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TUM-EFT 131/19

Lattice QCD constraints on the heavy quark diffusion coefficient

Nora Brambilla,1, 2, ∗ Viljami Leino,1, † Peter Petreczky,3, ‡ and Antonio Vairo1, §

(TUMQCD Collaboration)1Physik Department, Technische Universität München,

James-Franck-Strasse 1, 85748 Garching, Germany2Institute for Advanced Study, Technische Universität München,

Lichtenbergstrasse 2a, 85748 Garching, Germany3Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA

(Dated: November 11, 2020)

We report progress toward computing the heavy quark momentum diffusion coefficient from thecorrelator of two chromoelectric fields attached to a Polyakov loop in pure SU(3) gauge theory.Using a multilevel algorithm and tree-level improvement, we study the behavior of the diffusioncoefficient as a function of temperature in the wide range 1.1 < T/Tc < 10

4 in order to compare itto perturbative expansions at high temperature. We find that within errors the lattice results areremarkably compatible with the next-to-leading order perturbative result.

I. INTRODUCTION

The matter produced in heavy ion collisions can bedescribed as a nearly ideal fluid; see Ref. [1] for a re-cent review. Because of the high energy density, the cre-ated matter is deconfined and can be characterized as astrongly coupled quark-gluon plasma (sQGP) [2, 3]. Onerecently realized interesting feature of the quark gluonplasma is the fact that heavy quarks participate in thecollective behavior, see Ref. [4] for a recent review. Thisis interesting for the following reason: The relaxation

time of heavy quarks is expected to be ∼ (M/T )tlightrel ,with M being the heavy quark mass, T being the tem-

perature, and tlightrel being the relaxation time of the bulk(light) degrees of freedom in sQGP. The lifetime of thehot medium created in heavy ion collisions is about 5−10fm. Since the collectivity in the heavy quark sector im-plies that the relaxation time of the heavy quark is muchshorter than the lifetime of the medium despite the en-hancement factor of M/T , this in turn means that therelaxation time of the bulk degrees of freedom is veryshort, thus further corroborating the strongly couplednature of the matter produced in heavy ion collisions.

Because the relaxation time of heavy quarks is muchlarger than the relaxation time of light degrees of free-dom, the dynamics of heavy quarks can be understoodin terms of Langevin equations [5]. The drag coefficientη and the heavy quark momentum diffusion coefficient κthat enter into the Langevin equations describe the inter-action of the heavy quarks with the medium and are con-nected by the Einstein relation η = κ/(2MT ) in thermalequilibrium. The heavy quark diffusion coefficient hasbeen calculated in perturbation theory at leading order(LO) [5, 6], as well as at next-to-leading order (NLO) [7].

∗ nora.brambilla@ph.tum.de† viljami.leino@tum.de‡ petreczk@bnl.gov§ antonio.vairo@ph.tum.de

The NLO correction is very large, thus calling into ques-tion the validity of the perturbative expansion. Ana-lytic calculations for strong coupling are available onlyfor supersymmetric Yang-Mills theories [8, 9]. Therefore,lattice QCD calculations for the heavy quark diffusioncoefficient are needed.

It is well known, however, that lattice calculationsof the transport coefficients are very difficult. To ob-tain the transport coefficients one has to reconstruct thespectral functions from the appropriate Euclidean timecorrelation functions. At low energies, ω, the spectralfunction has a peak, called the transport peak, and thewidth of the transport peak defines the transport coef-ficient. Thus, one needs a reliable determination of thewidth of the transport peak in order to obtain the trans-port coefficient from lattice QCD calculations, which isdifficult [10, 11]. In the case of heavy quarks, this iseven more challenging because the width of the transportpeak is inversely proportional to the heavy quark mass.Moreover, Euclidean time correlators are rather insen-sitive to small widths [11–16]. Recently the problem ofheavy quark diffusion has also been studied out of equilib-rium with real-time lattice simulations in Refs. [17, 18].Moreover, the heavy quark momentum diffusion coeffi-cient is a crucial parameter entering the evolution equa-tions describing the out-of-equilibrium dynamics of heavyquarkonium in sQGP [19–21].

The above difficulty in the determination of the heavyquark diffusion coefficient can be circumvented by usingan effective field theory approach. Namely, by integrat-ing out the heavy quark fields, one can relate the heavyquark diffusion coefficient to the correlator of the chro-moelectric field strength [22]. The corresponding spectralfunction does not have a transport peak, and the smallω behavior is smoothly connected to the UV behavior ofthe spectral function [22]. The heavy quark diffusion co-efficient is given by the intercept of the spectral functionat ω = 0 and no determination of the width of the trans-port peak is needed. Lattice calculations of κ along theselines have been carried out in the SU(3) gauge theory in

http://arxiv.org/abs/2007.10078v2mailto:nora.brambilla@ph.tum.demailto:viljami.leino@tum.demailto:petreczk@bnl.govmailto:antonio.vairo@ph.tum.de

2

the deconfined phase i.e., for purely gluonic plasma [23–27]. The correlator of the chromoelectric field strengthis very noisy, making the lattice calculations extremelychallenging. To deal with this problem, it is mandatoryto use noise-reducing techniques such as the multilevelalgorithm by Lüscher and Weisz [28]. This algorithm isbased on the locality of the action and therefore is onlyavailable for the pure gauge theory. This is the reasonwhy the calculations of the heavy quark diffusion coeffi-cient are performed in the SU(3) gauge theory. Anotherchallenge in the determination of the heavy quark diffu-sion coefficient is the reconstruction of the spectral func-tion from the Euclidean time correlation function. Theabove lattice studies used a simple parameterization ofthe spectral function to extract κ. One has to explore thesensitivity of the results on the parameterization of thespectral function. More generally, one has to understandto what extent the Euclidean time correlation function ofthe chromoelectric field strength is sensitive to the smallω behavior of the corresponding spectral function.

At sufficiently high temperatures the perturbative cal-culations of the heavy quark diffusion coefficient shouldbe adequate. This suggests that κ/T 3 should decreasefrom large values at temperatures close to the transitiontemperature to smaller values when the temperature isincreasing. It would be interesting to see if contacts be-tween the lattice and the perturbative calculations canbe made for the heavy quark diffusion coefficient, as hasalready been done for the equation of state [29], quarknumber susceptibilities [30, 31], and static correlationfunctions [32–34]. If such contacts can be established,these would validate the methodology used in the latticeextraction of κ. Previous lattice studies focused on a nar-row temperature region [25] or only considered a singlevalue of the temperature [26]. In Ref. [25], no significanttemperature dependence of κ/T 3 was found. Large tem-peratures are needed in the lattice studies to establishthe temperature dependence of κ/T 3. The temperaturedependence of κ/T 3 is also important for phenomenol-ogy, as with a constant value of κ/T 3 it is impossibleto explain simultaneously the elliptic flow parameter, v2,for heavy quarks and the nuclear modification factor [4].Furthermore, the spectral function of the chromoelectricfield strength correlator is known at NLO [35]. Usingthis NLO result at high ω one can constrain the func-tional form of the spectral function used in the analysisof the lattice correlator.

