Lattice QCD constraints on the heavy quark diffusion coefficient
Nora Brambilla, Viljami Leino, Peter Petreczky, Antonio Vairo
TTUM-EFT 131/19
Lattice QCD constraints on the heavy quark diffusion coefficient
Nora Brambilla,
1, 2, ∗ Viljami Leino, † Peter Petreczky, ‡ and Antonio Vairo § (TUMQCD Collaboration) Physik Department, Technische Universität München,James-Franck-Strasse 1, 85748 Garching, Germany Institute for Advanced Study, Technische Universität München,Lichtenbergstrasse 2a, 85748 Garching, Germany Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: July 21, 2020)We report progress towards computing the heavy quark momentum diffusion coefficient from thecorrelator of two chromo-electric 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 . < T /T c < 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 relaxationtime of heavy quarks is expected to be ∼ ( M/T ) t lightrel ,with M being the heavy quark mass, T being the tem-perature, and t lightrel 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 − fm. 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 understood interms of Langevin equations [5]. The drag coefficient η and heavy quark momentum diffusion coefficient κ thatenter into the Langevin equations describe the interac-tion of the heavy quarks with the medium and are con-nected by the Einstein relation η = κ/ (2 M T ) 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]. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] The NLO correction is very large, thus questioning thevalidity of the perturbative expansion. Analytic calcu-lations for strong coupling are available only for super-symmetric Yang-Mills theories [8, 9]. Therefore, latticeQCD calculations for the heavy quark diffusion coefficientare 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 [11–15]. Recently the problem ofheavy quark diffusion has also been studied out of equilib-rium with real-time lattice simulations in Refs. [16, 17].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 [18–20].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 chromo-electric field strength [21]. The corresponding spectralfunction does not have a transport peak and the small ω behavior is smoothly connected to the UV behavior ofthe spectral function [21]. 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 a r X i v : . [ h e p - l a t ] J u l the deconfined phase, i.e. for purely gluonic plasma [22–26]. The correlator of the chromo-electric field strengthis very noisy making the lattice calculations extremelychallenging. To deal with this problem it is mandatoryto use noise reducing techniques such as the multi-levelalgorithm by Lüscher and Weisz [27]. 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 chromo-electric field strength is sensitive to the small ω 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 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 can bemade for the heavy quark diffusion coefficient as it hasalready be done for the equation of state [28], quark num-ber susceptibilities [29, 30] and static correlation func-tions [31–33]. If such contacts can be established thesewould validate the methodology used in the lattice ex-traction of κ . Previous lattice studies focused on a nar-row temperature region [24] or only considered a singlevalue of the temperature [25]. In Ref. [24] no significanttemperature dependence of κ/T was found. Large tem-peratures are needed in the lattice studies to establish thetemperature dependence of κ/T . The temperature de-pendence of κ/T is also important for phenomenology aswith a constant value of κ/T it is impossible to explainsimultaneously the elliptic flow parameter, v , for heavyquarks and the nuclear modification factor [4]. Further-more, the spectral function of the chromo-electric fieldstrength correlator is known at NLO [34]. Using thisNLO result at high ω one can constrain the functionalform of the spectral function used in the analysis of thelattice correlator.The aim of this paper is to study the correlator ofthe chromo-electric field strength in a wide temperaturerange in order to make contact with weak coupling calcu-lations of the Euclidean correlation function up to NLOin the spectral function, and also to constrain the tem-perature dependence of κ .The rest of the paper is organized as follows. In thenext section we go trough the procedure of calculating theEuclidean correlator of the chromo-electric field strengthon the lattice. The spectral function of the chromo-electric correlator and its relation to κ is discussed in section III. There we also review the perturbative resultsfor this spectral function. The short time behavior of thechromo-electric correlator and its proper normalizationis clarified in section IV. In section V, we discuss how tomodel the spectral functions of the chromo-electric corre-lator and to extract the value of κ from the lattice results.Finally, section VI contains our conclusions. II. LATTICE RESULTS FOR THECHROMO-ELECTRIC CORRELATOR
For a heavy quark of mass M (cid:29) 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 chromo-electric correlatorin Euclidean time [9, 21]: G E ( τ ) = − (cid:88) i =1 (cid:104) Re Tr [ U (1 /T, τ ) E i ( τ, ) U ( τ, E i (0 , )] (cid:105) (cid:104) Re Tr U (1 /T, (cid:105) , (1)where T is the temperature, U ( τ , τ ) is the temporalWilson line between τ and τ , and the chromo-electricfield, in which the coupling has been absorbed E i ≡ gE i ,is discretized on the lattice as [21]: E i ( x , τ ) = U i ( x , τ ) U ( x + ˆ i, τ ) − U ( x , τ ) U i ( x + ˆ4) . (2)This discretization is expected to be least sensitive toultraviolet effects [21].To calculate the discretized chromo-electric correlatordefined above on the lattice we use the standard Wil-son gauge action and the multilevel algorithm [27]. Weconsider N × N t lattices and vary the temperature ina wide range T = 1 . T c − T c by varying the latticegauge coupling β = 6 /g . Here T c is the deconfinementphase transition temperature. We use N t = 12 , , and at each temperature to check for lattice spacingeffects and perform the continuum extrapolation. In thisstudy we use N s = 48 , except for N t = 12 lattices, wheremultiple spatial volumes are used to check for finite vol-ume effects.To set the temperature scale as well as the lattice spac-ing we use the gradient flow parameter t [35] and thevalue T c √ t = 0 . [36]. We use the result ofRef. [36] to relate the temperature scale or the latticespacing to β . The parameters of the lattice calculationsincluding the statistics are given in Table I. In