The second order hydrodynamic transport coefficient κ for the gluon plasma from the lattice
PPrepared for submission to JHEP
The second order hydrodynamic transportcoefficient κ for the gluon plasma fromthe lattice Owe Philipsen, Christian Schäfer
Institut für Theoretische Physik, Goethe-Universität Frankfurt,Max-von-Laue Str. 1, 60438 Frankfurt am Main, Germany
E-mail: philipsen,[email protected]
Abstract:
The quark gluon plasma produced in heavy ion collisions behaves likean almost ideal fluid described by viscous hydrodynamics with a number of transportcoefficients. The second order coefficient κ is related to a Euclidean correlator of theenergy-momentum tensor at vanishing frequency and low momentum. This allowsfor a lattice determination without maximum entropy methods or modelling, butthe required lattice sizes represent a formidable challenge. We calculate κ in leadingorder lattice perturbation theory and simulations on × , lattices with a < . fm. In the temperature range T c − T c we find κ = 0 . T . The error coversboth a suitably rescaled AdS/CFT prediction as well as, remarkably, the result ofleading order perturbation theory. This suggests that appropriate noise reductionmethods on the lattice and NLO perturbative calculations could provide an accurateQCD prediction in the near future. a r X i v : . [ h e p - l a t ] J a n ontents κ κ in lattice QCD 5 κ to the lattice correlator 63.3 Lattice perturbation theory 73.4 Renormalisation 10 One of the major findings of the experimental heavy ion programme [1–4] is that QCDmatter at high temperatures and low densities behaves as a nearly ideal fluid withvery low viscosity. This conclusion is based on the fact that experimental data areexcellently described by relativistic hydrodynamics, with transport coefficients fittedto the data [5–10]. Unfortunately, theoretical predictions of transport coefficientsfrom the fundamental theory QCD remain very difficult [11]. Up to a few times thetransition temperature to the quark gluon plasma, the QCD coupling is not weakenough for perturbative methods to apply, which predict a less ideal fluid [12, 13].– 1 –n the other hand, results in the opposite strong coupling limit can be obtained byAdS/CFT duality methods in certain supersymmetric models [14, 15], but these donot correspond to QCD directly.Unfortunately, lattice simulations of real time quantities are in general severelylimited by the need for analytic continuation. Calculations of spectral functions onthe lattice based on maximum entropy methods [16, 17] or a model ansatz [18–20]require both functional input and high accuracy data to sufficiently constrain theresults. An exception to this conceptual difficulty are the three second-order hydro-dynamic coefficients κ, λ , λ [21–25], which can be related to Euclidean correlationfunctions through Kubo formulae. They have recently been computed to leadingorder in a weak coupling expansion in [26], where also possibilities for a lattice deter-mination were discussed. The coefficients λ , λ are related to three-point functions,which are still too costly to numerically evaluate.Here we present a first attempt to determine κ from the momentum expansionof a suitable two-point function in a lattice simulation. In order to approach the zeromomentum limit, very large lattices are required, demanding an enormous numericaleffort already in pure gauge theory. While the errors on our result are thus still toolarge to be satisfactory, our work demonstrates that the determination of the secondorder coefficients is possible without conceptual difficulties and should be improvedin the future with appropriate noise reduction methods. Interestingly and in contrastto the first order transport coefficients, the lattice result for κ is within error barscompatible with the perturbative weak coupling result. It is also compatible with asuitably rescaled AdS/CFT result.In section 2 we briefly summarise the relation between the transport coefficient κ and a Euclidean correlator of the energy-momentum tensor, in section 3 this is carriedover to the lattice formulation, including a leading order perturbative evaluation anda discussion of renormalisation. Section 4 contains the numerical results of oursimulations. κ The definition of transport coefficients is based on a gradient expansion of the energy-momentum tensor in relativistic hydrodynamics, but their respective values have tobe determined from experiment or an underlying theory. In the case of the quark-gluon plasma this underlying theory is QCD. In this section we review the connectionbetween the transport coefficient κ and a Euclidean correlator in QCD, which