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We discuss a recently introduced strategy to study non-perturbatively thermal QCD up to temperatures of the order of the electro-weak scale, combining step scaling techniques and shifted boundary conditions. The former allow to renormalize the theory for a range of scales which spans several orders of magnitude with a moderate computational cost. Shifted boundary conditions remove the need for the zero temperature subtraction in the Equation of State. As a consequence, the simulated lattices do not have to accommodate two very different scales, the pion mass and the temperature, at the very same spacing. Effective field theory arguments guarantee that finite volume effects can be kept under control safely. With this strategy the first computation of the hadronic screening spectrum has been carried out over more than two orders of magnitude in the temperature, from $T\sim 1$ GeV up to $\sim 160$ GeV. This study is complemented with the first quantitative computation of the baryonic screening mass at next-to-leading order in the three-dimensional effective theory describing QCD at high temperatures. Both for the mesonic and the baryonic screening masses, the known leading behaviour in the coupling constant is found to be not sufficient to explain the non-perturbative data over the entire range of temperatures. These findings shed further light on the limited applicability of the perturbative approach at finite temperature, even at the electro-weak scale., Comment: arXiv admin note: substantial text overlap with arXiv:2410.12626

We present a recently introduced strategy to study non-perturbatively thermal QCD up to temperatures of the order of the electro-weak scale, combining step scaling techniques and shifted boundary conditions. The former allow to renormalize the theory for a range of scales which spans several orders of magnitude with a moderate computational cost. Shifted boundary conditions avoid the need for the zero temperature subtraction in the Equation of State. As a consequence, the simulated lattices do not have to accommodate two very different scales, the pion mass and the temperature. Effective field theory arguments guarantee that finite volume effects can be kept under control safely. As a first application of this strategy, we present the results of the computation of the hadronic screening spectrum in QCD with $N_f=3$ flavours of massless quarks for temperatures from $T\sim 1$ GeV up to $\sim 160$ GeV., Comment: 6 pages, 3 figures, Proceedings of the 42nd International Conference on High Energy Physics (ICHEP2024)

In large enough volumes, translation-averaging for quark-line connected diagrams reduces the variance inversely proportional to the volume. Stochastic estimators which implement translation averaging however introduce new sources of fluctuations, which in some cases can be relatively large. In this work, we explore whether inexact deflation subspaces can be used to improve the precision of the isovector vector correlators. We perform numerical experiments with $N_\mathrm{f}=2$ non-perturbatively $\mathrm O(a)$-improved Wilson fermions and measure the relative contribution from the deflation subspace to the central value and the corresponding variance.

The thermal photon emissivity at the QCD chiral crossover is investigated using imaginary momentum correlators. These have been measured on a newly generated $20 \times 96^3$ lattice-QCD ensemble with $\mathcal{O}(a)$-improved Wilson quarks and physical up, down and strange quark masses at a temperature $T=154$\,MeV near the pseudo-critical temperature. In order to realize the photon on-shell condition, the spatially transverse Euclidean correlators have to be evaluated at imaginary spatial momenta. Employing a bounding method, we present a preliminary result on the quantity $H_E(\omega_1)$, which corresponds to an energy-moment of the photon spectral function $\sigma(\omega)/\omega$ defined by the weight function $1/(\omega^2+\omega_n^2)$, the $\omega_n$ being integer multiples of $2\pi T$., Comment: 12 pages, 4 Figures, 1 Table

The thermal photon emissivity of the quark-gluon plasma is determined by the in-medium spectral function of the electromagnetic current at lightlike kinematics, $\sigma(\omega)$. In this work, we present the first lattice QCD results on moments of $\sigma(\omega)/\omega$, defined by the weight function $1/(\omega^2+ (2\pi T n)^2)$, $n\in\mathbb{Z}$ and computed without encountering an inverse problem. We employ two dynamical flavours of O($a$)-improved Wilson fermions at a temperature $T\approx 250\;$MeV 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.

