Role of van der Waals interactions in hadron systems: from nuclear matter to lattice QCD

Transcription

1 Role of van der Waals interactions in hadron systems: from nuclear matter to lattice QCD Volodymyr Vovchenko a,b,c a Frankfurt Institute for Advanced Studies b Institute for Theoretical Physics, University of Frankfurt c Taras Shevchenko National University of Kiev Collaborators: Paolo Alba, Dmitry Anchishkin, Mark Gorenstein, and Horst Stoecker Erice School on Nuclear Physics 26 Erice, Sicily, Italy September 9, 26 HGS-HIRe Helmholtz Graduate School for Hadron and Ion Research Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 / 28

2 Strongly interacting matter Theory of strong interactions: Quantum Chromodynamics (QCD) Basic degrees of freedom: quarks and gluons At smaller energies confined into hadrons (baryons and mesons) Big bang Temperature T c hadron gas Quark Gluon Plasma Heavy Ion Collisions Where is it relevant? Early universe Neutron stars Heavy-ion collisions nucleon gas nuclei neutron stars net-baryon density First principles of QCD are rather established, but direct calculations are problematic Phenomenological tools are very useful Experiment: heavy-ion collisions Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 2 / 28

3 QCD equation of state at µ = Lattice simulations provide EoS at µ = 6 non-int. limit.2 χ 4 B /χ2 B hadron resonance gas T c HRG p/t 4 ε/t 4 s/4t T [MeV] stout cont. HISQ, N τ =6 8 data: BNL-Bielefeld-CCNU preliminary.2 free quark gas T [MeV] Common model for confined phase is ideal HRG: non-interacting gas of known hadrons and resonances Good description of thermodynamic functions up to 8 MeV Rapid breakdown in crossover region for description of susceptibilities 2 Often interpreted as clear signal of deconfinement... But what is the role of hadronic interactions beyond normal HRG? Bazavov et al., PRD 9, 945 (24); Borsanyi et al., PLB 7, 99 (24) 2 Ding, Karsch, Mukherjee, IJMPE 24, 57 (25) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 / 28

4 Van der Waals equation Van der Waals equation P(T, V, N) = NT V bn an2 V 2 Formulated in 87. Simplest model which contains attractive and repulsive interactions Contains st order phase transition and critical point Can elucidate role of fluctuations in phase transitions Two ingredients: ) Short-range repulsion: particles are hard spheres, V V bn, b = 4 4πr 2) Attractive interactions in mean-field approximation, P P a n 2 Nobel Prize in 9. Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 4 / 28

5 Van der Waals equation VDW isotherms show irregular behavior below certain temperature T C Below T C isotherms are corrected by Maxwell s rule of equal areas Results in appearance of mixed phase 2. P T=. T= T=.9 T=.8 T gas mixed phase liquid v n Critical point p v =, p C = 2 p v 2 =, v = V /N a 27b 2, n C = b, T C = 8a 27b Reduced variables p = p p C, ñ = n n C, T = T T C Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 5 / 28

6 Nucleon-nucleon interaction Nucleon-nucleon potential: Repulsive core at small distances Attraction at intermediate distances Suggestive similarity to VDW interactions Could nuclear matter described by VDW equation? Standard VDW equation is for canonical ensemble and Boltzmann statistics Nucleons are fermions, obey Pauli exclusion principle Unlike for classical fluids, quantum statistics is important Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 6 / 28

7 VDW equation originally formulated in canonical ensemble How to transform CE pressure P(T, n) into GCE pressure P(T, µ)? Calculate µ(t, V, N) from standard TD relations Invert the relation to get N(T, V, µ) and put it back into P(T, V, N) Consistency due to thermodynamic equivalence of ensembles Result: transcendental equation for n(t, µ) N V n(t, µ) = n id (T, µ ) + b n id (T, µ ), n T µ = µ b b n + 2a n Implicit equation in GCE, in CE it was explicit May have multiple solutions below T C Choose one with largest pressure equivalent to Maxwell rule in CE Advantages of the GCE formulation ) Hadronic physics applications: number of hadrons usually not conserved. 2) CE cannot describe particle number fluctuations. N-fluctuations in a small (V V ) subsystem follow GCE results. ) Good starting point to include effects of quantum statistics. Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 7 / 28

