The shear viscosity η and bulk viscosity ζ of the hot quark-gluon plasma characterize the dissipation which occurs due to nonuniform flow, such as occurs in heavy ion collisions. They have been a topic of intense study for the last two decades. Experimental results [1–5] suggest a small shear viscosity; indeed, based on the determined values of elliptic and higher-order flow as functions of momentum and impact parameter, the best extractions of the shear viscosity are in the range 1/(4π)<η/s<2/(4π) [6]. This is close to the claimed lower bound on η/s obtained from N=4 supersymmetric Yang-Mills theory at strong coupling, which predicts η/s=1/(4π) [7]. While leading-order weak-coupling calculations [8,9], extrapolated to the physical coupling strength, suggest a larger shear viscosity η/s∼0.5–1, the next-to-leading correction to this result at a physically interesting coupling and temperature reduces the tension, implying η/s∼0.2 [10]. The size of this difference implies that the perturbative series shows poor convergence. As for the bulk viscosity, its extraction from experiments shows that it is nonzero but somewhat smaller than the shear viscosity at temperatures of order 200 MeV [6]. At higher temperatures we have a leading-order perturbative calculation [11] which shows that, for 0.06<αs<0.3, ζ/s∼0.02αs2. That is, as the theory becomes more conformal at higher temperatures, the bulk viscosity is expected to become small, but it can nevertheless play a role at lower temperatures where QCD behaves strongly nonconformally.

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 SU(3) gauge theory, but one focus of our work is to develop tools which will be straightforward to extend to the theory with dynamical quarks.

The pioneering works [12–14] established the general approach for investigating shear viscosity via unequal Euclidean-time, zero space-momentum energy-momentum tensor (EMT) correlation functions. More recent studies [15,16] have extended this work to consider a range of temperatures. However, these works used rather coarse and small lattices, meaning that cutoff effects may be severe. Recently, a lattice calculation using the gradient flow method was conducted on a 643×16 lattice [17]. In that work, the shear viscosity is extracted at finite flow time, making the results difficult to interpret [18].

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. [14,16] the multilevel algorithm [19] was used; in this work we instead make use of the gradient-flow method [20–23] and the blocking method [24] which we proposed recently. In comparison to multilevel algorithms, the gradient flow approach has the advantages that it is straightforward to apply to the full theory with dynamical quarks, and it helps with the problem of operator renormalization. This paves the way for a future study in full QCD. The signal is improved further via the blocking method, up to a factor of 7 without additional computation cost, as we demonstrate in [24]. With these two methods we are able to achieve high precision for the desired correlators.

Our lattice setup consists of five large and fine lattices, of which the coarsest one (643×16) is already as large as the finest lattice used in previous literature. The largest and finest lattice in our study is of size 1443×36 at β=7.544 (a=0.0117  fm). With our setup, including such a fine lattice, the continuum extrapolation is well-behaved and, thanks to the large temporal extents of the underlying lattices, the results of the spectral reconstruction will be more reliable.

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. III we give the lattice setup used in this study. Sec. IV is devoted to the nonperturbative renormalization of the EMT correlators. After a short illustration to the temperature-correction and tree-level improvement in Sec. V we continue with the discussions of continuum extrapolation and flow-time extrapolation in Sec. VI. In Sec. VII we focus on the extraction of viscosities via spectral analysis and provide our estimates for the viscosities. The conclusion is given in Sec. VIII.

The fundamental object of our study is the energy-momentum tensor Tμν, defined as the Noether current of 4-translation symmetry (or equivalently as the variation of the action with respect to the spacetime metric). Shear viscosity is the response of Tij to shear flow, under which the traceless part of ∂ivj is nonzero. Shear flow also couples to the energy-momentum tensor, so the Kubo relation describing the shear viscosity involves a correlation function of two traceless energy-momentum tensors, η(T)=limω→0ρshear(ω,T)ω,(1)ρshear(ω,T)=110∫d3xdt eiωt⟨[πij(x,t),πij(0,0)]⟩,πij=Tij−13δijTkk.(2)Similarly, bulk viscosity is the response of the trace of the energy-momentum tensor to a divergent fluid flow, which also couples to the trace of the energy-momentum tensor, ζ(T)=19limω→0ρbulk(ω,T)ω,(3)ρbulk=∫d3xdt eiωt⟨[Tμμ(x,t),Tνν(0,0)]⟩.(4)