the simula-tions with the multilevel algorithm we divide the latticeinto four sub-lattices and update each sub-lattice 2000times to evaluate the chromo-electric correlator on a sin-gle gauge configuration. We use the simulation programdeveloped in a prior study [24].In order to obtain the heavy quark diffusion coeffi-cient, the lattice chromo-electric correlator needs to berenormalized and then extrapolated to the continuum. The renormalization coefficient Z E ( β ) ≡ Z E ( g ) of thechromo-electric correlator in case of the Wilson gaugeaction has been calculated at 1-loop [37]: Z − loopE = 1 + 0 . g . (3)We will use this 1-loop correction in the present study.However, we expect that the 1-loop result for Z E is notprecise enough. As it will be clear from the results ofthe lattice calculations this is indeed the case. The per-turbative error in Z E ( β ) affects both its absolute valuefor fixed β and its β -dependence. For the continuum ex-trapolation it is important to estimate the uncertainty inthe β -dependence of the renormalization constant. Theerror in the absolute value of Z E could be corrected af-ter the continuum extrapolation is done by introducingan additional multiplicative factor. We will postpone thediscussion of this multiplicative factor to section IV. Toestimate the error in the β dependence of Z E we considerthe tadpole improved result for Z E , namely Z tadE = 1 /u ,with u being the plaquette expectation value [24]. Thedifference in the β dependence of Z tadE and Z − loopE canbe used as an estimate of the error of the β -dependenceof Z E . Therefore, at each temperature we consider thevariation in Z − loopE · u in the β range that correspondsto N t = 12 − as an estimate of the systematic errorsin Z E for bare gauge couplings in that range.The chromo-electric correlator decays rapidly with in-creasing τ . This feature can be understood from theleading order (tree-level) result [21]: G LOE ( τ ) g C F ≡ G normE ( τ ) = π T (cid:20) cos ( πτ T )sin ( πτ T ) + 13 sin ( πτ T ) (cid:21) , (4)where C F = 4 / is the Casimir of the fundamental rep-resentation of SU(3). In Fig. 1 we show Z E G E /G normE fordifferent temperatures calculated on the largest, × lattice. We see a significant temperature dependence inthis ratio. Also shown in the figure are the numericalresults for the lowest temperature, T = 1 . T c calculatedfor different N t . As one can see from the figure, the cut-off ( N t ) dependence is significant even for relatively largevalues of τ T . We expect that the cutoff dependence in-creases with decreasing τ T , except when τ is of the orderof the lattice spacing because the cutoff dependence of Z E G E /G normE is proportional to ( a/τ ) . We see that ourlattice data follow this expectation for τ T > . . Thisobservation is important for estimating the reliability ofthe continuum extrapolations. A similar N t dependenceis observed at other temperatures. We will use the notation G E for both the lattice and the con-tinuum version of the chromo-electric correlator to keep the no-tation simple. It should be clear from the context which one weare referring to. We will use different notations for the contin-uum and the lattice version of the chromo-electric correlator onlywhen it is absolutely necessary. T /T c N t × N β N conf × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × In order to reduce discretization errors we turn toa tree-level improvement procedure [38, 39], where theleading order results in the continuum (4) and the lat-tice perturbation theory are matched. The LO latticeperturbation theory gives [23]: G LO , latE ( τ ) g C F = (cid:90) π − π d q (2 π ) ˜ q e ¯ qN t (1 − τT ) + ˜ q e ¯ qN t τT a ( e ¯ qN t −
1) sinh(¯ q ) , (5)where ¯ q = 2arsinh(˜ q/ , (6) ˜ q = (cid:88) i =1 ( q i / . (7)The improved distance τ is then defined so that G LOE ( τ ) = G LO , latE ( τ ) . In Fig. 2 we show our resultsfor Z E G impE ( τ ) /G normE = Z E G E ( τ ) /G normE . From the fig-ure we can observe that after the tree-level improvementthe ratio Z E G E /G normE appears monotonically increas-ing with increasing τ T and has a decreasing slope asa function of temperature. At the highest temperature T = 10 T c we see a nearly horizontal τ -independent line.Moreover, we observe a large reduction of cutoff effects τ T Z E G E / G n o r m E × T =1.1 T c × T =1.5 T c × T =3 T c × T =6 T c × T =10 T c × T =10000 T c τ T Z E G E / G n o r m E × T =1.1 T c × T =1.1 T c × T =1.1 T c × T =1.1 T c FIG. 1. The chromo-electric field correlator from Eq. (1) nor-malized with Eq. (4). Top: All measured temperatures for thebiggest lattice size. Bottom: All lattice sizes for the smallesttemperature. for all temperatures when tree-level improvement is used.As an example, we show this reduction at the bottom ofFig. 2 for the lowest temperature T = 1 . T c . A similarreduction in the N t dependence is seen at other tem-peratures. Due to its impact, we will use the tree levelimprovement for the rest of this paper and, therefore,unless otherwise indicated, drop the overline from τ andthe superscript imp from G impE .The normalized chromo-electric correlator shown inFig. 2 has a significant τ -dependence. We conclude thatthe LO perturbative result does not capture the key fea-tures of the chromo-electric correlator. Only at the high-est temperature, T = 10 T c , the τ -dependence of the cor-relator is well described by the leading order result. Onemay wonder whether the observed behaviour of the nor-malized chromo-electric 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 ordereffects at zero temperature. In order to answer this ques- τ T Z E G i m p E / G n o r m E × T =1.1 T c × T =1.5 T c × T =3 T c × T =6 T c × T =10 T c × T =10000 T c τ T Z E G i m p E / G n o r m E × T =1.1 T c × T =1.1 T c × T =1.1 T c × T =1.1 T c FIG. 2. The chromo-electric field correlator from Eq. (1) nor-malized with Eq. (4) and tree-level improved with (5). Top:All measured temperatures for the biggest lattice size. Bot-tom: All lattice sizes for the smallest temperature. τ / fm Z E G E / G n o r m E × T =1.1 T c × T =1.5 T c × T =3 T c × T =6 T c × T =10 T c × T =10000 T c FIG. 3. The data of Fig 2 in physical units. tion we show our lattice results in Fig. 2 as function of τ in physical units rather than of τ T . This figure showsthat the ratio of the chromo-electric correlator to the freetheory result is largely temperature independent imply-ing that the chromo-electric correlator is dominated bythe vacuum part of the spectral function. Even moreimportant it becomes to quantify the temperature