We outline a strategy to efficiently include the electromagnetic interactions of the sea quarks in QCD+QED. When computing iso-spin breaking corrections to hadronic quantities at leading order in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected diagrams which are challenging to compute precisely. An analysis of the variance of stochastic estimators for the relevant traces of quark propagators helps us to improve the situation for certain flavour combinations and space-time decompositions. We present preliminary numerical results for the variances of the corresponding contributions using an ensemble of $N_\mathrm{f}=2+1$ domain-wall fermions generated by the RBC/UKQCD collaboration., Comment: 9 pages, 3 figures; contribution to the 39th International Symposium on Lattice Field Theory, Lattice 2022, Bonn, Germany

The thermal photon emission rate is determined by the spatially transverse, in-medium spectral function of the electromagnetic current. Accessing the spectral function using Euclidean data is, however, a challenging problem due to the ill-posed nature of inverting the Laplace transform. In this contribution, we present the first results on implementing the proposal of directly computing the analytic continuation of the retarded correlator at fixed, vanishing virtuality of the photon via the calculation of the appropriate Euclidean correlator at imaginary spatial momentum. We employ two dynamical O(a)-improved Wilson fermions at a temperature of 250 MeV., Comment: 9 pages, 7 figures, contribution to the 39th International Symposium on Lattice Field Theory (LATTICE2022)

We report on the physical-point RBC/UKQCD calculation of the leading isospin-breaking corrections to light-meson leptonic decays. This is highly relevant for future precision tests in the flavour physics sector, in particular the first-row unitarity of the Cabibbo-Kobayashi-Maskawa matrix containing the elements $V_{us}$ and $V_{ud}$. The simulations were performed using Domain-Wall fermions for $2+1$ flavours, and with isospin-breaking effects included perturbatively in the path integral through order $\alpha$ and $(m_u - m_d)/\Lambda _{\mathrm{QCD}}$. We use QED$_{\mathrm{L}}$ for the inclusion of electromagnetism, and discuss here the non-locality of this prescription which has significant impact on the infinite-volume extrapolation., Comment: Proceedings for The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universit\"at Bonn, Bonn, Germany

We present a strategy to study QCD non-perturbatively on the lattice at very high temperatures. This strategy exploits a non-perturbative, finite-volume, definition of the strong coupling constant to renormalize the theory. As a first application we compute the flavor non-singlet mesonic screening masses in a wide range of temperature, from $T\sim 1 $ GeV up to $\sim 160 $ GeV with three flavors in the chiral limit of QCD. Our results show very interesting features of the screening spectrum at very high temperatures. On one hand the mass splitting between the vector and the pseudoscalar screening masses is clearly visible up to the electroweak scale and cannot be explained by the known 1-loop perturbative result. On the other hand the restoration of chiral symmetry manifests itself through the degeneracy of the pseudoscalar and the scalar channels and of the vector and the axial-vector ones. This degeneracy pattern is the one expected by Ward identities associated to the presence of chiral symmetry., Comment: 9 pages, 4 figures, contribution to the 39th International Symposium on Lattice Field Theory (LATTICE2022), August 8-13, 2022, Bonn (Germany)

We investigate several aspects of chiral symmetry in QCD at a temperature of $T = 128\,\text{MeV}$. The study is based on a $24\times 96^3$ lattice-QCD ensemble with O($a$)-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 $\sim2/3$ as compared to its vacuum counterpart., Comment: 27 pages, 9 Figures

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 $\mathrm{O}(1\%)$, 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 with $N_\mathrm{f}=2+1$ dynamical quarks close to the physical point and domain wall fermions in the M\"obius formulation are employed. Long-distance QED interactions are included according to the $\mathrm{QED_L}$ 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 $|V_\mathrm{us}|/|V_\mathrm{ud}|$ 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., Comment: 66 pages, 23 figures and 4 tables

We discuss a strategy to study non-perturbatively QCD up to very high temperatures by Monte Carlo simulations on the lattice. It allows not only the thermodynamic properties of the theory but also other interesting thermal features to be investigated. As a first concrete application, we compute the flavour non-singlet mesonic screening masses and we present the results of Monte Carlo simulations at 12 temperatures covering the range from T $\sim$ 1 GeV up to $\sim$ 160 GeV in the theory with three massless quarks. On the one side, 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, and, on the other side, we observe a clear splitting between the vector and the pseudoscalar screening masses up to the highest investigated temperature. A comparison with the high-temperature effective theory shows that the known one-loop order in the perturbative expansion does not provide a satisfactory description of the non-perturbative data up to the highest temperature considered., Comment: 6 pages, 3 figures; contribution to the proceedings of the International Conference on High Energy Physics (ICHEP 2022), July 6-13, 2022, Bologna (Italy)