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 ∫d3x), G(τ)=∫0∞dωπcosh[ω(1/2T-τ)]sinh(ω/2T)ρ(ω,T).(5)This expression can in principle be inverted to determine the spectral function, a task we will return to in Sec. VIII. Here G(τ) is the Euclidean function associated to the respective spectral function, that is, Gshear(τ)=110∫d3x⟨πij(0,0→)πij(τ,x→)⟩,Gbulk(τ)=∫d3x⟨Tμμ(0,0→)Tμμ(τ,x→)⟩.(6)Our main task will be evaluating the continuum limit of these correlation functions precisely.

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 πij renormalize differently, which presents a challenge. Both problems are ameliorated if we utilize gradient flow to generate our energy-momentum operators. Gradient flow is defined as the iterative replacement of the gauge fields Aμ(x) with fields containing less UV fluctuations, Bμ(x,τF), through the definitions [20] Bν(x,τF=0)=Aν(x),B˙μ=DνGνμ,Gμν=∂μBν-∂νBμ+[Bμ,Bν],Dμ=∂μ+[Bμ,·].(7)That is, at τF=0 the flowed field is the nonflowed field, but the field then evolves under a covariant heat equation which iteratively removes the most UV fluctuations of the field. Using the flowed field to construct operators such as the energy-momentum tensor leads to operators with well behaved renormalization properties and improved rotational invariance. In terms of the gradient-flowed field, we define the gradient-flowed squared field strength operator and the traceless tensor operator as E(τF,x)=14Fρσa(x,τF)Fρσa(x,τF),Uμν(x,τF)=Fμρa(x,τF)Fνρa(x,τF)-δμνE(τF,x).(8)The energy-momentum tensor can then be written in terms of these two operators and two not yet known coefficients as [25] Tμν(τF,x)=c1(τF)Uμν(τF,x)+4c2(τF)δμνE(τF,x).(9)Here c1, c2 are the coefficients on the traceless and pure-trace parts of the tensor, respectively. Arguably one should perform a vacuum subtraction from E(τF,x), but in practice we always compute connected correlation functions, which implements such a subtraction automatically.

There are two approaches to determining the coefficients c1,c2(τF). Suzuki has determined them up to 2-loop and 3-loop order in the MS¯-scheme [26]: c1(N2LO)(τF)=1g2(μ)∑n=02k1(n)(L(μ,τF))[g2(μ)(4π)2]n,(10)c2(N3LO)(τF)=1g2(μ)∑n=14k2(n)(L(μ,τF))[g2(μ)(4π)2]n,(11)where the coefficients k1(n), k2(n) can be found in [27,28]. Here L(μ,τF)≡log(2μ2eγEτF) and the running coupling can be evaluated in the MS¯-scheme at scale μ=1/8τF [29]. The series for c2 begins with a constant and is known to one higher order than for c1; therefore it suffers very little coupling and renormalization-point uncertainty, and is more accurate than any numerics-based nonperturbative estimate which we could develop. Therefore, we use the series expansion for c2. The error in this series expansion is negligible, below 0.1%. This will be swamped by statistical errors in our correlation functions and will play no role in our error analysis.

In contrast, since c1 depends on the coupling at leading order, the use of a series expansion is significantly less reliable. Instead, we will perform a nonperturbative renormalization on the lattice in Sec. IV, based on ideas developed by Giusti and Pepe [30].

According to small-flow time expansion [31], any composite operator at finite flow time can be expressed as superposition of renormalized operators with finite, flow-dependent coefficients [32]. That is, one can expand our stress-tensor operator in an operator product expansion, where the first term is the desired stress tensor and higher terms represent various higher-dimension operators with coefficients containing positive powers of τF. Therefore, one expects that the correlation functions we evaluate, at separation τ, correspond to the correct correlation functions, plus corrections which appear as a series expansion in (τF/τ2). Determining the desired correlation function therefore requires an extrapolation to τF→0 to eliminate the effects of these high-dimension contaminants. Only some finite range of τF values will actually be useful in this extrapolation; larger values of τF, such that (τF/τ2) is not small, will be outside of the range where an extrapolation is possible. Solving Eq. (7) perturbatively suggests that the flow smears the gauge field with a radius r≃8τF [20]. In general this radius should be larger than one lattice spacing to suppress the lattice effects and noise, and at the same time smaller than half the lattice extent so that the flow radius does not interact with the lattice periodicity. For a specific operator there can be further constraints on the flow radius. How much flow can be applied and what Ansatz should be used for the τF→0 extrapolation will be discussed in a later section.