8 Scaled variance in VDW equation New application from GCE formulation: particle number fluctuations Scaled variance is an intensive measure of N-fluctuations σ 2 N = ω[n] N2 N 2 = T ( ) n = T ( 2 ) P N n µ n µ 2 In ideal Boltzmann gas fluctuations are Poissonian and ω id [N] =. ω[n] in VDW gas (pure phases) [ ω[n] = ( bn) 2 2an T T ] T Repulsive interactions suppress N-fluctuations Attractive interactions enhance N-fluctuations N-fluctuations are useful because they Carry information about finer details of EoS, e.g. phase transitions Measurable experimentally Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 8 / 28

9 Scaled variance ω[n] = 9.5 [ ( ñ) ñ 2 4 T. ]. 2 T gas 2 liquid. mixed phase 2 Deviations from unity signal effects of interaction Fluctuations grow rapidly near critical point n V. Vovchenko et al., J. Phys. A 5, 48 (25) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 9 / 28..

10 VDW equation with quantum statistics in GCE Requirements for VDW equation with quantum statistics ) Reduce to ideal quantum gas at a = and b = 2) Reduce to classical VDW when quantum statistics are negligible ) s and s as T Ansatz: Take pressure in the following form,2 p(t, µ) = p id (T, µ ) an 2, µ = µ b p a b n 2 + 2an where p id (T, µ ) is pressure of ideal quantum gas. ( ) p n id (T, µ ) n(t, µ) = = µ + b n id (T, µ ) Algorithm for GCE ) Solve system of eqs. for p and n at given (T, µ) 2) Choose the solution with largest pressure V. Vovchenko, D. Anchishkin, M. Gorenstein, Phys. Rev. C 9, 644 (25) 2 Alternative derivation: K. Redlich, K. Zalewski, arxiv: (26) a= excluded-volume model, D. Rischke et al., Z.Phys. C5, 485 (99) T Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 / 28

11 VDW gas of nucleons: zero temperature How to fix a and b? For classical fluid usually tied to CP location. Different approach: Reproduce saturation density and binding energy From E B = 6 MeV and n = n =.6 fm at T = p = we obtain: a = 29 MeV fm and b =.42 fm E B (MeV) (b) n (fm - ) Mixed phase at T = is specific: A mix of vacuum (n = ) and liquid at n = n r (m ) C O 2 A rg o n W a te r N u c le a r m a tte r a /b (e V ) VDW eq. now at very different scale! Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 / 28

12 VDW gas of nucleons: pressure isotherms P (MeV/fm ) T=2 MeV T=9.7 MeV T=8 MeV T=5 MeV CE pressure p = p id[ T, µ id( n )] bn, T a n n (fm - ) (b) P (MeV/fm ) T=8 MeV T=5 MeV T=2 MeV T=9.7 MeV v (fm ) Behavior qualitatively same as for Boltzmann case Mixed phase results from Maxwell construction Critical point at T c = 9.7 MeV and nc =.7 fm Experimental estimate : T c = 7.9 ±.4 MeV, n c =.6 ±. fm J.B. Elliot, P.T. Lake, L.G. Moretto, L. Phair, Phys. Rev. C 87, (2) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 2 / 28 (a)

13 VDW gas of nucleons: (T, µ) plane T (MeV) Density in (T, µ) plane gas liquid (MeV) Crossover region at µ < µ C = 98 MeV is clearly seen n (fm - ) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 / 28

14 VDW gas of nucleons: (T, µ) plane T (MeV) Boltzmann CP Density in (T, µ) plane gas liquid (MeV) n (fm - ) Boltzmann: T C = 28.5 MeV. Classical VDW does not work! Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 / 28

15 VDW-HRG gas of nucleons: fluctuations T (MeV) Results for σ /hni, Sσ, and κσ 2 Fluctuations diverge at CP (as expected) For higher moments singularity is specific: sign depends on path of approach gas 5 liquid (MeV) S T (MeV) T (MeV) gas 5 liquid gas 5 liquid (MeV) (MeV) Vovchenko, Anchishkin, Gorenstein, Poberezhnyuk, PRC 9, 644 (25) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 4 / 28

16 VDW gas of nucleons: skewness and kurtosis VDW Kurtosis VDW Skewness S T (MeV) T (MeV) gas 5 liquid gas 5 liquid (MeV) (MeV) NJL, J.W. Chen et al., PRD 9, 47 (26) PQM, V. Skokov, QM22 Fluctuation patterns in VDW very similar to effective QCD models Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 5 / 28