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 flavours of dynamical Wilson fermions. We compare the next-to-leading order perturbative QCD correlators, as well as the ${\cal N}=4$ supersymmetric Yang-Mills correlators at infinite coupling, to the correlators from lattice QCD and find them to lie within $\sim10\%$ 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 $\omega\lesssim2.5\,$GeV we use fit ans\"atze 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, $\omega\gtrsim 2.5\,$GeV, 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 $k\simeq0.8\,$GeV, for which we find a differential photon emission rate per unit volume of $d\Gamma_\gamma/d^3k = (\alpha_{\rm em}/(\exp(k/T)-1))\times (2.2 \pm 0.8 ) \times 10^{-3}\,{\rm GeV}$., Comment: 26 pages, 13 figures, 1 table

In recent years, lattice determinations of non-perturbative quantities such as $f_K$ and $f_\pi$, which are relevant for $V_{us}$ and $V_{ud}$, have reached an impressive precision of $\mathcal{O}(1\%)$ or better. To make further progress, electromagnetic and strong isospin breaking effects must be included in lattice QCD simulations. We present the status of the RBC/UKQCD lattice calculation of isospin-breaking corrections to light meson leptonic decays. This computation is performed in a (2+1)-flavor QCD simulation using Domain Wall Fermions with near-physical quark masses. The isospin-breaking effects are implemented via a perturbative expansion of the action in $\alpha$ and $(m_u-m_d)$. In this calculation, we work in the electro-quenched approximation and the photons are implemented in the Feynman gauge and $\text{QED}_\text{L}$ formulation., Comment: 11 pages, 4 figures; Updated Author(s) metadata

We compute flavor non-singlet meson screening masses in the chiral limit of QCD with $N_f=3$ quarks. The calculation is carried out at 12 temperatures from $T\approx 1$ GeV up to the electroweak scale. For each temperature we simulated several lattice spacings, so as to be able to perform the continuum limit extrapolation with confidence at a few permille accuracy. In the entire range of temperatures explored, the meson screening masses deviate from the free theory result $2\pi T$ by at most a few percent. Their values, however, cannot be explained by one-loop perturbation theory up to the electroweak scale, where the pseudoscalar and the vector screening masses are still significantly different within our precision. Chiral symmetry restoration manifests itself through the degeneracy of the pseudoscalar and the scalar channels and of the vector and the axial ones., Comment: 8 pages, 4 figures, contribution to the 38th International Symposium on Lattice Field Theory, 26th-30th July 2021, Massachusetts Institute of Technology, USA

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 $T \sim 1$ GeV up to $\sim 160$ 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, $2 \pi T$, 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 $O(g^4)$ 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 $T \sim 1$ GeV with the analytic behaviour at asymptotically high temperatures., Comment: 32 pages, 5 figures. V2 typos corrected, minor modifications in the text, version to appear on JHEP

The first results from the Fermilab E989 experiment have confirmed the long-standing tension between the experimental determination of the muon anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ and its SM determination using the dispersive approach. In order to match the expected final precision from E989, the current uncertainty on ab initio determinations using lattice QCD must be decreased by a factor 5-15, a goal which is hampered by the signal-to-noise ratio problem of the electromagnetic current correlator. Multi-level Monte Carlo integration with fermions is a method which reduces the variance of correlators exponentially in the distance of the fields. Here we demonstrate that the variance reduction in a realistic two-level simulation with a pion mass of 270 MeV, linear size of 3 fm and lattice spacing around 0.065 fm is sufficient to compute the tail of the current correlator with the statistical accuracy required for the hadronic vacuum polarization contribution to $a_\mu$. An efficient estimator is also employed for computing the disconnected contribution., Comment: 8 pages, 5 figures, Proceedings of the 38th International Symposium on Lattice Field Theory (LATTICE2021). arXiv admin note: substantial text overlap with arXiv:2101.04642

We present results for the photon emission rate determined from the transverse channel vector correlator at fixed spatial momentum using two flavors of dynamical Wilson fermions at $T\sim$250 MeV. We estimate the transverse channel spectral function using the continuum extrapolated correlator by applying various fit ans\"atze with a smooth matching to the NLO perturbative result. We confront our estimate based on this channel with the latest results of our collaboration based on the difference of the transverse and longitudinal channels., Comment: Proceedings of the 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@MIT

The photon emissivity of quark-gluon plasma probes the interactions in the medium and differs qualitatively between a weakly coupled and a strongly coupled plasma in the soft-photon regime. The photon emissivity is given by the product of kinematic factors and a spectral function associated with the two-point correlator of the electromagnetic current at lightlike kinematics. A certain Euclidean correlator at imaginary spatial momentum can be calculated in lattice QCD and is given by an integral over the relevant spectral function at lightlike kinematics. I present a first exploratory lattice calculation of this correlator. Secondly, I show how Euclidean correlators at imaginary spatial momenta can also be used to probe the regime of deep inelastic scattering off quark-gluon plasma, which reveals its parton distribution function., Comment: 8 pages, 3 figures