allowsfor a direct computation of κ without resort to maximum entropy methods or func-tional input for the spectral function. – 2 – .1 Relativistic hydrodynamics The basic quantity in relativistic hydrodynamics is the energy momentum tensor (fora review, see [27]), which can be decomposed into an ideal part T µν and a dissipativepart Π µν T µν = T µν (0) + Π µν . (2.1)The ideal part is determined by the hydrodynamic degrees of freedom, wich are theenergy density (cid:15) , pressure p , the fluid’s four velocity u µ and the metric tensor g µν .Lorentz symmetry and the identifications T = (cid:15) , T i (0) = 0 and T ij (0) = p δ ij in thelocal rest frame restrict its form to T µν (0) = (cid:15)u µ u ν + p ( g µν + u µ u ν ) . (2.2)The dissipative contribution consists of a traceless part π µν and a part with non-vanishing trace Π Π µν = π µν + ( g µν + u µ u ν ) Π . (2.3)The former has been specified for a non-conformal fluid in a second order gradientexpansion within N = 4 Super-Yang-Mills theory [25] π µν = − ησ µν + ητ π (cid:18) (cid:104) Dσ µν (cid:105) + ∇ · u σ µν (cid:19) + κ (cid:0) R (cid:104) µν (cid:105) − u α u β R α (cid:104) µν (cid:105) β (cid:1) + . . . . (2.4)Besides the shear viscosity η and the relaxation time τ π , to second order the transportcoefficient κ enters the expansion and couples to the symmetrized Riemann curvaturetensor R , its contractions and the fluid’s four velocity u µ . For explanations of ∇ , σ µν , D and further terms in the expansion we refer to [25]. Note that even in flatspacetime the transport coefficient κ has a non-vanishing value [26, 28]. For the computation of the transport coefficient κ from QCD a relation between itsdefinition in relativistic hydrodynamics and thermal field theory is necessary. Thiscan be achieved by considering the fluid’s linear response to a metric perturbation[24] and establishes a connection between the transport coefficient κ and the retardedthermal correlator of the energy momentum tensor T ij in momentum space, G R ( x, y ) = (cid:104) [ T ( x ) , T ( y )] θ ( x − y ) (cid:105) , (2.5) G R ( ω, (cid:126)q ) = ∞ (cid:90) −∞ d t d x e − i( ωt − q i x i ) G R ( x, . (2.6)– 3 –he transport coefficient κ is identified as the leading low momentum coefficient atzero frequency with momentum aligned in z -direction, (cid:126)q = (0 , , q ) [24, 28], G R ( ω = 0 , (cid:126)q ) = G (0) + κ | (cid:126)q | + O ( | (cid:126)q | ) . (2.7)While the retarded correlator is a real time quantity, it is related by analytic contin-uation to the Euclidean correlator G E ( x, y ) = (cid:104) T ( x ) T ( y ) (cid:105) , (2.8) G E (i ω n , (cid:126)q ) = /T (cid:90) d τ ∞ (cid:90) −∞ d x e − i( ω n τ + q i x i ) G E ( x, , (2.9)with the discrete Matsubara frequencies ω n = n πT , n ∈ Z . This can be seen bywriting both correlators in their spectral representation G R ( ω, (cid:126)q ) = i ∞ (cid:90) −∞ d ω (cid:48) π ρ ( ω (cid:48) , (cid:126)q ) ω − ω (cid:48) + i η , (2.10) G E (i ω n , (cid:126)q ) = ∞ (cid:90) −∞ d ω π ρ ( ω, (cid:126)q ) ω − i ω n . (2.11)Appropriate boundary conditions render the analytic continuation unique [29], G R ( ω, (cid:126)q ) = G E ( ω + iη, (cid:126)q ) . (2.12)For vanishing frequency ω = 0 this can be written [11] G R ( ω = 0 , (cid:126)q ) = G E ( ω = 0 , (cid:126)q ) + B. (2.13)The contact term B arises from the missing commutator in the definition of theEuclidean correlator (2.8) compared to its retarded analogue (2.5) and corresponds tothe correlator evaluated at equal spacetime points, ∼ T (0) T (0) . An investigationof this contact term B by an operator product expansions shows that it is momentumindependent [30]. Hence equation (2.7) can be rewritten G E ( ω = 0 , (cid:126)q ) = G (cid:48) (0) + κ | (cid:126)q | + O ( | (cid:126)q | ) , (2.14)where we have absorbed the constant G (0) and the contact term B into a newconstant G (cid:48) (0) ≡ G (0) − B .The transport coefficient κ can now be obtained as the slope of the low momen-tum correlator G E ( q ) , which provides a possibility for a direct determination usinglattice QCD. This is in contrast to computations of the shear viscosity [19] or heat– 4 –onductivity [17, 20]. These are true dynamical quantities which cannot be related toEuclidean correlators without non-trivial analytic continuation. Their determinationby lattice calculations thus requires additional input, e.g. an ansatz for the spectralfunction or the maximum entropy method.So far the discussion was completely general. We now specify to Yang-Millstheory and its energy momentum tensor [11] T µν = θ µν + 14 δ µν θ, (2.15a) θ µν = 14 δ µν F aαβ F aαβ − F aµα F aνα (2.15b)where F aµν