The aim of this paper is to study the correlator of thechromoelectric field strength in a wide temperature rangein order to make contact with weak coupling calculationsof the Euclidean correlation function up to NLO in thespectral function, and also to constrain the temperaturedependence of κ.

The rest of the paper is organized as follows: In thenext section, we go through the procedure of calculat-ing the Euclidean correlator of the chromoelectric fieldstrength on the lattice. The spectral function of the chro-moelectric correlator and its relation to κ is discussed in

Sec III. There we also review the perturbative results forthis spectral function. The short-time behavior of thechromoelectric correlator and its proper normalization isclarified in Sec IV. In Sec V, we discuss how to model thespectral functions of the chromoelectric correlator and toextract the value of κ from the lattice results. Finally,Sec VI contains our conclusions.

II. LATTICE RESULTS FOR THE

CHROMOELECTRIC CORRELATOR

For a heavy quark of mass M ≫ πT , the heavy quarkeffective theory (HQEFT) provides a method of calcu-lating the heavy quark diffusion coefficient in the heavyquark limit by relating it to a chromoelectric correlatorin Euclidean time [9, 22]:

GE(τ) = −3

∑

i=1

〈Re Tr [U(1/T, τ)Ei(τ,0)U(τ, 0)Ei(0,0)]〉3 〈Re TrU(1/T, 0)〉 ,

(1)where T is the temperature, U(τ1, τ2) is the temporalWilson line between τ1 and τ2, and the chromoelectricfield, in which the coupling has been absorbed Ei ≡ gEi,is discretized on the lattice as [22]:

Ei(x, τ) = Ui(x, τ)U4(x+ î, τ)−U4(x, τ)Ui(x+ 4̂) . (2)

This discretization is expected to be the least sensitiveto ultraviolet effects [22].

To calculate the discretized chromoelectric correlatordefined above on the lattice, we use the standard Wilsongauge action and the multilevel algorithm [28]. We con-sider N3s ×Nt lattices and vary the temperature in a widerange T = 1.1Tc − 104 Tc by varying the lattice gaugecoupling β = 6/g20. Here Tc is the deconfinement phasetransition temperature. We use Nt = 12, 16, 20, and 24at each temperature to check for lattice spacing effectsand perform the continuum extrapolation. In this studywe use Ns = 48, except for Nt = 12 lattices, where mul-tiple spatial volumes are used to check for finite-volumeeffects.

To set the temperature scale as well as the lattice spac-ing, we use the gradient flow parameter t0 [36] and thevalue Tc

√t0 = 0.2489(14) [37]. We use the result of

Ref. [37] to relate the temperature scale or the latticespacing to β. The parameters of the lattice calculations,including the statistics, are given in Table I. In the sim-ulations with the multilevel algorithm, we divide the lat-tice into four sublattices and update each sublattice 2000times to evaluate the chromoelectric correlator on a sin-gle gauge configuration. We use the simulation programdeveloped in a prior study [25].

In order to obtain the heavy quark diffusion coeffi-cient, the lattice chromoelectric correlator needs to be

3

renormalized and then extrapolated to the continuum. 1

The renormalization coefficient ZE(β) ≡ ZE(g20) of thechromoelectric correlator in the case of the Wilson gaugeaction has been calculated at one loop [38]:

Z1−loopE = 1 + 0.1377185690942757(4)g20 . (3)

We will use this one loop correction in the present study.However, we expect that the one loop result for ZE is notprecise enough. As will be clear from the results of thelattice calculations, this is indeed the case. The pertur-bative error in ZE(β) affects both its absolute value forfixed β and its β-dependence. For the continuum extrap-olation, it is important to estimate the uncertainty in theβ-dependence of the renormalization constant. The er-ror in the absolute value of ZE could be corrected afterthe continuum extrapolation is done by introducing anadditional multiplicative factor. We will postpone thediscussion of this multiplicative factor to Sec IV. To es-timate the error in the β dependence of ZE we considerthe tadpole improved result for ZE, namely Z

tadE = 1/u0,

with u0 being the plaquette expectation value [25]. The

difference in the β dependence of ZtadE and Z1−loopE can

be used as an estimate of the error of the β-dependenceof ZE. Therefore, at each temperature we consider the

variation in Z1−loopE · u0 in the β range that correspondsto Nt = 12 − 24 as an estimate of the systematic errorsin ZE for bare gauge couplings in that range.

The chromoelectric correlator decays rapidly with in-creasing τ . This feature can be understood from theleading order (tree-level) result [22]:

GLOE (τ)

g2CF≡ GnormE (τ) = π2T 4

[

cos2(πτT )

sin4(πτT )+

1

3 sin2(πτT )

]

,

(4)where CF = 4/3 is the Casimir of the fundamental rep-resentation of SU(3). In Fig. 1 we show ZEGE/G

normE for

different temperatures calculated on the largest, 483×24lattice. We see a significant temperature dependence inthis ratio. Also shown in the figure are the numericalresults for the lowest temperature, T = 1.1Tc calculatedfor different Nt. As one can see from the figure, the cut-off (Nt) dependence is significant even for relatively largevalues of τT . We expect that the cutoff dependence in-creases with decreasing τT , except when τ is of the orderof the lattice spacing because the cutoff dependence ofZEGE/G

normE is proportional to (a/τ)

2. We see that ourlattice data follow this expectation for τT > 0.2. Thisobservation is important for estimating the reliability ofthe continuum extrapolations. A similar Nt dependenceis observed at other temperatures.

1 We will use the notation GE for both the lattice and the contin-

uum version of the chromoelectric correlator to keep the notation

simple. It should be clear from the context which one we are re-

ferring to. We will use different notations for the continuum and

the lattice version of the chromoelectric correlator only when it

is absolutely necessary.

