^{1}

^{1,2,3}

^{3}

^{3}

^{1}

^{4}

^{5}

^{1,2}

^{1}

^{3}.

We compute the Euclidean correlators of the stress tensor in pure

Since relativistic hydrodynamics is quite successful in the interpretation of heavy ion experiments

In classical transport theory, the shear viscosity to entropy density ratio for a dilute gas at temperature T is

One possible route to determine the viscosity is through the Kubo formula, relating transport coefficients to the zero-frequency behavior of spectral functions. The relevant Kubo formula for the shear viscosity is

In general, the correlator of the energy momentum tensor

Both early

Equation

For the particular case of the viscosity, the signal in the stress-energy tensor is strongly dominated by the high frequency part of the spectral function

To see an honest illustration of these problems for

The Euclidean correlator corresponding to the spectral function appearing in Eq.

Note that for Fig.

The bottom line of this discussion is that, for a credible lattice estimate of the viscosity, a high level of precision is necessary for the Euclidean correlators, especially if we want to use Eq.

The situation is much better for the correlators of conserved charges, appearing in the electric conductivity

Even with this knowledge, the calculation of the viscosity is still much more difficult than that of the electric conductivity. The source of the difficulty is the fact that the stress-energy tensor correlators

For the quenched case this problem can be ameliorated by using the multilevel algorithm

The study of cutoff effects of these correlators is rather limited in the literature. The tree level improvement coefficients for the plaquette action and two different discretizations of

In this paper, we take steps towards achieving the high precision necessary for the calculation of the shear viscosity, by investigating several technical aspects of such a calculation, namely,

Utilizing a different gauge action, the tree level Symanzik-improved action, as opposed to the plaquette action used in previous studies.

Studing the continuum limit behavior by simulating at different values of the lattice spacing

Calculating the

Using the Wilson flow for anisotropy tuning, as advertised in

Using shifted boundary conditions for the renormalization of the energy momentum tensor, a technique that was worked out for the isotropic case in

Calculating the tree-level improvement coefficients for the Symanzik-improved gauge action.

Throughout this paper, we will mostly focus on the calculation of the energy-momentum tensor, and not the inversion method for reconstructing the spectral function. We believe this to be an important first step. Before the inversion can be done, one needs to have reliable results for the correlator itself. Nevertheless, in the end we give an estimate of the viscosity, using a similar hydrodynamics motivated fit ansatz as some previous studies

Our calculation uses the tree-level Symanzik-improved gauge-action,

The anisotropy parameters are

We use a multilevel algorithm (more precisely, a two-level algorithm

We use the clover discretization of the energy momentum tensor, mainly because the center of the operator is always located on a site, therefore the separation of the operators is always an integer in lattice units. If one were to use the plaquette discretization, there would be a component that is defined for integer separations and one that is defined for half integer separations, and one would need an interpolation to add them together. This would lead to the appearance of a systematic error coming from the interpolation, that we want to avoid.

Following the line of previous studies, we use the two-level algorithm, but now with a tree-level Symanzik improvement.

We have ensembles at two different temperatures,

As was explained in the Introduction, the correlators are needed to a very high precision, if one wants to have useful information on transport. In the pure

The main difference is—apart from the higher statistical precision—that we are working with four different lattice spacings, which allows us to study the correlators in the continuum limit. Our statistics are summarized in Table

Number of measurements (millions) of the energy-momentum tensor correlators at the simulation points. Between every measurement, there are 100 regular updates and 500 inner multilevel updates.

Relative statistical error of the 1313 correlators at zero momentum for our different simulation points.

To fix the anisotropy we use the method introduced in

We simulate the

We calculate the gradient flow using

The

Top: Anisotropy tuning with simulations at different bare anisotropies. The tuned bare anisotropy corresponds to

In all cases,

Determination of

The translational symmetry is broken on the lattice. As a result, renormalization factors appear between the lattice definition of the energy momentum tensor

We use the clover discretization of

For an isotropic gauge action, the renormalization constants have been worked out with shifted boundary conditions in

The renormalization factors

The overall constant

The 13 channel correlators can be seen in Fig.

