We analyze the color-magnetic (or "$B$") field two-point function that encodes the finite-mass correction to the heavy quark momentum diffusion coefficient. The simulations are done on fine isotropic lattices in the quenched approximation at $1.5\,T_c$, using a range of gradient flow times for noise suppression and operator renormalization. The continuum extrapolation is performed at fixed flow time followed by a second extrapolation to zero flow time. Perturbative calculations to next-to-leading order of this correlation function, matching gradient-flowed correlators to MS-bar, are used to resolve nontrivial renormalization issues. We perform a spectral reconstruction based on perturbative model fits to estimate the coefficient $κ_B$ of the finite-mass correction to the heavy quark momentum diffusion coefficient. The approach we present here yields high-precision data for the correlator with all renormalization issues incorporated at next-to-leading order, and is also applicable for actions with dynamical fermions.
I discuss recent progress in calculating quarkonia correlators and spectral functions on the lattice in relation with the problem of quarkonia dissolution at high temperatures and heavy quark transport in Quark Gluon Plasma.
We study charmonia correlators at finite temperature. We analyze to what extent heavy quarkonia correlators are sensitive to the effect of heavy quark transport and whether it is possible to constrain the heavy quark diffusion constant by lattice calculations. Preliminary lattice calculations of quarkonia correlators performed on anisotropic lattices show that they are sensitive to the effect of heavy quark transport, but much detailed calculations are required to constrain the value of the heavy quark diffusion constant.
We compute the leading four physical terms in the low-energy expansions of heavy-light quark current correlators at four-loop order. As a by-product we reproduce the corresponding top-induced non-singlet correction to the electroweak rho parameter.
We study quarkonium correlators and spectral functions at zero and finite temperature using the anisotropic Fermilab lattice formulation with anisotropy xi=2 and 4. To control cutoff effects we use several different lattice spacings. The spectral functions were extracted from lattice correlators with Maximum Entropy Method based on a new algorithm. We find evidence for the survival of 1S quarkonium states in the deconfined medium till relatively high temperatures as well as for dissolution of 1P quarkonium states right above the deconfinement temperature.
We present the first measurement of a new correlation function of Fermion bilinears in finite temperature QCD with and without dynamical quarks in a quantum number channel in which non-trivial correlations are known to be present for purely gluonic operators. We find that the Fermion correlator vanishes for $T \ge 3T_c/2$, in agreement with the expectation for weakly interacting quarks in a quark-gluon plasma.
We consider the four point connected correlator representing a static quark-antiquark pair separated by a spatial distance R, propagating for a Euclidean time T. This function is computed by lattice Monte Carlo in SU(2) pure gauge theory at lattice couplings $β=2.2$ and $β=2.5$ in both Coulomb and Landau gauges. The Coulomb gauge correlator is well behaved, and is dominated at large T by a state whose energy grows linearly as $σR$, with $σ$ the known asymptotic string tension. The connected correlator in Landau gauge behaves differently. At intermediate R there is clear evidence of a linear po…
We present NLO results for thermal imaginary-time correlators in the vector and scalar channels as a function of the quark mass. The range of quark masses for which a non-relativistic approximation works in the temperature range considered is estimated, and charm quarks turn out to be a borderline case. Comparing with simulation data from fine lattices, we find good agreement in the vector channel but a substantial discrepancy in the scalar one. An explanation for the discrepancy is suggested in terms of physics of the quark-antiquark threshold region. Perturbative predictions for the bottom s…
In the investigation of in-medium modifications of quarkonia and for determining heavy quark diffusion coefficients, correlation functions play a crucial role. For the first time we perform a continuum extrapolation of charmonium and bottomonium correlators in the vector channel based on non-perturbatively clover-improved Wilson fermions in quenched lattice QCD. Calculations were done at 4 different lattice spacings, spatial extents between 96 and 192, aspect ratio from 1/6 to 1/2, for 5 temperatures between $T/T_c = 0.75$ and $T/T_c = 2.2$. We interpolate between different quark masses to mat…
We revisit previous determination of the strong coupling constant from moments of quarkonium correlators in (2+1)-flavor QCD. We use previously calculated moments obtained with Highly Improved Staggered Quark (HISQ) action for five different quark masses and several lattice spacings. We perform a careful continuum extrapolations of the moments and from the comparison of these to the perturbative result we determine the QCD Lambda parameter, $Λ_{\overline{MS}}^{n_f=3}=332 \pm 17 \pm 2(scale)$ MeV. This corresponds to $α_s^{n_f=5}(μ=M_Z)=0.1177(12)$.