Our lattice calculations are carried out in SU(3) Yang-Mills theory in four-dimensional spacetime with periodic boundary conditions for all directions. We summarize the settings in Table I. The gauge configurations are generated using the standard Wilson gauge action on five large, fine, isotropic lattices. On each lattice we generate 10,000 configurations. To ensure the gauge fields are fully thermalized the first 4,000 sweeps (each consists of one heat bath and four over-relaxation steps) are discarded. In the sampling procedure the configurations are stored after every 500 sweeps. This removes the autocorrelations in observables as we have confirmed. All the lattices are set to the same temperature ∼1.5Tc by tuning the β value. The scale is set via the Sommer parameter r0 [33] with state-of-the-art value r0Tc=0.7457 [34]. The parametrization form needed in scale setting is taken from [34] with updated coefficients from [35].

We use the clover definition of the energy-momentum tensor appearing in Eq. (8). The gradient flow is a Symanzik improved version [36]. We measure the EMT correlators at 140 discrete flow times in the range 8τFT∈{0.004,…,0.375} using an adaptive step-size method. In this method the step size is updated after each integration step such that the error in the integration does not exceed a certain tolerance [37]. The bin size used in the blocking method is given as nσ in Table I.

In this section we describe how we determine the renormalization constants appearing in Eq. (9). We determine the constant c1 using a method inspired by the work of Giusti and Pepe [30]. Namely, we observe that the enthalpy density is given by ⟨ε+P⟩τF=c1(τF)⟨13Uii(τF)-U00(τF)⟩,(12)where “0” denotes the time direction. Since ε+P has been measured at the subpercent level [38], we can determine c1 through the ratio c1(τF)=⟨ε+P⟩τF/⟨13Uii(τF)-U00(τF)⟩. We will explain below why this also determines the coefficients for the off-diagonal components of the stress tensor, to sufficient precision for this work.

Unfortunately the enthalpy density is proportional to T4 and therefore to Nτ-4, which leads to a poor signal-to-noise ratio for the finest lattices. We overcome this limitation by measuring ε+P at a range of Nτ values listed in Table II, not just the ones given in Table I. This is possible because the renormalization constant c1 depends on the lattice spacing but not on the temperature. However, after enough gradient flow, the gradient flow radius starts to interact with the periodicity radius and the result becomes contaminated and unreliable. A leading-order perturbative estimate of this effect is that [39] ⟨13Uii-U00⟩flowed(13Uii-U00)true=1-180π4e-1/x(1+1x+12x2),withx=8τFT2.(13)

We illustrate the method, and the effect of the different Nτ choices, in Fig. 1, which shows c1 for our finest lattice at different temperatures. It can be seen that, at very small flow times c1, measurements from different temperatures agree with each other, with smaller statistical errors for the smaller Nτ values. With increasing flow time, the higher-temperature c1 values start to deviate from the lower ones. The point where Eq. (13) implies a 1% correction is marked for each Nτ value by a vertical bar, and it corresponds well with the flow time value where a given lattice starts to deviate clearly from the larger-Nτ lattices.

Our final estimate for c1 will be based on a weighted average of the value determined from each Nτ value we explored. The weight is determined as 1/(σstat+σsyst)2 where σstat is the statistical uncertainty from the lattice data and σsyst is the systematic shift as determined from Eq. (13). The averaged c1 is the black curve labeled “combined” in Fig. 1.

We repeat this procedure for the other lattice spacings and summarize the final c1 in Fig. 2. The statistical error in c1 is small, ranging from 1.1% at the smallest flow time we use to 0.27% at the largest flow time we use. A table presenting the statistical uncertainties of c1 at each lattice spacing for a range of flow times is provided in Appendix A.

