UNIVERSITÀ DEGLI STUDI DI CATANIA INFN SEZIONE DI CATANIA

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

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.

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.

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).

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

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

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( 20 08 ) Chiral phase transition and dynamical generation of the mass. 0 0

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 22... 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

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).

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.

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.

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

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.

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

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

Using the QP-model (in collaboration with C. Ratti) C. Sasaki and K. Redlich, Phys. Rev. C 79, 055207 (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 - =

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...

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.

M T g T

1 s ln s

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.

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!

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)

Results with both freeze-out and no freeze-out No f.o. Au+Au@200 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

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.

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:1009.2664 Au+Au@200AGeV-b=8fm y <1 Hydrodynamics Effect of η /s(t) + f.o. Effect of finite η /s+f.o.

Transport coefficients: shear and bulk viscosity In the relaxation time approximation: C. Sasaki and K. Redlich, PRC 79, 055207 (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.

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 p2 1 2 2 E 2 d p'1 d p'2 2 2 E ' 1 2 2 E ' 2 ε-p=0, Collisions 2 η 0 4 f ' 1 f ' 2 M 1 ' 2 ' 1 2 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 71 064901 (2005).

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