EXPLORING THE QCD PHASE TRANSITION AT HIGHEST BARYON

Transcription

1 EXPLORING THE QCD PHASE TRANSITION AT HIGHEST BARYON DENSITIES WITH NICA-MPD AND FAIR-CBM David Blaschke Institute for Theoretical Physics, University of Wroclaw, Poland Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia QCD Phase Transition: QCD - Lattice vs. Models χsr, hadron dissociation Observable effects in HIC: Hadron yields, flow Moments of fluctuations Experiments at NICA & FAIR NICA - site and parameters Nuclotron, Collider

2 PARTITION FUNCTION FOR QUANTUM CHROMODYNAMICS (QCD) Partition function as a Path Integral (imaginary time τ = i t, 0 τ β = 1/T ) PS I { β } Z[T,V,µ] = D ψdψdaexp dτ d 3 x L QCD (ψ, ψ,a) QCD Lagrangian, non-abelian gluon field strength: F a µν (A) = µa a ν ν A a µ +g fabc [A b µ,ac ν ] L QCD (ψ, ψ,a) = ψ[iγ µ ( µ iga µ ) m γ 0 µ]ψ 1 4 Fa µν(a)f a,µν (A) Numerical evaluation: Lattice gauge theory simulations (hotqcd, Wuppertal-Budapest) 0 V ε/t 4, p/t ε Stefan-Boltzmann Borsanyi et al., arxiv: T [MeV] Equation of state: ε(t) = lnz[t,v,µ]/ β Phase transition at T c = 155 MeV Problem: Interpretation? ε/t 4 = π2 30 N π 1 (ideal pion gas) ε/t 4 = π2 30 (N G+ 7 8 N Q) 15.6 (quarks and gluons) Hadron resonance gas

3 CHIRAL MODEL FIELD THEORY FOR QUARK MATTER Partition function as a Path Integral (imaginary time τ = i t) { β } Z[T,V,µ] = D ψdψexp dτ d 3 x[ ψ[iγ µ µ m γ 0 (µ+λ 8 µ 8 +iλ 3 φ 3 ]ψ L int +U(Φ)] Polyakov loop: Φ = N 1 c Tr c [exp(iβλ 3 φ 3 )] V Order parameter for deconfinement Current-current interaction (4-Fermion coupling) and KMT determinant interaction L int = M=π,σ,... G M( ψγ M ψ) 2 + D G D( ψ C Γ D ψ) 2 K[det f ( q(1+γ 5 )q)+det f ( q(1 γ 5 )q)] Bosonization (Hubbard-Stratonovich Transformation) Z[T,V,µ] = DM M D D D D e M,D M 2 M 4G M D 2 4G D Tr lns 1 [{M M },{ D },Φ]+U(Φ)+V KMT Collective quark fields: Mesons (M M ) and Diquarks ( D ); Gluon mean field: Φ Systematic evaluation: Mean fields + Fluctuations Mean-field approximation: order parameters for phase transitions (gap equations) Lowest order fluctuations: hadronic correlations (bound & scattering states) Higher order fluctuations: hadron-hadron interactions

4 CHIRAL MODEL FIELD THEORY FOR QUARK MATTER Partition function as a Path Integral (imaginary time τ = i t) β Z[T,V,µ] = D qdqexp dτ d 3 x[ q(iγ µ µ m 0 γ 0 µ)q + Couplings: G π = G σ = G S (chiral symmetry) Vertices: Γ σ = 1 D 1 f 1 c ; Γ π = iγ 5 τ 1 c Bosonization (Hubbard-Stratonovich Transformation) exp [ G S ( qγ σ q) 2] [ ] σ 2 = const. Dσexp + qγ σ qσ 4G S V M=π,σ Integrate out quark fields bosonized partition function Z[T,V,µ] = DσDπ exp { σ2 +π } 4G S 2 Tr lns 1 [σ,π] Systematic evaluation: Mean fields + Fluctuations G M ( qγ M q) 2 ] Mean-field approximation: order parameters for phase transitions (gap equations) Lowest order fluctuations: hadronic correlations (bound & scattering states)