17 Net-baryon fluctuations and nuclear matter Are NN interactions relevant for observables in HIC region? Net-nucleon fluctuations within RMF (σ-ω model) of nuclear matter along line of chemical freeze-out Skewness Kurtosis.2. BES/STAR HRG RMF.2. BES/STAR HRG RMF.8.8 S¾.6 κ¾ Collision Energy [GeV] Collision Energy [GeV] K. Fukushima, PRC 9, 449 (25) A notable effect in fluctuations even at µ Reconciliation of HRG with nuclear matter can be interesting Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 6 / 28

18 Van der Waals interactions in HRG Simplest generalization of VDW nuclear matter model to full HRG: Similar VDW interactions between baryons and baryons The baryon-antibaryon, meson-meson, and meson-baryon VDW interactions are neglected Baryon VDW parameters extracted from ground state of nuclear matter (a = 29 MeV fm, b =.42 fm ) Three independent subsystems: mesons + baryons + antibaryons p(t, µ) = P M (T, µ) + P B (T, µ) + P B(T, µ), P M (T, µ) = j M p id j (T, µ j ) and P B (T, µ) = j B p id j (T, µ B j ) a nb 2 n B (T, µ) = ( b n B ) j B nj id (T, µ B j ). In this simplest setup model is essentially parameter-free Transcendental equations for P B and n B V. Vovchenko, M. Gorenstein, H. Stoecker, arxiv: Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 7 / 28

19 VDW-HRG at µ = : thermodynamic functions Comparison of VDW-HRG with lattice QCD at µ = Pressure p/t 4 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm p /T Energy density ε/t 4 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm VDW-HRG does not spoil existing agreement of Id-HRG with lqcd despite significant EV interactions between baryons Not surprising: matter meson-dominated at µ = ε/t 4 Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 8 / 28

20 VDW-HRG at µ = : speed of sound c 2 S Speed of sound c 2 s = dp dε L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm Monotonic decrease in Id-HRG, at odds with lattice Minimum for EV-HRG/VDW-HRG at 5-6 MeV No acausal behavior, often an issue in models with eigenvolumes Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 9 / 28

21 VDW-HRG at µ = : baryon number fluctuations Susceptibilities: χ BSQ lmn = Net-baryon χ B 2 L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm l+m+n p/t 4 (µ B /T ) l (µ S /T ) m (µ Q /T ) n Baryon-charge correlator χ BQ L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm χ B 2. 5 χ B Q Very different qualitative behavior between Id-HRG and VDW-HRG For χ B 2 lattice data is between Id-HRG and VDW-HRG at high T For χ BQ lattice data is below all models, closer to EV-HRG Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 2 / 28

22 VDW-HRG at µ = : baryon number fluctuations Higher-order of fluctuations are expected to be even more sensitive.4.2. χ B 4 /χb χ B 6 /χb 2 χ B 4 /χb χ B 6 /χb L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G.2 E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm Id -H R G -.5 E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm χ B 4 deviates from χb 2 at high enough T, stays equal in Id-HRG Cannot be related only to onset of deconfinemnet VDW-HRG predicts strong non-monotonic behavior for χ B 6 /χb 2 Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 2 / 28

23 VDW-HRG at µ = : net-light and net-strangeness Bazavov et al., PRL, 72 (24) 2 Alba et al., arxiv: and P. Alba s talk Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28 χ L L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm Net-strangeness χ S 2 Underestimated by HRG models, similar for χ BS Extra strange states? χ S 2 Net number of light quarks χ L 2 L = (u + d)/2 = (B + S)/2 Improved description in VDW-HRG Weaker VDW interactions for. strange baryons? L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm

24 VDW-HRG at µ = : net-light and net-strangeness χ L 4 /χl χ L 4 /χl 2 L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G E V -H R G, b =.4 2 fm.2 V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm χ S 4 /χs χ S 4 /χs 2 L a ttic e (W u p p e rta l-b u d a p e s t) Id -H R G E V -H R G, b =.4 2 fm.2 V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm Lattice shows peaked structures in crossover regions Not at all reproduced by Id-HRG, signal for deconfinement? Peaks at different T for net-l and net-s flavor hierarchy? 2 VDW-HRG also shows peaks and flavor hierarchy cannot be traced back directly to deconfinement S. Ejiri, F. Karsch, K. Redlich, PLB 6, 275 (26) 2 Bellwied et al., PRL, 222 (2) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, 26 2 / 28

