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 $\textit{N}$ = 2 + 1 flavors of O($\textit{a}$)-improved Wilson fermions, covering six lattice spacings between 0$\textit{.}$039 and 0$\textit{.}$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 $$ {a}_{\mu}^{\mathrm{hvp},\kern0.30em \mathrm{nlo}} $$ = −101.57(26) (54) × 10 . It lies below the estimate provided by the 2025 White Paper of the Muon ($\textit{g}$ – 2) Theory Initiative by 1.4$\textit{σ}$ but is two times more precise. It also exhibits a strong tension of 4.6$\textit{σ}$ with data-driven evaluations based on hadronic cross section measurements excluding the recent result by CMD-3.

Germany

Arnau Beltran

Alessandro Conigli

Switzerland

Simon Kuberski

Harvey Meyer

Konstantin Ottnad

Hartmut Wittig

Higher-order hadronic vacuum polarization contribution to the muon $\textit{g}$ − 2 from lattice QCD

<p>We present an update of our lattice QCD determination of the hadronic contribution to the running of the electromagnetic coupling, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msubsup><mi>α</mi><mi>had</mi><mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mo>−</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, and of the electroweak mixing angle in the spacelike momentum region up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. The calculation is based on coordinated lattice simulations 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> flavors of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math>-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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>7</mn><mi>σ</mi></math> at spacelike virtualities around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math> between our lattice results for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msubsup><mi>α</mi><mi>had</mi><mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mo>−</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> and the corresponding data-driven estimates based on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> cross section data. Combining our lattice data with perturbative QCD via the Euclidean split technique, we obtain at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Z</mi></math> pole <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msubsup><mi>α</mi><mi>had</mi><mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mi>M</mi><mi>Z</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0.027821</mn><mo stretchy="false">(</mo><mn>34</mn><msub><mo stretchy="false">)</mo><mi>lat</mi></msub><mo stretchy="false">(</mo><mn>35</mn><msub><mo stretchy="false">)</mo><mtext>pQCD</mtext></msub></math>, which is more than two times more precise than recent data-driven estimates. Our result deviatesslightly, by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>−</mo><mn>2</mn><mi>σ</mi></math>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi mathvariant="normal">Π</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></msup></math>, in good agreement with phenomenological models but with significantly higher precision.</p>

Dalibor Djukanovic

Georg von Hippel

CERN

Harvey B. Meyer

Japan

Kohtaroh Miura

Andreas Risch

Precision lattice calculation of the hadronic contribution to the running of the electroweak gauge couplings

<p>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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≲</mo><mn>12</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. At <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≃</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, our determination deviates by up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>7</mn><mi>σ</mi></math> from estimates based on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> measurements. Combining lattice QCD with perturbative QCD via the Euclidean split technique, we obtain for the electromagnetic coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>α</mi></mrow><mrow><mi>had</mi></mrow><mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0.027821</mn><mo stretchy="false">(</mo><mn>34</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>lat</mi></mrow></msub><mo stretchy="false">(</mo><mn>35</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mtext>pQCD</mtext></mrow></msub></mrow></math>, 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.</p>

Running of the Electroweak Gauge Couplings from First Principles

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

Nolan Miller

Julian Parrino

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

We present the first purely theoretical calculation of the weak mixing angle in the <math><mrow><mover><mrow><mi>MS</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> scheme at low energies by combining results from lattice QCD with perturbation theory. We discuss its correlation with the hadronic contribution to the anomalous magnetic moment of the muon and to the energy dependence of the electromagnetic coupling. We also compare the results with calculations using cross-section data as input. Implications for the standard model prediction of the mass of the <math><mi>W</mi></math> boson are also discussed.

Jens Erler

Rodolfo Ferro-Hernández

Theory-Driven Evolution of the Weak Mixing Angle

We present results for the leptonic decay constants of the D and Ds mesons from N f = 2 + 1 lattice QCD. We employ a set of 49 high statistics gauge ensembles generated by the Coordinated Lattice Simulations (CLS) effort utilising non-perturbatively improved Wilson fermions and the tree-level Symanzik improved gauge action at six values of the lattice spacing in the range a = 0.098 fm down to a = 0.039 fm, with pion masses varying from around 420 MeV down to below the physical point. The ensembles lie on three trajectories in the quark mass plane, two trajectories intersecting close to the physical quark mass point and the third one approaching the SU(3) chiral limit, enabling tight control of the light and strange quark mass dependence. We obtain   <math> <msub> <mi>f</mi> <msub> <mi>D</mi> <mi>s</mi> </msub> </msub> </math>  $$ {f}_{{\textrm{D}}_{\textrm{s}}} $$  = 246.8(1.3) MeV, f D = 208.4(1.5) MeV and   <math> <msub> <mi>f</mi> <msub> <mi>D</mi> <mi>s</mi> </msub> </msub> </math>  $$ {f}_{{\textrm{D}}_{\textrm{s}}} $$  /f D = 1.1842(36), where the precision of our results is mostly limited by the determination of the scale.