5 MEAN FIELD PLUS (GAUSSIAN) FLUCTUATIONS Separate the mean-field part of the quark determinant Mean-field quark propagator Tr lns 1 [σ,π] = Tr lns 1 MF [m]+tr ln[1+(σ +iγ 5 τ π)s MF [m]] S MF ( p,iω n ;m) = γ 0(iω n +µ) γ p+m (iω n +µ) 2 E 2 p Expand the logarithm: ln(1+x) = n=1 ( 1)n x n /n = x x 2 /2+... Thermodynamic potential in Gaussian approximation Ω(T,µ) = TlnZ(T,µ) = Ω MF (T,µ)+Ω (2) M (T,µ)+... Ω (2) M (T,µ) = N M d 2 p 1 e iνnη ln[1 2G 2 (2π) 3 S Π M ( p,iν n )], N σ = 1, N π = 3 β Mesonic polarization loop Π M ( p,iν n ) = 1 β n e iνn η n d 2 k [ (2π) 3Tr Γ M S MF ( k, iω n )Γ M S MF ( ] k + p,iω n +iν n )

6 GENERALIZED BETH-UHLENBECK EOS: NJL MODEL RESULTS Generalized Beth-Uhlenbeck approach: Schmidt, Röpke, Schulz, Ann. Phys. 202 (1990) 57 Hüfner, Klevansky, Zhuang, Voß, Ann. Phys. 234 (1994) 225

7 GEN. BETH-UHLENBECK EOS: NONLOCAL PNJL RESULTS Nonlocal PNJL model beyond meanfield: 2.5 Blaschke et al., Yad. Fiz. 71 (2008) Radzhabov et al., PRD 83 (2011) P/T MF+1/N c MF 1/N c χpt , Φ Φ MF χpt T [MeV] P/T T [MeV] MF+1/N c MF 1/N c χpt PL-Potential with T 0 = 270 MeV (upper panel), and T 0 = 208 MeV (lower panel) = T [MeV]

8 fm (p) S E = d 4 x NONLOCAL PNJL MODEL VS. LATTICE QCD { ψ(x)( iγ µ D µ + ˆm)ψ(x) G } S 2 [j a(x)j a (x) j P (x)j P (x)]+ U (Φ[A(x)]) S1 S2 S3 Lattice Nonlocal currents j a (x) = d 4 z g(z) ψ j P (x) = d 4 z f(z) ψ ( x+ z ) 2 Formfactors fitted to Lattice results g(q) = Γ a ψ, ( x z ) 2 ( x+ z ) i ( ψ x z ) 2 2 κ p 2 1+α z α m f m (q) m α z f z (q) 1+α z f z (q) α m m α z Z(p) p [GeV] 4 Parappilly et al., Phys. Rev. D 1+α z f(q) = 1+α z f z (q) f z(q) [ f m (q) = 1+ ( ) ] q 2 /Λ 2 3/2 1 0 f z (q) = [ 1+ ( q 2 /Λ 2 1)] 5/2. Noguera, Scoccola, PRD 78, (2008)

9 NONLOCAL PNJL MODEL VS. LATTICE QCD (II) Ω MFA = 4T π 2 c p,n ln [ (ρ c n, p ) 2 +M 2 (ρ c n, p ) ] Z 2 (ρ c n, p ) + σ2 1 +κ 2 p σ 2 2 2G S +U(Φ,T) 0.6 Mass function and WF renormalization M(p) = Z(p)[m+σ 1 g(p)] 0.5 Z(p) = [1 σ 2 f(p)] 1. P/T µ=0.8 T c µ=1.0 T c Finite T, µ formalism: Matsubara (ρ c n, p )2 = [(2n+1)πT iµ+φ c ] 2 + p 2, Lattice QCD, Allton et al. (2003) nonlocal rank-2 PNJL, set C, Log Potential T/T c Contrera, D.B., in preparation Polyakov-loop potential: Dexheimer-Schramm, arxiv: U 3 (Φ,T,µ) = (a 0 T 4 +a 1 µ 4 +a 2 T 2 µ 2 )Φ 2 +a 3 T 4 0 ln(1 6Φ 2 +8Φ 3 3Φ 4 ) Parameters: a 0 = 1.85, a 1 = 1.44x10 3, a 2 = 0.08, a 3 = 0.40

10 NONLOCAL PNJL MODEL: PHASE DIAGRAM AND CP CEP Parameters: T CP = MeV µ CP = MeV µ/t CEP = 1.74 Gustavo Contrera, D.B., in preparation (2011)

