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Nuclear Physics A NuclearPhysicsA00(2016)1–8 www.elsevier.com/locate/procedia Lattice QCD: bulk and transport properties of QCD matter Claudia Rattia aPhysicsDepartment,UniversityofHouston,HoustonTX77204,USA 6 1 0 2 n Abstract a J WepresentanoverviewofthemostrecentresultsonbulkandtransportpropertiesofQCDmatterinferredfromlattice 1 QCDsimulations. 1 Keywords: Quark-GluonPlasma,latticeQCD ] t a l 1. Introduction - p Lattice QCD is the most reliable first principle tool to address QCD in its non-perturbative regime. e h Givenenoughcomputerpower,bothstatisticalandsystematicuncertaintiescanbekeptundercontrol. Due [ toasteadyandcontinuousimprovementincomputerresources,numericalalgorithmsandourphysicalun- derstandingwhichmanifestsitselfinphysicaltechniques(e.g. theWilson-flowscalesettingintroducedin 1 v Ref. [1]),thelatticeresultswhicharebeingproducedtodayreachanunprecedentedlevelofaccuracy. This 7 allows a quantitative comparison to experimental observables for the first time in heavy ion physics. At 6 lowtemperatures,stronglyinteractingmattercanbewelldescribedbyanon-interactinggasofhadronsand 3 resonances; inthe infinitetemperaturelimit, the systembehaveslikean ideal, masslessgasof quarksand 2 gluons. Aswereducethetemperature,interactionsbetweenquarksandgluonsbecomerelevant: perturba- 0 . tion theory can be systematically used to calculate thermodynamic observables. Resummation techniques 1 improvetheconvergenceoftheperturbativeseriesandbringtheagreementwithlatticeQCDresultsdown 0 to ∼ 2.5T . The temperature range between these twoopposite regimes is the realmof lattice QCD: non- 6 c 1 perturbativemethodsareneededtoaddresstherelevantobservables. Thisisalsotherangeoftemperatures : whichcanbereachedinheavy-ioncollisionexperiments: anewsynergybetweenfundamentaltheoryand v i experimentistodaypossibleduetotheprecisionreachedinbothapproaches. Herewewillreviewthemost X recentresultsinthefield. r a 2. BulkpropertiesofQCDmatter The most reliable results obtained from lattice QCD simulations concern thermodynamic observables inequilibrium. Forexample,theequationofstateofQCDisnowavailableforasystemof2+1dynamical quarkflavorswithphysicalquarkmassesinthecontinuumlimit. In2014,theHotQCDcollaborationpub- lishedcontinuumresultsforpressure,energydensity,entropydensityandinteractionmeasureasfunctions of the temperature [2] which agree with the ones previously obtained by the Wuppertal Budapest (WB) QCD Equation of state at µ =0 B WB: S. Borsanyi et al.,1309.5258 2 /NuclearPhysicsA00(2016)1–8 collaboration[3,4]: theyareshowedintheleftpanelofFig. 1. Theseresultshavebeenindependentlyob- tainedwithtwodifferentstaggeredfermionactions(2stoutandHISQ):theagreementbetweenthetwoisa fundamentaltestofthevalidityofthediscretizedlatticemethodtosolveQCD.Recently,firstresultsforthe equationofstateobtainedfromotherapproachestolatticeQCDarebecomingavailable: theseincludethe Alternative methods for thermodynamics gradientflowmethod[5],whichextractsthethermodynamicquantitiesfromtheenergy-momentumtensor, andtwistedmassfermions[6];theformerarelimitedsofartothequenchedapproximation,thelattertotwo WB: S. Borsanyi et al.,1309.5258 flavorswithheavier-than-physicalquarkmasses. EoS available in the continuum   limit, with realistic quark masses Agreement between stout and   HISQ action for all quantities WB HotQCD WB: S. Borsanyi et al., 1309.5258, PLB (2014) HotQCD: A. Bazavov et al., 1407.6387, PRD (2014) 6/26 F. Burger et al., PRD (2015) Fig.1. Left:comparisonbetweenthelatticeresultsoftheWB[3,4]andHotQCD[2]collaborationsfortheequationofstateofQCD. Right:ComparisonbetweentheequationsofstateofpuregaugeQCDfroGmrRaedfs.ie[6n,t7 ]falondwth:e EonoeSof i2n+ 1thflaev orQCD[2].   