Cluster Expansion Model

Transcription

1 QCD equation of state at finite baryon density with Cluster Expansion Model Volodymyr Vovchenko Goethe University Frankfurt & Frankfurt Institute for Advanced Studies V.V., J. Steinheimer, O. Philipsen, H. Stoecker, , work in progress XQCD 2018, Frankfurt am Main, Germany May 21, 2018

2 QCD phase diagram: towards finite density μμ BB = 0? TT μμ BB plane? QCD equation of state at μμ BB = 0 available from lattice QCD No direct LQCD simulations at finite μμ BB but recently a lot of LQCD data which helps constrain/formulate phenomenological models 2/19

3 QCD thermodynamics with fugacity expansion No sign problem on the lattice at imaginary μμ BB ii μμ BB Observables obtain trigonometric Fourier series form Baryon density: 3/19

4 QCD thermodynamics with fugacity expansion No sign problem on the lattice at imaginary μμ BB ii μμ BB Observables obtain trigonometric Fourier series form Baryon density: Ideal (Boltzmann) HRG: Massless quarks (Stefan-Boltzmann limit): 3/19

5 Lattice QCD results on Fourier coefficients S. Borsanyi et al. [Wuppertal-Budapest collaboration], QM2017 Consistent with HRG at low temperatures Consistent with approach to the Stefan-Boltzmann limit bb 2 visibly departs from zero above T ~ 160 MeV 4/19

6 HRG with repulsive baryonic interactions Repulsive interactions with excluded volume (EV) [Rischke et al., Z. Phys. C 91] HRG with baryonic EV: V.V., A. Pasztor, Z. Fodor, S.D. Katz, H. Stoecker, Non-zero bb kk (TT) for kk 2 signal deviation from ideal HRG EV interactions between baryons (bb 1 fm 3 ) reproduce lattice trend 5/19

7 Higher-order coefficients from lower ones Feature of the EV-like models: temperature-independent ratios 6/19

8 Higher-order coefficients from lower ones Feature of the EV-like models: temperature-independent ratios Observation: αα 3 and αα 4 are T-independent in lattice data 6/19

9 Higher-order coefficients from lower ones Feature of the EV-like models: temperature-independent ratios Stefan-Boltzmann limit: αα 4 SSSS αα 3 SSSS Observation: αα 3 and αα 4 are T-independent in lattice data Ratios are consistent with Stefan-Boltzmann limit of massless quarks 6/19

10 Cluster Expansion Model CEM a model for QCD equation of state at finite baryon density V. Vovchenko, J. Steinheimer, O. Philipsen, H. Stoecker, , work in progress 7/19

11 Cluster Expansion Model (CEM) Model formulation: Fugacity expansion for baryon number density bb 1 (TT) and bb 2 (TT) are model input All higher order coefficients are predicted: Physical picture: Hadron gas with repulsion at moderate T, weakly interacting quarks and gluons at high T 8/19

12 Cluster Expansion Model (CEM) Model formulation: Fugacity expansion for baryon number density bb 1 (TT) and bb 2 (TT) are model input All higher order coefficients are predicted: Physical picture: Hadron gas with repulsion at moderate T, weakly interacting quarks and gluons at high T Resummed analytic form: Regular behavior at real μμ BB no-critical-point scenario 8/19

13 CEM: Baryon number susceptibility χχ 2 BB Model inputs used: Lattice data from (Wuppertal-Budapest), (HotQCD) CEM-LQCD: bb 1 (TT) and bb 2 (TT) from LQCD simulations at imaginary μμ BB CEM-HRG: bb 1 (TT) and bb 2 (TT) from excluded-volume HRG 9/19

14 CEM: Higher-order susceptibilities χχ 4 BB /χχ 2 BB χχ 6 BB /χχ 2 BB Lattice data from (Wuppertal-Budapest), & (HotQCD) Lattice data on higher-order susceptibilities validate CEM 10/19

15 CEM: Higher-order susceptibilities χχ 8 BB BB χχ 10 To be verified by future lattice data 11/19

16 CEM: Higher-order susceptibilities χχ 8 BB BB χχ 10 Preliminary lattice estimate from (Wuppertal-Budapest) To be verified by future lattice data 11/19

17 Radius of convergence Taylor expansion of the QCD pressure: Radius of convergence rr μμ/tt of the expansion is the distance to the nearest singularity of pp/tt 4 in the complex μμ BB /TT plane, which could point to the QCD critical point Lattice QCD strategy: Estimate rr μμ/tt from few leading terms [M. D'Elia et al., ; S. Datta et al., ; A. Bazavov et al., ] 12/19

18 Radius of convergence Taylor expansion of the QCD pressure: Radius of convergence rr μμ/tt of the expansion is the distance to the nearest singularity of pp/tt 4 in the complex μμ BB /TT plane, which could point to the QCD critical point Lattice QCD strategy: Estimate rr μμ/tt from few leading terms [M. D'Elia et al., ; S. Datta et al., ; A. Bazavov et al., ] CEM: Singularity in the complex plane what are the consequences? 12/19