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 (β=7.793, with Nτ=48 when T/Tc=1.5) so that we can compare to a still more continuumlike case. We can see that lattice cutoff effects are suppressed at large flow times but at small flow times they are noticeable. To see down to what flow time the c1 is free of lattice cutoff effects, we plot the ratio c1/c1(β=7.793) in the right panel. In order to see more clearly how the different lattice spacings differ from each other, we have plotted error bars based only on the statistical errors in the coarser lattices—that is, statistical errors in the β=7.793 lattice are treated as a common systematic error in the right plot. The figure shows that the lattices give compatible c1 values as long as the flow time is large enough, but each lattice starts to deviate at a flow time such that τF/a2 becomes order one. Specifically, in every case the deviation from continuum behavior reaches 2% when τF≃0.4a2. The deviation rapidly becomes more severe below this point. This deviation from continuum behavior indicates that the applied gradient flow is not sufficient to supply a continuumlike, well-renormalized stress-tensor operator. Since the statistical precision of our EMT correlator data is typically around 2% and since we want to keep systematic effects smaller than this, we will impose the condition τF≥0.4a2 when we perform the double extrapolation of shear correlators in the next section.

Now we calculate c2. According to Eq. (11), the running coupling in the MS¯ scheme is needed. For that we first calculate the coupling in the gradient flow-scheme and then convert it to the MS¯ scheme. In the gradient-flow scheme the running coupling can be calculated as [40,41] gflow2=128π23(Nc2-1)11+δ(τF)⟨τF2E⟩,(14)where Nc=3 and E is the energy density defined in Eq. (8). δ(τF) can be found in [40,41] as well. Note that the energy density should be measured at zero temperature. On the lattice we take large temporal extents to suppress the thermal effects. The lattices used to study this quantity are given in Table III. Because of high computation costs the two finest lattices have smaller-aspect ratios. However, based on the three coarse lattices, we have seen that finite volume effects are small compared to the statistical error of the correlators.

After obtaining τF2E in the gradient flow scheme, we can relate it to the one in the MS¯ scheme [29]. This requires solving a cubic equation, whose solution gives the running coupling in the MS¯ scheme. Inserting in Eq. (11), we get the final c2 shown in Fig. 3. The errors are not visible as they are tiny and in every case much smaller than 1%. We can see that unlike c1, the difference of c2 among different lattice spacings is always small. The ratio c2/c2(β=7.544) is always smaller than 1% at all flow times, suggesting that the cutoff effects can be ignored for c2.

Evaluating Eq. (6) involves computing a correlator with an integral over all values of the spatial separation. To improve signal-to-noise ratio, in practice one evaluates ∫d3xd3ydt⟨T(x,τ+t)T(y,t)⟩, that is, one performs an integral over the coordinates of each operator. The correlation function is dominated by small values of coordinate difference |x-y|. However, the fluctuations in the correlator, and therefore the noise, are approximately separation independent. Therefore, the inclusion of large separations makes the evaluation noisy without contributing meaningfully to the signal.

In Ref. [24] we proposed a way to reduce these noise contributions. The operator of interest (Tμμ or a component of πij) is first summed over small volumes called blocks, on a single τ sheet but with a cubic space extent given in Table I. We evaluate all block-to-block correlators and then average all correlators which have the same temporal and block-center spatial separation. Finally, we examine how the correlation function varies with the space separation between blocks, replacing the large-separation, small-signal values with an asymptotic fit as described in [24].

Each index combination of the ⟨πij(x,τ)πij(y,0)⟩ correlator has a distinctive angular structure as a function of the direction of the x→-y→ vector. For instance, from reflection positivity we know that ⟨Txy(r→)Txy(0)⟩<0 for r→ pointing along the x-axis or y-axis, but it is positive if r→ points along the z-axis or the line x=y. In contrast, the ⟨(Txx-Tyy)(r→)(Txx-Tyy)(0)⟩ correlator is positive along each lattice axis but is negative along the x=y line. In our blocking procedure, certain block separations primarily sample blocks which are separated along lattice axes, while others sample the directions along lattice diagonals or other combinations. Therefore, TxyTxy-type correlators will be larger for some blocks and smaller for others, while Txx-Tyy-type correlators will show the opposite trend. Including a single component or a subset of possible components leads to a correlation function which varies strongly with separation-direction and therefore jumps up and down as a function of the block separation. This effect goes away if we include all traceless ij combinations, which is therefore obligatory. We illustrate this in Fig. 4, which shows the ππ correlation function as a function of block separation.

