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Conserved Charge Fluctuations and Correlationsfrom Lattice QCD and the Beam Energy ScanFrithjof KarschBrookhaven National Laboratory & Bielefeld UniversityOUTLINE– the QCD phase diagram– EoS at non-zero baryon chemical potential– cumulant ratios of conserved charge fluctuations– the QCD critical pointF. Karsch, RHIC&AGS,BNL 20151

Chiral critical point and QCD critical endpointLHC: mayestablish contactwith the QCDchiral PHASEtransitionRHIC: may establishevidence for theQCD critical end pointIs physics on the freeze-outline sensitive to criticalbehavior?F. Karsch, RHIC&AGS,BNL 20152

Exploring the QCD phase diagram2.0 1.3 0.7X. Luo (STAR Collaboration),PoS CPOD2014 (2014) 019observableconsequences:conserved charge fluctuationcontrolled by(.and)F. Karsch, RHIC&AGS,BNL 20153

Exploring the QCD phase diagram2.0 1.3 0.7equilibriumthermodynamics:Where is thecritical point?conserved charge fluctuationcontrolled byF. Karsch, RHIC&AGS,BNL 20154

LGT attempts to find the critical pointP. deForcrand, O. Philipsen, 2002Z. Fodor, S. Katz. 2001, 2004these calculations were possiblebecause(I) the lattices were coarse,(II) the discretization schemeswere crudecritical point or breakdown of thereweighting approach (loosing the overlap) ?S. Ejiri, PRD69, 094506 (2004)since 10 years no progress along this lineF. Karsch, RHIC&AGS,BNL 20155

Exploring the QCD phase diagramMore moderate questions:Can we understand the systematicsseen in cumulants of charge fluctuationsin terms of QCD thermodynamics ?How far do we get with low orderTaylor expansions of QCD in explainingthe obvious deviations from HRG modelbehavior ?2.0 1.3 0.7equilibriumthermodynamics:Where is thecritical point?conserved charge fluctuationcontrolled byF. Karsch, RHIC&AGS,BNL 20156

Exploring the QCD phase diagramMore moderate questions:preliminary answer (modulo wellknown details like proton vs baryonfluctuations, acceptance cuts, .)ForStructure of net-electric charge andnet-proton cumulants is inconsistentwith HRG thermodynamics, but caneventually be understood2.0 1.3 0.7equilibriumthermodynamics:Where is thecritical point?conserved charge fluctuationcontrolled byF. Karsch, RHIC&AGS,BNL 20157

Exploring the QCD phase diagramMore moderate questions:preliminary answer (modulo wellknown details like proton vs baryonfluctuations, acceptance cuts, .)?2.0 1.3 0.7equilibriumthermodynamics:ForStructure of net-electric charge andnet-proton cumulants is inconsistentwith HRG thermodynamics, but caneventually be understood in terms ofQCD thermodynamics in a 6th-orderTaylor expansionWhere is thecritical point?conserved charge fluctuationcontrolled byF. Karsch, RHIC&AGS,BNL 20158

Taylor expansion of the pressuregeneralized susceptibilities:conserved charge fluctuations:F. Karsch, RHIC&AGS,BNL 20159

Strong interaction matter in the crossover region (close to the freeze-out line) isnot a Hadron Resonance Gas – although HRG sometimes is a good approximationSome 4th and 6th order cumulantsF. Karsch, RHIC&AGS,BNL 201510

Equation of state of (2 1)-flavor QCD:for simplicity:F. Karsch, RHIC&AGS,BNL 201511

Equation of state of (2 1)-flavor QCDpressure,pressure entropy & energy densityA. Bazavov et al. (hotQCD) ,Phys. Rev. D90 (2014) 094503– improves over earlier hotQCD calculations:A. Bazavov et al., Phys. Rev. D80, 014504 (2009)– consistent with results from Budapest-Wuppertal(stout): S. Borsanyi et al., PL B730, 99 (2014)– up to the crossover region the QCD EoS agrees quite well with hadron resonance gas(HRG) model calculations; However,However QCD results are systematically above HRGF. Karsch, RHIC&AGS,BNL 201512

Equation of state of (2 1)-flavor QCD:variance of net-baryonnumber distributionleading order correction agrees wellwith HRG in crossover regionkurtosis*variance 20% deviations from HRG incrossover regionF. Karsch, RHIC&AGS,BNL 201513

Equation of state of (2 1)-flavor QCD:estimating thecorrection:The EoS is well controlled forF. Karsch, RHIC&AGS,BNL 201514

Conserved charge fluctuations and freeze-outTaylor coefficients control also leading termsin several ratios of conserved charge fluctuationsF. Karsch, RHIC&AGS,BNL 201515

Conserved charge fluctuations and freeze-outfor simplicity:freeze-out lineratio of cumulants on ''a line'' in theplane (NLO Taylor expansion)alleventually need tobe expanded in TF. Karsch, RHIC&AGS,BNL 201516

Conserved charge fluctuations and freeze-outfor simplicity:freeze-out lineratio of cumulants on ''a line'' in theplane (NLO Taylor expansion)for the purposeof this talk:alleventually need tobe expanded in TF. Karsch, RHIC&AGS,BNL 201517

QCD vs. HRG and simple scaling argumentsa simple question: How do cumulant ratios change when we increase the baryonchemical potential, in particularor?F. Karsch, RHIC&AGS,BNL 201518

