<p>We develop a generalization of low-mode averaging in which the number of low quark modes of the Dirac operator required for a constant variance reduction can be kept independent of the volume by exploiting their local coherence. Typically in lattice QCD simulations, the benefit of translation averaging quark propagators over the space-time volume is spoiled by large fluctuations introduced by the approximations needed to estimate the average. For quark-line connected diagrams at large separations, most of this additional variance can be efficiently suppressed by the introduction of hierarchical subspaces, thanks to the reduced size of the coarse grid operators that act within the subspaces. In this work, we investigate the contributions to the variance of the isovector vector current correlator with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi mathvariant="normal">f</mi></msub><mo>=</mo><mn>2</mn></math> nonperturbatively <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math>-improved Wilson fermions on lattices approximately of size <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi><mo>=</mo><mn>2</mn></math>, 3 and 4 fm. The numerical results obtained confirm that the variance decreases as the volume is increased when a multigrid decomposition is used with a fixed number of low modes. We obtain a speedup of more than an order of magnitude compared with a plain stochastic estimator. Compared with state-of-the-art low-mode averaging we practically eliminate the cost of generating low modes and the associated memory and storage overheads. While the proposed decomposition can be applied to any quark propagator, it is expected to be especially effective for quark-line connected diagrams at large separations, for example, the isovector contribution to the hadronic vacuum polarization or baryonic correlators.</p>

Switzerland

Roman Gruber

Tim Harris

Marina Krstić Marinković

Multigrid low-mode averaging

We compute the baryonic screening masses of the nucleon and of its negative parity partner in thermal QCD with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>3</mn></math> massless quarks for a wide range of temperatures, from <math><mi>T</mi><mo>∼</mo><mn>1</mn></math> GeV up to ∼160 GeV. The computation is performed by Monte Carlo simulations of lattice QCD with <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math>-improved Wilson fermions by exploiting a recently proposed strategy to study non-perturbatively QCD at very high temperature. Very large spatial extensions are considered in order to have negligible finite volume effects. For each temperature we have simulated 3 or 4 values of the lattice spacing, so as to perform the continuum limit extrapolation with confidence at a few permille accuracy. The degeneracy of the positive and negative parity-state screening masses, expected from Ward identities associated to non-singlet axial transformations, provides further evidence for the restoration of chiral symmetry in the high temperature regime of QCD. In the entire range of temperatures explored, the baryonic masses deviate from the free theory result, <math><mn>3</mn><mi>π</mi><mi>T</mi></math>, by 4–8%. The contribution due to the interactions is clearly visible up to the highest temperature considered, and cannot be explained by the expected leading behavior in the QCD coupling constant g over the entire range of temperatures explored.

Italy

Leonardo Giusti

Davide Laudicina

Michele Pepe

Pietro Rescigno

Baryonic screening masses in QCD at high temperature

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.

Marco Cè

Germany

Ardit Krasniqi

Harvey B. Meyer

Csaba Török

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

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.

United Kingdom

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

The decreasing uncertainties in theoretical predictions and experimental measurements of several hadronic observables related to weak processes, which in many cases are now smaller than O(1%), require theoretical calculations to include subleading corrections that were neglected so far. Precise determinations of leptonic and semi-leptonic decay rates, including QED and strong isospin-breaking effects, can play a central role in solving the current tensions in the first-row unitarity of the CKM matrix. In this work we present the first RBC/UKQCD lattice calculation of the isospin-breaking corrections to the ratio of leptonic decay rates of kaons and pions into muons and neutrinos. The calculation is performed at fixed lattice spacing (a  −1 ≃ 1.730 GeV) on a 483 × 96 volume with N f = 2 + 1 dynamical quarks close to the physical point and domain wall fermions in the Möbius formulation are employed. Long-distance QED interactions are included according to the QEDL prescription and the crucial role of finite-volume electromagnetic corrections in the determination of leptonic decay rates, which produce a large systematic uncertainty, is extensively discussed. Finally, we study the different sources of uncertainty on |V us|/|V ud| and observe that, if finite-volume systematics can be reduced, the error from isospin-breaking corrections is potentially sub-dominant in the final precision of the ratio of the CKM matrix elements.

United States

Peter Boyle

Matteo Carlo

Felix Erben

Vera Gülpers

Maxwell Hansen

Sweden

Nils Hermansson-Truedsson

Raoul Hodgson

CERN

Andreas Jüttner

Fionn hÓgáin

Antonin Portelli

James Richings

Andrew Yong

Isospin-breaking corrections to light-meson leptonic decays from lattice simulations at physical quark masses

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 present a strategy based on the step-scaling technique to study non-perturbatively thermal QCD up to very high temperatures. As a first concrete application, we compute the flavour non-singlet meson screening masses at 12 temperatures covering the range from T ∼ 1 GeV up to ∼ 160 GeV in the theory with three massless quarks. The calculation is carried out by Monte Carlo simulations on the lattice by considering large spatial extensions in order to have negligible finite volume effects. For each temperature we have simulated 3 or 4 values of the lattice spacing, so as to perform the continuum limit extrapolation with confidence at a few permille accuracy. Chiral symmetry restoration manifests itself in our results through the degeneracy of the vector and the axial vector channels and of the scalar and the pseudoscalar ones. In the entire range of temperatures explored, the meson screening masses deviate from the free theory result, 2πT, by at most a few percent. These deviations, however, cannot be explained by the known leading term in the QCD coupling constant g up to the highest temperature, where other contributions are still very relevant. In particular the vector-pseudoscalar mass splitting turns out to be of O(g4) in the entire range explored, and it remains clearly visible up to the highest temperature, where the two screening masses are still significantly different within our numerical precision. The pattern of different contributions that we have found explains why it has been difficult in the past to match non-perturbative lattice results at T ∼ 1 GeV with the analytic behaviour at asymptotically high temperatures.

Mattia Dalla Brida

Non-perturbative thermal QCD at all temperatures: the case of mesonic screening masses

We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a “one-loop” analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.

Koushik Ganesan

Andrew Lucas

Leo Radzihovsky

Renormalization group in quantum critical theories with Harris-marginal disorder

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.

Harvey Meyer

Arianna Toniato

Vacuum correlators at short distances from lattice QCD

The hadronic contribution to the muon anomalous magnetic moment <math><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>−</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn></math> has to be determined at the per-mille level for the Standard Model prediction to match the expected final uncertainty from the ongoing E989 experiment. This is 3 times better than the current precision from the dispersive approach, and 5-15 times smaller than the uncertainty on the purely theoretical determinations from lattice QCD. So far the stumbling-block is the large statistical error in the Monte Carlo evaluation of the required correlation functions which can hardly be tamed by brute force. Here we propose to solve this problem by multi-level Monte Carlo integration, a technique which reduces the variance of correlators exponentially in the distance of the fields. We test our strategy by computing the Hadronic Vacuum Polarization on a lattice with a linear extension of 3 fm, a spacing of 0.065 fm, and a pion mass of 270 MeV. Indeed the two-level integration makes the contribution to the statistical error from long-distances de-facto negligible by accelerating its inverse scaling with the cost of the simulation. These findings establish multi-level Monte Carlo as a solid and efficient method for a precise lattice determination of the hadronic contribution to <math><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub></math>. As the approach is applicable to other computations affected by a signal-to-noise ratio problem, it has the potential to unlock many open problems for the nuclear and particle physics community.

Multi-level Monte Carlo computation of the hadronic vacuum polarization contribution to ( − 2)