11 FROM DIQUARKS TO BARYONS (I) The inverse diquark propagator is then obtained from (S A D ) 1 (k 0,k) = 1 4G D Π A D (k 0,k), Π A D (k 0,k) = d 4 q (2π) 4S Q(q)Σ A (k)s Q (q k)σ A (k) Propagator can be expressed via the spectral density after analytic continuation S A D (z,k) = dω 2π AD (ω,k) z ω, AD (ω,k) = lim ε 0 Similar, baryon propagator and spectral density Further details: S 1 B (P 0,P) = 1 G B Π B (P 0,P), Π B (P 0,P) = 8G 2 D ImΠA D (ω +iε,k) [1 2G D ReΠ A D (ω +iε,k)]2 +[2G D ImΠ A D (ω +iε,k)]2 A=2,5,7 Wang, Wang, Rischke, PLB (2011); arxiv: [nucl-th] Zablocki, Blaschke, Buballa, in preparation (2011) dk 4 (2π) 4 S11,A Q (P k)sa D (k)

12 THERMODYNAMIC POTENTIAL IN TERMS OF PHASE SHIFT Integrating the spectral density over the coupling constant leads to G dg 0 g 2ρg µ,t (ω,p) = i ( 1 2 log G χ ) µ,t(ω +iδ,p) 1 G χ δ µ,t (ω,p). µ,t(ω iδ,p) The in-medium phase shift δ µ,t (ω,p) is the argument of the dynamic pair susceptibility 1 G χ µ,t(ω ±iδ,p) 1 G χ µ,t(ω,p) = e iδ µ,t(ω,p). Thermodynamical potential in Beth-Uhlenbeck type form Ω NSR = d B Ω qfl = d B dω π dω π dp (2π) 3 f B (ω)δ µ,t (ω,p), dp ǫ(ω) (2π) 3 2 [δ µ,t(ω,p) δ 0,0 (ω,p)], where Ω NSR is exactly the Nozieres Schmitt-Rink result in J. Low Temp. Phys. 59 (1985) 195 See also Abuki, NPA 791 (2007) 117 and Zablocki, D.B., Buballa, in prep.

13 COMPOSE - COMPSTAR ONLINE SUPERNOVA EOS General Requirements: Densities: 10 8 n/n 0 10 Temperatures: 0 T 200 MeV Proton fractions: 0 Y p 0.6; β = 1 2Y p New Developments: Dissolution of clusters due to Pauli blocking Realistic high-density modeling: DD-RMF/3FSC PNJL Thermodynamics of 1 st order PT; pasta phases I. For Contributors: How to prepare EoS tables How to submit EoS tables Extending CompOSE II. For Users: Hadronic EoS: Statistical, Skyrme, DBHF,... Quark Matter EoS: Bag, PNJL,... Phase transition: Maxwell, Gibbs, Pasta,...

14 QUARKYONIC PHASE = CHIRAL SYMMETRY + CONFINEMENT

15 APPLICATION OF GBU APPOACH: MOTT-ANDERSON FREEZEOUT beach : hadron resonances QGP cliff : (unmodified) vacuum bound state energies fast chemical equilibration Explanation: Strong medium dependence of rates for flavor (quark) exchange processes quarkyonic "Happy Island" "beach" Reason: lowering of thresholds "cliff" increase of hadron size (Pauli principle) geometrical overlap (percolation) D.B., Berdermann, Cleymans, Redlich, Part. Nucl. Phys. Lett. (2011), arxiv: ; Few Body Syst. (2011) arxiv:

16 IDEA: FREEZE-OUT BY HADRON LOCALIZATION (INVERSE MOTT-ANDERSON MECHANISM) Kinetic freeze-out: τ exp (T,µ) = τ coll (T,µ) Reactive collisions: τ 1 coll (T,µ) = i,j σ ijn j Povh-Hüfner law: σ ij = λ ri 2 r2 j, λ 1 GeV/fm = 5 fm 2 Also for quark-exchange in hadron-hadron scatt. [Martins et al., PRC 51, 2723 (1995)] Pion swelling at χsr: r 2 π (T,µ) = 3 f 2 4π 2 π (T,µ), [Hippe & Klevansky, PRC 52, 2173 (1995)] Use GMOR relation f 2 π(t,µ) = m 0 qq T,µ /M 2 π to connect hadron radii and chiral restoration! r 2 π (T,µ) = 3M2 π 4π 2 m q qq T,µ 1, r 2 N (T,µ) = r2 0 +r2 π (T,µ) ; r π = 0.59 fm, r N = 0.74 fm, r 0 = 0.45 fm Expansion time scale: τ exp (T,µ) = a s 1/3 (T,µ), follows from S = s(t,µ) V(τ exp ) = const and τ exp (T,µ) = a s 1/3 (T,µ). D.B., J. Berdermann, J. Cleymans, K. Redlich, arxiv: (2011)