We present an NLO analysis of the massive vector current correlator at temperatures above a few hundred MeV. The physics of this correlator originates from a transport peak, related to heavy quark diffusion, and from the quark-antiquark threshold, related to quarkonium physics. In the bottom case both can be studied with separate effective theories, but for charm these may not be accurate, so a study within the full theory is needed. Working in imaginary time, the NLO correlator can be computed in unresummed perturbation theory; comparing with lattice data, we find good agreement. Subsequently…
We investigate the quarkonia correlators in QCD with no light quarks within a potential model with different screened potentials. Our results for the temperature dependence of the charmonium and bottomonium correlators are qualitatively consistent with existing and preliminary lattice results. We identify however, a much reacher structure in the correlators than the one seen on the lattice.
Gauge invariant correlators in QCD are studied on the lattice.A Systematic determination of the correlation lengths for gluon field strength correlators and quark correlators is made.The measurement of the gluon and quark condensates is discussed.
Taking the gauge invariant quark correlator as an example, in this work we perform a direct comparison of lattice QCD simulations with the QCD sum rule approach. The quark correlator is first investigated in the framework of QCD sum rules and the correlation length of the quark field is calculated. Comparing the phenomenological part of the sum rule with previous measurements of the quark correlator on the lattice, we are able to obtain an independent result for the correlation length. From a fit of the lattice data to the operator product expansion, the quark condensate and the mixed quark-gl…
We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-$β_0$ limit of QCD, extending the analysis of Grozin and Sturm beyond the vector current. Our results are used to study the higher-order behaviour of dimensionless ratios of vector and pseudo-scalar moments used for the precise extraction of the strong coupling, $α_s$, from relativistic quarkonium sum rules and lattice data, respectively. We show that these ratios benefit from a partial cancellation of the leading renormalon singularities. Our results can gu…
The quark-gluon mixed condensate $g<\bar q σ_{μν} F_{μν} q>$ is calculated in the Gaussian approximation of the Field Correlator Method. In the large N_c limit and for zero mass quarks one obtains a simple result, $m^2_0 \equiv \frac{g<\bar q σ_{μν} F_{μν} q>}{<\bar q q>} = \frac{16σ}π$, where $σ$ is the string tension. For a standard value $σ=0.18 GeV^2$ one obtains $m^2_0=1 GeV^2$ in good agreement with the QCD sum rules estimate $m^2_0 = (0.8\pm 0.2) GeV^2$ and the latest lattice result $m^2_0 \cong 1 GeV^2$.
We discuss a constant contribution to meson correlators at finite temperature. In the deconfinement phase of QCD, a colored single quark state is allowed as a finite energy state, which yields to a contribution of wraparound quark propagation to temporal meson correlators. We investigate the effects in the free quark case and quenched QCD at finite temperature. The ``scattering'' contribution causes a constant mode in meson correlators with zero spatial momentum and degenerate quark masses, which can dominate the correlators in the region of large imaginary times. In the free spectral function…
We calculate the low energy contribution to quarkonium correlators in Euclidean time in lattice QCD. This contribution was found to give the dominant source of the temperature dependence of the correlators. We have found that the low energy contribution is well described by a quasi-particle model and have determined the effective temperature dependent heavy quark mass.
We discuss different static quark correlators, including Wilson loops in 2+1 flavor QCD at non-zero temperature and their relation to in-medium quarkonium properties. We present lattice results on static correlation functions obtained with highly improved staggered fermion action and their implications for potential models.
