Thermodynamics. Quark-Gluon Plasma

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1 Thermodynamics of the Quark-Gluon Plasma Claudia Ratti Torino University and INFN, Italy Claudia Ratti 1

2 Quick review of thermodynamics In lectures I and II we saw... QCD and its symmetries Polyakov loop and deconfinement phase transition chiral condensate and chiral phase transition perturbation theory good for large T lattice QCD: non-perturbative method QCD thermodynamics for 2+1 quark flavors with physical masses estimate for QCD transition temperature Claudia Ratti 2

3 Limitations of lattice QCD Lattice QCD is computationally very involved and expensive It can only investigate Euclidean time quantities no dynamical processes It cannot go to finite chemical potential It is considered a black box input: quark masses and coupling output: data for thermodynamics and order parameters which are the relevant degrees of freedom? interpretation of lattice results Claudia Ratti 3

4 Need for phenomenological models capture the main features of QCD in a certain regime symmetries degrees of freedom easier to treat than full QCD do calculations in real time reach regions of the phase diagram where lattice QCD cannot go give an interpretation of lattice data in terms of effective degrees of freedom Claudia Ratti 4

5 What happens belowt c? At lowt andµ = 0, QCD thermodynamics is dominated by pions The interaction between pions is suppressed chiral perturbation theory: pion contribution to the thermodynamic potential the energy density of pions from 3-loop ChPT differs only less than 15% from the ideal gas value P. Gerber and H. Leutwyler (1989) as T increases, heavier hadrons start to contribute fort 140 MeV heavy states dominate the energy density their mutual interactions are proportional ton i n k exp[ (M i +M k )/T]: they are suppressed the virial expansion can be used to calculate the effect of the interaction Claudia Ratti 5

6 Why HRG? In the virial expansion, the partition function can be split into a non-interacting piece and a piece which includes all interactions virial expansion and experimental information on scattering phase shift Prakash and Venugopalan (1992) interplay between attractive and repulsive interaction Interacting hadronic matter can be well approximated by a non-interacting gas of resonances Claudia Ratti 6

7 Features of HRG model An increase of energy in the system has three consequences: a fixed temperature limitt T H ; the momenta of the constituents do not continue to increase; more and more species of heavier particles appear New, non-kinetic way to use energy: increasing the number of species not the momentum per particle T H MeV: critical temperature for transition to quark matter Hagedorn and Rafelsky (1981); Blanchard, Fortunato and Satz (2004). Claudia Ratti 7

8 The pressure can be written as Partition function of HRG model p HRG /T 4 = 1 VT VT 3 i mesons i baryons lnz M m i (T,V,µ X a) lnz B m i (T,V,µ X a), where lnz M/B m i = Vd i 2π 2 0 dkk 2 ln(1 z i e ε i/t ), with energiesε i = k 2 +m 2 i, degeneracy factorsd i and fugacities ( z i = exp ( ) Xi a µ X a)/t. a X a : all possible conserved charges, including the baryon numberb, electric chargeq, strangenesss. F. Karsch, A. Tawfik, K. Redlich; S. Ejiri, F. Karsch, K. Redlich Claudia Ratti 8

9 Hadronic species from the Particle Data Book hadron m i (GeV) d i B i S i I i hadron m i (GeV) d i B i S i I i π N (1535) /2 K /2 π 1 (1600) K /2 (1600) /2 η Λ (1600) ρ (1620) /2 ω η 2 (1645) K /2 N (1650) /2 K /2 ω (1650) N /2 Σ (1660) η Λ (1670) f Σ (1670) a ω 3 (1670) φ π 2 (1670) Λ Ω h N (1675) /2 Σ φ (1680) a K (1680) /2 b K (1680) / /2 N (1680) /2 f ρ 3 (1690) K /2 Λ (1690) K /2 Ξ (1690) /2 f ρ (1700) η (1295) N (1700) /2 π (1300) (1700) /2 Claudia Ratti 9

10 How many resonances do we include? With different mass cutoffs we can separate the contributions of different particles GeV < M < 2 GeV 0.5 GeV < M < 1 GeV Kaon Pion 2 p/t T [MeV] No visible difference between cuts at 2 GeV and 2.5 GeV in our temperature regime We include all resonances withm 2.5 GeV 170 different masses 1500 resonances Claudia Ratti 10

