Physics Letters B 689 (2010) 1822
a D lyb Ic Id P
nuchicstheo eu+re oe sy
03dollective owsear viscosity
section the v2 does not depend on the details of the collisional and/or eld dynamics and in particularit is not affected signicantly by the chiral phase transition.
2010 Elsevier B.V. All rights reserved.
The ultra-relativistic heavy-ion collisions at high energysNN
0 AGeV represent the main tool to study the formation ande properties of the quarkgluon plasma at high temperature. TheIC program at BNL has shown that the azimuthal asymmetry inomentum space, namely the elliptic ow v2, is the largest everen in HIC suggesting that an almost perfect uid with a veryall shear viscosity to entropy density ratio, /s, has been cre-ed . From simple quantum mechanical considerations  asell as from the study of supersymmetric YangMills theory ine innite coupling limit a lower bound for /s of about 101predicted . Such a value is much lower than any other knownid and in particular smaller than the one of water and even thane superuid He .First developments of relativistic viscous hydrodynamics [7,8]well as parton cascade models  indicate that even a smalls 0.10.2 affects signicantly the strength of v2(pT ) especiallypT > 1 GeV. Therefore it has become mandatory to determinee value of /s of the plasma created at RHIC through the studythe relation between /s and v2 . However viscous cor-
ctions to ideal hydrodynamics are indeed large and a simplelativistic extension of rst order NavierStokes equations is af-
Corresponding author at: INFN-LNS, Laboratori Nazionali del Sud, Via S. Soa, 95125 Catania, Italy.E-mail address: firstname.lastname@example.org (V. Greco).
fected by causality and stability pathologies [13,14]. It is thereforenecessary to go to second order gradient expansion, and in partic-ular the Israel-Stewart theory has been implemented to simulatethe RHIC collisions providing an upper bound for /s 0.4 .Such an approach, apart from the limitation to (2 + 1)D simula-tions, has the more fundamental problem that it is based on agradient expansion at second order that is not complete . Fur-thermore it cannot be sucient to describe correctly the dynamicsof a uid with large /s as the one in the cross-over region and/orhadronic phase which at least at RHIC still gives a non-negligiblecontribution to v2  that affects the determination of the /sitself .
A relativistic transport approach has the advantage to be a(3+ 1)D approach not based on a gradient expansion that is validalso for large viscosity and for out of equilibrium momentum dis-tribution allowing a reliable description also of the intermediatepT range where the important properties of quark number scaling(QNS) of v2(pT ) have been observed . In this pT region viscoushydrodynamics breaks its validity because the relative deviation ofthe equilibrium distribution function f / feq increases with p2T be-coming large already at pT 3T 1 GeV .
In this perspective transport approaches at cascade level havealready been developed [911,20], but they miss any effect of theeld interactions responsible for the chiral phase transition or con-nement. With this Letter we go one step further including atransport equation self-consistently derived from the NambuJona-Lasinio (NJL) Lagrangian. This allows to study microscopically thetransport behavior of a uid that includes the chiral phase tran-Contents lists availab
oes the NJL chiral phase transition affect
. Plumari a,b, V. Baran b,d, M. Di Toro a,c, G. Ferini a,b,
ipartimento di Fisica e Astronomia, Universit di Catania, Via S. Soa 64, 95125 Catania, ItaNFN, Sezione di Catania, 95125 Catania, ItalyNFN-LNS, Laboratori Nazionali del Sud, Via S. Soa 62, 95125 Catania, Italyhysics Faculty, University of Bucharest, and NIPNE-HH, Bucharest, Romania
r t i c l e i n f o a b s t r a c t
ticle history:ceived 15 January 2010cepted 14 April 2010ailable online 18 April 2010itor: J.-P. Blaizot
ywords:arkgluon plasmalativistic heavy-ion collisions
We have derived and solvedtwo-body collisions and theunderstand if the eld dynammeasure of the asymmetry inv2 and the shear viscosity tsection for the condition of Av2 due to the attractive natukey result is that if /s of th70-2693/$ see front matter 2010 Elsevier B.V. All rights reserved.i:10.1016/j.physletb.2010.04.034at ScienceDirect
he elliptic ow of a uid at xed /s?
