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

In order to reach a precision of 0$\textit{.}$2% on the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon, ($\textit{g}$ – 2) , such that the full Standard-Model prediction matches in precision the direct experimental measurement, it is crucial to include the leading, O($\textit{α}$) electromagnetic corrections to HVP. In this work, we determine an important contribution to the latter from a diagram comprised of two two-point quark-loops, connected by the internal photon and gluons. This ultraviolet-finite correction is calculated from lattice QCD using a coordinate-space formalism, where photons are treated in the continuum and infinite volume. Our result amounts to a (−0.89 ± 0.18)% correction to the leading-order HVP contribution to ($\textit{g}$ – 2) . To overcome the worsening statistical noise at large distances, our analysis is combined with phenomenological models featuring light pseudoscalar mesons with masses below 1 GeV. In particular, our data show a very steep mass dependence as dictated by the charged pion loop. Similarly, the other diagrams appearing in the O($\textit{α}$)-corrections to the HVP with the internal photon connecting two separate quark loops are also ultraviolet-finite and can be computed with the same formalism.

United States

En-Hung Chao

Computing the UV-finite electromagnetic corrections to the hadronic vacuum polarization in the muon ($\textit{g}$ − 2) 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.

Dalibor Djukanovic

Georg Hippel

Simon Kuberski

Nolan Miller

Konstantin Ottnad

Andreas Risch

Hartmut Wittig

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

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

In QCD at energies well below a heavy-quark threshold, the heavy-quark vector current can be represented via local operators made of the lighter quarks and of the gluon fields. We extract the leading perturbative matching coefficients for the two most important sets of operators from known results. As an application, we analytically determine the  $$\textrm{O}(\alpha _s^3)m_c^2/m_b^2$$  <math> <mrow> <mtext>O</mtext> <mrow> <mo>(</mo> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> <msubsup> <mi>m</mi> <mi>c</mi> <mn>2</mn> </msubsup> <mo>/</mo> <msubsup> <mi>m</mi> <mi>b</mi> <mn>2</mn> </msubsup> </mrow> </math>   effect of the bottom quark current on the R(s) ratio below the bottom but above the charm threshold. For the low-energy representation of the charm quark current, the two most important operators are given by the total divergence of dimension-six gluonic operators. We argue that the charm magnetic moment of the nucleon is effectively measuring the forward matrix elements of these gluonic operators and predict the corresponding bottom magnetic moment. Similarly, the contribution of the charm current to  $$R(s\approx 1\,\textrm{GeV}^2)$$  <math> <mrow> <mi>R</mi> <mo>(</mo> <mi>s</mi> <mo>≈</mo> <mn>1</mn> <mspace width="0.166667em"></mspace> <msup> <mtext>GeV</mtext> <mn>2</mn> </msup> <mo>)</mo> </mrow> </math>  , which is associated with quark-disconnected diagrams, is dominantly determined by the decay constants of the  $$\omega $$  <math> <mi>ω</mi> </math>   and  $$\phi $$  <math> <mi>ϕ</mi> </math>   mesons with respect to the two gluonic operators.