de-pendence of the chromo-electric correlator. This can bedone by considering the following ratio of the normalizedcorrelator at fixed value of β but at two temperaturescorresponding to temporal extent N t and N t . Latticeartifacts are canceled out in the double ratio: R ( N t ) = G E ( N t , β ) G normE ( N t ) (cid:30) G E (2 N t , β ) G normE (2 N t ) . (8)We present our results for this quantity in Fig. 4 and ob-serve that in the small τ T region there is no temperaturedependence. Instead the thermal effects become more no-ticeable as τ T is increased, being highest at the largest τ T . Also, these thermal effects become more prominentas the temperature is lowered towards the transition tem-perature. In particular, we see a near negligible temper-ature dependence in the T = 10 T c data. As we willsee in the next sections, these features of R ( N t ) can benaturally explained by a temperature dependent κ . Inany case thermal effects in the chromo-electric correla-tor, which also encode the value of κ , are small, at thelevel of few percent. This fact implies that extracting κ from lattice determinations of the chromo-electric corre-lator is challenging. τ T R ( ) β = 7.774, T N t =12 = 6 T c β = 7.193, T N t =12 = 3 T c β = 7.193, T N t =12 = 2.2 T c β = 7.193, T N t =12 = 2 × T c FIG. 4. The ratio (8) of simulations with same β at differenttemperatures with N t = 12 . Before extracting the heavy quark diffusion coefficientwe need to address finite volume effects and perform thecontinuum extrapolation of Z E G E . Most of our calcu-lations have been performed using N s = 48 . To checkfor finite volume effects for N t = 12 we have performedcalculations using spatial sizes N = 24 , , at twotemperatures, T = 1 . T c and T = 10 T c . The small-est spatial volume here corresponds to the aspect ratio N s /N t = 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 errors down to the aspect ratio N s /N t = 2 . Therefore,at the current level of precision using a N s = 48 latticeis sufficient even for N t = 24 .Next, we perform the continuum extrapolations of Z E G E . The systematic errors in the renormalization con-stant estimated above are combined with the statisti-cal errors of the chromo-electric correlator before per-forming the continuum extrapolation. In the interval . ≤ τ T ≤ . we have a sufficient number of datapoints to perform the continuum extrapolations. We firstinterpolate the data for each N t in τ T using ninth orderpolynomials to estimate Z E G E at common τ T values.We perform linear extrapolations in /N = ( aT ) of Z E G E at these τ T values using lattices with N t = 16 , and . As an example, we show the continuum extrap-olation for selected values of τ T in Fig. 5. One can seethat the N t = 12 data does not lie in the /N scal-ing region. Therefore, we also perform extrapolations to N t = 12 data with a ( aT ) term included. The differencebetween these continuum extrapolations is used as an es-timate of the systematic error of the continuum result.The slope of the a dependence is increasing with decreas-ing τ T as can be seen from Fig. 5. This is expected, thecutoff effects are larger at smaller τ T . However, at thesmallest value τ T = 0 . the slope of the a dependencebecomes smaller again contrary to the expectations. Wetake this as an indication that the cutoff effects in thisregion cannot be described by a simple a or a + a .As shown in Appendix A the slope of the a dependenceincreases monotonically only till τ T ≥ . . Therefore,we consider the continuum extrapolation to be reliableonly for τ T ≥ . . 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 chromo-electric correla-tor normalized by G normE is shown in Fig. 6 for all tem-peratures as function of τ T . The continuum extrapo-lated results share the general features of the tree-levelimproved results at non-zero lattice spacing in terms of τ and temperature dependence. In particular, we see astrong dependence on τ T , except for the highest tem-perature, indicating that the leading order result doesnot capture the τ T dependence of G E . We will try tounderstand these features of the correlator in the nextsections. III. SPECTRAL FUNCTIONS AND DIFFUSIONCOEFFICIENT IN PERTURBATION THEORY
In order to determine the heavy quark diffusion coeffi-cient κ from the chromo-electric correlator, G E one has touse the relation between this correlator and the spectral ( aT ) Z E G E / G n o r m E FIG. 5. The continuum extrapolation. The lines represent thefit done with the three largest lattices at temperature T =1 . T c at different values of τ T shown with different colors.The fitted line is extrapolated to 0 and to / 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. τ T G E / G n o r m E T c T c T c T c T c T c FIG. 6. The continuum extrapolation for all temperatures asa function of τ T . function ρ ( ω, T ) : G E ( τ ) = (cid:90) ∞ d ωπ ρ ( ω, T ) K ( ω, τ T ) , (9)where K ( ω, τ T ) = cosh (cid:0) ωT (cid:0) τ T − (cid:1)(cid:1) sinh (cid:0) ω T (cid:1) . The heavy quark diffusion coefficient is determined interms of ρ through the Kubo formula [40] κ ≡ lim ω → T ρ ( ω, T ) ω . (10)At leading order of perturbation theory the spectralfunction is given by [21]: ρ LO ( ω, T ) = g ( µ ω ) C F ω π , (11)where the coupling has been evaluated at the scale µ ω .We use the 5-loop running coupling constant in thiswork [41]. At LO the scale µ ω is arbitrary. A naturalchoice is µ simple ω = max( ω, πT ) [25]. The LO spectralfunction (11) gives κ = 0 .At NLO the perturbative calculation of ρ ( ω, T ) needsHard-Thermal-Loop (HTL) resummation for ω < ∼ m E ,with m E being the LO Debye mass: m E = (cid:112) N c / gT inthe pure gauge theory. The full NLO result of ρ ( ω, T ) has been calculated in [34]. The NLO spectral functionprovides the LO non-vanishing result for κ : κ LO T = g C F N c π (cid:20) ln 2 Tm E + ξ (cid:21) , (12)where ξ = − γ E + ζ (cid:48) (2) ζ (2) (cid:39) − . . For ω > ∼ T there isno need for resummation when calculating the spectralfunction at NLO; the naive (non-resummed) NLO resultfor ρ ( ω, T ) in the pure gauge case reads ρ naive ( ω, T ) = (13) g C F ω π (cid:26) g (4 π ) (cid:20) N c (cid:18)