, V. Greco a,c,
merically the BoltzmannVlasov transport equations that includes bothiral phase transition by mean of NJL-eld dynamics. The scope is tosupply new genuine effects on the build-up of the elliptic ow v2, amomentum space, and in particular if it can affect the relation between
ntropy ratio /s. Solving the transport equation with a constant crossAu collisions at
sNN = 200 AGeV it is shown a sizable suppression of
f the eld dynamics that generates the constituent mass. However thestem is kept xed by an appropriate local renormalization of the cross
trThorS. Plumari et al. / Physics Le
tion looking at its impact on the relation between the v2 ande /s of the system. The choice of the NJL is mainly drivenits wide and renowned application to study the QCD chiralase transition by mean of effective Lagrangians, even though theermodynamical properties of QCD can be reproduced only qual-tively as briey discussed in the following.The NJL Lagrangian is:
NJL = (i m
) + g
ith denoting a quark elds with N f avors = (u,d, . . .)t ,are the generators of the SU(N f ) group acting in avor space
ith = 1, . . . , (N2f 1). The m = diag(mu,md, . . .) is the cur-nt N f N f quark mass matrix in avor space. In the followinge will refer to the N f = 2,Nc = 3 for calculations. As is wellown the theory is non-renormalizable, hence a cut-off has tointroduced as a free parameter. The numerical results shownthe following are derived using the Buballa parametrization:= 588 MeV, g2 = 2.88, m = 5.6 MeV  that among the va-ety of parameterizations entails a behavior of , P , c2s closer toe lQCD results.A transport theory for the NJL model has been derived in the
osed-time-path formalism combined with the effective actionethod . The main steps of derivation are to perform a Wigneransformation of the Dirac equation of motion and of the relatedp-equation associated to the LNJL , Eq. (1). Then one exploitse semi-classical approximation widely used for applications inavy-ion collisions [23,24] evaluating the expectation value of theur-point fermion interaction in the Hartree approximation (i.e. atean eld level). Finally only the scalar and vector components ofe Wigner function are retained thanks to the spin saturated na-re of the systems we are interested in. One nally obtains theltzmannVlasov transport equations for the (anti-)quark phase-ace distribution function f :
f(x, p) + M(x)M(x)p f (x, p) = C(x, p) (2)
here C(x, p) is the Boltzmann-like collision integral, main ingre-ent of the several cascade codes already developed [25,26,11];(x) represents the local value of the scalar mass that is gen-ated by the chiral symmetry breaking, see Eq. (3). We noticeat respect to the already implemented cascades the NJL dynam-s introduces a new term associated to the mass generation. Also. (2) is formally the same as the widely used relativistic trans-rt approaches for hadronic matter like RBUU, uRQMD, RLV [24,,28], but with a vanishing vector eld. However a key differencethat particles do not have a xed mass and a self-consistentrivation couples Eq. (2) to the mass gap equation of the NJLodel that extended to the case of non-equilibrium can be written:
(2)31 f (x, p) f +(x, p)
d determines the local mass M(x) at the spacetime point x inrms of the distribution functions f (x, p).Eqs. (2) and (3) form a closed system of equations constitutinge BoltzmannVlasov equation associated to the NJL Lagrangianat allows to obtain self-consistently the local effective mass M(x)fecting the time evolution of the distribution function f (x, p).seminal work on the transport equation associated to the NJLnamics was done in Refs. [29,30], but without a collision term,t at nite /s and never applied to the physical conditions of
tra-relativistic heavy-ion collisions. pB 689 (2010) 1822 19
For the numerical solutions of Eqs. (2) and (3) we use aree-dimensional lattice that discretize the space as described inf. [11,26]. The standard test particle methods that sample thestribution function f by mean of an ensemble of points in thease-space is employed. The normalization condition is given by:d f = A( A) = Nq(Nq) with the phase space, A ( A ) thember of test particles (antiparticles) which are inside the con-dered cell and the proper normalization factor that relates thest particles to the real particle number.In such a way it is possible to get a solution of the transportuations propagating the momenta of the test (anti-)particles byean of the relativistic Hamiltons equation. For the numerical im-ementation they can be written in the discretized form as:
(t+)= pi(t) 2t M(ri, t)Ei(t) rM(ri, t) + coll.(
t+)= ri(t)+ 2t pi(t)
ith t = t t and t the numerical mesh time. The term coll.the right-hand side of Eq. (4) indicates the effects of the colli-
on integral as described in Ref. [26,11]. By mean of a reiteratingocedure on time steps one gets the solutions of the transportuation coupling Eq. (4) with the gap equation, Eq. (3) that dis-etized on lattice and for point-like test particles becomes:
here V = AT tanh is the volume of each cell of the spacettice given by AT = 0.5 fm2 the area in the transverse directiond the spacetime rapidity of the center of the cell. The inte-al is instead the vacuum contribution to the gap-equation whicha divergent quantity and it is regularized by a cutoff, and hassimple analytical expression.The spacetime dependence of the mass M(r, t) = m inuences the momenta of the particles because the -te gradient of the condensate generates a force which changese momentum of a particle proportionally to r, see Eq. (4).e last is negative because the phase transition occurs earlierthe surface of the expanding QGP reball. Therefore the phase
ansition which take place locally results in a negative contribu-n to the particle momenta that makes the system more stickyspect to a free massless gas.