Low-energy matrix elements of heavy-quark currents

Quantum Chromodynamics, the theory of quarks and gluons, whose interactions can be described by a local SU(3) gauge symmetry with charges called “color quantum numbers”, is reviewed; the goal of this review is to provide advanced Ph.D. students a comprehensive handbook, helpful for their research. When QCD was “discovered” 50 years ago, the idea that quarks could exist, but not be observed, left most physicists unconvinced. Then, with the discovery of charmonium in 1974 and the explanation of its excited states using the Cornell potential, consisting of the sum of a Coulomb-like attraction and a long range linear confining potential, the theory was suddenly widely accepted. This paradigm shift is now referred to as the November revolution. It had been anticipated by the observation of scaling in deep inelastic scattering, and was followed by the discovery of gluons in three-jet events. The parameters of QCD include the running coupling constant,  $$\alpha _s(Q^2)$$  <math> <mrow> <msub> <mi>α</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </math>  , that varies with the energy scale  $$Q^2$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   characterising the interaction, and six quark masses. QCD cannot be solved analytically, at least not yet, and the large value of  $$\alpha _s$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   at low momentum transfers limits perturbative calculations to the high-energy region where  $$Q^2\gg \varLambda _{{\textrm{QCD}}} ^2\simeq $$  <math> <mrow> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>≫</mo> <msubsup> <mi>Λ</mi> <mrow> <mtext>QCD</mtext> </mrow> <mn>2</mn> </msubsup> <mo>≃</mo> </mrow> </math>   (250 MeV) $$^2$$  <math> <msup> <mrow></mrow> <mn>2</mn> </msup> </math>  . Lattice QCD (LQCD), numerical calculations on a discretized space-time lattice, is discussed in detail, the dynamics of the QCD vacuum is visualized, and the expected spectra of mesons and baryons are displayed. Progress in lattice calculations of the structure of nucleons and of quantities related to the phase diagram of dense and hot (or cold) hadronic matter are reviewed. Methods and examples of how to calculate hadronic corrections to weak matrix elements on a lattice are outlined. The wide variety of analytical approximations currently in use, and the accuracy of these approximations, are reviewed. These methods range from the Bethe–Salpeter, Dyson–Schwinger coupled relativistic equations, which are formulated in both Minkowski or Euclidean spaces, to expansions of multi-quark states in a set of basis functions using light-front coordinates, to the AdS/QCD method that imbeds 4-dimensional QCD in a 5-dimensional deSitter space, allowing confinement and spontaneous chiral symmetry breaking to be described in a novel way. Models that assume the number of colors is very large, i.e. make use of the large  $$N_c$$  <math> <msub> <mi>N</mi> <mi>c</mi> </msub> </math>  -limit, give unique insights. Many other techniques that are tailored to specific problems, such as perturbative expansions for high energy scattering or approximate calculations using the operator product expansion are discussed. The very powerful effective field theory techniques that are successful for low energy nuclear systems (chiral effective theory), or for non-relativistic systems involving heavy quarks, or the treatment of gluon exchanges between energetic, collinear partons encountered in jets, are discussed. The spectroscopy of mesons and baryons has played an important historical role in the development of QCD. The famous X,Y,Z states – and the discovery of pentaquarks – have revolutionized hadron spectroscopy; their status and interpretation are reviewed as well as recent progress in the identification of glueballs and hybrids in light-meson spectroscopy. These exotic states add to the spectrum of expected  $$q{{\bar{q}}}$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   mesons and qqq baryons. The progress in understanding excitations of light and heavy baryons is discussed. The nucleon as the lightest baryon is discussed extensively, its form factors, its partonic structure and the status of the attempt to determine a three-dimensional picture of the parton distribution. An experimental program to study the phase diagram of QCD at high temperature and density started with fixed target experiments in various laboratories in the second half of the 1980s, and then, in this century, with colliders. QCD thermodynamics at high temperature became accessible to LQCD, and numerical results on chiral and deconfinement transitions and properties of the deconfined and chirally restored form of strongly interacting matter, called the Quark–Gluon Plasma (QGP), have become very precise by now. These results can now be confronted with experimental data that are sensitive to the nature of the phase transition. There is clear evidence that the QGP phase is created. This phase of QCD matter can already be characterized by some properties that indicate, within a temperature range of a few times the pseudocritical temperature, the medium behaves like a near ideal liquid. Experimental observables are presented that demonstrate deconfinement. High and ultrahigh density QCD matter at moderate and low temperatures shows interesting features and new phases that are of astrophysical relevance. They are reviewed here and some of the astrophysical implications are discussed. Perturbative QCD and methods to describe the different aspects of scattering processes are discussed. The primary parton–parton scattering in a collision is calculated in perturbative QCD with increasing complexity. The radiation of soft gluons can spoil the perturbative convergence, this can be cured by resummation techniques, which are also described here. Realistic descriptions of QCD scattering events need to model the cascade of quark and gluon splittings until hadron formation sets in, which is done by parton showers. The full event simulation can be performed with Monte Carlo event generators, which simulate the full chain from the hard interaction to the hadronic final states, including the modelling of non-perturbative components. The contribution of the LEP experiments (and of earlier collider experiments) to the study of jets is reviewed. Correlations between jets and the shape of jets had allowed the collaborations to determine the “color factors” – invariants of the SU(3) color group governing the strength of quark–gluon and gluon–gluon interactions. The calculated jet production rates (using perturbative QCD) are shown to agree precisely with data, for jet energies spanning more than five orders of magnitude. The production of jets recoiling against a vector boson,  $$W^\pm $$  <math> <msup> <mi>W</mi> <mo>±</mo> </msup> </math>   or Z, is shown to be well understood. The discovery of the Higgs boson was certainly an important milestone in the development of high-energy physics. The couplings of the Higgs boson to massive vector bosons and fermions that have been measured so far support its interpretation as mass-generating boson as predicted by the Standard Model. The study of the Higgs boson recoiling against hadronic jets (without or with heavy flavors) or against vector bosons is also highlighted. Apart from the description of hard interactions taking place at high energies, the understanding of “soft QCD” is also very important. In this respect, Pomeron – and Odderon – exchange, soft and hard diffraction are discussed. Weak decays of quarks and leptons, the quark mixing matrix and the anomalous magnetic moment of the muon are processes which are governed by weak interactions. However, corrections by strong interactions are important, and these are reviewed. As the measured values are incompatible with (most of) the predictions, the question arises: are these discrepancies first hints for New Physics beyond the Standard Model? This volume concludes with a description of future facilities or important upgrades of existing facilities which improve their luminosity by orders of magnitude. The best is yet to come!