17 PNJL BEYOND MF: PION (q q) AND NUCLEON (qqq) MEDIUM Idea: melting qq swelling hadrons flavor kinetics = quark percolation freeze-out qq (T,µ) = m 0 Ω(T,µ), Ω(T,µ) = Ω PNJL,MF (T,µ)+Ω meson (T,µ)+Ω baryon (T,µ) Ω meson (T,µ) = dω d 3 k { ω d M π (2π) 3 2 +T ln[ 1 e βω]} A M (ω,k), M=π,... Ω baryon (T,µ) = dω d 3 k { ω [ ] } d B π (2π) 3 2 +T ln 1+e β(ω µ B) +(µ B µ B ) A B (ω,k), B=N,... A M (ω,k) = πδ(ω E M (k))+continuum, A B (ω,k)...analoguous Remove vacuum terms; neglect continuum (for the freeze-out); use GMOR: Mπ 2 fπ 2 = m 0 qq and σ N = m 0 ( m N / m 0 ) = 45 MeV, EnforceM π (T,µ) = const by setting fπ 2(T,µ) = m 0 qq (T,µ)/M π 2, ( BRST, arxiv: ) qq (T,µ) = qq PNJL,MF (T,µ)+ M2 πt 2 8m 0 + σ N m 0 n s,n (T,µ) with the scalar nucleon density n s,n (T,µ) = 2 π 2 0 dpp 2 m N E N (p) {f N(T,µ)+f N (T, µ)} D.B., J. Berdermann, J. Cleymans, K. Redlich, arxiv: (2011)

18 PNJL MODEL BEYOND MF - RESULTS qq = qq PNJL,MF qq = qq PNJL,MF + M2 π T2 8m 0 + σ N m 0 n s,n (T,µ)+... D.B., J. Berdermann, J. Cleymans, K. Redlich, arxiv: (2011)

19 PNJL MODEL BEYOND MF - RESULTS qq = qq PNJL,MF +κ M M 2 π T2 8m 0 +κ B σ N m 0 n s,n (T,µ) qq = qq PNJL,MF + M2 π T2 8m 0 + σ N m 0 n s,n (T,µ)+... D.B., J. Berdermann, J. Cleymans, K. Redlich, arxiv: (2011)

20 PNJL MODEL BEYOND MF VS. PHENOMENOLOGICAL FIT n b = n(t,µ)+ n(t,µ) = i=n,,xb d i dp p 2 2π 2 [ 1 exp(β[e i (p) µ])+1 ] +(µ µ) 0.5 NICA energies T [MeV] ] 3 n+n [fm [MeV] µ B D.B., J. Berdermann, J. Cleymans, K. Redlich, arxiv: (2011)

21 PNJL MODEL BEYOND MF CONDENSATE VS. FREEZE-OUT COND. qq = qq MF [1 T 2 8fπ 2(T,µ) σ ] Nn s,n (T,µ) Mπ 2f2 π (T,µ), n s,π = d π M π T 2 / d π = 8, d N = < 50% T [MeV] [fm/c] τ exp [fm/c] τ coll T [MeV] = 70% > 90% T [MeV] [MeV] µ B 50 1/3 <qq> 50 d π =8, d =20 N d π =8, d =14 N d π =4, d =14 N [MeV] µ B [MeV] µ B D.B., J. Berdermann, J. Cleymans, K. Redlich, arxiv: (2011); Few Body Systems (2011); DOI: /s

22 ε R (T,{µ j }) = LATTICE QCD EOS AND MOTT-HAGEDORN GAS i=π,k,... ε i (T,{µ i })+ r=m,b g r dm m r dsρ(m)a(s,m;t) d 3 p (2π) 3 Hagedorn mass spectrum: ρ(m) p2 +s ( ) exp +δ r p 2 +s µ r T ε/t T H =1.2 T c resonances quarks + gluons T/T c Spectral function for heavy resonances: mγ(t) A(s,m;T) = N s (s m 2 ) 2 +m 2 Γ 2 (T) Ansatz with Mott effect at T = T H = 192 MeV: ( ) 2.5 ( ) 6 ( ) m T m Γ(T) = BΘ(T T H ) exp T H T H No width below T H : Hagedorn resonance gas Apparent phase transition at T c 160 MeV Blaschke & Bugaev, Fizika B13, 491 (2004) Prog. Part. Nucl. Phys. 53, 197 (2004) Blaschke & Yudichev (2006) T H