113 ln µ ω ω + 1499 − π (cid:19)(cid:21)(cid:27) + g C F π g π (cid:26) N c (cid:90) ∞ d q n B ( q ) (cid:20) ( q + 2 ω ) ln (cid:12)(cid:12)(cid:12)(cid:12) q + ωq − ω (cid:12)(cid:12)(cid:12)(cid:12) + qω (cid:18) ln | q − ω | ω − (cid:19) + ω q P (cid:18) q + ω ln q + ωω + 1 q − ω ln ω | q − ω | (cid:19)(cid:21)(cid:27) , where n B ( q ) = (exp( q/T ) − − is the Bose–Einstein dis-tribution, P takes the principal value, and g ≡ g ( µ ω ) .The first line of (13) gives the NLO T = 0 contribu-tion and the subsequent lines carry the thermal effects.For the NLO ρ ( ω, T ) , µ ω may be set such that the NLO T = 0 contribution vanishes [34]: ln( µ ω ) = ln(2 ω ) + (24 π − , (14)and the T = 0 part of (13) reduces to (11). This isa convenient choice of scale for ω (cid:29) T . For ω ∼ T or smaller a convenient choice of scale was proposed inRef. [42] ln( µ ω ) = ln(4 πT ) − γ E − , (15)in the pure gauge case. We switch between these twoscales when they become equal at ω (cid:39) . T [34].The heavy quark diffusion coefficient has been calcu-lated at NLO and the result reads [7]: κ NLO T = g C F N c π (cid:20) ln 2 Tm E + ξ + 2 . m E T (cid:21) . (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 result for κ are obtained under the weak couplingassumption m E (cid:28) T . This condition, however, is notsatisfied for most of the temperatures of interest. As aconsequence, one obtains an unphysical behavior at LO,i.e., that κ becomes negative for T < T c .One can also calculate κ using the kinetic theory. Thecorresponding expression reads [7, 43]: κ LO = g C F π (cid:90) ∞ q d q (cid:90) q p d p ( p + Π ) (17) × N c n B ( q )(1 + n B ( q )) (cid:18) − p q + p q (cid:19) . If we do not expand in the temporal gluon self-energy, Π ( p ) , which is formally of order g , the above expres-sion 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 momentait can be expanded as Π ( p ) = m − N c g T p + ... . (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 Π already leads to apositive result, while keeping also the second term leadsto an enhancement of the κ value. We present all thedifferent perturbative results for κ as a function of tem-perature in Fig. 7. The scale of the coupling is the onedefined in Eq. (15).At the highest temperature, T = 10 T c , considered inthis study we expect that the NLO result can providesome guidance on the properties of the spectral functionand on the τ dependence of the chromo-electric 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 κ LO ω/ (2 T ) form for ω < . T , whileit approximately agrees with the T = 0 result for ω > T .The full NLO result and the naive (unresummed) NLOresult agree for ω > . T . At small ω the naive NLO re-sult is logarithmically divergent. This divergence cancelsagainst contributions coming from the scale m E in the re-summed expression. We can model the spectral function T / T c κ / T LONLOΠ = m Π = m − g Tp FIG. 7. Perturbative estimates of κ for the pure gauge theoryas a function of temperature calculated at LO, NLO as wellas using the resummed leading order expression (17). − − − ω/ T − − − ρ / ( ω T ) κ at LO κ at NLO ρ NLO T =0 ρ NLOnaive ρ NLO ρ step FIG. 8. The perturbative spectral functions ρ ( ω, T ) at T =10 T c for different orders of perturbation theory. The dottedlines on the left indicate the perturbative estimates of κ givenby Eq. (12) (LO) and (16) (NLO). by smoothly matching the κ LO ω/ (2 T ) behaviour at small ω 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. Using the NLO spec-tral function we can calculate the corresponding chromo-electric correlator, which is shown in Fig. 9. The widthof the band corresponds to the variation of the scale by afactor two around the value given by Eq. (14). The scalevariation appears to be very small. We also calculate thechromo-electric correlator corresponding to the perturba-tive step form. The resulting correlator is indistinguish-able from the one obtained using the NLO spectral func-tion. This means that the additional structures in thespectral function in the region . < ω/T < . play nosignificant role when it comes to the correlator. We havealso considered a perturbative step model using κ NLO .While using the NLO result for κ significantly enhancesthe spectral function in the low ω region it only leads τ T G E / G n o r m E Continuum limit ρ NLO
Step
FIG. 9. The chromo-electric correlator at T = 10 T c 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 andit is hardly distinguishable from it. The errors for both orangeand green bands come from varying the scale by a factor oftwo. In blue we show the continuum limit of the T = 10 T c lattice data. to a . enhancement of the chromo-electric correlatorcompared to the one obtained using κ LO . Thus, the cor-relator is not sensitive to the small ω part of the spectralfunction at the highest temperature. At lower tempera-tures κ gets larger and the contribution of the low ω partof the spectral functions is more prominent. Therefore, itis at lower temperatures that the value of κ can be con-strained by accurate calculations of the chromo-electriccorrelator.While at T = 10 T c , 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, numerically m E > T . In particular, for T < T c 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 section Vwe will discuss the implications of this finding.In Fig. 9 we also show the continuum limit of thechromo-electric correlator at the high temperature T =10 T c for comparison. The continuum extrapolated lat-tice result of the chromo-electric correlator has the sameshape as the NLO calculation. We note, however, thatour continuum data differs from the perturbative curveby a factor 1.2, which indicates that the renormalizationconstant is not accurate. µ T c µ ω µ ω µ ω C N for three different renor-malization scales (rows) and for each measured temperature(columns) at τ T = 0 . . IV. SHORT TIME BEHAVIOR OF THELATTICE RESULTS ON THE ELECTRICCORRELATOR
The continuum results of G E normalized by G normE show significant dependence on τ . The analysis in sec-tion II implies that this cannot be caused by thermal ef-fects (cf. Figs. 3 and 4 ). The LO result does not take intoaccount the effect of the running of the gauge coupling,and this could be the reason why G LOE or equivalently G normE (which is the same up to a multiplicative factor)does not capture the τ dependence of the chromo-electriccorrelator. Therefore, as an alternative normalization weconsider a correlator obtained from Eq. (9) using the zerotemperature NLO result for the spectral function withrunning coupling constant evaluated at scale µ ω given byEqs. (14) and (15). We label the corresponding corre-lator as G NLO+E . The numerical results for G E /G NLO+E τ T G E / G N L O + E T c T c T c T c T c FIG. 10. Continuum extrapolation for all temperatures as afunction of τ T . are shown in Fig. 10. We see that this ratio increasesless with increasing τ and there is also some indicationof appearance of a plateau at