We propose a method to help control cutoff effects in the short-distance contribution to integrated correlation functions, such as the hadronic vacuum polarization (HVP), using the corresponding screening correlators computed at finite temperature. The strategy is investigated with Wilson fermions at leading order, which reveals a logarithmically-enhanced lattice artifact in the short-distance contribution, whose coefficient is determined at this order. We then perform a numerical study with $N_\mathrm{f}=2$ O($a$)-improved Wilson fermions and a temperature $T\approx250~\mathrm{MeV}$, with lattice spacings down to $a\approx0.03~\mathrm{fm}$, which suggests good control can be achieved on the short-distance contribution to the HVP and the Adler function at large virtuality. Finally, we put forward a scheme to compute the complete HVP function at arbitrarily large virtualities using a step-scaling in the temperature., Comment: 9 pages, 4 figures. Talk given at the 38th International Symposium on Lattice Field Theory (LATTICE2021)

The rapidly growing size of deep neural network (DNN) models and datasets has given rise to a variety of distribution strategies such as data, tensor-model, pipeline parallelism, and hybrid combinations thereof. Each of these strategies offers its own trade-offs and exhibits optimal performance across different models and hardware topologies. Selecting the best set of strategies for a given setup is challenging because the search space grows combinatorially, and debugging and testing on clusters is expensive. In this work we propose DistIR, an expressive intermediate representation for distributed DNN computation that is tailored for efficient analyses, such as simulation. This enables automatically identifying the top-performing strategies without having to execute on physical hardware. Unlike prior work, DistIR can naturally express many distribution strategies including pipeline parallelism with arbitrary schedules. Our evaluation on MLP training and GPT-2 inference models demonstrates how DistIR and its simulator enable fast grid searches over complex distribution spaces spanning up to 1000+ configurations, reducing optimization time by an order of magnitude for certain regimes.

Preliminary results are presented for nucleon isovector charges and twist-2 matrix elements which have been obtained employing an improved analysis strategy to deal with excited-state contamination. The set of CLS $N_f=2+1$ gauge ensembles in this study has been extended compared to our 2018 calculation, including an ensemble at physical quark masses. Besides the addition of new ensembles, the number of gauge configurations and measurements has been increased on several of the existing ensembles and the analysis has been extended to include additional source-sink separations. The ensembles cover a range of the light quark mass corresponding to $M_\pi\approx 0.130\,\mathrm{MeV} \ldots 350\,\mathrm{MeV}$, four values of the lattice spacing $a\approx0.05\,\mathrm{fm}\ldots0.09\,\mathrm{fm}$ and a large range of volumes. Results at the physical point are computed for each observable from a combined chiral, continuum and finite-volume extrapolation., Comment: 9 pages, 2 figures, 2 tables. Talk presented at the 38th International Symposium on Lattice Field Theory, 26th-30th July 2021, Zoom/Gather@Massachusetts Institute of Technology

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., Comment: 32 pages, 8 figures, 6 tables

Computers used for data analytics are often NUMA systems with multiple sockets per machine, multiple cores per socket, and multiple thread contexts per core. To get the peak performance out of these machines requires the correct number of threads to be placed in the correct positions on the machine. One particularly interesting element of the placement of memory and threads is the way it effects the movement of data around the machine, and the increased latency this can introduce to reads and writes. In this paper we describe work on modeling the bandwidth requirements of an application on a NUMA compute node based on the placement of threads. The model is parameterized by sampling performance counters during 2 application runs with carefully chosen thread placements. Evaluating the model with thousands of measurements shows a median difference from predictions of 2.34% of the bandwidth. The results of this modeling can be used in a number of ways varying from: Performance debugging during development where the programmer can be alerted to potentially problematic memory access patterns; To systems such as Pandia which take an application and predict the performance and system load of a proposed thread count and placement; To libraries of data structures such as Parallel Collections and Smart Arrays that can abstract from the user memory placement and thread placement issues when parallelizing code., Comment: 14 pages

The hadronic contribution to the muon anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ has to be determined at the per-mille level for the Standard Model prediction to match the expected final uncertainty of the ongoing E989 experiment. That is 3 times better than the current precision from the dispersive approach, and 5-15 times smaller than the uncertainty based 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. In this talk we present our proposal 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 report the results of our feasibility tests for the computation of 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 $a_\mu$., Comment: Proceedings of the 40th International Conference on High Energy physics (ICHEP2020). arXiv admin note: substantial text overlap with arXiv:2007.02973