In general, the lattice renormalization constant c1 is different for Txx-Tyy than for Txy, because the rotational symmetry which relates them in the continuum is absent on the lattice [30,42,43]. We have only evaluated the renormalization constant for the former operator type. However, the application of gradient flow should remove rotation-invariance violations in operator normalizations up to corrections suppressed by O(a2/τF). Therefore, any effects from this operator normalization issue should be removed in our fixed-τF continuum limit. A recent masters thesis¹

Jonas Winter, private communication.

In 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 ⟨Fμνa(r)Fαβb(0)⟩=g2δabπ2r4[δμαδνβ-δμβδνα-2r2(rμrαδνβ-rμrβδνα-rνrαδμβ+rνrβδμα)].(15)Applying gradient flow to a depth τF modifies this expression to [39] ⟨Gμνa(r)Gαβb(0)⟩τF=g2δabπ2r4[A(r,τF)(δμαδνβ-δμβδνα)+B(r,τF)r2(rμrαδνβ-rμrβδνα-rνrαδμβ+rνrβδμα)],(16)A(r,τF)=1-(1+r28τF)e-r2/8τF,B(r,τF)=-2+[2-2r28τF+(r28τF)2]e-r2/8τF.(17)Note that this is a continuum, not lattice, expression; but when τF/a2>0.5, the lattice-continuum difference for flowed correlators is small, and the use of a continuum limit at fixed flow depth based only on data which satisfies this criterion should avoid the need to include lattice spacing corrections as well.

Using these expressions, at finite τF,τ,|r→| and with periodic boundaries in the time direction, the leading-order stress tensor correlator summed over all transverse-traceless elements T^ij=Tij-δijTkk/3 relevant for shear viscosity and for bulk viscosity are ⟨T^ij(r→,τ)T^ij(0,0)⟩τF∝∑n1,n2∈ZA(r1)A(r2)r14r24+A(r1)B(r2)+A(r2)B(r1)2r14r24+B(r1)B(r2)6r16r26(3(r1·r2)2+r→2[r12+r22-4r1·r2+45r→2]),(18)⟨Tμμ(r→,τ)Tνν(0,0)⟩τF∝∑n1,n2∈ZA(r1)A(r2)r14r24+A(r1)B(r2)+A(r2)B(r1)2r14r24+B(r1)B(r2)6r16r26(2(r1·r2)2+r12r22),(19)where r1=(τ+n1β,r→) and r2=(τ+n2β,r→) are the 4-displacement with the temporal displacement shifted by independent integer multiples of the inverse temperature β.

From Table I it can be seen that the temperatures are not exactly 1.5Tc on all lattices. This setup is adopted for historical reasons [44,45], and the deviations of the temperature were only discovered after the correlators were measured. The temperature differences, though small, must be accounted for when performing a continuum extrapolation. Because the temperature differences are small and the lattices are fine enough that the continuum extrapolation is not very severe, we will content ourselves by evaluating the temperature dependence at the linearized level and at a single lattice spacing. We then assume that the established temperature correction also applies at the other lattice spacings. We choose to perform a linear temperature-dependence analysis on the lattice which has the largest deviation from T=1.5Tc, namely the 20×803 lattice with β≡β1=7.035 and T=1.4734Tc. For this lattice, we choose a second β value, β2=7.0767, corresponding to T=1.5501Tc, and we repeat our correlation function studies on this lattice. Since the renormalized correlators contain two parts, namely the renormalization constants c1 or c2 and the bare correlators, the corrections for both parts should be considered. The renormalization constants have been determined precisely in Sec. IV at β values listed in Table I. To obtain the one at β=7.0767 we linearly interpolate between β=7.035 and β=7.192. We then calculate the renormalized correlators, denoted as G1 and G2 for the lower and higher temperature, respectively, by multiplying the bare correlations functions and the squared renormalization constants. We then evaluate the difference, -1+G2(τ)/G1(τ), representing the temperature dependence of the correlation function, as a function of τ and τF. Statistical errors are computed using bootstrap sampling, and since G1, G2 arise from different ensembles, their errors are independent and can be propagated via Gaussian error propagation.