corresponds to the field strength tensor. The term θ = β ( g ) / (2 g ) F aαβ F aαβ includes the renormalisation group function β ( g ) and corresponds to the trace anomaly,which is caused by breaking of scale invariance. Since the transport coefficient κ isdefined in the shear channel, (cid:104) T T (cid:105) , it does not enter the computation.Equation (2.14) has been evaluated perturbatively in pure gluodynamics in theideal gas limit, i.e. at vanishing coupling, with the result [26, 28] κ = (cid:0) N − (cid:1) T . (2.16) κ in lattice QCD In this section we describe the discretisation of the action and the energy-momentumtensor and explain the need for renormalisation. Furthermore, we calculate κ inlattice perturbation theory and compare with the continuum result (2.16). We employ Wilson’s Yang-Mills action on an anisotropic lattice with different latticespacings in temporal and spatial direction, a σ and a τ , respectively, S [ U ] = βN c Re Tr (cid:34) ξ (cid:88) x,i 14 [ U µν ( x ) + U ν − µ ( x ) + U − µ − ν ( x ) + U − νµ ( x )] . (3.8)In contrast to an implementation with simple plaquette terms [34] the clover versionhas reduced discretisation errors and an improved signal-to-noise ratio [36], cf. figure2. κ to the lattice correlator In order to extract κ numerically from equation (2.14), we compute the Euclideancorrelator of the energy-momentum tensor within the lattice framework and perform– 6 – σ = ξa τ a τ Q µν ( x ) Figure 1 : Illustration of theclover plaquette on an anisotropiclattice. -0.06-0.04-0.0200.020.040.06 0 100 200 300 400 500 600 700 800 900 1000 a T heatbath stepsPlaquetteClover Figure 2 : Computation of θ on an isotropic × lattice for β = 7 . . We compare theclover and plaquette discretisations.a Fourier transform to momentum space with vanishing frequency. The determina-tion requires the momenta to be aligned orthogonally to the studied channel of theenergy-momentum tensor, i.e. (cid:126)q = (0 , , q ) for T . This is also the case for thecorresponding Kubo formula [11]. Thus the correlator in momentum space is givenby a σ a τ G E ( q ) = 1 V (cid:88) x,y e − i q ( x − y ) (cid:104) T ( x ) T ( y ) (cid:105) . (3.9)Additionally, we include the channels T and T with corresponding momenta inour analysis, since rotational invariance allows to average over all three channels.We need small momenta compared to temperature, which sets the relevant scale,i.e. q i /T < . With the discretised versions of temperature and momenta T = 1 a τ N τ , q i = 2 πa σ N σ n i , n i = 0 , , . . . , N σ − (3.10)we have for the ratio on the lattice q i T = 2 πN τ ξN σ n i < . (3.11)The temporal lattice extent N τ is fixed by the temperature and lattice spacing. Inorder to fit the transport coefficient κ to equation (2.14), we need at least threedifferent momenta satisfying this constraint (3.11). Thus the simulation requireslarge spatial lattice extents N σ , which makes the calculation costly. This can bepartly moderated by working with anisotropic lattices ξ > . In order to estimate lattice artefacts and check our numerics, we first compute thetransport coefficient κ in lattice perturbation theory on a lattice with anisotropy– 7 – in the case of vanishing coupling ( g = 0) . Definitions of relevant quantities andintermediate results can be found in appendix B, for an overview see e.g. [37].In the case of vanishing coupling the field strength tensor simplifies to F aµν = ∂ c µ A aν − ∂ c ν A aµ , (3.12)where we replace the differential operator by the central difference ∂ c µ A aν ( x ) = 1 a µ (cid:20) A aν ( x + a µ ˆ µ − A aν ( x − a µ ˆ µ (cid:21) (3.13)and define a lattice spacing a µ , which is excluded from Einstein’s sum convention a µ = (cid:40) a τ for µ = 0 a σ for µ = 1 , , . (3.14)In lattice perturbation theory the dynamical variables are the gauge fields A µ and wecan plug the energy-momentum tensor from equation (2.15b) together with the fieldstrength tensor (3.12) into the correlator (2.8). Then sixteen terms of the generalisedform C i i j j l l m m ( x, y ) = (cid:10) ∂ c i A ai ( x ) ∂ c j A aj ( x ) ∂ c l A bl ( y ) ∂ c m A bm ( y ) (cid:11) (3.15)have to be transformed to momentum space. After transforming the individual gaugefields A µ ( x ) to momentum space by (B.1b), we apply Wick’s theorem using the freegauge field propagator (B.3). Because of translational invariance it is sufficient toconsider y = 0 or C i i j j l l m m ( x, , and we obtain C i i j j l l m m ( ω, (cid:126)q ) = ( N − (cid:88)(cid:90) k (cid:101) k l ( (cid:93) k + q ) m (cid:101) k ( (cid:93) k + q ) × (cid:104) δ i l δ j m (cid:101) k i ( (cid:93) k + q ) j + δ i m δ j l (cid:101) k j ( (cid:93) k + q ) i (cid:105) (3.16)with the lattice momenta (cid:101) q , (cid:101) k as defined in appendix B. Evaluating the correlator(2.8) and aligning the outer momentum to q = (0 , , , q ) we find for its Fouriertransform G E ( q ) = (cid:0) N − (cid:1) (cid:88)(cid:90) k (cid:101) k ( (cid:93) q + k ) (cid:40) (cid:101) k x (cid:101) k y − (cid:101) k ( (cid:93) q + k )( (cid:101) k x + (cid:101) k y ) + (cid:101) k (cid:101) k x + ( (cid:93) q + k ) (cid:101) k y + (cid:104)(cid:101) k ( (cid:93) q + k ) (cid:105) (cid:41) . (3.17)We perform the finite Matsubara sums by the residue theorem using the formula [38] N τ N τ (cid:88) n =1 g ( z ) = − (cid:88) i Res ¯ z i (cid:0) z g ( z ) (cid:1) ¯ z N τ i − (3.18)– 8 –nd list the results for the individual terms in appendix C. As will be described insection 3.4 we subtract the temperature independent vacuum part to avoid ultravioletdivergences. The three-momentum integration can be performed after expanding theintegrals around the continuum limit. This step extends the integration measure toinfinite volume [ − π / a , π / a ] → R and produces correction terms in small lattice spac-ings a σ . Together with the expansion in small momenta q the remaining integralscan be solved analytically and one finds for the different terms (cid:88)(cid:90) k (cid:101) k x (cid:101) k y (cid:101) k ( (cid:93) q + k ) = π a τ N τ ) + π a σ ( a τ N τ ) (cid:18) ξ (cid:19) − q a τ N τ ) + π a σ q ( a τ N τ ) (cid:18) − − ξ (cid:19) , (3.19a) − (cid:88)(cid:90) k (cid:101) k ( (cid:93) q + k )( (cid:101) k x + (cid:101) k y ) (cid:101) k ( (cid:93) q + k ) = − π a τ N τ ) − π a σ ( a τ N τ ) (cid:18) ξ (cid:19) + q a τ N τ ) − π a σ q ( a τ N τ ) (cid:18) − − ξ (cid:19) , (3.19b) (cid:88)(cid:90) k (cid:101) k x ( (cid:93) q + k ) = (cid:88)(cid:90) k (cid:101) k y (cid:101) k = π a τ N τ ) + π a σ ( a τ N τ ) (cid:18) ξ (cid:19) , (3.19c) (cid:88)(cid:90) k (cid:104)(cid:101) k ( (cid:93) q + k ) (cid:105) (cid:101) k ( (cid:93) q + k ) = − q a τ N τ ) + π a σ q ( a τ N τ ) (cid:18) − ξ (cid:19) . (3.19d)For fixed temperature T = ( a τ N τ ) − we can rewrite the dependence on lattice spac-ings a τ and a σ as a dependence on the temporal lattice extent N τ and the anisotropy ξ = a σ /a τ . Combining the results of (3.19) we obtain the following expression forthe dimensionless energy-momentum tensor correlator in momentum space G E ( q ) T = ( N − (cid:40) π N τ (cid:18) ξ 945 + 4189 (cid:19) + q T (cid:20) 136 + π N τ (cid:18) − ξ 240 + 492160 (cid:19)(cid:21) (cid:41) + O (cid:0) q , N − τ (cid:1) , (3.20)from which we identify the dimensionless transport coefficient κ/T as κT = ( N − (cid:20) 118 + π N τ (cid:18) − ξ 120 + 491080 (cid:19)(cid:21) + O (cid:0) q , N − τ (cid:1) . (3.21)At fixed temperature the continuum limit a µ → is performed by taking N τ → ∞ ,where we reproduce the result of equation (2.16).Although the computation has been performed in the ideal gas limit and thuslacks corrections in the coupling g , it may serve as a check of our numerics at high– 9 –emperatures and helps to estimate the size of cut-off effects. The computed cor-rection in the inverse temporal lattice extent suggests an anisotropy of ξ ≈ . inorder to eliminate leading order lattice artefacts. In the case of other values for theanisotropy we can determine the required temporal lattice extent to decrease theleading discretisation error below a desired treshold. As stated in section 4 we use ξ = 2 in order to use previous results for the scale setting. Thus a temporal latticeextent of N τ ≥ is required in order to reduce the leading lattice artefacts below in the ideal gas limit. Note that an anisotropy larger than ξ > . causesa quadratic increase of the lattice artefacts, though it would milden the constraint(3.11). The correlator defined in (2.8) suffers from ultraviolet divergences. Although theybecome finite on the lattice, we have to correct the correlator by additive renormal-isation. Therefore we subtract the vacuum part, which is defined as the correlatorcomputed at vanishing temperature, from the measured correlator. We define a newvacuum corrected expectation value by (cid:104)O(cid:105) = (cid:104)O(cid:105) T − (cid:104)O(cid:105) T vac , (3.22)where (cid:104)O(cid:105) T is an observable evaluated at a given temperature T and (cid:104)O(cid:105) T vac itsvacuum contribution, i.e. evaluated at vanishing temperature T vac = 0 .The energy-momentum tensor is the Noether current corresponding to transla-tional invariance. In the continuum it is protected from renormalisation by Ward-identities [39]. However, on the lattice translations only form a discrete symmetrygroup