arXiv:1001.2736v1 [hep-ph] 15 Jan 2010 Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed η/s ? S. Plumari 1 , $^{2}$, V. Baran 2 , $^{4}$, M. Di Toro1 , $^{3}$, G. Ferini, and V. Greco 1 , 3 1 Dipartimento di Fisica e Astronomia, Universit´ a di Catania, Via S.Sofia 64, 95125 Catania, Italy $^{2}$INFN, Sezione di Catania, 95125 Catania, Italy $^{3}$INFN-LNS, Laboratori Nazionali del Sud, Via S.Sofia 62, 95125 Catania, Italy $^{4}$Physics Faculty, University of Bucharest and NIPNE-HH, Bucharest, Romania (Dated: November 16, 2018) We have derived and solved numerically the Boltzmann-Vlasov transport equations that includes both two-body collisions and the chiral phase transition by mean of NJL-field dynamics. The scope is to understand if the field dynamics supply new genuine effects on the build-up of the elliptic flow v 2 , a measure of the asymmetry in the momentum space, and in particular if it can affect the relation between v 2 and the shear viscosity to entropy ratio η/s . Solving the transport equation with a constant cross section for the condition of Au + Au collisions at √ s$_{NN}$ = 200 AGeV it is shown a sizable suppression of v 2 due to the attractive nature of the field dynamics that generates the constituent mass. However the key result is that if η/s of the system is kept fixed by an appropriate local renormalization of the cross section the v 2 does not depend on the details of the collisional and/or field dynamics and in particular it is not affected significantly by the chiral phase transition. PACS numbers: 25.75.-q , 25.75.Ld, 12.38.Mh, 24.85.+p Introduction - The ultra-relativistic heavy-ion colli- sions at high energy √ s$_{NN}$ ∼ 200 AGeV represent the main tool to study the formation and the properties of the quark-gluon plasma at high temperature. The RHIC program at BNL has shown that the azimuthal asymme- try in momentum space, namely the elliptic flow v 2 , is the largest ever seen in HIC suggesting that an almost perfect fluid with a very small shear viscosity to entropy density ratio, η/s , has been created [1–3]. From simple quantum mechanical considerations [4] as well as from the study of supersymmetric Yang-Mills theory in the infinite coupling limit a lower bound for η/s of about ∼ 10 − 1 is predicted [5]. Such a value is much lower than any other known fluid and in particular smaller than the one of water and even than the superfluid He [6]. mination of the η/s itself [17]. A relativistic transport approach has the advantage to be a 3+1D approach not based on a gradient ex- pansion that is valid also for large viscosity and for out of equilibrium momentum distribution allowing a reli- able description also of the intermediate p$_{T}$ range where the important properties of quark number scaling (QNS) of v 2 ( p$_{T}$ ) have been observed [18]. In this p$_{T}$ region viscous hydrodynamics breaks its validity because the relative deviation of the equilibrium distribution func- tion δf/f$_{eq}$ increases with p 2 T becoming large already at p$_{T}$ ≥ 3 T ∼ 1 GeV [19]. First developments of relativistic viscous hydrodynam- ics [7, 8] as well as parton cascade models [9–11] indicate that even a small η/s ∼ 0 . 1 − 0 . 2 affects significantly the strength of v 2 ( p$_{T}$ ) especially at p$_{T}$ > 1 GeV. Therefore it has become mandatory to determine the value of η/s of the plasma created at RHIC through the study of the relation between η/s and v 2 [12]. However viscous cor- rections to ideal hydrodynamics are indeed large and a simple relativistic extension of first order Navier Stokes equations is affected by causality and stability patholo- gies [13, 14]. It is therefore necessary to go to second order gradient expansion, and in particular the Israel- Stewart theory has been implemented to simulate the RHIC collisions providing an upper bound for η/s ≤ 0 . 4 [15]. Such an approach, apart from the limitation to 2+1D simulations, has the more fundamental problem that it is based on a gradient expansion at second or- der that is not complete [13]. Furthermore it cannot be sufficient to describe correctly the dynamics of a fluid with large η/s as the one in the cross-over region and/or hadronic phase which at least at RHIC still gives a non negligible contribution to v 2 [16] that affects the deter- with ψ denoting a quark fields with N$_{f}$ flavors ψ = ( u, d, ... ) $^{t}$, τ α are the generators of the SU ( N$_{f}$ ) group acting in flavor space with α = 1 , ..., ( N 2 f − 1) . The ˆ m = diag ( m$_{u}$, m$_{d}$, ... ) is the current N$_{f}$ × N$_{f}$ quark mass matrix in flavor space. In the following we will refer The NJL Lagrangian is: In this perspective transport approaches at cascade level have already been developed [9–11, 20], but they miss any effect of the field interactions responsible for the chiral phase transition or confinement. With this let- ter we go one step further including a transport equation self-consistently derived from the Nambu-Jona Lasinio (NJL) lagrangian. This allows to study microscopically the transport behavior of a fluid that includes the chiral phase transition looking at its impact on the relation be- tween the v 2 and the η/s of the system. The choice of the NJL is mainly driven by its wide and renowned applica- tion to study the QCD chiral phase transition by mean of effective lagrangians, even though the thermodynamical properties of QCD can be reproduced only qualitatively as briefly discussed in the following. L NJL = ¯ ψ ( iγ $^{µ}$∂$_{µ}$ − ˆ m ) ψ + g [ ( ¯ ψψ ) 2 +$^{N}$$^{2}$ $_{f}$− 1 ∑ α =1 ( ¯ ψτ $^{α}$iγ 5 ψ ) 2 ] (1)