small τ T . This indicatesthat G NLO+E captures the τ dependence of the chromo-electric correlator obtained on the lattice much better.However, even at the smallest τ the ratio G E /G NLO+E isdifferent from one. This is most likely due to the factthat the 1-loop result is not accurate for Z E . As shownin the previous section, even for the highest temperature, T = 10 T c , the NLO result is lower by a factor 1.2, asseen in Fig. 9, although its τ -dependence agrees well withthe continuum extrapolated lattice data. Therefore, weintroduce an additional normalization factor, C N by nor-malizing the ration G E /G NLO+E to one at τ T = 0 . . Tocheck the uncertainty of C N due to the choice of the nor-malization point, we also consider τ T = 0 . as possiblenormalization point. Furthermore, we vary the scale µ ω by a factor of two around the optimal value when eval-uating C N . The numerical values of C N are shown inTab. II for different temperatures. The dependence onthe normalization point is shown in the systematic errorand is of the same order as the scale dependence. Theadditional normalization constant C N decreases with in-creasing temperature. This is due to the fact that the β range used in the evaluation of the lattice correlator isincreasing with increasing temperature and the 1-loop re-sult is more reliable at large β values. We will normalize G E /G NLO+E with C N given in Tab. II before comparingwith the model spectral functions used for the extractionof κ . V. MODELING THE SPECTRAL FUNCTIONAND DETERMINATION OF κ To obtain the heavy quark diffusion coefficient fromthe continuum extrapolated lattice results we need to as-sume some model for the spectral function. We will usethe NLO results on the spectral function as well as κ toguide us in this process. We also need to consider howsensitive the Euclidean time chromo-electric correlator isto the spectral function in different ω regions. From theprevious sections it is clear that G E is dominated by thelarge ω part of the spectral function and thermal effectsin the spectral function contribute at the level of fewpercent to the correlator.It is reasonable to assume that at large enough ω per-turbation theory is reliable even if the condition m E (cid:28) T is 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 calculationof ρ ( ω, T ) is reliable for ω (cid:29) Λ QCD . Therefore, we as-sume that for ω > ω UV the spectral function is givenby ρ UV ( ω, T ) , which is calculated perturbatively. On theother hand for sufficiently small ω the spectral functionis given by ρ IR ( ω, T ) = ωκ T , (19)and we can assume that ρ ( ω, T ) = ρ IR ( ω, T ) for ω < ω IR .In the region ω IR < ω < ω UV the form of the spectralfunction is not known, in general, and this lack of knowl-edge will generate an uncertainty in the determinationof κ . We consider two possible forms of the spectralfunctions that are continuous and are based on simpleinterpolations between the small ω (IR) region and large ω (UV) region: ρ line ( ω, T ) = ρ IR ( ω, T ) θ ( ω IR − ω )+ (20) (cid:20) ρ IR ( ω, T ) − ρ UV ( ω, T ) ω IR − ω UV (cid:0) ω − ω IR (cid:1) + ρ IR ( ω, T ) (cid:21) × θ ( ω − ω IR ) θ ( ω UV − ω ) + ρ UV ( ω, T ) θ ( ω − ω UV ) , and ρ step ( ω, T ) = ρ IR ( ω, T ) θ (Λ − ω ) + ρ UV T =0 ( ω, T ) θ ( ω − Λ) . (21)The latter case corresponds to ω IR = ω UV = 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 ω IR and ω UV , which aredescribed below. We will refer to these two forms as theline model and the step model, respectively.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 section III. However, it can provide someguidance on how to choose ω IR and ω UV . 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 > . T . Furthermore,the thermal contribution to ρ ( ω, T ) is about the samefor ω > . T at the lowest and the highest temperaturewhen normalized by ωT . This indicates that the per-turbative calculations are reliable for these values of ω .Therefore, we choose ω UV = 2 . T . At the highest tem-peratures, the resummed NLO result is well describedby the linear form given by Eq. (19) with κ = κ LO for ω < . T . Therefore, ω IR = 0 . T 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 ρ UV ( ω, T ) at larger ω . We find that ρ IR ( ω, T ) and ρ UV ( ω, T ) are equal at around ω = 0 . T . Therefore,besides ω IR = 0 . T , we will also use ω IR = 0 . T and ω IR = 1 T in our analysis.In Fig. 11 we show the spectral functions obtained fromEq. (20) and Eq. (21) assuming κ = κ NLO in ρ step and ρ line , and three different ω IR at three representative tem-peratures, T = 1 . T c , T c and T c . From the fig-ure, we see that at the lowest temperature the ρ step ( ω, T ) model matches the UV behavior at larger ω without thedip around ω ∼ T of the ρ line ( ω, T ) model. The ρ line formwith ω IR = 0 . T and ρ step provide an upper and lowerbound for the spectral function at T = 1 . T c . The pic-ture is the same for T = 1 . T c and T = 3 T c . At T = 6 T c ,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 ρ line with ω IR = 0 . T and ω IR = T .Using the models for the spectral functions describedabove we have calculated the corresponding Euclidean0time chromo-electric correlators for different values of κ and compared these with the continuum extrapolated lat-tice results at each temperature to estimate the heavyquark diffusion coefficient. As discussed in the previoussection the continuum extrapolated lattice results needan additional renormalization because the 1-loop renor-malization constant, Z E , is not accurate. Therefore, wehave matched the correlator obtained from the modelspectral function to the continuum extrapolated latticedata at τ T = 0 . . The resulting multiplicative constants C N are slightly different form those shown in Tab. II.This is because the correlators obtained from the modelspectral functions are slightly different from G NLO+E at τ T = 0 . 