We discuss the calculations of quarkonium spectral functions in potential models and their implications for the interpretation of the lattice data on quarkonium correlators. In particular, we find that melting of different quarkonium states does not lead to significant change in the Euclidean time correlators. The large change of the quarkonium correlators above deconfinement observed in the scalar and axial-vector channels appears to be due to the zero mode contribution.
Recently an electric and magnetic field correlator appearing in the description of the heavy quarkonium system was evaluated on the lattice. Here, we give a nonperturbative analytical evaluation of this field correlator using a dual description of long distance Yang--Mills theory and using the stochastic vacuum model. The two predictions are both compatible with lattice data but show a different dependence on the quark separation. We discuss the analytic results in relation to the lattice data.
Point to point correlators of currents are computed in quenched QCD using a chiral lattice fermion action, the overlap action. I compare correlators made of exact quark propagators with correlators restricted to low (less than 500 MeV) eigenvalue eigenmodes of the Dirac operator. In many cases they show qualitative resemblence (typically at small values of the quark mass and distances larger than 0.4 fm) and they differ qualitatively at larger quark masses or at very short distance. Lattice results are in qualitative agreement (and in the difference of vector and axial vector channels, quantit…
We derive the Operator Product Expansion whose vacuum expectation value gives the time-moments of the pseudoscalar heavy-light current-current correlator up to and including terms in $α_s^2$ multiplying $\langle\overlineψψ\rangle/M^3$ and terms in $α_s$ multiplying $\langle α_s G^2 \rangle/M^4$, where $M$ is the heavy-quark mass. Using lattice QCD results for heavy-strange correlators obtained for a variety of heavy quark masses on gluon field configurations including $u$, $d$ and $s$ quarks in the sea at three values of the lattice spacing, we are able to show that the contribution of the str…
We study the role of diquarks in light baryons through point to point baryon correlators. We contrast results from quenched simulations with ones with two flavors of dynamical overlap fermions. The scalar, pseudoscalar and axial vector diquarks are combined with light quarks to form color singlets. The quenched simulation shows large zero mode effects in correlators containing the scalar and pseudoscalar diquark. The two scalar diquarks created by gamma_5 and gamma_0gamma_5 lead to different behavior in baryon correlators, showing that the interaction of diquarks with the third light quark mat…
We study the diffusion of heavy quarks in the Quark Gluon Plasma using the Langevin equations of motion and estimate the contribution of the transport peak to the Euclidean current-current correlator. We show that the Euclidean correlator is remarkably insensitive to the heavy quark diffusion coefficient and give a simple physical interpretation of this result using the free streaming Boltzmann equation. However if the diffusion coefficient is smaller than $\sim 1/(πT)$, as favored by RHIC phenomenology, the transport contribution should be visible in the Euclidean correlator. We outline a pro…
A combination of lattice and continuum data for the light-quark V-A correlator, supplemented by results from a chiral sum-rule analysis of the flavor-breaking flavor $ud$-$us$ V-A correlator difference, is shown to make possible a high-precision NNLO determination of the renormalized NLO chiral low-energy constant $L_{10}^r$. Key to this determination is the ability to simultaneously fix the two combinations of NNLO low-energy constants also entering the analysis. With current versions of the strange hadronic $τ$ branching fractions required as input to the flavor-breaking V-A sum rule, we fin…
We use lattice QCD simulations, with MILC gluon configurations and HISQ c-quark propagators, to make very precise determinations of moments of charm-quark pseudoscalar, vector and axial-vector correlators. These moments are combined with new four-loop results from continuum perturbation theory to obtain several new determinations of the MSbar mass of the charm quark and of the MSbar coupling. We find m_c(3GeV)=0.986(10)GeV, or, equivalently, m_c(m_c)=1.268(9)GeV, both for n_f=4 flavors; and alpha_msb(3GeV,n_f=4)=0.251(6), or, equivalently, alpha_\msb(M_Z,n_f=5)=0.1174(12). The new mass agrees …