25 VDW-HRG: extensions χ B 4 /χb Effect of reducing VDW interactions involving strange hadrons times smaller EV for strange baryons Small EV for mesons Illustrative calculation (preliminary!) Most observables improved L a ttic e Q C D Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm V D W -H R G, b l =.4 2 fm, a = 2 9 M e V fm, a l = 2 9 M e V fm , b S =. 7 fm, b M =. fm χ B 2 χ u s L a ttic e (W u p p e rta l-b u d a p e s t) L a ttic e (H o tq C D ) Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm V D W -H R G, b l =.4 2 fm L a ttic e Q C D Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm V D W -H R G, b l =.4 2 fm, a = 2 9 M e V fm, a l = 2 9 M e V fm, a = 2 9 M e V fm, a l = 2 9 M e V fm, b S =. 7 fm, b M =. fm , b S =. 7 fm, b M =. fm Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

26 VDW-HRG: influence on hadron ratios VDW interactions change relative hadron yields Thermal fit to ALICE hadron yields: from pions to Ω A L IC E -5 %, N d o f = 8 6 χ Id -H R G E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm Fit quality slightly better in EV-HRG/VDW-HRG vs Id-HRG but very different picture! All temperatures between 5 and 2 MeV yield similarly fair data description in VDW-HRG Results likely to be sensitive to further modifications, e.g for strangeness Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

27 VDW-HRG at finite µ B Net-baryon fluctuations in T -µ plane: χ B 4 /χb 2 6 Id-HRG κσ 2 6 VDW-HRG κσ µ B (M e V ) µ (M e V ) Almost no effect in Id-HRG, only Fermi statistics Rather rich structure for VDW-HRG Likely relevant for net-baryon fluctuations in HIC Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

28 Summary VDW equation provides simple insight on fluctuations near CP Nuclear matter can be described as VDW equation with Fermi statistics VDW interactions between baryons have strong influence on fluctuations of conserved charges in the crossover region within HRG VDW-HRG captures basic features of both lattice results at µ = and nuclear matter properties Freeze-out parameters extracted from thermal fits within ideal HRG are not unique, procedure sensitive to modeling of VDW interactions Interpretation of results obtained within standard ideal HRG should be done with extreme care Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

29 Summary VDW equation provides simple insight on fluctuations near CP Nuclear matter can be described as VDW equation with Fermi statistics VDW interactions between baryons have strong influence on fluctuations of conserved charges in the crossover region within HRG VDW-HRG captures basic features of both lattice results at µ = and nuclear matter properties Freeze-out parameters extracted from thermal fits within ideal HRG are not unique, procedure sensitive to modeling of VDW interactions Interpretation of results obtained within standard ideal HRG should be done with extreme care Thanks for your attention! Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

30 Backup slides Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

31 Skewness Higher-order (non-gaussian) fluctuations are even more sensitive Skewness: Sσ = ( N) σ 2 = ω[n] + T ( ) ω[n] asymmetry ω[n] µ T S T gas metastable - unstable metastable liquid 2 Skewness is Positive (right-tailed) in gaseous phase Negative (left-tailed) in liquid phase n Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, /

32 Kurtosis Kurtosis: κσ 2 = ( N)4 ( N) σ 2 - peakedness 4.. T 2 gas metastable - - unstable 2 n metastable liquid Kurtosis is negative (flat) above critical point (crossover), positive (peaked) elsewhere and very sensitive to the proximity of the critical point V. Vovchenko et al., J. Phys. A 5, 49 (26) Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, /

33 VDW gas of nucleons: scaled variance Scaled variance in quantum VDW: [ ω[n] = ω id (T, µ ) ( bn) 2 2an T ω id(t, µ ) ] T (MeV) gas liquid (MeV)..... Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

34 VDW gas of nucleons: skewness and kurtosis T (MeV) Skewness Sσ = ω[n] + T ( ) ω[n] ω[n] µ 5 - gas (MeV) - liquid T S T (MeV) Kurtosis ( ) [Sσ] κσ 2 = (Sσ) 2 + T µ (MeV) For skewness and kurtosis singularity is rather specific: sign depends on the path of approach V. Vovchenko et al., Phys. Rev. C 92, 549 (25) gas liquid T Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28

35 VDW-HRG at µ = : baryon number fluctuations χ B 6 /χb 2 BNL-Bielefeld-CCNU VDW-HRG χ B 6 /χb Id -H R G -.5 E V -H R G, b =.4 2 fm V D W -H R G, b =.4 2 fm, a = 2 9 M e V fm Volodymyr Vovchenko (FIAS, Frankfurt & Kiev University) Erice 26 September 6-24, / 28