T/Tc Nt ×N3s β Nconf

12× 483 6.407 13501.1 16× 483 6.621 2623

20× 483 6.795 203524× 483 6.940 253512× 483 6.639 180112× 323 6.639 1557

1.5 12× 242 6.639 100016× 483 6.872 277820× 483 7.044 208124× 483 7.192 2496

2.2 12× 483 6.940 153512× 483 7.193 1579

3 16× 483 7.432 155320× 483 7.620 140124× 483 7.774 166312× 483 7.774 1587

6 16× 483 8.019 155620× 483 8.211 125824× 483 8.367 143012× 483 8.211 180712× 323 8.211 1737

10 12× 243 8.211 100016× 483 8.458 276920× 483 8.651 207324× 483 8.808 2423

12× 483 14.194 103910000 16× 483 14.443 1157

20× 483 14.635 113924× 483 14.792 1375

20000 12× 483 14.792 1948

TABLE I. Parameters of the lattice calculations.

In order to reduce discretization errors we turn toa tree-level improvement procedure [39, 40], where theleading order results in the continuum (4) and the lat-tice perturbation theory are matched. The LO latticeperturbation theory gives [24]:

GLO,latE (τ)

g2CF=

∫ π

−π

d3q

(2π)3q̃2eq̄Nt(1−τT ) + q̃2eq̄NtτT

3a4 (eq̄Nt − 1) sinh(q̄) , (5)

where

q̄ = 2arsinh(q̃/2) , (6)

q̃2 =

3∑

i=1

4 sin2(qi/2) . (7)

The improved distance τ is then defined so that

GLOE (τ ) = GLO,latE (τ). In Fig. 2, we show our results

for ZEGimpE (τ)/G

normE = ZEGE(τ )/G

normE . From the fig-

ure, we can observe that after the tree-level improvement,the ratio ZEGE/G

normE appears monotonically increas-

ing with increasing τT and has a decreasing slope as afunction of temperature. At the highest temperature,T = 104 Tc, we see a nearly horizontal τ -independent

4

FIG. 1. The chromoelectric field correlator from Eq. (1) nor-malized with Eq. (4). Top: all measured temperatures forNt = 24. Bottom: all measured temporal extents Nt for thesmallest temperature.

line. Moreover, we observe a large reduction of cutoff ef-fects for all temperatures when tree-level improvement isused. As an example, we show this reduction at the bot-tom of Fig. 2 for the lowest temperature, T = 1.1Tc. Asimilar reduction in the Nt dependence is seen at othertemperatures. Due to its impact, we will use the treelevel improvement for the rest of this paper, and there-fore, unless otherwise indicated, drop the overline from τand the superscript "imp" from GimpE .

The normalized chromoelectric correlator shown inFig. 2 has a significant τ -dependence. We conclude thatthe LO perturbative result does not capture the key fea-tures of the chromoelectric correlator. Only at the high-est temperature, T = 104 Tc, is the τ -dependence of thecorrelator well described by the leading order result. Onemay wonder whether the observed behavior of the nor-malized chromoelectric correlator is due to thermal ef-fects that are not present at leading order, like the physicsof the heavy quark transport, or are due to higher-order

FIG. 2. The chromoelectric field correlator from Eq. (1) nor-malized with Eq. (4) and tree-level improved with (5). Top:all measured temperatures for the biggest Nt = 24. Bottom:all measured temporal extents Nt for the smallest tempera-ture.

effects at zero temperature. In order to answer this ques-tion, we show our lattice results in Fig. 3 as a function of τin physical units rather than a function of τT . This figureshows that the ratio of the chromoelectric correlator tothe free theory result is largely temperature independentimplying that the chromoelectric correlator is dominatedby the vacuum part of the spectral function. It thus be-comes even more important to quantify the temperaturedependence of the chromoelectric correlator. This can bedone by considering the following ratio of the normalizedcorrelator at a fixed value of β, but at two temperaturescorresponding to temporal extents Nt and 2Nt. Latticeartifacts are canceled out in the double ratio:

R2(Nt) =GE(Nt, β)

GnormE (Nt)

/

GE(2Nt, β)

GnormE (2Nt). (8)

Furthermore, if the LO result is a good approximationof the correlator and κ/T 3 is temperature independent,R2(Nt) should be 1 and independent of the temperature,

5

FIG. 3. The data of Fig 2 in physical units.

while the temperature dependence of κ/T 3 will make thisratio different from 1 and also temperature dependent.The amount by which R2(Nt) deviates from 1 also de-pends on the value of κ/T 3: small values of κ/T 3 willresult only in small deviations of R2(Nt) from 1. Ourresults for R2(Nt) are shown in Fig. 4. At the highesttemperature the double ratio is consistent with 1 withinerrors, perhaps not surprisingly as at high temperaturesthe temperature dependence of κ/T 3 is expected to belogarithmic, and thus rather mild. At lower tempera-tures, however, we see deviations from 1 in the doubleratio at the few-percent level, which increase with de-creasing temperature and increasing τT . On the otherhand, for τT < 0.2 the double ratio is close to 1, imply-ing that there the correlator is dominated by the T = 0part of the spectral function. In any case, thermal effectsin the chromoelectric correlator, which encode the valueof κ, are small, at the level of few percent. This factimplies that extracting κ from lattice determinations ofthe chromoelectric correlator is challenging.

Before extracting the heavy quark diffusion coefficientwe need to address finite-volume effects and perform thecontinuum extrapolation of ZEGE. Most of our calcu-lations have been performed using Ns = 48. To checkfor finite-volume effects for Nt = 12, we have performedcalculations using spatial sizes N3s = 24

3, 323, 483 at twotemperatures, T = 1.5Tc and T = 10Tc. The small-est spatial volume here corresponds to the aspect ratioNs/Nt = 2. The detailed study of finite-volume effects isdiscussed in Appendix A. We find that the finite-volumeeffects are small, considerably smaller than other sourcesof error down to the aspect ratio Ns/Nt = 2. Therefore,at the current level of precision, using a Ns = 48 latticeis sufficient even for Nt = 24.

Next, we perform the continuum extrapolations ofZEGE. The systematic errors in the renormalizationconstant estimated above are combined with the statis-tical errors of the chromoelectric correlator before per-

FIG. 4. The ratio Eq. (8) of simulations with the same β atdifferent temperatures with Nt = 12.

forming the continuum extrapolation. In the interval0.1 ≤ τT ≤ 0.45 we have a sufficient number of datapoints to perform the continuum extrapolations. We firstinterpolate the data for each Nt in τT using ninth-orderpolynomials to estimate ZEGE at common τT values. Weperform linear extrapolations in 1/N2t = (aT )

2 of ZEGEat these τT values using lattices with Nt = 16, 20, and24. As an example, we show the continuum extrapola-tion for selected values of τT in Fig. 5. One can see thatthe Nt = 12 data do not lie in the 1/N

2t scaling region.

