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Corresponding author.

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We calculate shear viscosity and bulk viscosity in

The shear viscosity

We want a first-principles theoretical determinations of shear and bulk viscosity, to accompany the values extracted from experiment. The temperatures achieved in real-world heavy ion collisions are in a range where perturbation theory does not appear to be applicable, and so truly nonperturbative methods are needed. Our best first-principles nonperturbative tool is lattice gauge theory, which we will pursue in this work. Like previous literature, we will work within pure

The pioneering works

The standard way to investigate transport coefficients on the lattice is through Kubo formulas, which relate these coefficients to spectral functions, which in turn are related to Euclidean correlators through analytic continuation. The biggest challenge is that the energy-momentum tensor correlators, from which the viscosities are extracted, are extremely noisy, such that a noise-reduction technique must be employed to obtain the necessary precision. In Refs.

Our lattice setup consists of five large and fine lattices, of which the coarsest one (

In the following we will start with the definition of the EMT under gradient flow and explain how shear and bulk viscosity can be obtained from the EMT correlators. In Sec.

The fundamental object of our study is the energy-momentum tensor

Our approach will be to use analyticity to relate these spectral functions to the Euclidean, time-dependent correlation (still at zero momentum or equivalently with

There are two principle challenges when treating energy-momentum tensor correlations on the lattice; the correlations are very noisy, and because of the lack of continuous translation symmetry on the lattice, there is no obvious choice for the energy-momentum tensor operator. In particular, different components of

There are two approaches to determining the coefficients

In contrast, since

According to small-flow time expansion

Our lattice calculations are carried out in

We use the clover definition of the energy-momentum tensor appearing in Eq.

In this section we describe how we determine the renormalization constants appearing in Eq.

Unfortunately the enthalpy density is proportional to

We illustrate the method, and the effect of the different

Our final estimate for

We repeat this procedure for the other lattice spacings and summarize the final

Left: combined

Let us now focus on the small flow-time region, to establish how much flow time is enough to eliminate lattice spacing effects. We have added one more, still finer lattice (

Now we calculate

The lattices with smaller temporal extents for the determination of

The lattices at

After obtaining

Evaluating Eq.

In Ref.

Each index combination of the

Traceless spatial stress tensor (shear-channel) correlator between lattice blocks, at a fixed temporal separation, as a function of box separation, together with statistical error bars. The black points contain all traceless stress tensor components, while the red data points contain only the diagonal-type contributions. Some block separations only occur along lattice axes where the diagonal-type contributions are largest, while other block separations occur along lattice diagonals where some diagonal-type contributions are negative. Hence, the red points jump around, while the black points follow a smooth curve until the statistical errors become large.

In general, the lattice renormalization constant

Jonas Winter, private communication.

explores both renormalization constants as a function of flow and finds that they are consistent with each other within 2% error bars already forIn order to remove the large-separation data and therefore its noise, it is necessary to fit the large-separation tail to a physically-motivated Ansatz. The fitted value is then used instead of the data at those separations where the block-by-block signal-to-noise ratio is poor. For our Ansatz we will use the leading-order perturbative behavior of the correlation function, accounting for time periodicity, gradient flow, and our blocking procedure. In vacuum, the leading-order correlator of two field strength tensors is

Using these expressions, at finite

From Table

Figure

The temperature correction when going from

Next, consider discretization effects associated with computing on a lattice rather than in continuous space. To suppress the lattice discretization effects, we apply tree level improvement to the bare correlators. Specifically, if we assume that the lattice correlation functions will deviate from the continuum ones in the same way as occurs at lowest-perturbative order, then we can remove this effect by rescaling by the ratio of leading-order continuum to lattice-correlation functions

After temperature corrections, tree-level improvement and renormalization, in Fig.

Tree-level-improved EMT correlators in the shear channel (left) and bulk channel (right) normalized by the leading-order correlator on the

The double extrapolation contains two steps: first we perform the continuum extrapolation

Figure

The continuum extrapolation of EMT correlators in shear channel (

The

Now we consider the

The more fitting coefficients we use, the larger the errors in the resulting fit. Therefore, we want to avoid using two extrapolation coefficients, e.g., a fit of form

For the extrapolation of the bulk viscosity correlators we have taken a slightly different approach, based on the work of

This section is devoted to the spectral extraction from the extrapolated correlators. We first reconstruct the spectral function using

The spectral reconstruction performed here is mathematically ill-posed

In the case of the spectral analysis via fit this issue presents itself as the difficulty in finding a global, well-determined minimum. In principle, if the “correct” Ansatz were known, with enough data points and without considering any noise the analysis should yield a global minimum in the

According to Eqs.

Using this relation the coupling is fixed to the value

For the bulk channel the LO and NLO spectral function are also available

The infrared behavior of the spectral function is not known

For the bulk channel our fit has two parameters on 13 data points, leaving 11 degrees of freedom. The leading-order fit shows a poor

For the shear channel we find that when using the LO spectral function the

As one attempts to capture possibly missing structure, we have considered amending the UV part of the spectral function with an anomalous dimension, namely changing Eq.

The technical difficulty in performing the spectral reconstruction can be traced in part to two issues, the finiteness of the number of data points and their noise. The first implies a discretization of the integral transform

Consider an estimator

Keeping this in mind, one recipe to evaluate

In our implementation we further consider the rescaling function

One key difficulty in the BGM, or any spectral reconstruction, is the determination of its errors, both statistical and systematic. The number of points, the rescaling function, the regularization parameter and the noise of the data all feed into the estimator result. Here, we focus on the impact of the regularization parameter

Choosing

Double-extrapolated correlators in the shear channel (

The comparison of fit correlators and lattice correlators (left) and the fit spectral function in the shear channel. In M3 the width of the Lorentzian peak

The comparison of fit correlators and lattice correlators (left) and the fit spectral function in the bulk channel.

The resolution function (left) and output spectral function (right) at

Nevertheless, a robust result over a broad range in

One could imagine using a criterion for

We have calculated the energy-momentum tensor correlators in both the shear and the bulk channel at

In fitting our data, we find that the statistical errors are significantly smaller than the difference in fit values found from various-fit Ansätze choices, despite relatively little difference in the fit quality from the different Ansätze choices. This is summarized in Table

Bulk and shear viscosity fit results for three models, described in the previous section. The errors are statistical only; the difference between different fit models represents a systematic error. In each case, the NLO spectral function at large momentum was used in the fit.

In our opinion, there are two pressing tasks to further improve on this work. The first is to find better models for the spectral function’s behavior at low to intermediate frequencies

All authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 “Strong-interaction matter under extreme conditions”—Project No. 315477589—TRR 211. A. F. acknowledges support by the Ministry of Science and Technology Taiwan (MOST) under Grant No. 111-2112-M-A49-018-MY2. The computations in this work were performed on the GPU cluster at Bielefeld University using

In Sec.

The coefficient

In correcting for the slight temperature variation between our lattices, we made the assumption that the temperature dependence in the spectral function is approximately separation-independent. Looking at Fig.

The difference of correlators obtained in two different ways of treating the slope of the correlators with respect to

The errors in the continuum extrapolation, shown in Fig.

We also tried continuum extrapolation excluding the coarsest lattice

The difference of continuum-extrapolated correlators in shear channel with (top) and without (bottom) the coarsest lattice.

Next consider the extrapolation to zero flow depth. In the main text we argue that the operator product expansion predicts flow-depth effects which are polynomial in

In Table

Flow-extrapolated results using different kinds of models.

Statistical errors arise both in our determined

The (un)importance of the statistical error in