and thus multiplicative renormalisation becomes necessary. (The lattice per-turbation theory computation in section 3.3 does not require multiplicative renor-malisation because it is the non-interacting case).For an isotropic lattice the finite renormalisation factor only depends on thelattice coupling β whereas on an anisotropic lattice it also depends on the anisotropy ξ . Additionally, temporal and spatial direction (3.6) require separate renormalisationfactors Z σ ( β, ξ ) and Z τ ( β, ξ ) . Then the renormalised energy-momentum tensor inthe diagonal channel reads θ ii = Z τ ( β, ξ ) (cid:20) θ τii + Z σ ( β, ξ ) Z τ ( β, ξ ) θ σii (cid:21) . (3.23)Applying the cubic symmetry (3.4) we rewrite the correlator (3.9) using the abovenotation and find a σ a τ G E ( q ) = 12 V (cid:88) x,y e − i q ( x − y ) (cid:2) Z τ G τ ( x, y ) + Z τ Z σ G τσ ( x, y )+ Z σ G σ ( x, y ) (cid:3) (3.24)– 10 –ith the newly defined bare correlators G τ ,T ( x, y ) ≡ (cid:104) θ τ ( x ) θ τ ( y ) − θ τ ( x ) θ τ ( y ) (cid:105) T (3.25a) G τσ ,T ( x, y ) ≡ (cid:104) θ τ ( x ) θ σ ( y ) + θ σ ( x ) θ τ ( y ) − θ τ ( x ) θ σ ( y ) − θ σ ( x ) θ τ ( y ) (cid:105) T (3.25b) G σ ,T ( x, y ) ≡ (cid:104) θ σ ( x ) θ σ ( y ) − θ σ ( x ) θ σ ( y ) (cid:105) T , (3.25c)and their vacuum subtracted versions G i ( x, y ) = G i ,T ( x, y ) − G i ,T vac ( x, y ) , i ∈ { τ, τ σ, σ } . (3.26)Performing the renormalisation procedure we need the ratio Z σ ( β, ξ ) /Z τ ( β, ξ ) and the absolute scale Z τ ( β, ξ ) . The former can be obtained from renormalisationgroup invariant quantities [40]. To this end one introduces three differently sizedlattices (cid:104)O(cid:105) (cid:98) = 2 L × L × L × L, (cid:104)O(cid:105) (cid:98) = L × L × L × L, (cid:104)O(cid:105) (cid:98) = L × L × L × L, (cid:104)O(cid:105) (cid:98) = L × L × L × L, (3.27)and the renormalisation group invariant quantities F = L (cid:104) T (cid:105) , F = L (cid:104) T (cid:105) , F = L (cid:104) T (cid:105) , F = L (cid:104) T (cid:105) . (3.28)Since the renormalisation factors do not depend on the temperature, all directionsare symmetric and it follows F = F , F = F , F = F . (3.29)Applying equation (3.23) one can solve for the ratio of renormalisation factors. Forinstance the equation F = F translates to Z σ ( β, ξ ) Z τ ( β, ξ ) = (cid:104) θ τ (cid:105) − (cid:104) θ τ (cid:105) (cid:104) θ σ (cid:105) − (cid:104) θ τ (cid:105) , (3.30)where the expectation values are computed by lattice simulations of (3.28). Wecompute the ratio Z σ ( β, ξ ) /Z τ ( β, ξ ) from all three equations in (3.29) and averagethe results. The simulations have to be performed for every lattice coupling β andanisotropy ξ .We obtain the absolute renormalisation factor by utilising the physical interpreta-tion of the energy-momentum tensor, whose diagonal spatial elements are equivalentto the pressure (cid:104) θ ii (cid:105) = p. (3.31)The absolute renormalisation factor enters the energy-momentum tensor correlatorquadratically. Therefore the renormalisation procedure is very sensitive to the exactvalue of the pressure and encourages us to use a highly precise value for it. For thisreason we use the continuum extrapolated lattice data from [41]. Figure 3 illustratesthe difference between the continuum value of the pressure and the not multiplica-tively renormalised energy-momentum tensor. The difference between them at agiven temperature corresponds to the absolute renormalisation factor.– 11 –0.511.52 0 2 4 6 8 10 p / T , (cid:10) θ b a r e ii (cid:11) / T T /T c (cid:10) θ bare ii (cid:11) p continuum Figure 3 : Comparison of the not multiplicatively renormalised energy-momentumtensor (cid:10) θ bare ii (cid:11) /T for N τ = 6 and ξ = 2 to the continuum extrapolated pressure p/T from the lattice [41], where the line is obtained by a cubic spline interpolation.The difference between them at a given temperature corresponds to the absoluterenormalisation factor. We create the gauge field configurations using the standard heatbath algorithm [42–44] adapted to an anisotropic lattice. Our implementation is based on the library QDP++ [45].In order to compute the vacuum part necessary for additive renormalisation, werun extra simulations with increased temporal lattice extent N τ . For our fine andspatially large lattices this is very costly. We therefore choose T vac ≈ . T c , withthe critical temperature T c ≈ MeV for Yang-Mills theory [46]. For our purposesthis temperature is low enough since firstly the vacuum divergence is temperatureindependent, and secondly it is well known that the pressure or the deviation ofscreening masses from their vacuum values are exponentially small in the confinedphase (see [41, 46–48] for numerical evidence and [49] for an analytic explanation).The set of momenta has to fulfil the constraint (3.11), which basically dictates thesimulation parameters. An anisotropy ξ > benefits this constraint. As discussedin section 3.3 a value for the anisotropy of ξ ≈ . minimizes the first order latticecorrections. However, we choose an anisotropy of ξ = 2 , which allows to set the scaleby taking the lattice spacing as a function of the lattice coupling a = a ( β ) from [32].