quenched approximation ThelatticeresultsforQCDthermodynamicsaresofarlimitedtozeroorsmallchemicalpotentials,dueto the“signproblem”,whichmakesdirectsimulationsofQ CTDwaitsfitenidte mdeansssit yWnoiltsfoenas fieblrem. Sioonmse: promising alternative meFthloowds QarCeDb Cecoolml.,i nPgRDav (a2il0a1bl4e), but their appEliocaSti oanvatoilaQbCleD swoi tfharp hfoysri chaelapvarieamr-eters and controlleddiscretizationhasnotyetbeenachieved[8,9]. Oththaenrm-pehthyosdicsahal vqeubaerekn mpraopsosseeds taoncdir cumvent thesignproblem: herewewillfocusontheTaylorexpansioNn=o2fthermodynamicobservablesaroundµB =0 f [10,11](whichcanbeconsideredasatruncatedversionofthemultiparameterreweighting[12])anda9n/a26ly tic continuationfromimaginarychemicalpotentials[13,14,15]. OnecanexpandtheQCDpressureinTaylorseriesaroundµ =0: B p(µ ) (cid:18)µ (cid:19)2 (cid:18)µ (cid:19)4 (cid:18)µ (cid:19)6 B =c (T)+c (T) B +c (T) B +c (T) B +O(µ8); (1) T4 0 2 T 4 T 6 T B thisexpansioncontainsthecoefficientsc(T),extractedfromlatticeQCDsimulations.Aftertheearlyresults i forc ...c [16],thefirstcontinuumextrapolatedresultsforc werepublishedinRef.[17];inRef.[18]c was 2 6 2 4 showed,butonlyatfinitelatticespacing. InFig. 2weshowthepreliminary,continuumextrapolatedresults forc , c andc asfunctionsofthetemperature. SuchresultshavebeenobtainedbytheWBcollaboration 2 4 6 fromimaginaryµ simulations:c ,...c havebeenfittedontheµ −derivativesofp/T4forfixedtemperature B 2 6 B [19]. Atµ = 0,theQCDphasetransitionisananalyticcrossover[20];apseudocriticaltemperatureT can B c bedefinedbylookingattheinflectionpointorpeakofsomespecificobservables[21,22,23,24]. Onecan followthechangeintheirpositionasthechemicalpotentialincreases: thisgivesrisetoaµ −dependence B ofT whichcanbeexpressedas: c T (µ ) (cid:32) µ (cid:33)2 (cid:32) µ (cid:33)4 c B =1−κ B +λ B +.... (2) T (µ =0) T (µ ) T (µ ) c B c B c B Theparameterκintheaboveexpansionisthecurvatureofthephasediagramanditcanbeextractedfrom latticeQCDsimulations: bylookingatthreedifferentobservables(chiralcondensate, chiralsusceptibility andstrangequarksusceptibility)theWBcollaborationrecentlypublishedavalueofκ = 0.0149±0.0021 [25];thishasbeenobtainedbyfixingthestrangequarkchemicalpotentialtoimposestrangenessneutrality. /NuclearPhysicsA00(2016)1–8 3 0.003 0.09 0.0001 0.08 8x10-5 0.07 6x10-5 0.06 Wpurpepleirmtianla-rByudapest 0.002 Wpurpepleirmtianla-rByudapest 24xx1100--55 Wpurpepleirmtianla-rByudapest c2 00..0045 c4 c6 -2x10 -05 0.03 0.001 --64xx1100--55 0.02 -8x10-5 0.01 WB lattice HdRatGa WB lattice HdRatGa -0.0001 WB lattice HdRatGa 0 100 150 200 250 300 100 150 200 250 300 100 150 200 250 300 T [MeV] T [MeV] T [MeV] Fig.2. PreliminaryresultsfortheTaylorcoefficientsc2...c6asfunctionsofthetemperaturefromtheWBcollaboration,obtainedfrom imaginaryµBsimulations.Thedataarecontinuumextrapolated;theerror-barsareonlystatistical:thesystematicsoftheµBfittingare notincluded[19]. The phase diagram corresponding to this value of κ is showed in the left panel of Fig. 3, together with a compilation of freeze-out parameters obtained with different methods. Similar results have been obtained QCD phase diagraremcent ly by two other groups: P. Cea et al. obtain a vQalueCofDκ = 0p.02h0(a4)sbyefix indg µia=gµr[2a6]m, wh ile s l Bonatietal. findκ = 0.0135(20)bothwithµ = 0andµ = µ [27]. IntherightpanelofFig. 