We provide an interpretation of the structure functions of a thermal medium such as the quark-gluon plasma in terms of the scattering of an incoming electron on the medium via the exchange of a spacelike photon. We then focus on the deep-inelastic scattering (DIS) regime, and formulate the corresponding moment sum rules obeyed by the structure functions. Accordingly, these moments are given by the thermal expectation value of twist-two operators, which is computable from first principles in lattice QCD for the first few moments. We also show how lattice QCD calculations can be used to probe how large the photon virtuality needs to be in order for the Bjorken scaling of structure functions to set in. Finally, we provide the parton-model interpretation of the structure functions in the Bjorken limit and test its consistency. As in DIS on the proton, the kinematic variable $x$ is proportional to the longitudinal momentum carried by the partons, however $x$ ranges from zero to infinity. Choosing the parton momentum parametrization to be $ x T u$ where $u$ is the fluid four-velocity and $T$ its temperature in the rest frame, the parton distribution function for a plasma of non-interacting quarks is proportional to $ x \log(1+e^{-x/2}) $., Comment: 22 pages, 4 figures

The hadronic contribution to the muon anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ 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 $a_\mu$. 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., Comment: 6 pages, 5 figures; added Eq. (6), improved text, version accepted for publication on PLB

In these proceedings we address the computation of quark-line disconnected diagrams in lattice QCD. The evaluation of these diagrams is required for many phenomenologically interesting observables, but suffers from large statistical errors due to the vacuum and random-noise contributions to their variances. Motivated by a theoretical analysis of the variances, we introduce a new family of stochastic estimators of single-propagator traces built upon a frequency splitting combined with a hopping expansion of the quark propagator, and test their efficiency in two-flavour QCD with pions as light as 190 MeV. The use of these estimators reduces the cost of the computation by one to two orders of magnitude over standard estimators depending on the fermion bilinear. As a concrete application, we show the impact of these findings on the computation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment., Comment: 7 pages, 4 figures; Proceedings of the 37th annual International Symposium on Lattice Field Theory (Lattice 2019), 16-22 June 2019, Wuhan, China

We calculate the thermal rate of real-photon production in the quark-gluon plasma at a temperature of $T=254$ MeV using lattice QCD. The calculation is based on the difference between the spatially transverse and longitudinal parts of the polarization tensor, which has the advantage of falling off rapidly at large frequencies. We obtain this linear combination in the time-momentum representation from lattice QCD with two flavors of quarks in the continuum limit with a precision of about two parts per mille. Applying a theoretically motivated fit ansatz for the associated spectral function, we obtain values for the photon rate that are in line with QCD weak-coupling calculations; for photon momenta $ 1.0\leq k[{\rm GeV}]\leq 1.4$, our non-perturbative results constrain the rate to be no larger than twice the weak-coupling prediction. We also provide a physics interpretation of the electromagnetic spectral functions valid for all frequencies and momenta., Comment: 7 pages, 4 figures, one table

We present a computation of the photon emission rate of the quark-gluon plasma from two-flavor lattice QCD at a temperature of 254 MeV, which follows up on the work presented in [1]. We perform a continuum extrapolation of the vector-current correlator, and consider a linear combination of the Lorentz indices corresponding to a UV-finite spectral function. To extract the spectral function from the lattice correlators, an ill-posed inverse problem, we model the spectral function with a Pad\'e ansatz. We further constrain our analysis by simultaneously fitting data with different momenta. We present results for a multi-momentum fit including the three smallest momenta available from our lattice analysis., Comment: 7 pages, 4 figures. Contribution to the 37th International Symposium on Lattice Field Theory - Lattice2019, 16-22 June 2019, Wuhan, China