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 errorin the determination of κ . We have also studied the de-pendence of κ on the choice of the normalization point in τ and the choice of the renormalization scale. Choosingthe normalization point in the range . ≤ τ T ≤ . leads to a 8% variation in the resulting κ . Varying therenormalization scale by a factor two results in a similarvariation.Putting everything together we obtain the followingestimates for the heavy quark diffusion coefficient fromthe analysis: . < κT < . for T = 1 . T c , (22) . < κT < . for T = 1 . T c , (23) . < κT < . for T = 3 T c , (24) . < κT < . for T = 6 T c , (25) < κT < . for T = 10 T c , (26) < κT < . for T = 10 T c , (27)although it should be reminded that, as discussed at theend of section III, the lattice data are weakly sensitive to κ at 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, 22–25] in terms of the spatial diffu-sion coefficient D s which is given by the relation κ/T =2 / ( D s T ) , in the temperature range T c − T c . This isshown in Fig. 12. We see that our results agree well withthe other lattice determinations with the exception of theone in Ref. [13] that is based on charmonium correlators.This is likely due to the fact that the determination of D s from the quarkonium correlators is not accurate sincethe width of the transport peak is difficult to determine[11, 12].The temperature dependence of the heavy quark diffu-sion coefficient in the entire temperature region is shown − − ω/ T − − − − − − ρ / ( ω T ) ρ NLO T =0 ρ NLO ρ step ρ line [0.01,2.2] ρ line [0.4,2.2] ρ line [1,2.2] − − ω/ T − − − − − − ρ / ( ω T ) ρ NLO T =0 ρ NLO ρ step ρ line [0.01,2.2] ρ line [0.4,2.2] ρ line [1,2.2] − − ω/ T − − − ρ / ( ω T ) ρ NLO T =0 ρ NLO ρ step ρ line [0.01,2.2] ρ line [0.4,2.2] ρ line [1,2.2] FIG. 11. The shapes of different spectral function models ρ ( ω, T ) at (from top to bottom) T = 1 . T c , T = 6 T c , and T = 10 T c . The arguments of ρ line in square brackets standfor [ ω IR , ω UV ] . in Fig. 13. We clearly see the temperature dependenceof κ /T . The κ obtained on the lattice is not incompat-ible with the NLO result given the large errors. Inspired1by this we fitted the temperature dependence of the lat-tice result by modeling it on Eq. (16) but keeping thecoefficient of m E /T as a free parameter C . From the fitwe obtain C = 3 . . , which is larger than the NLOperturbative result C ≈ . .We note that our result is significantly larger than theholographic estimate [44]: πD s T = 1 . Finally, com-paring to more experimental quantities, we note thatour result for D s at the lowest temperature is muchsmaller than the calculation from the pion gas [45], whichfinds, πD s T ≈ for T ≈ T c . Experimental determi-nations of the D-meson azimuthal anisotropy coefficient ν at ALICE [46] and STAR [47] estimate at T ≈ T c κ/T ≈ . − . and κ/T ≈ . − . , respectively.These are in agreement with our findings. All these ex-perimental determinations include mass dependent con-tributions, while our determination of κ is in the heavyquark limit. Therefore the two should agree up to /m corrections. T / T c π D s T NLOBanarjee 2011Francis 2015 Ding 2012Brambilla 2019Our result
FIG. 12. Our results compared to existing lattice studies.The shaded band shows the perturbative behavior (16) andthe effect of the scale µ ω being varied by a factor 2. T / T c κ / T fitNLOOur result FIG. 13. Temperature dependence of our results compared tothe NLO result. The shaded bands include the errors comingfrom varying the scale by a factor 2. The blue band alsoincludes the statistical error.
VI. CONCLUSIONS
In the paper, we have studied the chromo-electric cor-relator, G E 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 statisti-cal errors with the help of the multi-level algorithm. Wehave studied the dependence of the chromo-electric cor-relator on the Euclidean time, τ , in a wide temperaturerange 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 correspondingspectral function results in a correlation function, G NLO+E that can capture the τ -dependence of the lattice resultmuch better.To fully describe the τ -dependence of G E calculatedon the lattice, the effect of κ encoded in the low ω partof the chromo-electric spectral function has to be con-sidered. At high ω , we have used forms of the spectralfunction that are motivated by the next-to-leading orderperturbative results. Fitting the lattice results on G E , wehave obtained values of κ at different temperatures. Weobserve that the sensitivity of the chromo-electric cor-relator to κ is small, varying from few percent at thelowest temperatures to sub-percent at the highest tem-peratures. This finding is corroborated by a model in-dependent analysis of the chromo-electric correlator, c.f.Figs. 3 and 4. It is this small sensitivity that makes thelattice determination of κ quite challenging. Our mainresult is summarized in Fig. 13, which shows the temper-ature dependence of the heavy quark diffusion coefficient.For T < T c our results agree with other lattice determi-nations, while at higher temperatures they appear con-sistent with the NLO result. ACKNOWLEDGMENTS
N.B., V.L., and A.V. acknowledge the support from theBundesministerium für Bildung und Forschung projectno. 05P2018 and by the DFG cluster of excellenceORIGINS funded by the Deutsche Forschungsgemein-schaft under Germany’s Excellence Strategy - EXC-2094-390783311. P.P. has been supported by the U.S. Depart-ment of Energy under Contract No. de-sc0012704. Thesimulations have been carried out on the computing facil-ities of the Computational Center for Particle and Astro-physics (C2PAP) of the cluster of excellence ORIGINS.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.2
Appendix A: Infinite volume limit and continuumextrapolation
To check to what extent using lattices with aspect ra-tio N s /N t smaller than four leads to visible finite volumeeffects we have performed calculations at two tempera-tures, T = 1 . T c and T = 10 T c on N × lattices with N s = 24 , and . The numerical results are shown inFig. 14 for some representative values of τ T . 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 /N s form. 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 between theinfinite volume result and the lattice results with different N s are of the order of the statistical errors. Therefore,the use of N s = 48 is justified.As discussed in the main text, to obtain the continuumresult for the chromo-electric correlator we first performthe interpolation in τ T and then for each value of τ T we perform the continuum extrapolation using the a/N form without N t = 12 data or using the a/N + b/N form with N t = 12 data included ( a and b are fitconstants). We have demonstrated this procedure inFig. 