– 12 –un i ii iii iv β . . . 68 6 . N τ N σ 120 120 120 120 N vac τ 72 72 42 24 ξ a σ [ fm ] 0 . 026 0 . 026 0 . 044 0 . T / T c . . . . T vac / T c . . . . T T vac Table 1 : Simulation parameters for four evaluations of κ . The lower temperature T vac is required for renormalisation.Adjusting the temporal lattice extent to N τ ≥ reduces the computed lattice errorsin (3.21) below . A numerical analysis of the relevant correlators in lattice QCD[36] even suggests values for the temporal lattice extent of N τ ≥ .Extracting the transport coefficient κ from (2.14) by performing a linear fit in q requires at least three different momenta q , where the highest momentum stillhas to fulfil the constraint (3.11). More momenta would be favourable improvingthe fit’s quality. Thus we choose for the temporal lattice extent N τ = 6 and forthe spatial lattice extent N σ = 120 at a given anisotropy ξ = 2 . All simulationparameters are listed in table 1. In the deconfined phase topological fluctuationsare suppressed [50] and we expect no difficulties in using very fine lattices. Due tothe large computational effort creating gauge fields on × N τ lattices, we do notexclude any configurations but account for existing correlations by jackknife errorsampling, see e.g. [51].The multiplicative renormalisation procedure requires knowledge of the renor-malisation factor ratio Z σ ( β, ξ ) /Z τ ( β, ξ ) . As described in section 3.4 we determineit from computing the quantities (3.28) on lattices (3.27) with L = 48 . The simula-tions must be performed for every lattice coupling β of table 1. Intermediate resultsfor the computation of the renormalisation factors are shown in table 6 and table 7in appendix D with reference to run (i) of table 1. Our first simulation aims at making contact to lattice peturbation theory, section3.3. The weak coupling regime is reached by increasing the temperature. Adoptingthe parameters from the previous section 4.1 we choose for the lattice coupling β =7 . , corresponding to a temperature of T = 9 . T c , and a spatial lattice spacing of a σ = 0 . fm (see column (i) in table 1).– 13 – .60.650.70.750.80.850.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G E / T ( q/T ) T = 9 . T c LPT Figure 4 : Correlator G E ( q ) /T for momenta q /T < compared to results from lattice perturbation theory(LPT). The slope of the linear fit gives κ/ . q /T G ( q ) /T . . . . . Table 2 : Intermediatenumerical results for run(i) of table 1.Figure 4 shows the correlator G E ( q ) /T for five momenta compared to the resultfrom lattice perturbation theory and table 2 the corresponding data points. The largeerrors of the correlator are almost entirely due to the additive renormalisation proce-dure. Table 4 lists the data of the bare correlators (3.25) regarding this simulation,whereas table 5 lists the data of the additively renormalised correlators (3.26). Thevacuum subtraction causes a significant loss of accuracy. Computing the pressureby means of the interaction measure [46] suffers from the same phenomenon. Thus,we create a large amount of statistics (see table 1) to provide a significant signal forthe correlators. In terms of error reduction it is highly favourable to perform theadditive renormalisation before the multiplicative one. Otherwise, the propagatederrors entering from the multiplicative renormalisation add to the described loss ofprecision.Fitting the datapoints of the correlator to a line G E (cid:16) q T (cid:17) T = G (cid:48) (0) T + κT q T (4.1)yields for the y-intercept G (cid:48) (0) /T = 0 . and for the transport coefficient κ/T =0 . , which is consistent with the leading order lattice perturbation theory result κ LPT /T = 0 . . Note that full agreement is not yet expected since at T = 9 . T c there are still significant corrections due to interactions, i.e. we are still far from theideal gas limit. In principle, the temperature can be varied at fixed β and lattice spacing by changing N τ , where lower temperature implies larger N τ . However, due to the constraint on– 14 –he momenta from equation (3.11) this would require a similar increase of the spatialvolume and thus a