <math><mrow><mo>⟨</mo><mi>x</mi><msub><mrow><mo>⟩</mo></mrow><mrow><mi>δ</mi><mi>u</mi><mo>−</mo><mi>δ</mi><mi>d</mi></mrow></msub><mo>=</mo><mn>0.195</mn><mo>(</mo><mn>17</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>15</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo>[</mo><mn>23</mn><msub><mrow><mo>]</mo></mrow><mrow><mtext>total</mtext></mrow></msub></mrow></math>. While our results for the isovector charges are in excellent agreement with the FLAG 21 averages, we note that our error for the tensor charge <math><msubsup><mi>g</mi><mi>T</mi><mrow><mi>u</mi><mo>−</mo><mi>d</mi></mrow></msubsup></math> is considerably smaller.

Improved analysis of isovector nucleon matrix elements with <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math> improved Wilson fermions

We present results for the short-distance window observable of the hadronic vacuum polarization contribution to the muon g – 2, computed via the time-momentum representation (TMR) in lattice QCD. A key novelty of our calculation is the reduction of discretization effects by a suitable subtraction applied to the TMR kernel function, which cancels the leading  $${x}_{0}^{4}$$ -behaviour at short distances. To compensate for the subtraction, one must substitute a term that can be reliably computed in perturbative QCD. We apply this strategy to our data for the vector current collected on ensembles generated with 2 + 1 flavours of O(a)-improved Wilson quarks at six values of the lattice spacing and pion masses in the range 130 – 420 MeV. Our estimate at the physical point contains a full error budget and reads  $${\left({a}_{\mu }^{{\text{hvp}}}\right)}^{{\text{SD}}}$$  = 68.85(14)stat (42)syst·10 −10, which corresponds to a relative precision of 0.7%. We discuss the implications of our result for the observed tensions between lattice and data-driven evaluations of the hadronic vacuum polarization.

CERN

Italy

Marco Cè

Hadronic vacuum polarization in the muon g − 2: the short-distance contribution from lattice QCD

The isovector axial form factor of the nucleon plays a key role in interpreting data from long-baseline neutrino oscillation experiments. We perform a lattice-QCD based calculation of this form factor, introducing a new method to directly extract its <math><mi>z</mi></math> expansion from lattice correlators. Our final parametrization of the form factor, which extends up to spacelike virtualities of <math><mrow><mn>0.7</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> with fully quantified uncertainties, agrees with previous lattice calculations but is significantly less steep than neutrino-deuterium scattering data suggests.

Tobias Schulz

Isovector axial form factor of the nucleon from lattice QCD

We compute the hadronic running of the electromagnetic and weak couplings in lattice QCD with N f = 2 + 1 flavors of   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (a) improved Wilson fermions. Using two different discretizations of the vector current, we compute the quark-connected and –disconnected contributions to the hadronic vacuum polarization (HVP) functions   <math> <mover> <mi>Π</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\varPi} $$   γγ  and   <math> <mover> <mi>Π</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\varPi} $$   γZ  for Euclidean squared momenta Q 2 ≤ 7 GeV2. Gauge field ensembles at four values of the lattice spacing and several values of the pion mass, including its physical value, are used to extrapolate the results to the physical point. The ability to perform an exact flavor decomposition allows us to present the most precise determination to date of the SU(3)-flavor-suppressed HVP function   <math> <mover> <mi>Π</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\varPi} $$  08 that enters the running of sin2 θ  W . Our results for   <math> <mover> <mi>Π</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\varPi} $$   γγ ,   <math> <mover> <mi>Π</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\varPi} $$   γZ  and   <math> <mover> <mi>Π</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\varPi} $$  08 are presented in terms of rational functions for continuous values of Q 2 below 7 GeV2. We observe a tension of up to 3.5 standard deviation between our lattice results for   <math> <mtext>∆</mtext> <msubsup> <mi>α</mi> <mi>had</mi> <mfenced> <mn>5</mn> </mfenced> </msubsup> </math>  $$ \Delta {\alpha}_{\mathrm{had}}^{(5)} $$ (−Q 2) and estimates based on the R-ratio for space-like momenta in the range 3–7 GeV2. The tension is, however, strongly diminished when translating our result to the Z pole, by employing the Euclidean split technique and perturbative QCD, which yields   <math> <mtext>∆</mtext> <msubsup> <mi>α</mi> <mi>had</mi> <mfenced> <mn>5</mn> </mfenced> </msubsup> <mfenced> <msubsup> <mi>M</mi> <mi>Z</mi> <mn>2</mn> </msubsup> </mfenced> </math>  $$ \Delta {\alpha}_{\mathrm{had}}^{(5)}\left({M}_Z^2\right) $$  = 0.02773(15) and agrees with results based on the R-ratio within the quoted uncertainties.

France

Antoine Gérardin

Japan

Kohtaroh Miura

Teseo José

Jonas Wilhelm

The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD

We present the first study of baryon-baryon interactions in the continuum limit of lattice QCD, finding unexpectedly large lattice artifacts. Specifically, we determine the binding energy of the <math><mi>H</mi></math> dibaryon at a single quark-mass point. The calculation is performed at six values of the lattice spacing <math><mi>a</mi></math>, using <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>-improved Wilson fermions at the SU(3)-symmetric point with <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>≈</mo><mn>420</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. Energy levels are extracted by applying a variational method to correlation matrices of bilocal two-baryon interpolating operators computed using the distillation technique. Our analysis employs Lüscher’s finite-volume quantization condition to determine the scattering phase shifts from the spectrum and vice versa, both above and below the two-baryon threshold. We perform global fits to the lattice spectra using parametrizations of the phase shift, supplemented by terms describing discretization effects, then extrapolate the lattice spacing to zero. The phase shift and the binding energy determined from it are found to be strongly affected by lattice artifacts. Our estimate of the binding energy in the continuum limit of three-flavor QCD is <math><mrow><msubsup><mrow><mi>B</mi></mrow><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>SU</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mi>f</mi></mrow></msub></mrow></msubsup><mo>=</mo><mn>4.56</mn><mo>±</mo><mn>1.1</mn><msub><mrow><mn>3</mn></mrow><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>0.6</mn><msub><mrow><mn>3</mn></mrow><mrow><mi>syst</mi></mrow></msub><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>.