The renormalized shear correlator

The renormalized shear correlator

We note here, that trying a simple featureless ansatz

For the 01 channel, in the continuum, the correlator for zero spatial momentum should be a constant, equal to the entropy. The renormalization condition we used for this correlator is simply that at the middle point,

For the purpose of this paper we focus our discussion of the continuum limit extrapolation to the middle point of the correlators

This is the most IR sensitive part of the correlators, therefore the most interesting part for studying transport.

This is the part of the correlator with the least amount of cutoff effects, therefore one has to control the continuum extrapolation of this first, before attempting to go to smaller separations in imaginary time.

We will attempt a continuum limit extrapolation both with and without tree-level improvement. The tree-level improvement coefficients are the result of a tedious, but straightforward computation. The numerical values of the improvement coefficients are summarized in Appendix

In the 0101 channel, since one applies the renormalization condition (11) after the tree-level improvement, the continuum extrapolation is quite flat, regardless of whether one uses tree-level improvement or not.

A linear fit to the

A quadratic fit to the

Linear versus quadratic fits to the data obtained without tree-level improvement are not consistent within

Linear versus quadratic fits to the tree-level improved data are closer, but still not consistent within

Continuum limit extrapolation of

Continuum limit extrapolation of

Continuum limit extrapolation of

From this analysis, we conclude that from our present data, the continuum extrapolation has error bars on the few percent level, for both channels. The results for the three-point linear fits are summarized in Table

Values of the correlators in the continuum. The error bar includes statistical errors, as well as systematic errors coming from the linear vs quadratic continuum fit, and continuum extrapolation with and without tree-level improvement.

From the tree-level calculation, we can estimate the finite volume effects on the UV contribution to the correlators. For the volumes used for our simulations, i.e.,

Finite volume corrections at tree level for the different correlators.

To get an educated guess on the viscosity, one needs to assume an ansatz. Here, we assume a very simple hydrodynamics plus tree-level ansatz for the spectral function, corresponding to the featureless scenario in Fig.

We introduce a constant

This model has two free parameters

Before describing the fitting procedure let us show how the different model parameters influence the observables considered here. Since it is the more interesting quantity, our discussion here will focus on

Effect of changing the UV parameter

Effect of changing the hydrodynamic parameter

We will present two different fits for the parameter

The viscosity appears to be temperature independent from our analysis. Here we have to mention a serious drawback of our fit ansatz. It assumes that the hydrodynamic prediction for the spectral function, strictly valid only for

For the second fit we consider the

This is the first estimate of

In this paper we studied the continuum behavior of the energy-momentum tensor correlators in pure

The achieved percent level precision of the data does not yet allow us to distinguish different scenarios for the spectral function. The statistical precision was already boosted by the multilevel algorithm on an anisotropic lattice. Despite recent efforts of using the gradient flow to measure energy-momentum tensor correlator

Once, however, a model is postulated, it is possible to give a model-dependent estimate of the shear viscosity to entropy ratio. We gave the first estimate of this phenomenologically important quantity from continuum extrapolated lattice data. Our estimate is in the same ballpark as earlier estimates based on finite

This project was funded by the DFG Grant No. SFB/TR55. This work was partially supported by the Hungarian National Research, Development and Innovation Office—NKFIH Grants No. KKP126769 and No. K113034. An award of computer time was provided by the INCITE program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DE-AC02-06CH11357. We acknowledge computer time on the QPACE machines

The combination of linearized relativistic hydrodynamics and linear response theory allows for a derivation of the low frequency behavior of the energy-momentum tensor correlators. For a nice derivation of these formulas, see the Appendix of

The leading-order perturbative result for the spectral function at high frequency is

In our analysis, we only take the first part

The formulas for the tree-level improvement are the result of a tedious but straightforward leading-order calculation. The resulting formulas still contain Matsubara sums, that can be easily evaluated numerically. For reference, we include here the numerical values of the tree-level improvement coefficients relevant for our study.

See Appendix

Note, that this combination was already used in the literature for the calculation of static quark potentials

See Appendix

The entropy is fixed from previous calculations of the equation of state.