5 for T = 1 . T c . In Fig. 15 we show this procedurefor other temperatures: T = 1 . T c , . T c , T c and T = 10 T c . We do not show the analysis for T c as itlooks similar to the one for T = 10 T c . From the figurewe see that the slope of the /N dependence increaseswith decreasing τ T as expected, since the cutoff depen-dence is larger for smaller τ T . But for the smallest τ T we do not see this tendency. To understand the situationbetter we show the coefficient of the /N term in thecontinuum extrapolation as a function of τ T in Fig. 16.We see that the coefficient of the /N term increases inabsolute value with decreasing τ T till about τ T = 0 . and then either flattens off or decreases if τ T is furtherdecreased. We take this as an indication that thecontinuum limit is not reliable for τ T < . . We alsoperformed continuum extrapolations using lattice datawithout tree-level improvement and the correspondingresults are also shown in Fig. 15 as open symbols. Inthis case, the continuum limit is always approached fromabove. The continuum extrapolated result from treelevel improved lattice data and the unimproved latticedata agree within errors for τ T ≥ . . In absence oftree-level improvement the continuum extrapolations forsmaller τ T are not reliable. Appendix B: Modeling of the spectral function and κ determination In order to understand the main features of the per-turbative spectral function corresponding to the chromo-electric correlator at NLO, in Fig. 17 we show the fol- lowing quantity ρ ( ω, T ) − ρ ( ω ) NLO T =0 ) / ( ωT ) calculatedwith and without HTL resummation at the lowest andhighest temperature. The plotted quantity gives κ in the ω → limit. The naive (unresummed) result is loga-rithmically divergent at small ω . On the other hand for ω > . T the resummed and the naive result agree well.This indicates that the NLO calculation is valid in this ω range. We also see that for . < ω/T < the naiveand resummed NLO expressions are negative and theirshapes independent of the temperature.In Fig. 17 we also show the two model spectralfunctions (line model and step model), where we use κ = 2 . T for the lowest temperature and κ = 0 . T for the highest one. At the lowest temperature the stepmodel has a larger finite temperature part than the linearmodel, while at the highest temperature the opposite istrue. The two models also have somewhat different UVbehavior. The step model is matched to the zero tem-perature spectral function and thus ignores the thermalcorrection in the region . < ω/T < , while the linemodel incorporates this. The two models thus allow toextract κ using a set of reasonable assumptions about thelarge ω -behavior of the spectral function.We match the chromo-electric correlator, G modelE ob-tained from the above model spectral functions at τ T =0 . 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 temperature divided by thecorresponding G modelE . For a given spectral function andthe appropriately chosen κ this ratio should be close toone. Since the errors of the continuum extrapolated lat-tice result are sizable we get a range of κ that is compati-ble with the lattice result. In Fig. 18 we show the resultsfor κ = 2 . T and κ = 0 . T , and for T = 1 . T c and T = 10 T c , respectively. These κ values are cho-sen to present the extreme values that fit the lattice datawithin the step model. At T = 1 . T c , κ = 2 . T isin the middle of our range for κ (see Eq. (22)), whereasat T = 10 T c , κ = 0 . T is at the upper edge of ourrange (see Eq. (27)).3 / / / / N s Z E G E / G n o r m E / / / / N s Z E G E / G n o r m E FIG. 14. Finite volume effects for several τ T values presented with different colors at T = 1 . T c (left) and T = 10 T c (right).The lines and bands correspond to the /N fits and their uncertainties. ( aT ) Z E G E / G n o r m E ( aT ) Z E G E / G n o r m E ( aT ) Z E G E / G n o r m E ( aT ) Z E G E / G n o r m E FIG. 15. The continuum extrapolation of the chromo-electric correlator for T = 1 . T c and T = 3 . T c (top panels) and for T = 6 . T c and T = 10 T c (bottom panels). The filled symbols and solid bands correspond to the extrapolation of the tree-levelimproved lattice data, while the open symbols and patterned bands to the extrapolations of the lattice data without tree-levelimprovement. The different colors correspond to different τ T values. -80-60-40-20 0 20 40 60 80 0.1 0.15 0.2 0.25 0.3 0.35 0.4 s l ope (cid:111) 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 s l ope (cid:111) T FIG. 16. The coefficient of the /N dependence as function of τ for T = 1 . T c (left) and T = 10 T c (right). ω/ T ( ρ − ρ N L O T = ) / ( ω T ) ρ NLO ρ NLOnaive ρ step ρ line [0.01,2.2] ω/ T ( ρ − ρ N L O T = ) / ( ω T ) ρ NLO ρ NLOnaive ρ step ρ line [0.01,2.2] FIG. 17. The NLO T = 0 spectral function subtracted from different models or perturbative curves at T = 1 . T c (left) and T = 10 T c (right). See the text for further specifications. τ T G E / ( C N G m o d e l E ) ρ step ρ line [0.01,2.2] τ T G E / ( C N G m o d e l E ) ρ step ρ line [0.01,2.2] FIG. 18. The ratio of the continuum extrapolated chromo-electric correlator obtained on the lattice and the correlator obtainedwithin the line model (orange band) and the step model (blue band) for . T c and κ = 2 . T (left), and T c and κ = 0 . T (right). The bands show the errors originating from lattice effects and from varying the scale by a factor 2.[1] W. Busza, K. Rajagopal, and W. van der Schee, Ann.Rev. Nucl. Part. Sci. , 339 (2018), arXiv:1802.04801[hep-ph].[2] E. V. Shuryak and I. Zahed, Phys. Rev. C70 , 021901(2004), arXiv:hep-ph/0307267 [hep-ph].[3] E. Shuryak,
Heavy ion reaction from nuclear to quarkmatter. Proceedings, International School of NuclearPhysics, 25th Course, Erice, Italy, September 16-24,2003 , Prog. Part. Nucl. Phys. , 273 (2004), arXiv:hep-ph/0312227 [hep-ph].[4] A. Beraudo et al. , Nucl. Phys. A979 , 21 (2018),arXiv:1803.03824 [nucl-th].[5] G. D. Moore and D. Teaney, Phys. Rev.
C71 , 064904(2005), arXiv:hep-ph/0412346 [hep-ph].[6] B. Svetitsky, Phys. Rev.
D37 , 2484 (1988).[7] S. Caron-Huot and G. D. Moore, JHEP , 081 (2008),arXiv:0801.2173 [hep-ph].[8] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G.Yaffe, JHEP , 013 (2006), arXiv:hep-th/0605158 [hep-th].[9] J. Casalderrey-Solana and D. Teaney, Phys. Rev. D74 ,085012 (2006), arXiv:hep-ph/0605199 [hep-ph].[10] G. Aarts and J. M. Martinez Resco, JHEP , 053(2002), arXiv:hep-ph/0203177 [hep-ph].[11] P. Petreczky and D. Teaney, Phys. Rev. D73 , 014508(2006), arXiv:hep-ph/0507318 [hep-ph].[12] P. Petreczky, Eur. Phys. J. C , 85 (2009),arXiv:0810.0258 [hep-lat].[13] H. T. Ding, A. Francis, O. Kaczmarek, F. Karsch,H. Satz, and W. Soeldner, Phys. Rev. D86 , 014509(2012), arXiv:1204.4945 [hep-lat].[14] H.-T. Ding, O. Kaczmarek, A.-L. Kruse, R. Larsen,L. Mazur, S. Mukherjee, H. Ohno, H. Sandmeyer, andH.-T. Shu,
Proceedings, 27th International Conference onUltrarelativistic Nucleus-Nucleus Collisions (Quark Mat-ter 2018): Venice, Italy, May 14-19, 2018 , Nucl. Phys.