Shear viscosity to entropy density ratio
The effect of the NJL mean-eld can be evaluated looking at the-called interaction measure normalized by T 4, 3P
T 4, that gives
e deviation from the free gas relation between the energy den-ty and the pressure P that is also a measure of the breakdownconformal symmetry. In lattice QCD it is known to be quite largeith a peak at Tc and a non-negligible value up to T 23Tc . Ing. 1(left) its behavior is shown for three different NJL parameter-ations. Quantitatively the QCD behavior is correctly reproduced.the following we will use the Buballa parametrization that givese largest values closer to lQCD where however the peak is foundbe about a factor two larger at Tc . This means that the numer-
al results on the impact of the mean eld in the dynamics ofIC collisions is reduced respect to a more realistic case.Our nal goal is to study a uid at xed shear viscosity to en-
opy ratio /s extending the study started at cascade level [11,10].ere the strategy was to normalize locally the cross section inder to x the /s according to the simple relation /s =
/15n valid for a massless gas. Here because of the NJL eld
Fi entfu renLe
m/s of the system:e problem to the evaluation of ,n, T . These are easily calcu-ted analytically (as for Fig. 1) but also numerically summing upe number of the test-particles and their energy in each -cell.evaluate the temperature we exploit the general formula for a
ith e the energy per particle, /n, and Kn(z) the modied Besselnctions of the second kind, z = MT . We know e and M directlyom the code hence we can use Eq. (7) to extract z = MT anderefore the temperature. We have checked that the procedureorks well performing calculation in a box with particles dis-ibuted according to a Boltzmann equilibrium distribution.The behavior of /s for a thermodynamical system is shown in
g. 1(right) for a massless free gas (dashed line) and for the NJLolid line). In both cases the cross section is xed to = 10 mb.e dot-dashed line indicates the lower bound for /s. We seeat with a constant cross section = 10 mb the /s is even
tr, = 115
4p2/E + M2p2/E3( + nT)/s (8)
for a massless gas M 0 (p = E , 3nT ) and Eq. (8) reduce tothe simple relation tr, = 415 p/n used in Refs. [11,34,10,14]for /s = 1/4 . Eq. (8) will allow to extend such studies of a uidat nite viscosity to the case of partons with nite mass.
3. Elliptic ow
We have run the simulations for Au+ Au at sNN = 200 AGeVand b = 7 fm. The density distribution in coordinate space is givenby the standard Glauber model. The maximum initial temperatureis T = 340 MeV and the initial time is 0 = 0.6 fm/c as usuallydone also in hydrodynamical calculations. We follow the dynami-cal evolution of quarks, anti-quarks and gluons. The last has beenincluded, even if they are not explicitly present in the NJL model,with the aim of using a realistic density for both the total andthe (anti-)quark density in the simulation of the collisions. How-S. Plumari et al. / Physics Le
g. 1. Left: Interaction measure shown as a function of temperature T for three differnction of T for various cases as indicated in the legend. (For interpretation of the refetter.)