23 P[T 4 ], ε[t 4 ], c s 2 HYBRID APPOACH: PNJL & MOTT-HAGEDORN RESONANCE GAS ε hybrid (T,{µ j }) = ε PNJL (T,{µ i })+ 5 p/t 4 ε/t 4 c s 2 *30 Borsanyi et al. [Wu-Bu]; r=m,b Hadron resonance gas with broadened resonances (Blaschke, Prorok, Turko) g r ds A(s,m r ;T) d 3 p (2π) 3 p2 +s ( ) exp +δ r p 2 +s µ r T Spectral function for heavy resonances: mγ(t) A(s,m;T) = N s (s m 2 ) 2 +m 2 Γ 2 (T) Ansatz with Mott effect at T = T H = 198 MeV: ( ) 2.5 ( ) 6 ( ) m T m Γ(T) = BΘ(T T H ) exp T H T H Apparent phase transition at T c 165 MeV T H T [MeV] Blaschke & Bugaev, Fizika B13, 491 (2004) Prog. Part. Nucl. Phys. 53, 197 (2004) Blaschke, Prorok & Turko, in preparation

24 HYBRID APPOACH: PNJL & MOTT-HAGEDORN RESONANCE GAS P tot (T,{µ j }) = P PNJL (T,{µ i })+ r=m,b δ r g r dsa r (s,m r ;T) d 3 p (2π) 3Tln { 1+δ r exp ( p2 +s µ r T )} P[T 4 ], ε[t 4 ] Lattice data: Borsanyi et al. ε QGP = ε PNJL + ε 2 P QGP =P PNJL +P 2 P MHRG,21 ε MHRG,21 P tot = P MHRG,21 + P QGP ε tot = ε MHRG,21 + ε QGP Spectral function for hadronic resonances: mγ r (T) A r (s,m;t) = N s (s m 2 ) 2 +m 2 Γ 2 r(t) Ansatz motivated by chemical freeze-out model: Γ r (T) = τ 1 r (T) = h λ < r 2 r > T < r 2 h > T n h (T) Apparent phase transition at T c 165 MeV 3 Hadron resonances present up to T max 250 MeV Blaschke & Bugaev, Fizika B13, 491 (2004) Prog. Part. Nucl. Phys. 53, 197 (2004) T [MeV] Hadronic states abovet c! See also: Ratti, Bellwied et al., arxiv: [hep-ph] Blaschke, Prorok & Turko, in preparation

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30 INVITATION: CONTRIBUTE TO THE NICA WHITE PAPER

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32 VISIT OF RF PRESIDENT V. PUTIN AT NICA SITE,

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38 SUMMARY Generalized Beth-Uhlenbeck approach as microphysical basis to account for hadron dissociation (Mott effect) at extreme temperatures and densities Benchmark: pion and sigma Mott effect within NJL model, revised within nonlocal PNJL model Nonlocal PNJL model calibrated with lattice quark propagator data, EoS at finite T andµ, Phase diagram with critical point Application of GBU to interprete chemical freeze-out as Mott-Anderson localization Effective GBU model description: Mott-Hagedorn resonance gas + PNJL model describes LAttice QCD thermodynamics OUTLOOK: NEXT STEPS... Walecka model as limit of PNJL model: chiral transition effects in nuclear EoS Prospects for HIC (CBM & NICA) and Supernovae: color superconducting (quarkyonic) phases accessible!

39 Invitations: Karpacz Winter School on Theoretical Physics Cosmic Matter in Heavy-Ion Collision Laboratories La dek Zdròj, Poland, February 4-11, karp48 International Conference CompStar: the physics and astrophysics of compact stars Tahiti, June 4-8, Helmholtz International Summer School Dense Matter in HIC & Astrophysics Dubna, Russia, CompStar School & Workshop EoS in Compact Star Astrophysics & HIC Zadar, Croatia, September

40 Invitations: Karpacz Winter School on Theoretical Physics Cosmic Matter in Heavy-Ion Collision Laboratories La dek Zdròj, Poland, February 4-11, karp48 International Conference CompStar: the physics and astrophysics of compact stars Tahiti, June 4-8, Helmholtz International Summer School Dense Matter in HIC & Astrophysics Dubna, Russia, CompStar School & Workshop EoS in Compact Star Astrophysics & HIC Zadar, Croatia, September