We present results from a lattice QCD study of nucleon matrix elements at vanishing momentum transfer for local and twist-2 isovector operator insertions. Computations are performed on gauge ensembles with non-perturbatively improved $N_f=2+1$ Wilson fermions, covering four values of the lattice spacing and pion masses down to $M_\pi\approx200$MeV. Several source-sink separations (typically ~1.0fm to ~1.5fm) allow us to assess excited-state contamination. Results on individual ensembles are obtained from simultaneous two-state fits across all observables and all available source-sink separations with the energy gap as a common fit parameter. Renormalization has been performed non-perturbatively using the Rome-Southampton method for all but the finest lattice spacing for which an extrapolation has been used. Physical results are quoted in the $\overline{MS}$ scheme at a scale of $\mu=2$GeV and are obtained from a combined chiral, continuum and finite-size extrapolation. For the nucleon isovector axial, scalar and tensor charges we find physical values of $g_A^{u-d}=1.242(25)_\text{stat}(\genfrac{}{}{0pt}{2}{+00}{-31})_\text{sys}$, $g_S^{u-d}=1.13(11)_\text{stat}(\genfrac{}{}{0pt}{2}{+07}{-06})_\text{sys}$ and $g_T^{u-d}=0.965(38)_\text{stat}(\genfrac{}{}{0pt}{2}{+13}{-41})_\text{sys}$, respectively, where individual systematic errors in each direction from the chiral, continuum and finite-size extrapolation have been added in quadrature. Our final results for the isovector average quark momentum fraction and the isovector helicity and transversity moments are given by $\langle x\rangle_{u-d}=0.180(25)_\text{stat}(\genfrac{}{}{0pt}{2}{+14}{-06})_\text{sys}$, $\langle x\rangle_{\Delta u-\Delta d}=0.221(25)_\text{stat}(\genfrac{}{}{0pt}{2}{+10}{-00})_\text{sys}$ and $\langle x\rangle_{\delta u-\delta d}=0.212(32)_\text{stat}(\genfrac{}{}{0pt}{2}{+20}{-10})_\text{sys}$, respectively., Comment: 35 pages, 13 figures; matching version accepted for publication in PRD

We study the thermal transition of QCD with two degenerate light flavours by lattice simulations using $\mathcal{O}(a)$-improved Wilson quarks. Particular emphasis lies on the pattern of chiral symmetry restoration, which we probe via the static screening correlators. On $32^3$ volumes we observe that the screening masses in transverse iso-vector vector and axial-vector channels become degenerate at the transition temperature. The splitting between the screening masses in iso-vector scalar and pseudoscalar channels is strongly reduced compared to the splitting at zero temperature and is actually consistent with zero within uncertainties. In this proceedings article we extend our studies to matrix elements and iso-singlet correlation functions. Furthermore, we present results on larger volumes, including first results at the physical pion mass., Comment: 10 pages, 9 figures, invited contribution to the 9th International Workshop on Chiral Dynamics, Sept. 17-21, 2018, Duke University, Durham, NC, USA

Single-propagator traces are the most elementary fermion Wick contractions which occur in numerical lattice QCD, and are usually computed by introducing random-noise estimators to profit from volume averaging. The additional contribution to the variance induced by the random noise is typically orders of magnitude larger than the one due to the gauge field. We propose a new family of stochastic estimators of single-propagator traces built upon a frequency splitting combined with a hopping expansion of the quark propagator, and test their efficiency in two-flavour QCD with pions as light as 190 MeV. Depending on the fermion bilinear considered, the cost of computing these diagrams is reduced by one to two orders of magnitude or more with respect to standard random-noise estimators. As two concrete examples of physics applications, we compute the disconnected contributions to correlation functions of two vector currents in the isosinglet omega channel and to the hadronic vacuum polarization relevant for the muon anomalous magnetic moment. In both cases, estimators with variances dominated by the gauge noise are computed with a modest numerical effort. Theory suggests large gains for disconnected three and higher point correlation functions as well. The frequency-splitting estimators and their split-even components are directly applicable to the newly proposed multi-level integration in the presence of fermions., Comment: 26 pages, 8 figures, LaTeX

We report on our ongoing project to determine the leading-order hadronic vacuum polarisation contribution to the muon $g-2$, using ensembles with $N_f=2+1$ flavours of O($a$) improved Wilson quarks generated by the CLS effort, with pion masses down to the physical value. We employ O($a$) improved versions of the local and conserved vector currents to compute the contributions of the light, strange and charm quarks to $(g-2)_\mu$, using the time-momentum representation. We perform a detailed investigation of the systematic effects arising from constraining the long-distance regime of the vector correlator. To this end we make use of auxiliary calculations in the iso-vector channel using distillation and the L\"uscher formalism. Our results are corrected for finite-volume effects by computing the timelike pion form factor in finite and infinite volume. For certain parameter choices, the corrections computed in this way can also be confronted with results determined on different volumes. Currently, the overall precision of our results is limited by the uncertainties in the lattice scale., Comment: To appear in: Proceedings of the 36th International Symposium on Lattice Field Theory (Lattice 2018), Michigan State University, East Lansing, USA; 7 pages