It is argued that strong dynamics in the quark-gluon plasma and bound states of quarks and gluons is mostly due to nonperturbative effects described by field correlators. The emphasis in the paper is made on two explicit calculations of these effects from the first principles: one analytic using gluelump Green's functions and another using independent lattice data on correlators. The resulting hadron spectra are investigated in the range T_c < T < 2T_c. The spectra of charmonia, bottomonia, light s-sbar mesons, glueballs and quark-gluon states calculated numerically are in general agreement wi…
We study on the lattice the correlator of heavy-quark currents in the vicinity of vanishing momentum. The renormalized charmed quark mass, the renormalized strong coupling constant and gluon condensate can be defined in terms of the derivatives of that correlator at zero momentum. We analyze quenched Monte-Carlo data on a small lattice $8^3*16$ for $β=6$. We generalize dispersion relations to the lattice theory in a simple way and use them successfully to fit the correlator at both small and large distances. We fit the short-distance part of the correlator with the relevant expressions of pert…
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 c…
Many lattice studies of heavy quark diffusion originate from a colour-electric correlator, obtained as a leading term after an expansion in the inverse of the heavy-quark mass. In view of the fact that the charm quark is not particularly heavy, we consider subleading terms in the expansion. Working out correlators up to $O(1/M^2)$, we argue that the leading corrections are suppressed by $O(T/M)$, and one of them can be extracted from a colour-magnetic correlator. The corresponding transport coefficient is non-perturbative already at leading order in the weak-coupling expansion, and therefore r…
The current-current correlator method has been used successfully to obtain very accurate results for quark masses and the coupling alpha_s. The calculations were done using Highly Improved Staggered Quarks (HISQ) and heavy-heavy meson correlators. We now extend this work to the significantly more challenging heavy-light case, reporting the first results here. The aim is to determine nonperturbative Z factors for NRQCD heavy-light currents, but first we test the method in the HISQ case where Z=1.
The static three-quark potential in arbitrary configuration of quarks is calculated analytically. It is shown to be in a full agreement with the precise numerical simulations in lattice QCD. The results of the work have important application in nuclear physics, as they allow to perform accurate analytic calculations of spectra of the baryons.
I discuss the temperature dependence of quarkonium correlators calculated in lattice QCD. I show that the dominant source of the temperature dependence comes from the zero mode contribution, while the the temperature dependence associated with melting of bound states is quite small. I study the zero mode contribution quantitatively for different quark masses and show that it is well described by a quasi-particle model with temperature dependent heavy quark mass. As a byproduct an estimate of the medium dependence of the heavy quark mass is obtained.
A method for computing all-point quark propagators is applied to a variety of processes of physical interest in lattice QCD. The method allows, for example, efficient calculation of disconnected parts and full momentum-space 2 and 3 point functions. Examples discussed include: extraction of chiral Lagrangian parameters from current correlators, the pion form factor, and the unquenched eta-prime.
We report our progress in simulating Neuberger valence fermions on N_f=2 Wilson O(a)-improved sea quarks. We compute correlators with valence quark masses both in the p- and in the epsilon-regime, and we match the results with the predictions of the Chiral Effective Theory in the mixed regime. This allows us to extract the Low Energy Couplings (LECs) of the N_f=2 theory and to test the validity of the approach.