Franz Gross

Eberhard Klempt

Stanley Brodsky

Andrzej Buras

Volker Burkert

Gudrun Heinrich

Karl Jakobs

Curtis Meyer

Kostas Orginos

Michael Strickland

Johanna Stachel

Giulia Zanderighi

Nora Brambilla

Peter Braun-Munzinger

Daniel Britzger

Simon Capstick

Tom Cohen

Volker Crede

Martha Constantinou

United Kingdom

Christine Davies

Luigi Debbio

Achim Denig

Carleton DeTar

Alexandre Deur

Russia

Latvia

Yuri Dokshitzer

Hans Dosch

Jozef Dudek

Monica Dunford

Evgeny Epelbaum

Spain

Miguel Escobedo

Harald Fritzsch

Japan

Kenji Fukushima

Paolo Gambino

Canada

Sweden

Dag Gillberg

Steven Gottlieb

Per Grafstrom

Massimiliano Grazzini

Boris Grube

JINR

Alexey Guskov

Toru Iijima

Xiangdong Ji

Frithjof Karsch

Stefan Kluth

John Kogut

Frank Krauss

Shunzo Kumano

Australia

Derek Leinweber

Heinrich Leutwyler

China

Hai-Bo Li

Yang Li

Bogdan Malaescu

Chiara Mariotti

Pieter Maris

Simone Marzani

Wally Melnitchouk

Johan Messchendorp

Ryan Mitchell

Chandan Mondal

Frank Nerling

Sebastian Neubert

Marco Pappagallo

Saori Pastore

José Peláez

Andrew Puckett

Jianwei Qiu

Klaus Rabbertz

Alberto Ramos

Patrizia Rossi

Azerbaijan

Anar Rustamov

Andreas Schäfer

Stefan Scherer

Matthias Schindler

Steven Schramm

Mikhail Shifman

Edward Shuryak

Torbjörn Sjöstrand

George Sterman

Iain Stewart

Joachim Stroth

Eric Swanson

Costa Rica

Guy Téramond

Ulrike Thoma

Antonio Vairo

Danny Dyk

James Vary

Javier Virto

Marcel Vos

Christian Weiss

Markus Wobisch

Sau Wu

Christopher Young

Feng Yuan

Xingbo Zhao

Xiaorong Zhou

50 Years of quantum chromodynamics

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  .

Nils Asmussen

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.

Kohtaroh Miura

Teseo José

Jonas Wilhelm

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