United Kingdom

Fabian Joswig

Sara Collins

Jochen Heitger

Wolfgang Söldner

D and Ds decay constants in N f = 2 + 1 QCD with 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.

Italy

Marco Cè

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

Günther Dissertori

Christian Caron

Editorial: Simon Eidelman (1948–2021)

We present a determination of the charm quark mass in lattice QCD with three active quark flavours. The calculation is based on PCAC masses extracted from N f = 2 + 1 flavour gauge field ensembles at five different lattice spacings in a range from 0.087 fm down to 0.039 fm. The lattice action consists of the O(a) improved Wilson-clover action and a tree-level improved Symanzik gauge action. Quark masses are non-perturbatively O(a) improved employing the Symanzik-counterterms available for this discretisation of QCD. To relate the bare mass at a specified low-energy scale with the renormalisation group invariant mass in the continuum limit, we use the non-pertubatively known factors that account for the running of the quark masses as well as for their renormalisation at hadronic scales. We obtain the renormalisation group invariant charm quark mass at the physical point of the three-flavour theory to be M c = 1486(21) MeV. Combining this result with five-loop perturbation theory and the corresponding decoupling relations in the   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  scheme, one arrives at a result for the renormalisation group invariant charm quark mass in the four-flavour theory of M c(N f = 4) = 1548(23) MeV, where effects associated with the absence of a charmed, sea quark in the non-perturbative evaluation of the QCD path integral are not accounted for. In the   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{MS}} $$  scheme, and at finite energy scales conventional in phenomenology, we quote   <math> <msubsup> <mi>m</mi> <mi>c</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msubsup> </math>  $$ {m}_{\mathrm{c}}^{\overline{\mathrm{MS}}} $$ (  <math> <msubsup> <mi>m</mi> <mi>c</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msubsup> </math>  $$ {m}_{\mathrm{c}}^{\overline{\mathrm{MS}}} $$ ; N f = 4) = 1296(19) MeV and   <math> <msubsup> <mi>m</mi> <mi>c</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msubsup> </math>  $$ {m}_{\mathrm{c}}^{\overline{\mathrm{MS}}} $$ (3 GeV; N f = 4) = 1007(16) MeV for the renormalised charm quark mass.

Determination of the charm quark mass in lattice QCD with 2 + 1 flavours on fine lattices

We determine non-perturbatively the normalisation parameter $$Z_\mathrm{m}Z_\mathrm{P}/Z_\mathrm{A}$$ <math><mrow><msub><mi>Z</mi><mi>m</mi></msub><msub><mi>Z</mi><mi>P</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>A</mi></msub></mrow></math>  as well as the Symanzik coefficients $$b_\mathrm{m}$$ <math><msub><mi>b</mi><mi>m</mi></msub></math>  and $$b_\mathrm{A}-b_\mathrm{P}$$ <math><mrow><msub><mi>b</mi><mi>A</mi></msub><mo>-</mo><msub><mi>b</mi><mi>P</mi></msub></mrow></math> , required in $$\mathrm{O}(a)$$ <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>  improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in $$N_\mathrm{f}=3$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></mrow></math>  QCD with $$\mathrm{O}(a)$$ <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>  improved massless sea and non-degenerate valence quarks in the finite-volume Schrödinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume $$N_\mathrm{f}=2+1$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>  QCD simulations with Wilson fermions at lattice spacings below $$0.1\,\mathrm{fm}$$ <math><mrow><mn>0.1</mn><mspace width="0.166667em"></mspace><mi>fm</mi></mrow></math> , are of particular use for the non-perturbative calculation of $$\mathrm{O}(a)$$ <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>  improved renormalised quark masses.

Giulia Divitiis

Patrick Fritzsch

Carl Köster

Anastassios Vladikas

Non-perturbative determination of improvement coefficients $$b_\mathrm{m}$$ <math><msub><mi>b</mi><mi>m</mi></msub></math>  and $$b_\mathrm{A}-b_\mathrm{P}$$ <math><mrow><msub><mi>b</mi><mi>A</mi></msub><mo>-</mo><msub><mi>b</mi><mi>P</mi></msub></mrow></math>  and normalisation factor $$Z_\mathrm{m}Z_\mathrm{P}/Z_\mathrm{A}$$ <math><mrow><msub><mi>Z</mi><mi>m</mi></msub><msub><mi>Z</mi><mi>P</mi></msub><mo>/</mo><msub><mi>Z</mi><mi>A</mi></msub></mrow></math>  with $$N_\mathrm{f}= 3$$ <math><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn></mrow></math>  Wilson fermions