A982 , 715 (2019), arXiv:1807.06315 [hep-lat]. [15] A.-L. Lorenz, H.-T. Ding, O. Kaczmarek, H. Ohno,H. Sandmeyer, and H.-T. Shu, (2020), arXiv:2002.00681[hep-lat].[16] K. Boguslavski, A. Kurkela, T. Lappi, and J. Peu-ron, in (2020) arXiv:2001.11863[hep-ph].[17] K. Boguslavski, A. Kurkela, T. Lappi, and J. Peuron,(2020), arXiv:2005.02418 [hep-ph].[18] N. Brambilla, M. A. Escobedo, J. Soto, and A. Vairo,Phys. Rev.
D96 , 034021 (2017), arXiv:1612.07248 [hep-ph].[19] N. Brambilla, M. A. Escobedo, J. Soto, and A. Vairo,Phys. Rev.
D97 , 074009 (2018), arXiv:1711.04515 [hep-ph].[20] N. Brambilla, M. A. Escobedo, A. Vairo, andP. Vander Griend, Phys. Rev.
D100 , 054025 (2019),arXiv:1903.08063 [hep-ph].[21] S. Caron-Huot, M. Laine, and G. D. Moore, JHEP ,053 (2009), arXiv:0901.1195 [hep-lat].[22] H. B. Meyer, New J. Phys. , 035008 (2011),arXiv:1012.0234 [hep-lat].[23] A. Francis, O. Kaczmarek, M. Laine, and J. Lan-gelage, Proceedings, 29th International Symposium onLattice field theory (Lattice 2011): Squaw Valley, LakeTahoe, USA, July 10-16, 2011 , PoS
LATTICE2011 ,202 (2011), arXiv:1109.3941 [hep-lat].[24] D. Banerjee, S. Datta, R. Gavai, and P. Majumdar,Phys. Rev.
D85 , 014510 (2012), arXiv:1109.5738 [hep-lat].[25] A. Francis, O. Kaczmarek, M. Laine, T. Neuhaus,and H. Ohno, Phys. Rev.
D92 , 116003 (2015),arXiv:1508.04543 [hep-lat].[26] L. Altenkort, O. Kaczmarek, L. Mazur, and H.-T. Shu, ,(2019), arXiv:1912.11248 [hep-lat]. [27] M. Lüscher and P. Weisz, JHEP , 010 (2001),arXiv:hep-lat/0108014 [hep-lat].[28] A. Bazavov, P. Petreczky, and J. H. Weber, Phys. Rev. D97 , 014510 (2018), arXiv:1710.05024 [hep-lat].[29] A. Bazavov, H. T. Ding, P. Hegde, F. Karsch, C. Miao,S. Mukherjee, P. Petreczky, C. Schmidt, and A. Velytsky,Phys. Rev.
D88 , 094021 (2013), arXiv:1309.2317 [hep-lat].[30] H. T. Ding, S. Mukherjee, H. Ohno, P. Petreczky,and H. P. Schadler, Phys. Rev.
D92 , 074043 (2015),arXiv:1507.06637 [hep-lat].[31] A. Bazavov, N. Brambilla, H. T. Ding, P. Petreczky, H. P.Schadler, A. Vairo, and J. H. Weber, Phys. Rev.
D93 ,114502 (2016), arXiv:1603.06637 [hep-lat].[32] A. Bazavov, N. Brambilla, P. Petreczky, A. Vairo, andJ. H. Weber (TUMQCD), Phys. Rev.
D98 , 054511(2018), arXiv:1804.10600 [hep-lat].[33] A. Bazavov et al. , (2019), arXiv:1908.09552 [hep-lat].[34] Y. Burnier, M. Laine, J. Langelage, and L. Mether,JHEP , 094 (2010), arXiv:1006.0867 [hep-ph].[35] M. Lüscher, JHEP , 071 (2010), [Erratum: JHEP 03,092 (2014)], arXiv:1006.4518 [hep-lat].[36] A. Francis, O. Kaczmarek, M. Laine, T. Neuhaus,and H. Ohno, Phys. Rev. D91 , 096002 (2015),arXiv:1503.05652 [hep-lat]. [37] C. Christensen and M. Laine, Phys. Lett.
B755 , 316(2016), arXiv:1601.01573 [hep-lat].[38] R. Sommer, Nucl. Phys.
B411 , 839 (1994), arXiv:hep-lat/9310022 [hep-lat].[39] H. B. Meyer, JHEP , 077 (2009), arXiv:0904.1806 [hep-lat].[40] J. Kapusta and C. Gale, Finite-temperature field the-ory: Principles and applications , Cambridge Mono-graphs on Mathematical Physics (Cambridge UniversityPress, 2011).[41] M. Tanabashi et al. (Particle Data Group), Phys. Rev.
D98 , 030001 (2018).[42] K. Kajantie, M. Laine, K. Rummukainen, and M. E.Shaposhnikov, Nucl. Phys.
B503 , 357 (1997), arXiv:hep-ph/9704416 [hep-ph].[43] S. Caron-Huot and G. D. Moore, Phys. Rev. Lett. ,052301 (2008), arXiv:0708.4232 [hep-ph].[44] P. Kovtun, D. T. Son, and A. O. Starinets, JHEP ,064 (2003), arXiv:hep-th/0309213 [hep-th].[45] L. M. Abreu, D. Cabrera, F. J. Llanes-Estrada, andJ. M. Torres-Rincon, Annals Phys. , 2737 (2011),arXiv:1104.3815 [hep-ph].[46] S. Acharya et al. (ALICE), Phys. Rev. Lett. , 102301(2018), arXiv:1707.01005 [nucl-ex].[47] L. Adamczyk et al. (STAR), Phys. Rev. Lett.118