drastical growth of the numerical effort. Hence the fixed scaleapproach is not practical for temperatures approaching the phase transition.We therefore investigate the temperature dependence of κ at fixed N τ /N σ byrepeating the simulations at various lattice couplings β . In this case the differenttemperatures are evaluated at different lattice spacings, and consequently also dif-ferent spatial volumes in physical units. However, since our lattice spacings are all a σ < . fm, we expect the lattice artefacts on the temperature dependence of thetransport coefficient κ/T to be negligible. As a consistency check for this, we alsoperform simulations at different temperatures but the same lattice spacings (simula-tions (i) and (ii) in table 1).The results are shown in figure 5. The datapoint at T = 7 . T c suffers from largeerrorbars since the spatial lattice extents have been kept fixed while increasing thetemporal lattice extent N τ . This corresponds to less momenta fulfilling the constraint(3.11) and generates a loss of accuracy in the fit (4.1). Within the errorbars, thevalues of κ/T at T = 9 . T c and T = 7 . T c agree (c.f. table 3) and thus justify thecomparison at different lattice spacings and temperatures.The numerical values for the transport coefficient κ are also summarised in table3. Within errorbars, the temperature dependence of the transport coefficient isconsistent with that of the ideal gas, κ ∼ T , which is also the prediction of AdS/CFT[24] for the opposite strong coupling limit. Assuming this functional dependence, wemay increase the accuracy by averaging the data points with N τ = 6 to give our finalresult, κ avr = 0 . T . (4.2)The prediction from AdS/CFT correspondence for this coefficient is [27] κT = ηs × sπT , (4.3)where η is the shear viscosity and s the entropy density. The latter is proportional tothe number of degrees of freedom of the theory, which is higher in the SUSY Yang-Mills used for the correspondence . In order to compare with the QCD calculations,we thus use the AdS/CFT results η/s = 1 / π and eq. (4.3), but take the QCDentropy density from a lattice calculation [41]. The result is about a third of theperturbative prediction and also consistent with the simulation results. We have calculated the second order hydrodynamic transport coefficient κ for theYang-Mills plasma using lattice perturbation theory and Monte Carlo simulations. We missed this point in the first version of the manuscript and thank the referee and editor fortheir suggestions. – 15 –0.500.51 1 2 3 4 5 6 7 8 9 10 11 κ / T T /T c β = 7 . β = 6 . β = 6 . ADS/CFTLPT N τ = 6 N τ = 8 N τ = 6 N τ = 6 Figure 5 : Temperature dependence of the transport coefficient κ/T . The linesmark the result from ADS/CFT correspondence [24] and lattice perturbation theory(3.21), respectively. T / T c . . . . a σ [ fm ] 0 . 026 0 . 026 0 . 044 0 . κ / T . . . . Table 3 : Lattice results for the transport coefficient κ/T at different spatial latticespacings a σ and temperatures T /T c .This is possible because the retarded correlator of the energy momentum tensor atzero frequency has a trivial analytic continuation to a corresponding Euclidean cor-relator. The transport coefficient parametrises the low momentum behaviour of thiscorrelator, whose realisation requires large spatial lattice directions, making a numer-ical calculation very challenging and thus leaving large statistical errors. Their mainsource are the vacuum subtractions leading to similar problems in calculations of theequation of state at low temperatures. One might hope that alternative methodsavoiding this step [52] may improve this situation.In the investigated temperature range T c < T < T c our data are consistentwith κ ∝ T , as predicted both by weak and strong coupling methods. Becauseof still large errorbars, our result also quantitatively covers both the leading orderperturbative as well as the AdS/CFT prediction rescaled by the QCD entropy. Thiswould suggest that, besides improved simulation methods, next-to-leading order an-alytic calculations should be able to give a result with improved accuracy.