e particles acquire a mass hence both the viscosity and the en-opy density are modied respect to the simple massless case.e briey discuss the and s for a system of massive particlesriving the pertinent formula to renormalize the cross section order to keep xed locally the /s. Both and s have beenrived for a thermodynamical system and has been studied alsor the case of the NJL model . Here we derive expressions inrms of quantities that can be used easily also in the numericallution of the transport equations. A widely used formula for isduced from the relaxation time approximation, like in Ref. .ter integration by parts it is possible to write the shear viscos-y for the general case of massive relativistic particles in terms oferage quantities that can be easily evaluated numerically:
here = [ntr vrel]1 is the relaxation time, i.e. the time inter-l between two collisions, and n is the total local density. Onen easily see that in the ultra-relativistic limit (M 0) the well-own formula for the shear viscosity, = 415 p/tr , is recovered.The entropy density cannot be related simply to the local den-
ty, s 4n, as for the massless case. A suitable method to evaluatecally s during the dynamical evolution of the collision is basedthe use of the thermodynamical relation sT = + nT that shiftwer than the 1/4 at high T . The lower light lines show the evB 689 (2010) 1822
NJL parameter sets. Right: The /s and /s for the Buballa parametrization as aces to color in this gure legend, the reader is referred to the web version of this
havior of bulk viscosity to entropy ratio /s in the NJL modelr two cases one with = 10 mb (green solid line) as above ande other for an /s xed at the 1/4 value (dot-dashed line). Ofurse for the massless case the bulk viscosity is zero, while forn-vanishing masses there is a link between the and throughe relaxation time . The last results in a smaller growth of /shen /s is xed respect to the case when tr is xed. Howeverboth cases we can see that only at T < 1.1Tc we have a non-nishing /s. This is due to the fact that the is expected toproportional to the deviation of the sound velocity from 1/3,
2s 1/3)2 that in the NJL model is known to occur only veryose to Tc . More importantly for our purposes is that in the NJLe /s remains order of magnitudes lower that rst extrapola-on from lQCD  and also much lower than the smaller valuesed for rst studies with viscous hydrodynamics [19,33]. Accord-g to this study /s has a small impact on elliptic ow  evenalready moderate values, /s 0.04, can better reproduce thee structure of v2(pT ) mass-ordering . However for NJL es-cially at 4/s = 1 the value is so small that we can judgefe to discard any role of /s in the following results for theliptic ow considering also that we will not discuss the mass-dering.We notice that Eqs. (6) and (7) supply the formula for the nor-
alization of the cross section in each -cell in order to keep xeder gluons do not actively participate in the evaluation of the
Fi -ram to c
antendensate 300 MeV. This is due to the fact that a high-T particle collides mainly with the much more abundant parti-es in the bulk. These have an average momentum comparable toe strength of the scalar eld: pT 2T Mc . Therefore the ef-ct of the scalar eld extends thanks to collisions into a rangeite larger than one would naively think and the interplay be-een collisions and mean eld is fundamental. We nd that theesence of an NJL-eld that drives the chiral phase transition sup-ess the v2(pT ) by about 20% at pT > 1 GeV. This would implye need of a parton scattering cross section tr even larger thanat estimated with the cascade model which was already quiterger than the pQCD estimates [25,37]. On the other hand theean eld modies both the local entropy density reduced by theass generation, and the shear viscosity that increases as shownFig. 1(right). Therefore even if the cross section is the same withd without the NJL eld the system evolves with a different /s.nsidering that one of the main goal is to determine the /s ofe QGP we have investigated what is the action of the mean eldce the /s of the system is xed to be the same by mean of theoss section renormalization according to Eq. (8). Therefore weve run simulations with and without the NJL-eld but keeping
eld. We can see that once the /s is xed there is essentially nodifference in the calculations with and without a eld dynamicsincluded. This is seen for both the average v2 and the v2(pT )in Figs. 2(left) and 3 respectively. This is a key result that showsthat even in a microscopic approach that distinguishes betweenthe mean eld and the collisional dynamics the v2(pT ) is mainlydriven by the /s of the uid. In other words we have foundthat in a microscopic approach the /s is the pertinent parame-ter and the language of viscous hydrodynamics is appropriate. Ofcourse this does not mean that v2(pT ) in the transport theory isthe same of the viscous hydrodynamics one, but that, and evenmore importantly, the direct relation between v2(pT ) and /s isquite general and their relation is not modied by the NJL elddynamics. We have checked that this is valid also at other impactparameter (b = 3,5,9 fm) and for larger /s up to 1. This is ofcourse very important for the determination of /s by mean of thedata on elliptic ow and conrms the validity of the studies pur-sued till now even if they miss an explicit mean eld dynamicsand/or the chiral phase transition.