3,thephase s s l diagramwiththecurvaturefromRef. [26]isshown. R. Bellwied et al., 1507.07510 R. Bellwied et al., 1507.07510 P. Cea et al., 1508.07599 P. Cea et al., 1508.07599 C. Bonati et al., 1507.03571 Curvature κ defined as: Curvature κ defined as: Fig.3. Left: ThephasediagrambasedontheµB−dependentTcfromthechiralcondensate,analyticallycontinuedfromimaginary chemicalpotential[25].Thebluebandindicatesthewidthofthetransition.Theshadedblackregionshowsthetransitionlineobtained fromthechiralcondensate.Thewideningaround300MeViscomingfromtheuncertaintyofthecurvatureandfromthecontribution ofhigherorderterms,thustheapplicationrangeoftheresultsisrestrictedtosmallervalues. Wealsoshowsomeselectednon-lattice Recent results: results:theDyson-Schwingerresult[28],andthefreeze-outdataofRefs.[29]-[35].Right:analogousplotfromRef.[26]. Recent results: Amongthemostinterestingobservableswhichcanbesimulatedonthelatticearefluctuationsofcon- P. Cea et al., 1508.07599 served charges; they are defined as derivatives of the pressure with respect to the chemical potentials of 10/2P6. Cea et al., 1508.07599 conservedcharges(baryonnumberB,electricchargeQ,strangenessS): C. Bonati et al., 1507.03571 10/26 ∂l+m+np/T4 χBQS(T,µ )= . (3) lmn B ∂(µ /T)l∂(µ /T)m∂(µ /T)n B Q S The µ = 0 diagonal second-, fourth- and sixth-order baryon number fluctuations are the Taylor expan- B sion coefficients of Eq. (1), shown in Fig. 1. Their interest resides in the fact that the lattice results can be compared to experimental measurements, to the purpose of extracting information on the QCD matter created in heavy-ion collisions: while higher order fluctuations can be used to gain information about the positionofthecriticalpointintheQCDphasediagram[36,11],thelowerorderonescanleadtothedeter- mination of the freeze-out temperature and chemical potential in the evolution of the system, at which all inelasticreactionscease[37]-[39]. Indeed, thefluctuationsofagivenconservedchargearethecumulants of its event-by-event distribution; volume-independent ratios can conveniently be defined, which allow to 4 /NuclearPhysicsA00(2016)1–8 determinethefreeze-outtemperatureandchemicalpotentialbycomparingthelatticeQCDcurvestotheex- perimentalvalue. Forameaningfulcomparison,allnon-thermalsourcesoffluctuationsmustbeunderstood and kept under control, and a variety of effects has been identified and studied in the literature [40]-[46]. In2014theWBcollaborationfoundthat,analyzingthefluctuationsofelectricchargeandbaryonnumber independently,thereisaconsistencybetweenthefreeze-outchemicalpotentialscorrespondingtothehigh- est RHIC energies [47, 48, 49]. Recently, the authors of Ref. [50] performed a fit to the ratio of ratios ofχ /χ (mean/variance)forelectricchargeanprotonnumberandwereabletoobtainboththefreeze-out 1 2 temperatureandthecurvatureofthefreeze-outline.Thevalueofthefreeze-outtemperature(T =(147±2) f MeV)isinagreementwiththeoneobtainedinRef. [47]. TheleftpanelofFig. 