11 Discretization effects M N [GeV] M Ω [GeV] physical point 0.18fm 0.12fm 0.09fm 0.06fm experiment physical point 0.18fm 0.12fm 0.09fm 0.06fm experiment m 2 π [GeV2 ] 1.3 M φ [GeV] M ρ [GeV] m 2 π [GeV2 ] fm 0.09fm experiment physical point 0.18fm 0.15fm 0.12fm 0.09fm 0.06fm experiment m 2 π [GeV2 ] m 2 π [GeV2 ] C. W. Bernard et al., PRD (2001), C. Aubin et al., PRD (2004), A. Bazavov et al., Claudia Ratti 11

12 Non-strange baryons and mesons: Strange baryons and mesons: Hadron masses r 1 m = r 1 m 0 + a 1(r 1 m π ) 2 1+a 2 x + b 1x 1+b 2 x, x = ( a r 1 ) 2 r 1 m Λ (a,m π ) = r 1 m phys Λ r 1 m Σ (a,m π ) = r 1 m phys Σ r 1 m Ξ (a,m π ) = m phys Ξ r 1 m Ω (a,m π ) = r 1 m phys Ω a 1 (r 1 m π ) 2 1+a 2 x + b 1x 1+b 2 x + r 1 (m phys Λ 1+a 2 x a 1 (r 1 m π ) 2 1+a 2 x + b 1x 1+b 2 x + r 1 (m phys Σ a 1 (r 1 m π ) 2 1+a 2 x + b 1x 1+b 2 x + r 1 (m phys Ξ m phys p ) 1+a 2 x 1+a 2 x +a 1 (r 1 m π ) 2 a 1 (r 1 m phys π ) 2 +b 1 x+(m phys Ω Distorted spectrum implemented in the HRG model Assumption: all resonances behave as their fundamental states P. Huovinen and P. Petreczky (2009). ( m phys p ) m phys p ) m s m phys s ), ( ms m phys s ( ms m phys s ), ) m phys ) 1.02x Claudia Ratti 12

13 Results: strangeness susceptibilities χ S n = T n n p(t,µ B,µ S,µ I ) µ n S µx =0 Χ 2 s T Continuum N t N t 12 N t TMeV Χ 2 s T stout continuum HRG physical HRG distorted N t 8 HRG distorted N t 12 asqtad N t 8 asqtad N t 12 p4 N t 8 hisq N t TMeV HRG results in good agreement with stout action asqtad and p4 results show similar shape but shift in temperature HRG results with corresponding distorted spectrum reproduce asqtad and p4 results S. Borsanyi et al., JHEP1009 (2010) Claudia Ratti 13

14 Results: subtracted chiral condensate l,s = ψψ l,t m l ms ψψ s,t ψψ l,0 m l ms ψψ s,0 l, s 1.0 Continuum 0.8 N t 16 N t 12 N t N t TMeV l, s stout continuum HRG physical HRG distorted N t 8 HRG distorted N t 12 asqtad N t 12 asqtad N t 8 p4 N t TMeV ψψ s = ψψ s,0 + ψψ K + i mesons lnz M m i m i m i m s + i baryons lnz B m i m i m i m s. m i m s from fit to lattice data Camalich, Geng and Vacas (2010) S. Borsanyi et al., JHEP1009 (2010) Claudia Ratti 14

15 Results: pressure Nt=8 lattice Nt=6 lattice HRG 2 p/t T [MeV] Very good agreement between HRG and lattice results Claudia Ratti 15

16 What happens abovet c? Claudia Ratti 16

17 Strongly coupled quark-gluon plasma (sqgp) pqcd:η/s one order of magnitude larger than the exp. one N = 4 supersymmetric Yang-Mills theory at strong coupling predicts η/s = 1/4π (Policastro, Son and Starinets, PRL87 (2001)) In the same theory, they predicts/s SB 3/4 Conjecture: this is the lowest possible value No running coupling constant May or may not have qualitative relationship to QCD No limit where theory is QCD Can generate new ways of thinking Claudia Ratti 17

