<p>Determining the spectrum of photons emitted by the quark-gluon plasma nonperturbatively remains an open computational challenge. In this paper, we calculate two moments of that spectrum at a tem-perature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mo>≈</mo><mn>254</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, employing lattice QCD with two flavors of <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, without facing an inverse problem. Our central value for the difference of these two moments, which is sensitive to photon energies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi><mo>≳</mo><mi>π</mi><mi>T</mi></math>, is lower than, but compatible with that obtained by integrating the leading-order weak-coupling photon spectrum. This study informs the “direct photon puzzle” in heavy-ion collision phenomenology, where it has proved difficult to simultaneously explain the yield and azimuthal anisotropy of photons not originating from final-state hadronic decays.</p>

Germany

Ardit Krasniqi

Italy

Marco Cè

Switzerland

Tim Harris

United States

Renwick J. Hudspith

Harvey B. Meyer

Probing how bright the quark-gluon plasma glows in lattice QCD

We determine the hyperfine splitting in the QCD flavour non-singlet mesonic screening masses at asymptotically large temperatures. The analytic calculation is carried out in the dimensionally-reduced effective theory where the first non-zero contribution is of $\textit{O}$($\textit{g}$ ) in the QCD coupling constant $\textit{g}$. Apart for its own theoretical interest, this result provides instrumental information to interpret and to parameterize non-perturbative data that are being produced at very high temperatures by numerical simulations of lattice QCD. Indeed, the comparison with existing non-perturbative results shows that higher order (non-perturbative) contributions in $\textit{g}$ are needed to explain the data up to the highest temperature explored, which is of the order of the electroweak scale.

Leonardo Giusti

Davide Laudicina

Michele Pepe

Pietro Rescigno

The hyperfine splitting in QCD mesonic screening masses at asymptotically large temperatures

Using lattice QCD simulations with <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math>-improved Wilson quarks and physical up, down, and strange quark masses, we investigate the properties of thermal QCD matter at the temperatures <math><mi>T</mi><mo>=</mo><mo>{</mo><mn>128</mn><mo>,</mo><mn>154</mn><mo>,</mo><mn>192</mn><mo>}</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> with a fixed lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.064</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and volume <math><mi>V</mi><mo>=</mo><mo>(</mo><mn>6.1</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mo>)</mo><mn>3</mn></msup></math>. We find that the pion quasiparticle, defined as the low-energy pole in the two-point function of the axial charge, becomes lighter as the temperature increases, and we give an argument based on hydrodynamics as to why the pole becomes purely diffusive above the chiral crossover. We study the thermal modification of the isovector vector spectral function using the Backus-Gilbert method, finding an enhancement at low energies and a depletion at energies around 1 GeV. The analogous study of the axial-vector channel reveals a larger enhancement at energies below 1 GeV, and we show that these findings are consistent with rigorous spectral sum rules. The difference between vector and axial-vector correlators, an order parameter for chiral symmetry, turns out to be overall suppressed by more than an order of magnitude at the crossover.

Hot QCD matter around the chiral crossover: A lattice study with <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

Simon Kuberski

Georg Hippel

Harvey Meyer

Konstantin Ottnad

Andreas Risch

Hartmut Wittig

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

The thermal photon emissivity of the quark-gluon plasma is determined by the in-medium spectral function of the electromagnetic current at lightlike kinematics, <math><mi>σ</mi><mo>(</mo><mi>ω</mi><mo>)</mo></math>. In this work, we present the first lattice QCD results on moments of <math><mi>σ</mi><mo>(</mo><mi>ω</mi><mo>)</mo><mo>/</mo><mi>ω</mi></math>, defined by the weight function <math><mn>1</mn><mo>/</mo><mo>(</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>2</mn><mi>π</mi><mi>T</mi><mi>n</mi><msup><mo>)</mo><mn>2</mn></msup><mo>)</mo></math>, <math><mi>n</mi><mo>∈</mo><mi>Z</mi></math> and computed without encountering an inverse problem. We employ two dynamical flavors of <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>-improved Wilson fermions at a temperature <math><mi>T</mi><mo>≈</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and perform the continuum limit. We compare our results for the first two moments to those obtained dispersively by integrating over the spectral function computed at weak coupling by Arnold, Moore and Yaffe.

Csaba Török

Probing the photon emissivity of the quark-gluon plasma without an inverse problem in lattice QCD

Because of the mass gap, lattice QCD simulations exhibit stochastic locality: distant regions of the lattice fluctuate independently. There is a long history of exploiting this to increase statistics by obtaining multiple spatially-separated samples from each gauge field; in the extreme case, we arrive at the master-field approach in which a single gauge field is used. Here we develop techniques for studying hadronic observables using position-space correlators, which are more localized, and compare with the standard time-momentum representation. We also adapt methods for estimating the variance of an observable from autocorrelated Monte Carlo samples to the case of correlated spatially-separated samples.