QCD vs. HRG and simple scaling argumentsa simple question: How do cumulant ratios change when we increase the baryonchemical potential, in particular?ori) HRG:for allii) 3-d, Z(2) Ising universality,e.g. sigma models with a critical point:M. Stephanov, PRL 102 (2009) 032301;S. Mukherjee, R. Venugopalan, Y. Yin,arXiv:1506.00645correlation length diverges at the critical point,higher moments diverge with larger exponents200 62 39 27 20117.7F. Karsch, RHIC&AGS,BNL 201519

QCDa simple QCD prediction:for smallQCD:LO Taylor expansion:F. Karsch, RHIC&AGS,BNL 201520

QCDa simple QCD prediction:for smallQCD:LO Taylor expansion:atfit:caution: net-proton is only aproxy for net-baryonF. Karsch, RHIC&AGS,BNL 201521

QCDa simple QCD prediction:for smallQCD:LO Taylor expansion:– What does the curvature inthe data tell us?– Is it consistent with QCDexpectations?caution: net-proton is only aproxy for net-baryonF. Karsch, RHIC&AGS,BNL 201522

Skewness in Next-to-Leading OrderNLO Taylor expansioncurvature is negativeL.Chen (STAR Collaboration),Nucl. Phys. A904-905 (2013) 471c,F. Karsch, RHIC&AGS,BNL 201523

Consequences for the kurtosis on the freeze-out linein NLO Taylor expansionF. Karsch, RHIC&AGS,BNL 201524

Consequences for the kurtosis on the freeze-out linein NLO Taylor expansiontwice the curvature term ofF. Karsch, RHIC&AGS,BNL 201525

Consequences for the kurtosis on the freeze-out linein NLO Taylor expansionF. Karsch, RHIC&AGS,BNL 201526

Generic structure of 6th order cumulants6-th order net ''up-ness'' fluctuations – unlike baryon number fluctuationsstatistically under control;– qualitatively similar6th order cumulantschange rapidly in thecrossover region6th order fluctuations change sign in the transition region at– it is conceivable that they stay small on the freeze-out lineat small values of the chemical potential– they are expected to be large and positive close to the critical pointF. Karsch, RHIC&AGS,BNL 201527

Consequences for the kurtosis on the freeze-out lineintercept atquadratic fit:determines freeze-out temperaturesmaller ratio largerF. Karsch, RHIC&AGS,BNL 201528

Cumulant ratios of conserved net-charge fluctuationswhereare fixed through strangenessneutrality and charge/baryon numberconstraints realized in a HICF. Karsch, RHIC&AGS,BNL 201529

Freeze-out parameter from conserved charge fluctuationscumulant ratios of electric charge fluctuationsconstraints freeze-out temperaturedetermines freeze-out chemical potentialBI-BNL, PRL 109, 192302 (2012)S. Mukherjee, M. Wagner, PoS CPOD2013 (2013) 039F. Karsch, RHIC&AGS,BNL 201530

Mean net-charge over variance ratioslattice QCD (preliminary)NLO LQCD correction assumesnegligible contribution fromcurvature of the freeze-out linenet-proton fluctuations as a proxy for net-baryonfluctuations:systematic effects in some observables;may shift freeze-out temperature 5 MeVF. Karsch, RHIC&AGS,BNL 201531

Taylor expansion of the pressure and critical pointestimator for the radius of convergence:for simplicity :– radius of convergence correspondsto a critical point only, iffif not:– radius of convergencedoes not determinethe critical point– Taylor expansion can not beused close to the critical pointforcesandto be monotonically growing withatF. Karsch, RHIC&AGS,BNL 201532

Estimates of the radius of convergencea challenging prediction fromsusceptibility series forstandard staggered fermions:suggests large deviations fromHRG in the hadronic phasehuge deviationsfrom HRG inth6 order cumulants!HRGHRGS. Datta et al.,PoS Lattice2013 (2014) 202suggests a criticalpoint forat present, wecannot rule it out!BNL-Bielefeld-CCNUF. Karsch, RHIC&AGS,BNL 201533

Kurtosis*variance on the freeze-out linechanges sign incrossover region,can get large at lower T(larger)sketchconsistent treatment requiresknowledge of T-dependenceofon the freeze-out lineansatz:and control over NNLO correctionF. Karsch, RHIC&AGS,BNL 201534

Conclusionsin the rangethe pattern seen in the beam energy dependence ofup to 4th order cumulants of electric charge and proton (baryon)number fluctuations can be understood in terms ofQCD equilibrium thermodynamicsexperimental data atwith this patternmay still be consistentQCD equilibrium thermodynamics sets a baseline for thediscussion of systematic effects which, of course, need to betaken into account for a more quantitative comparison– in particular when one wants to understand thesystematic difference between STAR and PHENIX dataF. Karsch, RHIC&AGS,BNL 201535

preliminary PHENIX resultsF. Karsch, RHIC&AGS,BNL 201536

STAR net-proton data (preliminary)X. Luo (STAR Collaboration,PoS CPOD2014 (2014) 019,arXiv:1503.02558F. Karsch, RHIC&AGS,BNL 201537

decreases increases V. Skokov,Quark Matter 2012F. Karsch, RHIC&AGS,BNL 201538

– the QCD phase diagram – EoS at non-zero baryon chemical potential – cumulant ratios of conserved charge fluctuations – the QCD critical point Frithjof Karsch Brookhaven National Laboratory & Bielefeld University Conserved Charge Fluctuations and Correlations from Lattice QCD and the Beam Energy

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