Figure 5 shows the thermal correction for the largest gradient flow depth we use (and therefore the least noisy data). The figure shows that the temperature effect is nearly τ independent except at the smallest τ values (which are contaminated by lattice effects). Based on this result, we treat -1+G2/G1 as a function of τF only, determining its value based on the weighted average of all the points at τ/a≥4 at each τF. As the figure shows, the thermal corrections are relatively small, considering that the temperature difference 1.5501-1.4734=0.0767Tc is significantly larger than any of the individual deviations from 1.5Tc shown in Table I. We will therefore use the determined slope P=(G2/G1-1)/(T2-T1), averaged over τ values, and apply it as a linearly interpolated correction to all data. For instance, data at temperature T can be interpolated to the temperature T0 through G(T0)≃G(T)(1+P(T0-T)). A detailed analysis on the uncertainties in the temperature correction can be found in Appendix B. The appendix also presents an alternative model, which gives a consistent result.

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 [46,47], Gt.l.(τT)=Glat(τT)·GcontLO(τT)GlatLO(τT).(20)The leading-order continuum correlators in shear channel and bulk channel can be found in [13,14] GcontLO,shear(τT)T5=32dA5π2(f(x)-π472),GcontLO,bulk(τT)T5=484dA16π6g4(f(x)-π460),(21)where x=1–2τT, f(x)=∫0∞ds s4cosh2(xs)/sinh2s and dA=8 counting the number of gluons. The leading-order lattice correlator for clover discretization is available in [47]. For better visibility we always normalize the tree-level improved correlators with a normalization correlator Gnorm calculated at τF=0, where for shear channel we use Gnorm≡GcontLO,shear and for bulk we use Gnorm≡GcontLO,bulk/g4.

After temperature corrections, tree-level improvement and renormalization, in Fig. 6 we show the lattice correlators normalized by the free continuum correlators on 1443×36 lattice at different flow times, in both the shear and the bulk channels. We have not plotted data down to small flow times because it has large errors. We can see that as flow time increases the signal-to-noise ratio improves. At very large flow times the signal is strongly modified by flow effects and we leave the regime where an extrapolation τF→0 can be performed.

The double extrapolation contains two steps: first we perform the continuum extrapolation a→0, and then we perform a flow-time-to-zero extrapolation. As we pointed out in Ref. [18], this has the advantage that the continuum extrapolation eliminates terms of form a2/τF, so that the τF extrapolation will consist only of positive powers. Before the continuum extrapolation, the correlators on coarse lattices have to be interpolated to the separations of the finest lattice, for details see, for example, references [18,48]. In the continuum extrapolation we use the Ansatz Gt.l.(Nτ)Gnorm(Nτ)=m·Nτ-2+b,(22)because the lattice action has leading discretization errors of order a2. Here m and b are fit parameters that can be different for each temporal separation and flow time. The continuum estimates for the (normalized) correlators are given by b≡Gcont/Gnorm.

Figure 7 shows how good the fit Ansatz, Eq. (22), works at an intermediate flow time τFT2=0.00416. We can see for the bulk channel that in some cases the fit is poor in the sense that χ2/dof>1. Our procedure is to enlarge the error bars by χ2/dof in these cases. After the continuum extrapolation we collect the continuum estimates for each flow time and show them in gray bands in Fig. 8.

Now we consider the τF→0 extrapolation. To perform the extrapolation, we need to understand the functional dependence on τF, and we need to determine over what range of τF values to perform the extrapolation. For general values of τF/τ2, the correlator is a complicated function of this ratio, in some cases even taking on a different sign than the small-τF value [39]. However, if τF/τ2 is small, then as discussed near the end of Sec. II, we expect the flowed stress tensor to be described in terms of an operator product expansion, with the leading coefficient equaling the stress tensor and with higher-dimension operators suppressed by powers of τF. As a result, in this regime the small-τF expansion of the correlation function should approach τF→0 with polynomial-in-τF corrections. (We will ignore possible anomalous dimensions in this discussion.)