Mattia Bruno

Taiwan

Anthony Francis

Ireland

Patrick Fritzsch

Jeremy Green

United Kingdom

Maxwell Hansen

Denmark

Antonio Rago

Exploiting stochastic locality in lattice QCD: hadronic observables and their uncertainties

We investigate several aspects of chiral symmetry in QCD at a temperature of <math><mi>T</mi><mo>=</mo><mn>128</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The study is based on a <math><mn>24</mn><mo>×</mo><mn>9</mn><msup><mn>6</mn><mn>3</mn></msup></math> lattice-QCD ensemble with <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>-improved Wilson quarks and physical up, down and strange quark masses. The pion quasiparticle turns out to be significantly lighter than the zero-temperature pion mass, even though the corresponding static correlation length is shorter. We perform a quantitative comparison of our findings to predictions of chiral perturbation theory. Among several order parameters for chiral symmetry restoration, we compute the difference of the vector- and axial-vector time-dependent correlators and find it to be reduced by a factor <math><mo>∼</mo><mn>2</mn><mo>/</mo><mn>3</mn></math> as compared to its vacuum counterpart.

Aspects of chiral symmetry in QCD at <math><mi>T</mi><mo>=</mo><mn>128</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>

We present results for the thermal photon emissivity of the quark-gluon plasma derived from spatially transverse vector correlators computed in lattice QCD at a temperature of 250 MeV. The analysis of the spectral functions, performed at fixed spatial momentum, is based on continuum-extrapolated correlators obtained with two flavors of dynamical Wilson fermions. We compare the next-to-leading-order perturbative QCD correlators, as well as the <math><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></math> supersymmetric Yang-Mills correlators at infinite coupling, to the correlators from lattice QCD and find them to lie within <math><mo>∼</mo><mn>10</mn><mo>%</mo></math> of each other. We then refine the comparison, performing it at the level of filtered spectral functions obtained model-independently via the Backus-Gilbert method. Motivated by these studies, for frequencies <math><mi>ω</mi><mo>≲</mo><mn>2.5</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> we use fit Ansätze to the spectral functions that perform well when applied to mock data generated from the NLO QCD or from the strongly coupled SYM spectral functions, while the high-frequency part, <math><mi>ω</mi><mo>≳</mo><mn>2.5</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, is matched to NLO QCD. We compare our results for the photon emissivity to our previous analysis of a different vector channel at the same temperature. We obtain the most stringent constraint at photon momenta around <math><mi>k</mi><mo>≃</mo><mn>0.8</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, for which we find a differential photon emission rate per unit volume of <math><mi>d</mi><msub><mi>Γ</mi><mi>γ</mi></msub><mo>/</mo><msup><mi>d</mi><mn>3</mn></msup><mi>k</mi><mo>=</mo><mo>(</mo><msub><mi>α</mi><mrow><mi>em</mi></mrow></msub><mo>/</mo><mo>(</mo><mi>exp</mi><mo>(</mo><mi>k</mi><mo>/</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>×</mo><mo>(</mo><mn>2.2</mn><mo>±</mo><mn>0.8</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>.

Photon emissivity of the quark-gluon plasma: A lattice QCD analysis of the transverse channel

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

Non-perturbatively computing the hadronic vacuum polarization at large photon virtualities and making contact with perturbation theory enables a precision determination of the electromagnetic coupling at the Z pole, which enters global electroweak fits. In order to achieve this goal ab initio using lattice QCD, one faces the challenge that, at the short distances which dominate the observable, discretization errors are hard to control. Here we address challenges of this type with the help of static screening correlators in the high-temperature phase of QCD, yet without incurring any bias. The idea is motivated by the observations that (a) the cost of high-temperature simulations is typically much lower than their vacuum counterpart, and (b) at distances x 3 far below the inverse temperature 1/T, the operator-product expansion guarantees the thermal correlator of two local currents to deviate from the vacuum correlator by a relative amount that is power-suppressed in (x 3 T). The method is first investigated in lattice perturbation theory, where we point out the appearance of an O(a 2 log(1/a)) lattice artifact in the vacuum polarization with a prefactor that we calculate. It is then applied to non-perturbative lattice QCD data with two dynamical flavors of quarks. Our lattice spacings range down to 0.049 fm for the vacuum simulations and down to 0.033 fm for the simulations performed at a temperature of 250 MeV.

Arianna Toniato

Vacuum correlators at short distances from lattice QCD