UNIVERSITÀ DEGLI STUDI DI CATANIA INFN SEZIONE DI CATANIA

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1 UNIVERSITÀ DEGLI STUDI DI CATANIA INFN SEZIONE DI CATANIA MOMENTUM ANISOTROPIES IN A TRANSPORT APPROACH V. BARAN, M. DI TORO, V. GRECO, S. PLUMARI

2 Transport Theory with a Mean Field at fixed η/s. Effective field theory: Nambu-Jona Lasinio model. Quasi-particle model to describe a HIC. Conclusions and outlook.

3 Motivation for a kinetic approach: p { p [ p F M M ] } f x, p =C 22 C 2 Free streaming Field Interaction ε P Collisions η 0 It is not a gradient expansion in η/s. Valid at intermediate pt out of equilibrium. Valid at high η/s (to study the effect when η/s increase in the cross over region). Extension to Bulk viscosity ζ (instabilities in hydrodynamics). Include hadronization by coalescence + fragmentation.

4 The Nambu-Jona Lasinio model The chiral symmetry SU(Nf )R SU(Nf )L is exact in the chiral limit. The parameters m, g, are fixed to reproduce m, f, at T=0. Y. Nambu and G. Jona-Lasinio, Phys.Rev.122, 45 (1961); Phys.Rev.124, 246 (1961).

5 T, /V = 2 N f N c 2 N f Nc E p 2 d p d p 2 E ln[ 1 e 1 e E ] M m 2 c 4g M T, =m 4 g N f N c M T, d p 1 [1 f T, f T, ] E 2 p Two effects: ε-p 0 EoS different from the massless free gass (CS<1/). Chiral phase transition and dynamical generation of the mass. 0 0

6 T, /V = 2 N f N c 2 N f Nc E p 2 d p d p 2 E ln[ 1 e 1 e E ] M m 2 c 4g M T, =m 4 g N f N c M T, d p 1 [1 f T, f T, ] E 2 p Two effects: ε-p 0 EoS different from the massless free gass (CS<1/). Chiral phase transition and dynamical generation of the mass. 0 0

7 T, /V = 2 N f N c 2 N f Nc E p 2 d p d p 2 E ln[ 1 e 1 e E ] M m 2 c 4g M T, =m 4 g N f N c M T, d p 1 [1 f T, f T, ] E 2 p Two effects: ε-p 0 EoS different from the massless free gass (CS<1/). Fo do r, J ET P( ) Chiral phase transition and dynamical generation of the mass. 0 0

8 Transport theory Parton Cascade p f X, p =C=C 22 C 2 Model ε-p=0 To start the investigation of the role of the field interaction we use the Boltzmann-Vlasov equation for the NJL model p f X, p M X M X p f X, p =C=C M X =m 4 g N c M X d p 1 [1 f X, p f X, p ] 2 E p X A. Abada and J. Aichelin, Phys.Rev.Lett. 74 (1995) 10. ε-p 0 χ-transition

9 The test particle method A The phase-space distribution function can be written as a sum of delta functions. f r, p, t = r r i t p p i t i =1 Gap-equation M cell =m 2 g M cell [ I, M cell 2 a A cell A cell 1 1 E E i =1 i=1 i i ] I, M cell =2 N f N c d p 1 2 E Hamilton equations for the test paticles. paticles r i = pi / E i p i= r E i coll.=2 g M /E i r coll. Contribution of the NJL mean field i=1,, A Take into account the effects of the collision integral, we use the stochastic algorithm (Z. Xu and C. Greiner).

10 Test of the thermodynamics Cubic box with periodic boundary conditions. At t = 0 fm/c the test particles are randomly distributed in the box. The momenta of the test particles are chosen according to the FermiDirac distribution.

11 Heavy ion collisions at 200 AGeV Simulating a constant η /s with a NJL mean field Massive gas in relaxation time approximation [ ] 2 2 n tot x, t p 2 p x, t = x, t 4 M 15 E E M 0 x, t = 4 x, t n tot x, t p 15 1 T 4 p 2 / E M 2 p 2 / E = 15 v rel n T /s σ is evaluated in such way to keep the η/s of the medium costant during the dynamics. In r Glauber model In p: For pt<2 GeV thermal distribution (at 2 TC). For pt> 2 GeV spectra of minijets.

12 Dynamical evolution of Heavy ion collisions in the NJL model 200 AGeV N g =N q N q Central collision b=0 fm. Non central collision b=7 fm.

13 Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed η /s? S. Plumari et al., PLB689(2010) The NJL mean field reduce the v2: attractive field (the effect is about 15 %). if η/s is fixed the effect of NJL mean field is compensed by the increase of σ. V2 η/s not modified by the NJL mean field dynamics.

