We determine the low-energy constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>5</mn><mi>r</mi></msubsup></math> of SU(3) chiral perturbation theory from a lattice QCD calculation of the scalar form factors of the pion with fully controlled systematics. Lattice results are computed on a large set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><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> gauge ensembles covering four lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0.049</mn><mo>,</mo><mn>0.086</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math>, pion masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>130</mn><mo>,</mo><mn>350</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, and various large physical volumes. By determining the notorious quark-disconnected contributions with unprecedented precision and using a large range of source-sink separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>sep</mi></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>1.0</mn><mo>,</mo><mn>3.25</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math>, we are able for the first time to obtain the scalar radii from a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi></math>-expansion parametrization of the form factors rather than a simple linear approximation at small momentum transfer. The low-energy constants are obtained from the physical extrapolation of the radii using next-to-leading-order SU(3) next-to-leading-order chiral perturbation theory to parametrize the quark mass dependence. Systematic uncertainties are estimated via model averages based on the Akaike information criterion. Our determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></math> is the first lattice determination to obtain a result not compatible with zero.

Germany

Georg von Hippel

Konstantin Ottnad

Low-Energy Constants of Chiral Perturbation Theory from Pion Scalar Form Factors in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><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>-Flavor Lattice QCD with Controlled Errors

We present a lattice QCD calculation of the pion scalar form factor and associated radii with fully controlled systematics. Lattice results are computed on a large set of 17 gauge ensembles with <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> Wilson Clover-improved sea quarks. These ensembles cover four values of the lattice spacing between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.049</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.086</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, a pion mass range of 130–350 MeV, and various physical volumes. A precise determination of the notorious quark-disconnected contributions facilitates an unprecedented momentum resolution for the form factor, particularly on large and fine ensembles in the vicinity of physical quark mass. A large range of source-sink separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1.0</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><mo>≲</mo><msub><mi>t</mi><mi>sep</mi></msub><mo>≲</mo><mn>3.25</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> is used to reliably extract the relevant ground state matrix elements at vanishing and nonvanishing momentum transfer. This allows us for the first time to obtain the scalar radii from a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi></math>-expansion parametrization of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> dependence of the resulting form factors rather than a simple, linear approximation at small momentum transfer. The physical extrapolation for the radii is carried out using three-flavor NLO chiral perturbation theory to parametrize the quark mass dependence in terms of three low-energy constants, including the first lattice determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>4</mn><mi>r</mi></msubsup></math>. Systematic uncertainties on the final results related to the ground state extraction, form factor parametrization as well as the physical extrapolation are accounted for via model averages based on the Akaike information criterion. Our physical results for the pion singlet and octet scalar radii are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mo stretchy="false">⟩</mo></mrow><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mn>0.550</mn><mo stretchy="false">(</mo><mn>32</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>39</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>50</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mo stretchy="false">⟩</mo></mrow><mrow><mi>π</mi></mrow><mrow><mn>8</mn></mrow></msubsup><mo>=</mo><mn>0.443</mn><mo stretchy="false">(</mo><mn>16</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>45</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>47</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, respectively. The values of the relevant low energy constants are given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mspace linebreak="goodbreak"></mspace><mn>116.5</mn><mo stretchy="false">(</mo><mn>5.5</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>13.7</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>14.8</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>4</mn><mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>μ</mi><mo>=</mo><mn>770</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>0.38</mn><mo stretchy="false">(</mo><mn>09</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>15</mn><msub><mo stretchy="false">)</mo><mi>sys</mi></msub><mo stretchy="false">[</mo><mn>18</mn><msub><mo stretchy="false">]</mo><mtext>total</mtext></msub><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>5</mn><mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>μ</mi><mo>=</mo><mn>770</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>0.58</mn><mo stretchy="false">(</mo><mn>0.38</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>1.08</mn><msub><mo stretchy="false">)</mo><mi>sys</mi></msub><mo stretchy="false">[</mo><mn>1.14</mn><msub><mo stretchy="false">]</mo><mtext>total</mtext></msub><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>. For the light quark contribution to the pion scalar radius, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mo stretchy="false">⟩</mo></mrow><mrow><mi>π</mi></mrow><mrow><mi>l</mi></mrow></msubsup><mo>=</mo><mn>0.515</mn><mo stretchy="false">(</mo><mn>28</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>32</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>43</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, resulting in a value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mover accent="true"><mo>ℓ</mo><mo stretchy="true">¯</mo></mover><mn>4</mn></msub><mo>=</mo><mn>3.99</mn><mo stretchy="false">(</mo><mn>15</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>17</mn><msub><mo stretchy="false">)</mo><mi>sys</mi></msub><mo stretchy="false">[</mo><mn>23</mn><msub><mo stretchy="false">]</mo><mtext>total</mtext></msub></math> for the pertinent low-energy constant in two-flavor chiral perturbation theory.

Scalar size of the pion from lattice QCD

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.

Alessandro Barone

Dalibor Djukanovic

Jonna Koponen

Harvey B. Meyer

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

We calculate the double-virtual <math><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> transition form factor <math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></msub><mo>(</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></math> from first principles using a lattice QCD simulation with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> quark flavors at the physical pion mass and at one lattice spacing and volume. The kinematic range covered by our calculation is complementary to the one accessible from experiment and is relevant for the <math><mi>η</mi></math>-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment <math><msub><mi>a</mi><mi>μ</mi></msub><mo>=</mo><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn></math> of the muon. From the form factor calculation we extract the partial decay width <math><mrow><mi>Γ</mi><mo>(</mo><mi>η</mi><mo>→</mo><mi>γ</mi><mi>γ</mi><mo>)</mo><mo>=</mo><mn>338</mn><mo>(</mo><mn>87</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>17</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub><mtext> </mtext><mtext> </mtext><mi>eV</mi></mrow></math> and the slope parameter <math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>=</mo><mn>1.34</mn><mo>(</mo><mn>28</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>14</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math>. For the <math><mi>η</mi></math>-pole contribution to <math><msub><mi>a</mi><mi>μ</mi></msub></math> we obtain <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>η</mi><mtext>-pole</mtext></mrow></msubsup><mo>=</mo><mn>13.8</mn><mo>(</mo><mn>5.2</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>1.5</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>syst</mi></mrow></msub><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>.

Cyprus

Constantia Alexandrou

Simone Bacchio

Sebastian Burri

Jacob Finkenrath

Andrew Gasbarro

Kyriakos Hadjiyiannakou

Karl Jansen

Gurtej Kanwar

Bartosz Kostrzewa

Marcus Petschlies

Ferenc Pittler

Carsten Urbach

Urs Wenger

<math><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> transition form factor and the hadronic light-by-light <math><mi>η</mi></math>-pole contribution to the muon <math><mi>g</mi><mo>−</mo><mn>2</mn></math> from lattice QCD