– 16 – cknowledgments We thank H. Meyer and D. Rischke for their comments on the manuscript. Thisproject is supported by the Helmholtz International Center for FAIR within theLOEWE program of the State of Hesse. C.S. acknowledges travel support by theHelmholtz Research School H-QM. The calculations have been performed on LOEWE-CSC at Goethe-University Frankfurt, we thank the team of administrators for sup-port. A Cubic symmetry of the energy momentum tensor The correlator (cid:104) T ( x ) T ( y ) (cid:105) can be expressed in terms of diagonal energy-momentumtensor elements by exploiting rotation invariance (cid:104) T (cid:48) ( x ) T (cid:48) ( y ) (cid:105) = (cid:104) T ( x ) T ( y ) (cid:105) , (A.1)on a spatially isotropic lattice (and medium) under rotations by α = π/ about the z -direction. The transformation of a second rank tensor reads T (cid:48) µν ( x ) = (cid:0) M − z (cid:1) µα (cid:0) M − z (cid:1) νβ T αβ ( x ) , (A.2)and the corresponding transformation matrix is given by M − z = α sin α − sin α cos α 00 0 0 1 = 1 √ √ − √ . (A.3)For the energy-mometum tensor components of interest, this means T (cid:48) ( x ) = 12 [ T − T ] , (A.4)where we used T = T . With the definition of the energy-momentum tensor (2.15a)we find for the correlator T (cid:48) ( x ) T (cid:48) ( y ) = 14 [ T ( x ) T ( y ) − T ( x ) T ( y ) − T ( x ) T ( y ) + T ( x ) T ( y )]= 14 [ θ ( x ) θ ( y ) + θ ( x ) θ ( y ) − θ ( x ) θ ( y ) − θ ( x ) θ ( y )] . (A.5)Note that the trace anomaly θ cancels completely. From rotational invariance follows (cid:104) θ ( x ) θ ( y ) (cid:105) = (cid:104) θ ( x ) θ ( y ) (cid:105) , (cid:104) θ ( x ) θ ( y ) (cid:105) = (cid:104) θ ( x ) θ ( y ) (cid:105) , (A.6)and the correlator expressed in diagonal elements reads (cid:104) T ( x ) T ( y ) (cid:105) = 12 [ (cid:104) θ ( x ) θ ( y ) (cid:105) − (cid:104) θ ( x ) θ ( y ) (cid:105) ] . (A.7)– 17 – Definitions in lattice perturbation theory The Fourier transforms of the gauge field A µ to momentum space and back aredefined by A µ ( q ) = a σ a τ N τ (cid:88) n =1 (cid:88) (cid:126)x e − i (cid:16) x + aµ (cid:98) µ (cid:17) q A µ ( x ) , (B.1a) A µ ( x ) = (cid:88)(cid:90) q e i (cid:16) x + aµ (cid:98) µ (cid:17) q A µ ( q ) , (B.1b)where we introduce (cid:88)(cid:90) q ≡ a τ N τ N τ (cid:88) n =1 πaσ (cid:90) − πaσ d q (2 π ) . (B.2)The shift to the center of the link variables x + a µ (cid:98) µ/ in the Fourier transformsimplifies the computation. The free gauge field propagator is given by ∆ ABµν ( q ) = 1 (cid:101) q (cid:18) δ µν − (1 − ξ ) (cid:101) q µ (cid:101) q ν (cid:101) q (cid:19) δ AB , (B.3)where we use Feynman-’t Hooft gauge with ξ = 1 . The momenta in lattice pertur-bation theory are given by (cid:101) q µ = 2 a µ sin (cid:16) a µ q µ (cid:17) , (B.4) (cid:16) (cid:94) k µ + q µ (cid:17) = 2 a µ sin (cid:18) a µ ( k µ + q µ )2 (cid:19) , (B.5) (cid:88) µ (cid:101) q µ = 4 a µ (cid:88) µ sin (cid:16) a µ q µ (cid:17) (B.6)with a ≡ a τ and a i ≡ a σ . We do not imply a sum over the index µ . C Results for finite Matsubara sums The evaluation of the finite Matsubara sums gives with the definitions E ( k i ) ≡ ξ − | k i | − a σ | k i | (cid:34) ξ − (cid:88) i k i + ξ − | k i | (cid:35) + O ( a σ ) (C.1a) E ≡ E ( k i ) (C.1b) E ≡ E ( k i + q i ) (C.1c)– 18 –nd A ≡ a σ E ) − cosh( a σ E ) (C.2a) B ≡ a σ E ) 1e a σ N τ E − − a σ E ) 1e a σ N τ E − (C.2b) C ≡ a σ E ) (C.2c) C ≡ a σ E ) (C.2d)the following results N τ N τ (cid:88) n =1 (cid:93) k + q ) = a τ (cid:20) a σ N τ E − (cid:21) C (C.3a) N τ N τ (cid:88) n =1 (cid:101) k ( (cid:93) k + q ) = a τ (cid:104) C e a σ E − C e a σ E (cid:105) A + a τ AB (C.3b) N τ N τ (cid:88) n =1 (cid:101) k ( (cid:93) k + q ) (cid:101) k ( (cid:93) k + q ) = a τ e a σ N τ E − C + a τ (cid:104)(cid:101) k i ( (cid:94) k i + q i ) − (cid:101) k i (cid:105) AB (C.3c)(C.3d) N τ N τ (cid:88) n =1 (cid:104)(cid:101) k ( (cid:93) k + q ) (cid:105) (cid:101) k ( (cid:93) k + q ) = − a τ e a σ N τ E − (cid:104) ( (cid:94) k i + q i ) − (cid:101) k i (cid:105) C + a τ (cid:104)(cid:101) k i ( (cid:94) k i + q i ) − (cid:101) k i (cid:105) AB (C.3e) N τ N τ (cid:88) n =1 (cid:101) k = a τ (cid:20) a σ N τ E − (cid:21) C . (C.3f) D Numerical intermediate results In this section we present numerical intermediate results for run (i) of table 1.– 19 – G τ ,T ( q ) G τ ,T vac ( q ) G σ ,T ( q ) G σ ,T vac ( q ) G τσ ,T ( q ) G τσ ,T vac ( q )0 − . . (2) 0 . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4 : Simulation results for the bare correlators G τ ,T , G σ ,T and G τσ ,T and theirvacuum parts G τ ,T vac , G σ ,T vac , G τσ ,T vac in momentum space for six momentum modes n fulfilling the constraint (3.11). n G τ ( q ) G σ ( q ) G τσ ( q )0 − . (3) 0 . − . . − . . − . − . . − . − . . − . − . . − . − . . − . Table 5 : According to (3.26) vacuum subtracted correlators of table 4. i < θ i > < θ i > < θ i > < θ i > τ − . . . . σ . − . − . − . Table 6 : Diagonal energy-momentum tensor elements evaluated on lattices (3.27)in order to compute the renormalisation ratio Z σ ( β, ξ ) /Z τ ( β, ξ ) . T T vac (cid:104) θ τ (cid:105) − . − . (cid:104) θ τ (cid:105) − . − . 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