14 Next step - use a quasiparticle model with a realistic EoS for a quantitative estimate of η /s to compare with Hydro but still missing the -body collisions and also hadronization

15 Using the QP-model (in collaboration with C. Ratti) U. Heinz and P. Levai, Phys. Rev. C 57, 1879 (1998). T = i =g,q, q i i i p Mi M(T) and B(T) are fitted to reproduce lqcd data on ε. =0, B T Di Mi d k Mi f i k i =0 2 Ei i=u,d,... T, S.Plumari,V.Greco and C.Ratti in preparation. NJL el i= g, q, q 2 D d k E k f k B T 2 0 Thermodynamic consistency mo d 2 k dk f i k B T E i k 0 lqc D-F od QP or - p T = Di

16 Using the QP-model (in collaboration with C. Ratti) C. Sasaki and K. Redlich, Phys. Rev. C 79, (2009). i 1 = Di T i 4 d p p f 1 f i i 2 i 2 Ei M i2 Ei d p p2 2 f 1 f C E T i i S i Ei i Ei T 2 [ ] S.Plumari,V.Greco and C.Ratti in preparation. el Di mo d 1 15 T lqc D-F od QP or - =

17 Test in a Box at equilibrium K 1 M /T e= T M K 2 M /T R =n tot v rel =n tot 4 s0 d s s K 1 s 2 2 M a M b K 2 M b s =[ s M a M b 2 ][ s M a M b 2 ] Work in progress...

18 Conclusions and Outlook NJL chiral phase transition do not modify v2 < > η /s At fixed σ the effect is to reduce the saturation value of the average elliptic flow v2 and of the differential elliptic flow v2(pt). The approach proposed can be generalized to quasi-particle models which are fitted to reproduce the energy density and pressure of lqcd.

19

20 M T g T

21 1 s ln s

22 Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed η /s? S. Plumari et al., PLB689(2010) The NJL mean field reduce the v2: attractive field (the effect is about 15 %). if η/s is fixed the effect of NJL mean field is compensed by the increase of σ. The effect does not depend on the value of η/s. The effect does not depend on the centrality of the collision.

23 Two kinetic freeze-out scheme collisions switched off for ε <ε c=0.7 GeV/fm No f.o. η /s increases in the cross-over region, with a smooth transition between the QGP and the hadronic phase 1 p 1 σ tr = 15 n η / s At 4π η /s ~ 8 viscous hydrodynamics is not applicable!

24 Parton Cascade without a freeze-out v2/ε and v2/<v2> as a function of pt 4π η /s=1 Au+Au & Cu+Cu@200 AGeV Scaling for both v2/<v2> and v2/ε for both Au+Au and Cu+Cu Agreement with PHENIX data for v2/<v2> Bhalerao et al., Phys. Lett. B 627 (2005). G. Ferini et al., Phys. Lett. B 670 (2009)

25 Results with both freeze-out and no freeze-out No f.o. AGeV v 2 / ε scaling broken No f.o. Agreement with Hydrodynamics v 2 /<v 2 > scaling kept! Cascade at finite η /s + freeze-out : V2/ε broken in a way similar to STAR data Agreement with PHENIX and STAR scaling of v2/<v2> Freeze-out + η /s lowers the v2(pt) at higher pt

26 Results for the v4 Relevance of time scale! v4 generated later than v2 : more sensitive to the properties at T Tc. v4 more sensitive to both η/s and the f.o. G. Ferini et al. Phys.Lett. B 670 (2009) 25.

27 Effect of η/s(t) on the anisotropies 4 πη /s V2 develops earlier at higher η/s. 2 QGP 1 1 T/Tc 2 V4 develops later at lower η/s. This gives an higher value of the ratio V4/(V2)2. S. Plumari et al., arxiv: Au+Au@200AGeV-b=8fm y <1 Hydrodynamics Effect of η /s(t) + f.o. Effect of finite η /s+f.o.

28 Transport coefficients: shear and bulk viscosity In the relaxation time approximation: C. Sasaki and K. Redlich, PRC 79, (2009). σ=10 mb NJL M=0 NJL σ=10 mb η/s=1/(4 π) For T>TC the ratios η/s and ζ/s approach the massless gas limit.

29 Parton PartonCascade Cascademodel model If we neglect the field interaction Parton Cascade Model (only collisions) p f X, p =C=C 22 C 2 1 C 22= 2 E1 d p E 2 d p'1 d p'2 2 2 E ' E ' 2 ε-p=0, Collisions 2 η 0 4 f ' 1 f ' 2 M 1 ' 2 ' p ' 1 p ' 2 p 1 p 2 4 For the numerical implementation of the collision integral we use the stochastic algorithm (Z. Xu and C. Greiner). P 22= N 2 2 coll N1 N 2 =vrel 22 t x t 0 x x 0 right solution Z. Xu and C. Greiner, Phys.Rev. C (2005).

30 Test of the collision algorithm R =n tot v rel =n tot 4 s0 d s s K 1 s 2 2 M a M b K 2 M b s =[ s M a M b 2 ][ s M a M b 2 ] n 2 n! n K n z = z 2 n! z 2 2 n 1 /2 d z e