We compute the isospin-violating part $$ {a}_{\mu}^{\textrm{HVP},38} $$ of the hadronic-vacuum-polarization (HVP) contribution to the muon ($\textit{g}$ − 2) in lattice QCD at the SU(3) -symmetric point where $\textit{M}$ = $\textit{M}$ ≃ 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 ($\textit{m}$ – $\textit{m}$ ), 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.

Germany

Dominik Erb

France

Antoine Gérardin

Switzerland

Harvey Meyer

Julian Parrino

Volodymyr Biloshytskyi

Vladimir Pascalutsa

Isospin-violating vacuum polarization in the muon ($\textit{g}$ − 2) with SU(3) flavour symmetry from lattice QCD

We present a lattice quantum chromodynamics (QCD) calculation of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon using <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><mo>+</mo><mn>1</mn></math> flavors of staggered quarks with masses tuned to their physical values. Our final result, in the continuum limit, reads <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>hlbl</mi></mrow></msubsup><mo>=</mo><mn>125.5</mn><mo stretchy="false">(</mo><mn>11.5</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>1.9</mn><msub><mo stretchy="false">)</mo><mrow><mi>syst</mi></mrow></msub><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup></math> where the first error is statistical and the second is systematic. Light, strange, and charm-quark contributions are considered. In addition to the connected and leading disconnected contributions, we also include an estimate of the subleading disconnected diagrams. Our result is compatible with previous lattice QCD and data-driven dispersive determinations.

Hungary

United States

Zoltan Fodor

Laurent Lellouch

Kalman K. Szabo

Balint C. Toth

Christian Zimmermann

Hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon at the physical pion mass

We present a lattice calculation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> meson masses at the physical point and in the continuum limit, based 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><mo>+</mo><mn>1</mn></math> flavors of rooted staggered quarks. Our analysis includes gauge ensembles at the physical pion and kaon masses spread over six lattice spacings in the range [0.064–0.1315] fm. Our main results read <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>η</mi></msub><mo>=</mo><mn>543.5</mn><mo stretchy="false">(</mo><mn>5.6</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><msup><mi>η</mi><mo>′</mo></msup></msub><mo>=</mo><mn>986</mn><mo stretchy="false">(</mo><mn>38</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, consistent with the experimental values. This is an important numerical test that supports the validity of the fourth root procedure used in the staggered quark formalism. This calculation was the first step toward extracting the pseudoscalar transition form factors of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> mesons that play a crucial role in the hadronic light-by-light contribution to the muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>.

Willem E.A. Verplanke

Jana N. Guenther

Lukas Varnhorst

Lattice QCD calculation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> meson masses at the physical point using rooted staggered fermions

In this paper we present a first ab initio calculation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mn>0</mn></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> transition form factors performed with physical light-quark masses. We provide a complete parametrization of the form factors that includes both single and double-virtual kinematics. Our results are compared with experimental measurements of the form factors in the spacelike region and with the measured two-photon decay widths. In a second step, our parametrizations of the transition form factors are used to compute the dominant pseudoscalar-pole contributions to the hadronic light-by-light scattering in the muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>. Our final result reads <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>hlbl</mi><mo>,</mo><mtext>ps-pole</mtext></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mn>85.1</mn><mo>±</mo><mn>5.2</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>. Although the pion-pole is dominant, we confirm that, together, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> provide roughly half of its contribution.

Gen Wang

Lattice calculation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mn>0</mn></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> transition form factors and the hadronic light-by-light contribution to the muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>

The precision with which hadronic vacuum polarization (HVP) is obtained determines how accurately important observables, such as the muon anomalous magnetic moment <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub></mrow></math> or the low-energy running of the electromagnetic coupling <math><mi>α</mi></math>, are predicted. The two most precise approaches for determining HVP are dispersive relations combined with <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mtext>hadrons</mtext></mrow></math> cross section data and lattice QCD. However, the results obtained in these two approaches display significant tensions, whose origins are not understood. Here we present a framework that sheds light on this issue and—if the two approaches can be reconciled—allows them to be combined. Via this framework, we test the hypothesis that the tensions can be explained by modifying the R-ratio in different intervals of center-of-mass energy <math><msqrt><mi>s</mi></msqrt></math>. As ingredients, we consider observables that have been precisely determined in both approaches. These are the leading hadronic contributions to <math><msub><mi>a</mi><mi>μ</mi></msub></math>, to the so-called intermediate window observable, and to the running of <math><mi>α</mi></math> between spacelike virtualities 1 and <math><mn>10</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mn>2</mn></msup></math> (for which only a preliminary lattice result exists). Our tests take into account all uncertainties and correlations, as well as uncertainties on uncertainties in the lattice results. For instance, using this framework we show that results obtained in the two approaches can be made to agree, for all three observables, by modifying the <math><mi>ρ</mi></math> peak in the experimental spectrum. More specifically, we show that this requires a common <math><mo>∼</mo><mn>5</mn><mo>%</mo></math> increase in the contributions of the peak to each of the three observables. This result is robust against the presence or absence of the running of <math><mi>α</mi></math> in the comparison. However, such an increase is much larger than the uncertainties on the measured R-ratio. We also discuss a variety of generalizations of the methods used here, as well as the limits in the information that can be extracted from the R-ratio via a finite set of observables.