However we notice that while the v2(pT ) appears to be to-tally independent on the presence of the NJL-eld once the /sS. Plumari et al. / Physics Le
g. 2. Left: Average elliptic ow as a function of time for Au + Au collisions in the midomentum for the same case as in the left panel. (For interpretation of the references
iral phase transition, but they simply acquire the mass of thearks not contributing to its determination according to the NJLodel. The justication for this choice relies on the quasi-particleodels that are tted to lQCD thermodynamics [35,36]. One ndssimilar behavior of M(T ) for both gluons and quarks approxi-ately. Of course for a more quantitative calculation a more carefuleatment would be needed but it is not relevant to the mainjective of the present seminal work, considering also that any-ay the NJL model cannot be used for an accurate quantitativeudy.In Fig. 2(left) it is shown the time evolution of the average ellip-ow v2 for a constant transport cross section of tr = 10 mbtypical value that is able to reproduce the amount of v2 ob-rved in experiments . Comparing the two solid lines (blackd green) we can see that the NJL mean eld cause a decrease of2 of about 15%. The reduction of v2 can be expected con-dering that the NJL eld produce a scalar attractive eld thatthe phase transition results in a gas of massive particles. In
g. 3, we show also the elliptic ow at freeze-out as a func-n of the transverse momentum pT . One can see that the rolethe mean eld even increases with momentum affecting alsorticles at a pT quite larger than the energy scale of the scalarnstant locally the /s. isB 689 (2010) 1822 21
pidity region |y| < 1 at b = 7 fm. Right: time evolution of the average transverseolor in this gure legend, the reader is referred to the web version of this Letter.)
g. 3. As in Fig. 2 for the elliptic ow as a function of pT . (For interpretation of theferences to color in this gure legend, the reader is referred to the web version ofis Letter.)
The results for 4/s = 1 are shown by dashed lines in Figs. 2d 3, the (black) dashed line is the case with only the collisionrm (cascade) while the (green) dashed line is the case with thekept xed, the time evolution of v2 still shows a slightly re-
22 S. Plumari et al. / Physics Letters B 689 (2010) 1822
duced elliptic ow at t 35 fm/c. A similar difference can beobserved also in the time evolution of the transverse momentumpT shown in Fig. 2(right). One may ask what is the physicalorigin of such differences. In principle there are two parametersaffecting the v2(pT ) and the pT : the sound velocity cs and thebulk viscosity . As discussed previously it is safe to discard thepossibility of any signicant inuence of the nite /s on our re-sults considering its tiny value in the NJL, see Fig. 1(right). It isinstead reasonable that the weak decrease is due to the decreaseof the sound velocity for NJL at T < 1.1Tc . It is well known thatc2s decreases from 1/3 that is the value of a massless free gas andthat this cause already in ideal hydrodynamics a decrease of theelliptic ow . On the other hand when the bulk of the systemreaches this region most of the v2(pT ) has already been built-uphence the effect of a moderate decrease of cs is quite weak andcould explain the small difference still visible in the time evolu-tion of v2 and pT .
This work for V. Baran is supported in part by the RomanianMinistry for Education and Research under the CNCSIS contractPNII ID-946/2007.
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[3[3[3[3The novelty of the present work is to be the rst study withintransport approach of a uid at nite /s that includes the eldnamics of the chiral phase transition. Generally we nd that ated cross section the effect of the NJL eld is to reduce the el-tic ow by about a 20%. More importantly we can state thate presence of the NJL dynamics does not change the relation be-een the elliptic ow and the /s that remains the same as ine cascade models and at low pT is very close to the one fromdrodynamics [9,11,10,34]. If such a nding is conrmed also formore general class of interacting quasi-particle models it willake much safer and solid the determination of /s by v2(pT ). Inct as we have shown the relation is independent on the micro-opic details of the interaction once the EoS and/or the c2s (T ) hasen xed. This will be investigated in the next future, in fact thenetic theory and the numerical implementation presented heren be easily extended to quasi-particle models that are tted toproduce the energy density and pressure of lQCD results. In suchcase it will be possible also to study the elliptic ow with a real-tic behavior of cs(T ) and the effect of a nite and sizable /s one elliptic ow complementing the study from viscous hydrody-mics that are subject to problems for not too small /s and/orr pT > 3T .4] P. Huovinen, D. Molnar, Phys. Rev. C 79 (2009) 014906.5] H. Song, U.W. Heinz, arXiv:0812.4274, 2008.6] T. Hirano, U.W. Heinz, D. Kharzeev, R. Lacey, Y. Nara, Phys. Lett. B 636 (2006)
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Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed eta/s?IntroductionShear viscosity to entropy density ratioElliptic flowConclusionAcknowledgementsReferences