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 G(τF/τ2)=A+BτF/τ2+CτF2/τ4. And if we use a wide enough data range that the τF2/τ4 coefficient is really relevant, then there is a danger that we also need still higher-order coefficients. Therefore, we will restrict ourselves to a region where the total variation in G(τF/τ2) appears to be at most 20% from its extrapolated value. In this range, within the few % accuracy which is our goal, we expect that a linear extrapolation, e.g., G(τF/τ2)=A+BτF/τ2, should be sufficient. Based on our previous experience with the topological density operator [48], we expect that a fitting range out to 8τFmax=0.5220τ should remain in this small-correction regime. We will fit a range of τF from this maximum down to half this value, because the correlator becomes so noisy at smaller τF that extending the range further is not helpful. In addition, to prevent lattice spacing effects of form a2/τF, we restrict to values with τF/a2≥0.4 as already discussed. For small τ values this constraint excludes too much of the τF range over which we want to extrapolate, which prevents us from determining the correlator at small temporal separations. The resulting correlators within the range [0.5τFmax,τFmax] are shown as colored bands in Fig. 8.

For the extrapolation of the bulk viscosity correlators we have taken a slightly different approach, based on the work of [25,26,49]. A recent three-loop calculation of the flow-dependence of the EMT trace suggests a finite-τF fitting function of form [26] θ(τF)=(1-c(g2(μ(τF))(4π))3)θ(τF=0),(23)where c and θ(τF=0) are fit parameters. Since what we measured in this study is the correlators of θ, we take the square root of the correlators and fit it to Eq. (23). The fitted curves are shown as dashed black lines in Fig. 8 and the extrapolated correlators are shown as colored points at τFT2=0. It can be seen that the fit function is almost linear, indicating that a fit to an Ansatz linear in flow time (as used in [18,48]) would give similar results. Appendix C presents more details on both the continuum and the small flow-time extrapolations. The double extrapolated correlators in both channels are shown in Fig. 9.

This section is devoted to the spectral extraction from the extrapolated correlators. We first reconstruct the spectral function using χ2-fits with models based on perturbative calculations and then determine the viscosities using the Backus-Gilbert (BG) method [50].

The spectral reconstruction performed here is mathematically ill-posed [51]. One feature of this is the difficulty in quoting a robust spectral function since uniqueness of any solution is a priori not given.

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 χ2-plane. Without this knowledge and with noise included, however, this minimum is less well determinable and a fit often yields χ2-values that are not very sensitive to the parameter choices. Consequently, it becomes difficult to choose with confidence which solution and Ansatz is the best description. In the following we address this difficulty by augmenting our study with a spectral analysis using a method that does not rely on an Ansatz per se in form of the BG method.

We have calculated the energy-momentum tensor correlators in both the shear and the bulk channel at 1.5Tc in the quenched approximation on five large and fine lattices. To improve the signal-to-noise ratio we have applied both the gradient flow method and the blocking method. We thoroughly studied the temperature corrections and the renormalization of the operators. The correlators have been extrapolated first to the continuum limit and then to the τF→0 limit. The final correlators are used to extract the shear and bulk viscosity based on perturbative models. For the bulk channel, we find that the NLO spectral function can describe our lattice data when adding a transport part with appropriate interpolation. For the shear channel we were unable to find a fit with better than χ2/d.o.f.=2. To further improve the fit quality, we need either a more flexible model or a better theoretical understanding of the expected spectral function.

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 IV. Therefore we will estimate the lowest and highest value of viscosity to be the extreme values we found among the fit functions. Using s/T3=5.098 from [38], our shear and bulk results become η/s=0.15-0.48,T=1.5Tc,ζ/s=0.017-0.059,T=1.5Tc.(34)The lower estimates are above the lower bounds from the Backus-Gilbert analysis. The upper bounds are based on a model which assumes that there is a relatively narrow feature near ω=0, namely a Lorentzian-type peak with a width of 1T. If a strongly-coupled medium does not support long-lived excitations, this assumption appears unlikely and the lower limit is more likely to be correct. However, the data cannot definitively prove or disprove this theoretical prejudice. The shear viscosity we obtained in Eq. (34) is close to the hydrodynamic estimate 1<(4π)η/s<2.5 [61].

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 ω∼[1-5]T. This will allow a fitting extraction which makes maximal use of the high-quality data which is now available. The second task is to extend these results to the unquenched case. This is not just a matter of performing much more expensive unquenched simulations. It is also necessary to understand the renormalization of the more-complicated unquenched stress tensor operator at the percent level, which appears to be possible but quite challenging. Some progress in this direction has been made recently by Dalla Brida et al. [62], but precision studies including gradient flow do not yet exist. We leave these developments for future work. All data from our calculations, presented in the figures of this paper, can be found in [63].