Therefore, we also perform extrapolations to Nt = 12data with a (aT )4 term included. The difference betweenthese continuum extrapolations is used as an estimate ofthe systematic error of the continuum result. The slopeof the a2 dependence is increasing with decreasing τT ,as can be seen from Fig. 5. This is expected; the cutoffeffects are larger at smaller τT . However, at the small-est value, τT = 0.1, the slope of the a2 dependence be-comes smaller again contrary to expectations. We takethis as an indication that the cutoff effects in this re-gion cannot be described by a simple a2 or a2 + a4. Asshown in Appendix A the slope of the a2 dependence in-creases monotonically only till τT ≥ 0.175. Therefore,we consider the continuum extrapolation to be reliableonly for τT ≥ 0.175. For an additional cross-check, wealso perform the continuum extrapolation of the latticedata without tree-level improvement. This is discussedin Appendix A, where further details of the continuumextrapolations can be found.

The continuum-extrapolated chromoelectric correlatornormalized by GnormE is shown in Fig. 6 for all tempera-tures as function of τT . The continuum-extrapolated re-sults share the general features of the tree-level-improvedresults at nonzero lattice spacing in terms of τ and tem-perature dependence. In particular, we see a strong de-pendence on τT , except for the highest temperature, in-dicating that the leading-order result does not capture

6

FIG. 5. The continuum extrapolation. The lines represent thefit performed with the three largest Nt’s at temperature T =1.1 Tc at different values of τT , shown with different colors.The fitted line is extrapolated to 0 and to 1/122 to show thequality of the fit compared to points at those locations. Thepoint at zero includes the systematic error coming from theinclusion of the smallest lattice.

FIG. 6. The continuum extrapolation for all temperatures asa function of τT .

the τT dependence of GE. We will try to understandthese features of the correlator in the next sections.

III. SPECTRAL FUNCTIONS AND DIFFUSION

COEFFICIENT IN PERTURBATION THEORY

In order to determine the heavy quark diffusion coeffi-cient κ from the chromoelectric correlator GE, one has touse the relation between this correlator and the spectral

function ρ(ω, T ):

GE(τ) =

∫ ∞

0

dω

πρ(ω, T )K(ω, τT ) , (9)

where

K(ω, τT ) =cosh

(

ωT

(

τT − 12))

sinh(

ω2T

) .

The heavy quark diffusion coefficient is determined interms of ρ through the Kubo formula [41]

κ ≡ limω→0

2Tρ(ω, T )

ω. (10)

At the leading order of the perturbation theory, thespectral function is given by [22]:

ρLO(ω, T ) =g2(µω)CFω

3

6π, (11)

where the coupling has been evaluated at the scale µω.We use the five-loop running coupling constant in thiswork [42]. At LO the scale µω is arbitrary. A naturalchoice is µsimpleω = max(ω, πT ) [26]. The LO spectralfunction (11) gives κ = 0.

At NLO, the perturbative calculation of ρ(ω, T ) needsHard-Thermal-Loop (HTL) resummation for ω ∼ T , there

is no need for resummation when calculating the spectralfunction at NLO; the naive (nonresummed) NLO resultfor ρ(ω, T ) in the pure gauge case reads

ρnaive(ω, T ) = (13)

g2CFω3

6π

{

1 +g2

(4π)2

[

Nc

(

11

3ln

µ2ω4ω2

+149

9− 8π

2

3

)]}

+g2CF6π

g2

2π2

{

Nc

∫ ∞

0

dq nB(q)

[

(q2 + 2ω2) ln

∣

∣

∣

∣

q + ω

q − ω

∣

∣

∣

∣

+qω

(

ln|q2 − ω2|

ω2− 1

)

+ω4

qP

(

1

q + ωln

q + ω

ω+

1

q − ω lnω

|q − ω|

)]}

,

where nB(q) = (exp(q/T )−1)−1 is the Bose–Einstein dis-tribution, P takes the principal value, and g2 ≡ g2(µω).The first line of Eq. (13) gives the NLO T = 0 contribu-tion, and the subsequent lines carry the thermal effects.

7

For the NLO ρ(ω, T ), µω may be set such that the NLOT = 0 contribution vanishes [35]:

ln(µω) = ln(2ω) +(24π2 − 149)

66, (14)

and the T = 0 part of Eq. (13) reduces to Eq. (11). Thisis a convenient choice of scale for ω ≫ T . For ω ∼ Tor smaller a convenient choice of scale was proposed inRef. [43]

ln(µω) = ln(4πT )− γE −1

22, (15)

in the pure gauge case. We switch between these twoscales when they become equal at ω ≃ 0.8903T [35].

The heavy quark diffusion coefficient has been calcu-lated at NLO, and the result reads [7]:

κNLO

T 3=

g4CFNc18π

[

ln2T

mE+ ξ + 2.3302

mET

]

. (16)

The NLO result for κ cannot be replicated from currentlyknown spectral functions as that would require ρ(ω, T ) tobe available at NNLO, which it is not. Both the LO andNLO results for κ are obtained under the weak couplingassumption mE ≪ T . This condition, however, is notsatisfied for most of the temperatures of interest. As aconsequence, one obtains an unphysical behavior at LOi.e., that κ becomes negative for T < 103 Tc.

One can also calculate κ using the kinetic theory. Thecorresponding expression reads [7, 44]:

κLO =g4CF12π3

∫ ∞

0

q2 dq

∫ 2q

0

p3 dp

(p2 +Π00)2(17)

×NcnB(q)(1 + nB(q))(

2− p2

q2+

p4

4q4

)

.

If we do not expand in the temporal gluon self-energy,Π00(p), which is formally of order g

2, the above ex-pression contains higher-order contributions to κ as well.Therefore, the above expression can be considered as theresummed leading-order result. The temporal gluon self-energy depends on the gauge choice. For small momenta,it can be expanded as

Π00(p) = m2E −

Nc4g2Tp+ ... . (18)

The first two terms in this expansion are gauge indepen-dent. We can take either the first term or the first andsecond terms in the above expression and evaluate theintegral in Eq. (17) numerically. Only keeping the firstterm in the above expression for Π00 already leads toa positive result, while keeping the second term as wellleads to an enhancement of the κ value. We present allthe different perturbative results for κ as a function oftemperature in Fig. 7. The scale of the coupling is theone defined in Eq. (15).