We represent the Polyakov loop correlator as a spectral sum of correlators of eigenvectors of the lattice Dirac operator. This spectral representation is studied numerically using quenched SU(3) configurations below and above the deconfinement temperature. We analyze whether the individual Dirac eigenvector correlators differ in the confined and deconfined phases. The decay properties of the normalized Dirac eigenvector correlators turn out to be essentially identical in the two phases, but the amplitudes change. This change of the amplitudes shifts the relative contributions of the individual…
Brambilla, Escobedo, Soto, and Vairo have derived an effective description of quarkonium with two parameters; a momentum diffusion term and a real self-energy term. We point out that there is a similar real self-energy term for a single open heavy flavor and that it can be expressed directly in terms of Euclidean electric field correlators along a Polyakov line. This quantity can be directly studied on the lattice without the need for analytical continuation. We show that Minkowski-space calculations of this correlator correspond with the known NLO Euclidean value of the relevant electric fiel…
I review recent progress in relating meson spectral function to imaginary time correlation function at finite temperature calculated on isotropic as well as on anisotropic lattices. Special attention is payed for the lattice artifacts present in calculation of meson spectral functions. Results in the case of light quarks as well as heavy quarks are reviewed which indicate in particular that even in the chiral limit meson spectral functions have non-trivial structure and the ground state quarkonia survive up to temperature 1.5T_c
We extract the charm quark mass and the strong coupling constant from the charmonium current correlators calculated with $n_f = 2 + 1$ Mobius domain wall fermions. We match our lattice calculation for the temporal moments of the correlator with perturbative result known up to four-loop order, and extract the charm quark mass with uncertainty less than 1%. Using the temporal moments, we also confirm the correlators in the vector channel to be consistent with the experimental data for the R-ratio. We used the ensembles generated by the JLQCD collaboration at lattice spacings $a$ = 0.080 fm, 0.05…
We describe a new technique (presented in arXiv:1403.1778) to determine the contribution to the anomalous magnetic moment (g-2) of the muon coming from the hadronic vacuum polarisation using lattice QCD. Our method uses Padé approximants to reconstruct the Adler function from its derivatives at $q^2=0$. These are obtained simply and accurately from time-moments of the vector current-current correlator at zero spatial momentum. We test the method using strange quark correlators calculated on MILC Collaboration's $n_f$ = 2+1+1 HISQ ensembles at multiple values of the lattice spacing, multiple vo…
We study the volume dependence of the left-current correlator with non-degenerate quark masses to next-to-leading order in the chiral expansion. We consider three possible regimes: all quark masses are in the $ε$-regime, all are in the $p$-regime and a mixed-regime where the lighest quark masses satisfy $m_v ΣV \leq 1$ while the heavier $m_s ΣV \gg 1$. These results can be used to match lattice QCD and the Chiral Effective Theory in a large but finite box in which the Compton wavelength of the lightest pions is of the order of the box size. We consider both the full and partially-quenched resu…
We calculate charmonium correlators on the lattice with 2+1-flavors of sea quarks and charm valence quark both described by the Möbius domain-wall fermion. Temporal moments of the correlators are calculated and matched to perturbative QCD formulae to extract the charm quark mass $m_c(μ)$ and strong coupling constant $α_s(μ)$. Lattice data at three lattice spacings, 0.044, 0.055, and 0.080~fm, are extrapolated to the continuum limit. The correlators in the vector channel are confirmed to be consistent with the experimental data for $e^+e^-\to c\bar{c}$, while the pseudo-scalar channel is used t…
We present non-perturbative first-principle results for quark-, gluon- and meson $1$PI correlation functions of two-flavour Landau-gauge QCD in the vacuum. These correlation functions carry the full information about the theory. They are obtained by solving their Functional Renormalisation Group equations in a systematic vertex expansion, aiming at apparent convergence. This work represents a crucial prerequisite for quantitative first-principle studies of the QCD phase diagram and the hadron spectrum within this framework. In particular, we have computed the gluon, ghost, quark and scalar-pse…
Hadronic correlation functions at finite temperature in QCD, with four flavours of dynamical quarks, have been analyzed both above and below the chiral symmetry restoration temperature. We have used both point and extended sources for spatial as well as temporal correlators. The effect of periodic temporal boundary conditions for the valence quarks on the spatial meson correlators has also been investigated. All our results are consistent with the existence of individual quarks at high temperatures. A measurement of the residual interaction between the quarks is presented.