19 CEM: Structure of Taylor coefficients Negative coefficients appear eventually 13/19

20 CEM: Structure of Taylor coefficients Negative coefficients appear eventually They never settle into a regular (same- or alternate-sign) pattern 13/19

21 Using estimators for radius of convergence a) Ratio estimator: Ratio estimator is unable to determine the radius of convergence, nor to provide an upper or lower bound, so use it with care!! b) Mercer-Roberts estim.: 14/19

22 CEM: Radius of convergence Radius of convergence of Taylor expansion sees Roberge-Weiss transition? At TT > TT RRRR expected μμ BB = ±iiππ [Roberge, Weiss, NPB 86] TT cc Puts CEM in contrast to various critical point estimates TT RRRR ~ 208 MeV [C. Bonati et al., ] 15/19

23 Extracting bb 1 (TT) and bb 2 (TT) from susceptibilities CEM: All χχ kk BB determined by bb 1 and bb 2 at a given temperature Reverse prescription: Extract bb 1 (TT) and bb 2 (TT) from two independent (combinations of) χχ kk BB, assuming that CEM is valid 16/19

24 Extracting bb 1 (TT) and bb 2 (TT) from susceptibilities CEM: All χχ kk BB determined by bb 1 and bb 2 at a given temperature Reverse prescription: Extract bb 1 (TT) and bb 2 (TT) from two independent (combinations of) χχ kk BB, assuming that CEM is valid Example: bb 1 TT, bb 2 (TT) from HotQCD data for χχ 2 BB and χχ 4 BB /χχ 2 BB at μμ BB = 0 HotQCD collaboration, & /19

25 Extracting bb 1 (TT) and bb 2 (TT) from susceptibilities CEM: All χχ kk BB determined by bb 1 and bb 2 at a given temperature Reverse prescription: Extract bb 1 (TT) and bb 2 (TT) from two independent (combinations of) χχ kk BB, assuming that CEM is valid Example: bb 1 TT, bb 2 (TT) from HotQCD data for χχ 2 BB and χχ 4 BB /χχ 2 BB at μμ BB = 0 Implies accuracy of CEM and consistency between LQCD data of different groups 16/19

26 CEM: Observables at finite μμ BB χχ 4 BB /χχ 2 BB χχ 6 BB /χχ 2 BB Non-monotonic μμ BB dependence of χχ 4 BB /χχ 2 BB and χχ 6 BB /χχ 2 BB Ratios consistent with free Fermi gas in the limit of large μμ BB χχ 6 BB /χχ 2 BB 0 in the STAR-BES range 17/19

27 CEM outlook: Full equation of state Integrating the baryon number density one obtains the scaled pressure pp(tt, μμ BB )/TT 4 in CEM which provides the full equation of state within the model Full model input: Fourier coefficients bb 1 (TT) and bb 2 (TT) μμ BB -independent part of pressure pp 0 TT LQCD at imaginary μμ BB LQCD at μμ BB = 0 Useful for hydro at finite baryon density 18/19

28 Summary Lattice QCD data at imaginary μμ constrain phenomenological models Initial deviations from uncorrelated gas of hadrons can be understood in terms of repulsive baryonic interactions Cluster expansion model (CEM) combines hadron gas with deconfinement and is consistent with presently available lattice data, both at μμ = 0 and imaginary μμ BB no signal of CP Radius of convergence of Taylor expansion at μμ = 0 is sensitive to the Roberge-Weiss transition in the complex μμ BB /TT plane 19/19

29 Summary Lattice QCD data at imaginary μμ constrain phenomenological models Initial deviations from uncorrelated gas of hadrons can be understood in terms of repulsive baryonic interactions Cluster expansion model (CEM) combines hadron gas with deconfinement and is consistent with presently available lattice data, both at μμ = 0 and imaginary μμ BB no signal of CP Radius of convergence of Taylor expansion at μμ = 0 is sensitive to the Roberge-Weiss transition in the complex μμ BB /TT plane Thanks for your attention! 19/19

30 Backup slides

31 CEM and effective model Recent CEM developments: [Almasi et al., ] Deviations from CEM ansatz when applied to effective QCD model with chiral criticality (PQM) although model setup is not realistic compared to lattice (TT cc PPPPPP ~ 230 MeV) Deviations of high order Fourier coefficients or susceptibilities from CEM ansatz may signal chiral CP, if there is one

32 CEM and effective model Recent CEM developments: [Almasi et al., ] Deviations from CEM ansatz when applied to effective QCD model with chiral criticality (PQM) Lattice QCD although model setup is not realistic compared to lattice (TT cc PPPPPP ~ 230 MeV) Deviations of high order Fourier coefficients or susceptibilities from CEM ansatz may signal chiral CP, if there is one

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