At the highest temperature considered in this, T =104 Tc, study we expect that the NLO result can provide

FIG. 7. Perturbative estimates of κ for the pure gauge theoryas a function of temperature calculated at LO and NLO, aswell as using the resummed leading-order expression (17).

FIG. 8. The perturbative spectral functions ρ(ω,T ) at T =104 Tc for different orders of perturbation theory. The dottedlines on the left indicate the perturbative estimates of κ givenby Eqs. (12) (LO) and (16) (NLO).

some guidance on the properties of the spectral functionand on the τ dependence of the chromoelectric correla-tor. Therefore, in Fig. 8 we show different versions of theNLO spectral function, including the zero-temperatureone. The full NLO spectral function can be describedwell by the simple κLOω/(2T ) form for ω < 0.02T , whileit approximately agrees with the T = 0 result for ω > 2T .The full NLO result and the naive (unresummed) NLOresult agree for ω > 0.6T . At small ω, the naive NLO re-sult is logarithmically divergent. This divergence cancelsagainst contributions coming from the scale mE in the re-summed expression. We can model the spectral functionby smoothly matching the κLOω/(2T ) behavior at smallω with the zero-temperature spectral function at large ω.We call this the perturbative step form. It is also shownin Fig. 8 by the blue dotted line. By using the NLO spec-tral function evaluated for µ = µω, we can calculate thecorresponding chromoelectric correlator, which is shown

8

FIG. 9. The chromoelectric correlator at T = 104 Tc calcu-lated from the NLO spectral function (orange band) and theperturbative step form of the spectral function (green band).The orange band completely overlaps with the green band,and it is hardly distinguishable from it. The errors for bothorange and green bands come from varying the scale by afactor of 2. In blue, we show the continuum limit of theT = 104 Tc lattice data.

in Fig. 9. Varying the renormalizarion scale by a factorof 2 around µω leads only to very small changes of thecorrelator, roughly corresponding to the width of the linein Fig. 9. We also calculate the chromoelectric correlatorcorresponding to the perturbative step form. The result-ing correlator is indistinguishable from the one obtainedusing the NLO spectral function. This means that theadditional structures in the spectral function in the re-gion 0.02 < ω/T < 0.6 play no significant role when itcomes to the correlator. We have also considered a per-turbative step model using κNLO. While using the NLOresult for κ significantly enhances the spectral functionin the low ω region it only leads to a 0.2% enhancementof the chromoelectric correlator compared to the one ob-tained using κLO. Thus, the correlator is not sensitive tothe small ω part of the spectral function at the highesttemperature. At lower temperatures, κ/T 3 gets larger,and the contribution of the low ω part of the spectralfunctions is more prominent. Therefore, it is at lowertemperatures that the value of κ can be constrained byaccurate calculations of the chromoelectric correlator.

While at T = 104 Tc, one may expect the resummedNLO result to provide an adequate description of thespectral function, this is not expected at lower tem-peratures, because, as pointed out above, numericallymE > T . In particular, for T < 10

3 Tc the resummedspectral function turns negative at some point in the re-gion ω < T , thus implying that the resummed perturba-tive result is not applicable in this ω range. In Sec. V wewill discuss the implications of this finding.

In Fig. 9, we also show the continuum limit of the chro-moelectric correlator at the high temperature T = 104 Tc

for comparison. The continuum-extrapolated lattice re-sult of the chromoelectric correlator has the same shapeas the NLO calculation. We note, however, that our con-tinuum data differ from the perturbative curve by a factor1.2, which indicates that the renormalization constant isnot accurate. If we normalize the above lattice result tothe correlator obtained from the NLO spectral functiondiscussed above at τT = 0.19, we find that the two agreewithin errors.

IV. SHORT-TIME BEHAVIOR OF THE

LATTICE RESULTS ON THE ELECTRIC

CORRELATOR

The continuum results of GE normalized by GnormE

show significant dependence on τ . The analysis in Sec. IIimplies that this cannot be caused by thermal effects (cf.Figs. 3 and 4 ). The LO result does not take into ac-count the effect of the running of the gauge coupling, andthis could be the reason why GLOE , or equivalently G

normE

(which is the same up to a multiplicative factor) does notcapture the τ dependence of the chromoelectric correla-tor. Therefore, as an alternative normalization we con-sider a correlator obtained from Eq. (9) using the zero-temperature NLO result for the spectral function witha running coupling constant evaluated at scale µω givenby Eqs. (14) and (15). We label the corresponding cor-

relator as GNLO+E . The numerical results for GE/GNLO+E

FIG. 10. Continuum extrapolation for all temperatures as afunction of τT .

are shown in Fig. 10. We see that this ratio increasesless with increasing τ . We expect that for small enoughτT , one would see a plateau in GE/G

NLO+E . This is not

the case, however, because the continuum extrapolationis not reliable for τT < 0.17, and for larger τT there aresome small thermal effects. On the other hand, given theuncertainties, we could still fit GE/G

NLO+E with a con-

stant for 0.17 ≤ τT ≤ 0.19. This indicates that GNLO+E

9

µTc 1.1 1.5 3 6 10 104

0.5µω 1.82(5) 1.74(5) 1.61(3) 1.52(3) 1.47(2) 1.20(1)1µω 1.81(5) 1.73(5) 1.60(3) 1.51(3) 1.46(2) 1.20(1)2µω 1.84(5) 1.76(5) 1.62(3) 1.53(3) 1.48(2) 1.20(1)

TABLE II. Normalization factor CN for three different renor-malization scales (rows) and for each measured temperature(columns) at τT = 0.19.

captures the τ dependence of the chromoelectric correla-tor obtained on the lattice much better. However, evenat the smallest τ the ratio GE/G

NLO+E is different from 1.

This is most likely due to the fact that the one-loop resultis not accurate for ZE. As shown in the previous section,even for the highest temperature, T = 104 Tc, the NLOresult is lower by a factor of 1.2, as seen in Fig. 9, al-though its τ -dependence agrees well with the continuumextrapolated lattice data. Therefore, we introduce anadditional normalization factor, CN by normalizing theratio GE/G

NLO+E to 1 at τT = 0.19. To check the uncer-

tainty of CN due to the choice of the normalization point,we also consider τT = 0.175 as a possible normalizationpoint. Furthermore, we vary the scale µω by a factor of2 around the optimal value when evaluating CN. Thenumerical values of CN are shown in Tab. II for differ-ent temperatures. The dependence on the normalizationpoint is shown in the systematic error and is of the sameorder as the scale dependence. The additional normaliza-tion constant CN decreases with increasing temperature.This is due to the fact that the β range used in the evalu-ation of the lattice correlator is increasing with increasingtemperature, and the one-loop result is more reliable atlarge β values. We will normalize GE/G

NLO+E with CN

given in Table II before comparing with the model spec-tral functions used for the extraction of κ.