18 Back to sqgp Which are the effective degrees of freedom in the vicinity oft c? Is the interaction between quasiparticles strong enough to bind them? Lattice calculations show that there is aj/ψ abovet c In QGP there is no confinement hundreds of colored channels may have bound states! Claudia Ratti 18

19 How can we test the QGP? Simple QGP: strangeness is carried by strange quarks Baryon number and strangeness are correlated Hadron gas: strangeness is carried mostly by mesons Baryon number and strangeness are uncorrelated Bound state QGP: strangeness is carried mostly by partonic bound states Baryon number and strangeness are uncorrelated We define the following object C BS = 3 BS S 2 V. Koch, A. Majumder, J. Randrup, PRL95 (2005). Claudia Ratti 19

20 Simple estimates In a QGP phase: 3 BS = n s + n s S 2 = n s + n s at allt andµ C BS = 1 In hadron gas phase: 3 BS = 3[Λ + Λ + Σ + Σ +...]+6[Ξ+ Ξ+...]+9[Ω+ Ω+...] S 2 = K + +K +K 0 +Λ+ Λ+... att T c andµ = 0 C BS = 0.66 In bound state QGP: heavy quark, antiquark quasiparticle contribute both to BS and to S 2 bound states of the formsg or sg contribute both to BS and to S 2 bound states of the forms q or sq contribute only to S 2 att = 1.5T c MeV andµ = 0 C BS = 0.61 Claudia Ratti 20

21 Estimates from the lattice C BS = 1+ cus 2 +cds 2 χ s 2 C BS rules out the possibility of bound states soon abovet c quark flavors are uncorrelated the presence of pure gluon clusters cannot be ruled out Claudia Ratti 21

22 Simplest possible choice: Quark and gluon quasiparticle scenario for QGP gas of non-interacting quarks and gluons with effective mass the (T dependent) mass is obtained from a fit to the lattice data d 3 p p 2 P qp (m u,m d,...,t) = d i (2π) 3 3E i (p) f i(p) B(T), i=u,d,s,g wheref i (p) = [1 exp(βe i (p)] 1 are the Bose and Fermi distribution functions, with E i (p) = p 2 +m 2 i ;d i = 2 2 N C for quarks andd i = 2 (NC 2 1) for gluons. Claudia Ratti 22

23 Calculation of transport coefficients Applications to real-time physics bulk and shear viscosities use quark and gluon quasiparticles with effective masses in transport code Claudia Ratti 23

24 The Nambu Jona-Lasinio model J a µ (x) = ψ(x)γ µ λ a 2 ψ(x) δs g2 d 4 x d 4 yj(x)d(x y)j(y) ASSUMPTION : short correlation range for color transport between quarks L int = G c J a µ(x)j µ a(x) LocalSU(N C ) gauge symmetry of QCD GlobalSU(N C ) gauge symmetry of NJL Gap equation: M = m 0 G ψψ Chiral condensate: ψψ = 2iN f N c d 4 p Mθ(Λ 2 p 2 ) (2π) 4 p 2 M 2 +iǫ Spontaneous chiral symmetry breaking...but no confinement Claudia Ratti 24

25 Polyakov loop extended Nambu Jona-Lasinio (PNJL) model Synthesis of 3 Polyakov loop dynamics Aspects of CONFINEMENT Nambu Jona-Lasinio model CHIRAL phase transition Identify order parameters as collective degrees of freedom which drive dynamics and thermodynamics Fix parameter in hadronic and pure gauge sectors make predictions for QCD thermodynamics Pisarski PRD62 (2000); Fukushima PLB591 (2004); C.R., Rößner, Thaler, Weise PRD73 (2006); PRD75 (2007). Claudia Ratti 25

26 Entanglement of Confinement and Spontaneous chiral symmetry breaking Claudia Ratti 26

27 Model predictions for different observables Claudia Ratti 27

28 Need for phenomenological models interpretation of lattice results explore phase diagram calculations in REAL TIME Conclusions belowt c : HRG model abovet c : possibility of bound states ruled out soon abovet c quasiparticle model PNJL model Claudia Ratti 28