4showstheratioofratios of χ /χ for electric charge and proton number used for this fit. Along the same lines, the WB collabo- 1 C2 urvature of the freeze-out line ration performed a combined fit of χ /χ for electric charge and proton number and found the freeze-out 1 2 temperatureandchemicalpotentialforthehighestRHICenergies. Thesepreliminaryresultsareshownin therightpanelofFig. 4,togetherwiththeisentropiclineswhichmatchthefreeze-outdata,thecontoursfor constantmean/varianceofnet-electricchargefromthelattice, andtheresultsofapreviousanalysisbased ontheHRGmoPdealr[3a5m]. eThteriWzaBtrieosnul tsoafg rteheew iftrhetheezoene-soouftR elifn. [e5:0 ]andwiththeHRGmodelones.   Talk by F. Karsch on Monday Unfortunately,fluctuationdataarenotyetavailableattheLHC.HowevertheauthorsofRef.[53],assuming thatthelow ermomentsfollowaSkellamdistribution,expressedthesecondmomentsintermsoftheparticle yieldsandcomparedthelatticeresultstotheALICEexperimentaldata,findingaslightlyhigherfreeze-out Taylor expansion of the “ratio of ratios” R QB= temperaturet hanthenonesobtainedatRHIC.Recently,thestudyoffluctuation1s2hasbeenextendedtovery largetemperatures[54,55]toextracttheonsetoftheHTLperturbativeexpansion[56,57],whichisfound tobeT (cid:39)250MeV.FluctuationsarealsousefultoinferthedegreesoffreedomwhichpopulatetheQuark- 222222222222222222222200000000000 Matching Wuppertal-Budapest lattice T [MeV] 112211221122112211221122112211221122112211228901890189018901890189018901890189018901890100000000000000000000000000000000000000000000 200 GeV ; S/N=420 62.4 GeV ; S/N=144 39 GeV ; S/N=94 27 GeV ; S/N=68 19 GeV ; S/N=48.5 RRRRrReQ1Q1P1P1P1s22222u=====l0000t0s.....00 145t13o60626 057245(((0244((111)))4)) S ((((66 22t(22a030..r09044 f lGGGGGuceeeeetVVVVVua)))))tion data 111111111117777777777700000000000 RQ12=0.0570(3) (39 GeV) RP=0.728(4) (27 GeV) 12 111111111116666666666600000000000 RQ=0.0779(6) (27 GeV) 12 RQ=0.1105(15) (19 GeV) 111111111115555555555500000000000 12 S/N=const trajectories 111111111114444444444400000000000 from lattice EOS [WB 2015] Wuppertal-Budapest 111111111113333333333300000000000 preliminary HRG analysis [Alba et al] T from lattice [WB 1507.07510] c 00000000000 5555555555500000000000 111111111110000000000000000000000 111111111115555555555500000000000 222222222220000000000000000000000 222222222225555555555500000000000 A. Bazavov et al., 1509.05786 STAR2.0: X. Luo, PoS CPOD 2014 Fig.S4.TALeRft0:.F8ro: mPRReLf. ([2500]1:3th)e ratioofratioso fχ1/χ2fornet electriccharge andnet-protonfl u c tu aPtioHnEsmNeIaXsu: r1ed5b0y6t.h0e7S8T3A4R 15/26 andPHENIXCollaborations[48,49,51,52]. Right: PreliminaryresultsoftheWBcollaboration. Thecoloredfullanddashedlines arethecontoursatconstantmean/varianceratiosofthenetelectricchargefromlatticesimulations. Thecontoursthatcorrespondto STARdataintersectinthefreeze-outpointsofRef.[35].TheredbandistheQCDphasediagramshowninFig.3.Alsoshownarethe isentropiccontoursthatmatchthechemicalfreeze-outdata[19]. Gluon Plasma (QGP) around the transition temperature. For example, studying the correlations between charmandbaryonnumber,itispossibletoextractthetemperatureatwhichthecharmquarksareliberated. Arecentstudy[58]showsthat,eveniftheonsetofdeconfinementforthecharmquarktakesplacearound T (cid:39)165MeV,itbecomesthedominantdegreeoffreedominthethermodynamicsofthecharmsectoronly atT (cid:39)200MeV,whilebetweenthesetwotemperaturesthedominantcontributiontothecharmedpressure isgivenbyopencharmmeson-andbaryon-likeexcitationswithintegralbaryoniccharge. /NuclearPhysicsA00(2016)1–8 