V. MODELING THE SPECTRAL FUNCTION

AND DETERMINATION OF κ

To obtain the heavy quark diffusion coefficient fromthe continuum-extrapolated lattice results, we need toassume some model for the spectral function. We willuse the NLO results on the spectral function as well as κto guide us in this process. We also need to consider howsensitive the Euclidean-time chromoelectric correlator isto the spectral function in different ω regions. From theprevious sections it is clear that GE is dominated by thelarge ω part of the spectral function and thermal effectsin the spectral function contribute at the level of a fewpercent to the correlator.

It is reasonable to assume that at large enough ω, per-turbation theory is reliable even if the condition mE ≪ Tis not satisfied. This is because for large ω HTL resum-mation is not important, as will be detailed later. Cer-tainly at zero temperature the perturbative calculation

of ρ(ω, T ) is reliable for ω ≫ ΛQCD. Therefore, we as-sume that for ω > ωUV the spectral function is givenby ρUV(ω, T ), which is calculated perturbatively. On theother hand, for sufficiently small ω, the spectral functionis given by

ρIR(ω, T ) =ωκ

2T, (19)

and we can assume that ρ(ω, T ) = ρIR(ω, T ) for ω < ωIR.In the region ωIR < ω < ωUV, the form of the spectralfunction is not known, in general, and this lack of knowl-edge will generate an uncertainty in the determination ofκ. Based on these considerations, we adopt the followingprocedure to estimate κ:

• For a given value of κ, we construct the model spec-tral function that is given by the NLO result at highenergy, ω > ωUV, and by Eq. (19) at low energy,ω < ωIR.

• For intermediate ω, namely ωIR ≤ ω ≤ ωUV, weconsider various forms of the spectral function suchthat the total spectral function is smooth, as de-scribed below.

• We match the continuum-extrapolated lattice re-sult for the chromoelectric correlator to the corre-lator obtained from the model spectral function atsmall τT and adjust the value of κ to obtain thebest description of the lattice result. We take carethat τT is not too small, so that the procedure isnot affected by lattice artifacts.

• We estimate the uncertainties due to modeling thespectral function for ωIR ≤ ω ≤ ωUV, due to thenormalization point in τT , and due to the choice ofthe renormalization scale in the NLO result.

We consider two possible forms of the spectral func-tions that are continuous and are based on simple inter-polations between the small ω (IR) region and the largeω (UV) region:

ρline(ω, T ) = ρIR(ω, T )θ(ωIR − ω)+ (20)

[

ρIR(ω, T )− ρUV(ω, T )ωIR − ωUV

(

ω − ωIR)

+ ρIR(ω, T )

]

×

θ(ω − ωIR)θ(ωUV − ω) + ρUV(ω, T )θ(ω − ωUV) ,

and

ρstep(ω, T ) = ρIR(ω, T ) θ(Λ− ω) + ρUVT=0(ω, T ) θ(ω − Λ) .

(21)The latter case corresponds to ωIR = ωUV = Λ and thevalue of Λ is self-consistently determined by the conti-nuity of the spectral function for a given κ. Thus, thismodel depends only on κ. In the former case additionalconsiderations are needed to fix ωIR and ωUV, which aredescribed below. We will refer to these two forms as theline model and the step model, respectively.

10

The NLO result for the spectral function naturally in-terpolates between the IR and UV regions, but it is notreliable for small ω even at the highest temperature asdiscussed in Sec. III. However, it can provide some guid-ance on how to choose ωIR and ωUV. As mentionedabove, for ω > T , HTL resummation may not be im-portant, and the naive and resummed NLO result forthe spectral function should agree. As discussed in Ap-pendix B, the resummed and naive NLO results for thespectral function agree well for ω > 2.2T . Furthermore,the thermal contribution to ρ(ω, T ) is about the samefor ω > 2.2T at the lowest and the highest temperaturewhen normalized by ωT 2. This indicates that the per-turbative calculations are reliable for these values of ω.Therefore, we choose ωUV = 2.2T . At the highest tem-peratures, the resummed NLO result is well describedby the linear form given by Eq. (19) with κ = κLO

for ω < 0.02T . Therefore, ωIR = 0.01T appears tobe a reasonable choice. The NLO result for κ is sig-nificantly larger than the LO result, implying that thespectral function at low ω is also larger and thereforewill match ρUV(ω, T ) at larger ω. We find that ρIR(ω, T )and ρUV(ω, T ) are equal at around ω = 0.4T . Therefore,besides ωIR = 0.01T , we will also use ωIR = 0.4T andωIR = 1T in our analysis.

In Fig. 11 we show the spectral functions obtained fromEqs. (20) and (21), assuming κ = κNLO in ρstep and ρline,and three different ωIR at three representative tempera-tures, T = 1.1Tc, 6Tc, and 10

4 Tc. From the figure, wesee that at the lowest temperature, the ρstep(ω, T ) modelmatches the UV behavior at larger ω without the diparound ω ∼ T of the ρline(ω, T ) model. The ρline formwith ωIR = 0.01T and ρstep provide upper and lowerbounds for the spectral function at T = 1.1Tc. The pic-ture is the same for T = 1.5Tc and T = 3Tc. At T = 6Tc,all forms of the spectral functions provide nearly identi-cal results. At the highest two temperatures, the possiblechoices of the spectral functions are limited by ρline withωIR = 0.01T and ωIR = T .

Using the models for the spectral functions describedabove, we have calculated the corresponding Euclidean-time chromoelectric correlators for different values of κand compared these with the continuum-extrapolatedlattice results at each temperature to estimate the heavyquark diffusion coefficient. As discussed in the previ-ous section, the continuum-extrapolated lattice resultsneed an additional renormalization because the one-looprenormalization constant, ZE , is not accurate. Therefore,we have matched the correlator obtained from the modelspectral function to the continuum-extrapolated latticedata at τT = 0.19. The resulting multiplicative constantsCN are slightly different from those shown in Table II.This is because the correlators obtained from the modelspectral functions are slightly different from GNLO+E atτT = 0.19 due to the thermal contribution. We demon-strate this procedure in Appendix B for different modelspectral functions. Different forms give different valuesof κ, and this is the dominant source of systematic error

in the determination of κ. We have also studied the de-pendence of κ on the choice of the normalization point inτ and the choice of the renormalization scale. Choosingthe normalization point in the range 0.17 ≤ τT ≤ 0.19leads to an 8% variation in the resulting κ. Varying therenormalization scale by a factor of 2 results in a similarvariation.