Michel Davier

Zoltán Fodor

Bogdan Malaescu

Finn M. Stokes

Kálmán K. Szabó

Zhiqing Zhang

Hadronic vacuum polarization: Comparing lattice QCD and data-driven results in systematically improvable ways

Hadronic light-by-light scattering is one of the virtual processes that causes the gyromagnetic factor g of the muon to deviate from the value of two predicted by Dirac’s theory. This process makes one of the largest contributions to the uncertainty of the Standard Model prediction for the muon (g − 2). Lattice QCD allows for a first-principles approach to computing this non-perturbative effect. In order to avoid power-law finite-size artifacts generated by virtual photons in lattice simulations, we follow a coordinate-space approach involving a weighted integral over the vertices of the QCD four-point function of the electromagnetic current carried by the quarks. Here we present in detail the semi-analytical calculation of the QED part of the amplitude, employing position-space perturbation theory in continuous, infinite four-dimensional Euclidean space. We also provide some useful information about a computer code for the numerical implementation of our approach that has been made public at  https://github.com/RJHudspith/KQED  .

United Kingdom

Nils Asmussen

En-Hung Chao

Jeremy Green

Renwick Hudspith

Andreas Nyffeler

Hadronic light-by-light scattering contribution to the muon g − 2 from lattice QCD: semi-analytical calculation of the QED kernel

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 ∗ → π 0 process.

Ireland

Franziska Hagelstein

Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

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.

CERN

Marco Cè

Georg Hippel

Japan

Kohtaroh Miura

Konstantin Ottnad

Andreas Risch

Teseo José

Jonas Wilhelm

Hartmut Wittig

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

We compute the hadronic light-by-light scattering contribution to the muon $$g-2$$ <math> <mrow> <mi>g</mi> <mo>-</mo> <mn>2</mn> </mrow> </math> from the charm quark using lattice QCD. The calculation is performed on ensembles generated with dynamical (u, d, s) quarks at the SU(3) $$_\mathrm{f}$$ <math> <msub> <mrow></mrow> <mi>f</mi> </msub> </math> symmetric point with degenerate pion and kaon masses of around 415 MeV. It includes the connected charm contribution, as well as the leading disconnected Wick contraction, involving the correlation between a charm and a light-quark loop. Cutoff effects turn out to be sizeable, which leads us to use lighter-than-physical charm masses, to employ a broad range of lattice spacings reaching down to 0.039 fm and to perform a combined charm-mass and continuum extrapolation. We use the $$\eta _c$$ <math> <msub> <mi>η</mi> <mi>c</mi> </msub> </math> meson to define the physical charm-mass point and obtain a final value of $$a_\mu ^\mathrm{HLbL,c}= (2.8\pm 0.5) \times 10^{-11}$$ <math> <mrow> <msubsup> <mi>a</mi> <mi>μ</mi> <mrow> <mi>HLbL</mi> <mo>,</mo> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mn>2.8</mn> <mo>±</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>11</mn> </mrow> </msup> </mrow> </math> , whose uncertainty is dominated by the systematics of the extrapolation. Our result is consistent with the estimate based on a simple charm-quark loop, whilst being free of any perturbative scheme dependence on the charm mass. The mixed charm–light disconnected contraction contributes a small negative amount to the final value.

The charm-quark contribution to light-by-light scattering in the muon $$(g-2)$$ <math> <mrow> <mo>(</mo> <mi>g</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> </math> from lattice QCD

We compute the hadronic light-by-light scattering contribution to the muon $$g-2$$ <math> <mrow> <mi>g</mi> <mo>-</mo> <mn>2</mn> </mrow> </math> from the up, down, and strange-quark sector directly using lattice QCD. Our calculation features evaluations of all possible Wick-contractions of the relevant hadronic four-point function and incorporates several different pion masses, volumes, and lattice-spacings. We obtain a value of $$a_\mu ^{\text {Hlbl}}= 106.8(15.9) \times 10^{-11}$$ <math> <mrow> <msubsup> <mi>a</mi> <mi>μ</mi> <mtext>Hlbl</mtext> </msubsup> <mo>=</mo> <mn>106.8</mn> <mrow> <mo>(</mo> <mn>15.9</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>11</mn> </mrow> </msup> </mrow> </math> (adding statistical and systematic errors in quadrature), which is consistent with current phenomenological estimates and a previous lattice determination. It now appears conclusive that the hadronic light-by-light contribution cannot explain the current tension between theory and experiment for the muon $$g-2$$ <math> <mrow> <mi>g</mi> <mo>-</mo> <mn>2</mn> </mrow> </math> .

Hadronic light-by-light contribution to $$(g-2)_\mu $$ <math> <msub> <mrow> <mo>(</mo> <mi>g</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>μ</mi> </msub> </math> from lattice QCD: a complete calculation