5 3. TransportpropertiesofQCDmatter Intheregionaround1-2T ,QCDmatterishighlynon-perturbativeandsignificantmodificationsofits c transportpropertiesareexpected. Unfortunately,theobservablesthatarerelatedtothetransportproperties ofmatterallhaveonecommonfeaturethatmakesitdifficulttoextractthemfromlatticeQCDsimulations. Thelattercaninvestigateacertainsetofcurrent-currentcorrelatorsonadiscretesetofpoints. Suchcorre- latorshaveaspectralrepresentationwhichinvolvesintegralsofspectralfunctionsweightedbyappropriate integrationkernels.Extractingthedesiredobservables(thelow-frequencyandlow-momentumlimitofsuch spectralfunctions)requirestheapplicationofinversionmethodsoramodelingofthespectralfunctionsat lowfrequenciesinordertointegrateoveradiscretesetoflatticepoints. Inspiteofthesedifficulties,several resultshavebeenobtainedrecentlyonthetransportpropertiesofmatter. One of the most interesting points concerns the properties of quarkonia. Usually, this problem is ad- dressedbymeansofthreedistinctapproaches: • extractthequark-antiquarkpotentialandplugitintoSchro¨dinger’sequationfortheboundstatetwo- pointfunction • extractthequarkoniaspectralfunctionsfromeuclideantemporalcorrelators • studyspatialcorrelatorsandtheirin-mediumscreeningproperties. HereIwillconcentrateonthefirsttwo. Forthefirstapproach,wehavecontinuumextrapolatedresultsfor the qq¯ free energy obtained from correlators of two Polyakov loops: these results have been obtained for a sysItenm tofe2+r1-flqavoursaat trhke p hpysicoaltmeassn[5t9]i:athley are shown in the left panel of Fig. 5. Also, the qq¯ potentialhasbeenobtainedinasystemof2+1dynam icaIlqnuatrkeflrav-oqrsuusinagarnkew pBaoyestiaeninnfetrieancel prescription[60]: theseresultsareshownintherightpanelofFig. 5. Otherresultshavebeenobtainedby assumingthevalidityofSchro¨dinger’sequationforcharmquarksandextractingthepotentialdirectlyfrom charmonium correlators [61]. Both methods agree with each other and show the typical Debye-screening Static quark-antiquark free-energy flatte ningofthepotentialathightemperatures. Quark-antiquark potential in N=2+1 QCD   f Burnier et al. (2014) Allton et al. (2015) Borsanyi et al. JHEP(2015) Fwiigth.5p .hyLCseifcota:lnFmrtoaimsnseRuse.uf.Rm[i5g9h ]te::cxfornottmrinauRpuemfo.vl[aa6l0ut]ee:stfdhoer rtrheeeaslstpauatirlcttq owqf¯fthrieteehset anteir cgyinRateterdqiauffaler rkpenpatotrteetmn topiaeflr a(ttouhpreeens fcsoyromambsoylspst)elmecoxomf p2pa+roe1dtqetuoanrtkthieflaaclvo olloriser s   pCseenutrdaol spcoatleanr taianl:d c voemctboinr ation of singletfrNeee=ne2rg+ie1si nflCaouvloomrbsg aaugte t(hgreay cpirhcleys)s.ical mass close to the color singlet free energy f potentials: Asforthecharmoniumspectralfunctions, boththequenchedapproximationresultsandtheoneswith dynamicalquarksshowthatallcharmoniumstatesaredissociatedatT ≥ 1.5T [62]-[64]. Thesituationis 22/26 c different for the bottomonium, for which there is a discrepancy between different analyses. G. Aarts and 22/26 his collaborators, using the Maximum Entropy Method to reconstruct the spectral function, show that the s-wavestatesurvivesuptoT (cid:39)1.9T whilethep-waveonemeltsjustaboveT [65]. ByusingtheBayesian c c methodtoreconstructthespectralfunction,S.Kimetal. findthatboths-andp-wavessurviveintheplasma uptoT (cid:39)250MeV[66]: thesefindingsareshowninFig. 