Putting everything together we obtain the followingestimates for the heavy quark diffusion coefficient fromthe analysis:

1.91 <κ

T 3< 5.4 for T = 1.1Tc , (22)

1.31 <κ

T 3< 3.64 for T = 1.5Tc , (23)

0.63 <κ

T 3< 2.20 for T = 3Tc , (24)

0.43 <κ

T 3< 1.05 for T = 6Tc , (25)

0 <κ

T 3< 0.72 for T = 10Tc , (26)

0 <κ

T 3< 0.10 for T = 104 Tc , (27)

although one should be reminded that, as discussed at theend of Sec. III, the lattice data are weakly sensitive to κat the highest temperature. The dominant uncertaintyin the above result comes from the form of the spectralfunction used in the analysis and the uncertainty of thecontinuum-extrapolated lattice results.

We compare our result on κ with the results of otherlattice studies [13, 23–26] in terms of the spatial dif-fusion coefficient Ds, which is given by the relationκ/T 3 = 2/(DsT ), in the temperature range Tc − 3Tc.This is shown in Fig. 12. We see that our results agreewell with the other lattice determinations, with the ex-ception of the one in Ref. [13] that is based on charmo-nium correlators. This is likely due to the fact that thedetermination of Ds from the quarkonium correlators isnot accurate, since the width of the transport peak isdifficult to determine [11, 12].

The temperature dependence of the heavy quark diffu-sion coefficient in the entire temperature region is shownin Fig. 13. We clearly see the temperature dependenceof κ3/T . The κ obtained on the lattice is not incompat-ible with the NLO result given the large errors. Inspiredby this, we fit the temperature dependence of the lat-tice result by modeling it on Eq. (16) but keeping thecoefficient of mE/T as a free parameter C. From the fit,we obtain C = 3.81(1.33), which is larger than the NLOperturbative result C ≈ 2.3302.

We note that our result is significantly larger than thesimple holographic estimate [45]: 2πDsT = 1. However,more recent holographic estimates [46] are close to our re-sults. Finally, comparing with more experimental quanti-ties, we note that our result for Ds at the lowest temper-ature is in agreement with the calculations of D [47] andB̄ [48] mesons propagating in a medium of light hadrons,which find 2πDsT ∼ 5 for T ≈ Tc, but much smaller thanan earlier pion gas study [49] that found 2πDsT ≈ 17 for

11

FIG. 11. The shapes of different spectral function modelsρ(ω,T ) at (from top to bottom) T = 1.1Tc, T = 6Tc, andT = 104 Tc. The arguments of ρline in square brackets standfor [ωIR, ωUV].

T ≈ Tc. Experimental determinations of the D-mesonazimuthal anisotropy coefficient ν2 at ALICE [50] andSTAR [51] estimate at T ≈ Tc that κ/T 3 ≈ 1.8 − 8.38

and κ/T 3 ≈ 1.05−6.28, respectively. These are in agree-ment with our findings. All these experimental determi-nations include mass-dependent contributions, while ourdetermination of κ is in the heavy quark limit. Thereforethe two should agree up to 1/m corrections.

FIG. 12. Our results compared to existing lattice studies.The shaded band shows the perturbative behavior (16) andthe effect of the scale µω being varied by a factor of 2.

FIG. 13. Temperature dependence of our results compared tothe NLO result. The shaded bands include the errors comingfrom varying the scale by a factor of 2. The blue band alsoincludes the statistical error.

VI. CONCLUSIONS

In this paper, we have studied the chromoelectric cor-relator, GE at finite temperature on the lattice with theaim of extracting the heavy quark diffusion coefficient κ.The calculations have been performed in quenched QCD(SU(3) gauge theory) in order to obtain small statisticalerrors with the help of the multilevel algorithm. We havestudied the dependence of the chromoelectric correlatoron the Euclidean time, τ , in a wide temperature rangein order to enable the comparison with weak coupling re-sults. It turned out that the τ -dependence of the electriccorrelator is poorly captured by the leading order result.Going beyond the leading-order result and incorporatingthe effect of the running coupling in the corresponding

12

spectral function results in a correlation function GNLO+Ethat can capture the τ -dependence of the lattice resultmuch better.

To fully describe the τ -dependence of GE calculated onthe lattice, the effect of κ encoded in the low ω part of thechromoelectric spectral function has to be considered. Athigh ω, we have used forms of the spectral function thatare motivated by the next-to-leading-order perturbativeresults. Fitting the lattice results on GE, we have ob-tained values of κ at different temperatures. We observethat the sensitivity of the chromoelectric correlator to κis small, varying from a few percent at the lowest temper-atures to sub-percent at the highest temperatures. Thisfinding is corroborated by a model-independent analysisof the chromoelectric correlator, cf. Figs. 3 and 4. It isthis small sensitivity that makes the lattice determina-tion of κ quite challenging. Our main result is summa-rized in Fig. 13, which shows the temperature dependenceof the heavy quark diffusion coefficient. For T < 2Tc,our results agree with other lattice determinations, whileat higher temperatures, they appear consistent with theNLO result.

One of the shortcomings of the present analysis is theuse of the one-loop result for the renormalization con-stant, ZE, which, as we argued, is not reliable. The useof tadpole-improved one-loop perturbation theory willnot help, since it will make ZE even larger, while in or-der to achieve agreement of the lattice and NLO resultsat small τ , we need a smaller ZE than the one-loop re-sult. Clearly, a nonperturbative renormalization proce-dure will be needed, but this is beyond the scope of thepresent paper.

ACKNOWLEDGMENTS

We thank Saumen Datta for providing the code for thelattice calculation of the chromo-electric correlator. V.L.thanks Mikko Laine for clarifying details on the numeri-cal evaluation of the NLO spectral function. N.B., V.L.,and A.V. acknowledge the support from the Bundesmin-isterium für Bildung und Forschung project no. 05P2018and by the DFG cluster of excellence ORIGINS funded bythe Deutsche Forschungsgemeinschaft under Germany’sExcellence Strategy - EXC-2094-390783311. P.P. hasbeen supported by the U.S. Department of Energy underContract No. DE-SC0012704. The simulations have beencarried out on the computing facilities of the Computa-tional Center for Particle and Astrophysics (C2PAP) ofthe cluster of excellence ORIGINS.