We investigate the properties of heavy quarkonia at finite temperature in detail using QCD sum rules. Extending previous analyses, we take into account a temperature dependent effective continuum threshold and derive constraints on the mass, the width, and the varying effective continuum threshold. We find that at least one of these quantities of a charmonium changes abruptly in the vicinity of the phase transition. We also calculate the ratio of the imaginary time correlator to its reconstructed one, $G/G_{\text{rec}}$, by constructing a model spectral function and compare it to the correspon…
In the high-temperature phase of QCD, the heavy quark momentum diffusion constant determines, via a fluctuation-dissipation relation, how fast a heavy quark kinetically equilibrates. This transport coefficient can be extracted from thermal correlators via a Kubo formula. We present a lattice calculation of the relevant Euclidean correlators in the gluon plasma, based on a recent formulation of the problem in heavy-quark effective field theory (HQET). We find a $\approx20%$ enhancement of the Euclidean correlator at maximal time separation as the temperature is lowered from $6T_c$ to $2T_c$, po…
Temporal meson correlators and their spectral functions are calculated in the deconfined phase using the hard thermal loop resummation technique. The spectral functions exhibit strong medium effects coming from the hard thermal loop approximation for the quark propagator. The correlators, on the other hand, do not differ significantly from free correlators, for which bare quark propagators are used. This is in contrast to lattice calculations showing a clear deviation from the free correlations functions.
We discuss color screening in 2+1 flavor QCD in terms of free energies of a static quark-antiquark pair. Thermal modifications of long distance correlations in quark-antiquark systems are studied in terms of static meson correlators. We calculate the Polyakov loop correlator, the color-singlet Wilson line correlator in Coulomb gauge and the cyclic Wilson loop on lattices using the HISQ/Tree action and almost physical quark mases with $N_τ= 4, 6, 8, 10, 12$. We present results in the continuum limit for temperatures up to $T \lesssim 650$ MeV and discuss the linear divergence of the cyclic Wils…
The open-charm Euclidean correlators have been computed for the first time using the thermal spectral functions extracted from a finite-temperature self-consistent unitarized approach based on a chiral effective field theory that implements heavy-quark spin symmetry. The inclusion of the full-energy dependent open-charm spectral functions in the calculation of the Euclidean correlators leads to a similar behaviour as the one obtained in lattice QCD for temperatures well below the transition deconfinement temperature. The discrepancies at temperatures close or above the transition deconfinement…
We study singlet and triplet correlation functions of static quark anti-quark pair defined through gauge invariant time-like Wilson loops and Polyakov loop correlators in finite temperature SU(2) gauge theory. We use the Luescher-Weisz multilevel algorithm, which allows to calculate these correlators at very low temperatures. We observe that the naive separation of singlet and triplet states in general does not hold non-perturbatively, however, is recovered in the limit of small separation and the temperature dependence of the corresponding correlators is indeed very different.
We provide the analytic predictions for the flavor non-singlet scalar correlator, which will enable determination of the scalar meson mass from the lattice scalar correlator. We consider simulations with 2+1 staggered sea quarks and staggered or chiral valence quarks. At small u/d masses the correlator is dominated by the bubble contribution, which is the intermediate state with two pseudoscalar mesons. We determine the bubble contribution within Staggered and Mixed Chiral Perturbation Theory. Its effective mass is smaller than the mass M_pi+M_eta, which is the lightest intermediate state in p…
We present our methods to fit the two point correlators for light, strange, and charmed pseudoscalar meson physics with the highly improved staggered quark (HISQ) action. We make use of the least-squares fit including the full covariance matrix of the correlators and including Gaussian constraints on some parameters. We fit the correlators on a variety of the HISQ ensembles. The lattice spacing ranges from 0.15 fm down to 0.06 fm. The light sea quark mass ranges from 0.2 times the strange quark mass down to the physical light quark mass. The HISQ ensembles also include lattices with different …
We study the fate of P wave bottomonium states in the quark-gluon plasma, using a spectral function analysis of euclidean lattice correlators. The correlators are obtained from lattice QCD simulations with two light quark flavours on highly anisotropic lattices, treating the bottom quark nonrelativistically. We find clear indications of melting immediately after the deconfinement transition.