6[67]. Quarkonia spectral functions Charmonium spectral functions in quenched approximation and preliminary   studies with dynamical quarks yield consistent results: all charmonium states are dissociated for T(cid:1)1.5T c H. Ding et al., PRD (2012) G. Aarts et al., PRD (2007) 6 /NuclearPhysicsA00(2016)1–8 WB: S. Borsanyi et al., JHEP (2014) Bottomonium (N=2+1, m =160 MeV), Bayesian method:   f π S. Kim et al. PRD (2015) S-wave groFuign.6d. Sspteacttreal faunnctdio nPsf-owrthaevs-e( legftr)oanudnp-d(r isghtat)tweav esbuorttvomivoeni uumpst attoes [T67~].250 MeV   Talk by A. Rothkopf on Tuesday 23/26 Oneofthetransportcoefficientsthathavebeenextensivelystudiedonthelatticeistheelectricconduc- tivity,σ,whichmeasurestheresponseofthemediumtosmallperturbationsinducedbyanelectromagnetic field.Severalresultsareavailableforthisquantity[68,69],butsofaronlyonehasbeenobtainedinasystem of 2+1 quark flavors in Ref. [70], by means of the Maximum Entropy Method. The electric conductivity increasesbyafactor6intherangeoftemperaturesbetween140and350MeV.Thechargediffusioncoeffi- cienthasalsobeenobtained,bydividingtheelectricconductivitybythesecondorderfluctuationχQ. The 2 diffusion coefficient has a dip close to T , which is consistent with the expectations of a strongly coupled c system. TheseresultsareshownintheleftpanelofFig. 7. The right panel of Fig. 7 shows a compilation of all available lattice QCD results on the pure gauge shearviscosityoverentropyratioasafunctionofthetemperature. Forthisobservable,besidesthedifficulty of inverting the energy-momentum tensor correlator, an additional problem arises: this correlator itself is extremelynoisy,andnotechniqueisavailabletoreduceitifquarksareintroducedinthesimulations. This is the reason why so far only quenched results are available. An algorithm which allows to increase the Viscosity signal-to-noiseratioisneededtoextractthisobservablealsointhefullQCDcase.   Shear viscosity in the pure gauge sector of QCD Nakamura & Sakai ‘05 S. Borsanyi et al. ‘14   Challenge: very low signal-to noise ratio for the Euclidean energy- momentum correlator Fig.7. Left: FromRef. [70]: diffusioncoefficientDmultipliedby2πT asafunctionofthetemperature,usingD=σ/χQ. Right: 2 compilationofallavailablelatticeQCDresultsonthepuregaugeshearviscosityoverentropyasafunctionofthetemperature: H. Meyer(blacksquaresandcircles)[71,72],Christiansenetal.(verticallines)[73],NakamuraandSakai(emptyredcircles)[74],S.W. Magesetal.(greenfullcircles)[75]. 25/26 /NuclearPhysicsA00(2016)1–8 7 4. Conclusions Asshownbythelargeamountofnewresultssummarizedintheseproceedings,whichbecameavailable sincethe2014QuarkMatterconference,theprogressandtheprecisionachievedbylatticeQCDsimulation is really impressive. Precise results are available for QCD thermodynamics at zero and small chemical potentials,whichallowaquantitativecomparisonwithexperimentalresultsforthefirsttime. Progresshas beenmadeinthedeterminationofrealtimedynamics. Thisshouldenableustoachieveacomprehensive understandingofbulkandtransportpropertiesofQCDmatterfromlatticeQCDsimulations. Acknowledgements IwouldliketothankallmylatticeQCDcolleagueswhosentmetheircontributionsandcommentsto my plenary talk. This work is supported by the National Science Foundation through grant number NSF PHY-1513864andtheDOEINCITEprogramsupportedunderContractDE-AC02-06CH11357. 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