Appendix A: Infinite-volume limit and continuum

extrapolation

To check to what extent using lattices with an aspectratio Ns/Nt smaller than 4 leads to visible finite-volumeeffects, we have performed calculations at two tempera-

tures, T = 1.5Tc and T = 10Tc, on N3s ×12 lattices with

Ns = 24, 32, and 48. The numerical results are shown inFig. 14 for some representative values of τT . As one cansee from the figure, the finite-volume effects are small.We have also attempted to perform an infinite-volumeextrapolation by fitting the lattice results with a 1/N3sform. The corresponding fits are shown in the figure aslines and bands together with the infinite-volume result.It is clear from the figure that the differences betweenthe infinite-volume result and the lattice results with dif-ferent Ns values are of the order of the statistical errors.Therefore, the use of Ns = 48 is justified.

As discussed in the main text, to obtain the continuumresult for the chromoelectric correlator, we first performthe interpolation in τT , and then for each value of τTwe perform the continuum extrapolation using the a/N2tform without Nt = 12 data or using the a/N

2t + b/N

4t

form with Nt = 12 data included (a and b are fit con-stants). We have demonstrated this procedure in Fig. 5for T = 1.1Tc. In Fig. 15, we show this procedurefor other temperatures: T = 1.5Tc, 3.0Tc, 6Tc, andT = 104 Tc. We do not show the analysis for 10Tc, as itlooks similar to the one for T = 104 Tc. From the figure,we see that the slope of the 1/N2t dependence increaseswith decreasing τT as expected, since the cutoff depen-dence is larger for smaller τT . But for the smallest τT ,we do not see this tendency. To understand the situationbetter, we show the coefficient of the 1/N2t term in thecontinuum extrapolation as a function of τT in Fig. 16.The coefficient of the 1/N2t term decreases monotonicallyas τT decreases. For this coefficient, the minimum of theabsolute value is reached at τT = 0.32 instead of at thelargest available τT . Note, however, that this is some-what accidental, as the coefficient changes sign aroundthis value of τT . Also, the errors for 0.3 < τ < 0.4 arequite large. More importantly for us, the absolute valueof the coefficient increases when we decrease τT from0.3 to 0.175, and then either flattens off or decreases ifτT is further decreased. We take this as an indicationthat the continuum limit is not reliable for τT < 0.175.We also perform continuum extrapolations using latticedata without tree-level improvement and the correspond-ing results are also shown in Fig. 15 as open symbols. Inthis case, the continuum limit is always approached fromabove. The continuum-extrapolated result from tree-level-improved lattice data and the unimproved latticedata agree within errors for τT ≥ 0.25. In the absenceof tree-level improvement, the continuum extrapolationsfor smaller τT are not reliable.

Appendix B: Modeling of the spectral function and

κ determination

In order to understand the main features of the pertur-bative spectral function corresponding to the chromoelec-tric correlator at NLO, in Fig. 17 we show the quantity2(ρ(ω, T )−ρ(ω)NLOT=0 )/(ωT 2) calculated with and without

www.origins-cluster.de

13

FIG. 14. Finite-volume effects for several τT values presented with different colors at T = 1.5 Tc (left) and T = 10Tc (right).The lines and bands correspond to the 1/N2s fits and their uncertainties.

FIG. 15. The continuum extrapolation of the chromoelectric correlator for T = 1.5 Tc and T = 3.0 Tc (top panels) and forT = 6.0 Tc and T = 10

4 Tc (bottom panels). The filled symbols and solid bands correspond to the extrapolation of the tree-level-improved lattice data, while the open symbols and patterned bands correspond to the extrapolations of the lattice datawithout tree-level improvement. The different colors correspond to different τT values.

14

-80

-60

-40

-20

0

20

40

60

80

0.1 0.15 0.2 0.25 0.3 0.35 0.4

slo

pe

!T

-40

-30

-20

-10

0

10

20

30

40

50

0.1 0.15 0.2 0.25 0.3 0.35 0.4

slo

pe

!T

FIG. 16. The coefficient of the 1/N2t dependence as a function of τ for T = 1.5 Tc (left) and T = 10 Tc (right).

15

HTL resummation at the lowest and highest tempera-tures. The plotted quantity gives κ in the ω → 0 limit.The naive (unresummed) result is logarithmically diver-gent at small ω. On the other hand, for ω > 2.2T , theresummed and the naive results agree well. This indi-cates that the NLO calculation is valid in this ω range.We also see that for 2.2 < ω/T < 10, the naive and re-summed NLO expressions are negative, and their shapesare independent of the temperature.

In Fig. 17 we also show the two model spectral func-tions (line model and step model), where we use κ =2.75T 3 for the lowest temperature and κ = 0.088T 3 forthe highest one. At the lowest temperature, the stepmodel has a larger finite-temperature part than the lin-ear model, while at the highest temperature, the oppo-site is true. The two models also have somewhat dif-ferent UV behavior. The step model is matched to thezero-temperature spectral function and thus ignores thethermal correction in the region 2.2 < ω/T < 10, whilethe line model incorporates this. The two models thusallow us to extract κ using a set of reasonable assump-tions about the large-ω behavior of the spectral function.

We match the chromoelectric correlator GmodelE , ob-tained from the above model spectral functions at τT =0.19 to the continuum-extrapolated lattice result to findthe optimal value of κ. We demonstrate this procedurein Fig. 18 where we show the continuum lattice result forthe lowest and the highest temperatures divided by thecorresponding GmodelE . For a given spectral function andthe appropriately chosen κ this ratio should be close to1. Since the errors of the continuum-extrapolated latticeresult are sizable we get a range of κ that is compatiblewith the lattice result. In Fig. 18 we show the resultsfor κ = 2.75T 3 and κ = 0.05T 3, and for T = 1.1Tc andT = 104 Tc, respectively. These κ values are chosen tobe in the middle of the quoted ranges (see Eqs. (22) and(27)). At the lowest temperature for the given κ the stepform and the line form seem to be on the opposite sidecompared to the lattice result, while at the highest tem-perature the step form clearly gives a better descriptionof the lattice data on average. We see that the optimalvalue of κ strongly depends on the assumed form, espe-cially at high temperatures.

16

FIG. 17. The NLO T = 0 spectral function subtracted from different models or perturbative curves at T = 1.1 Tc (left) andT = 104 Tc (right). See the text for further specifications.

FIG. 18. The ratio of the continuum-extrapolated chromoelectric correlator obtained on the lattice and the correlator obtainedwithin the line model (orange band) and the step model (blue band) for 1.1 Tc and κ = 2.75 T

3 (left), and for 104 Tc andκ = 0.05 T 3 (right). The bands show the errors originating from lattice effects and from varying the scale by a factor of 2.

17

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