Thermalization of Color Glass Condensate within Partonic Cascade BAMPS and Comparison with Bottom-Up Scenario.

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1 Thermalization of Color Glass Condensate within Partonic Cascade BAMPS and Comparison with Bottom-Up Scenario. Shear viscosity from BAMPS Andrej El Zhe Xu Carsten Greiner Institut für Theoretische Physik Frankfurt am Main 18 th July 2007

2 Outline Saturation scenario: gluon distribution in hadrons at high energies Color Glass Condensate Bottom-Up scenario Parton Cascade BAMPS. Initial conditions and technical details Results: Thermalization process and time Results: Shear viscosity from BAMPS Summary and Outlook

3 Motivation v 2 can be well understood if ideal hydro is applied. This suggests a strongly interacting and local thermalized matter. By what mechanisms can thermalization on a short timescale be achieved? Solve Boltzmann Equation in a Partonic Cascade Model with expanding geometry and pqcd calculated crossections for both elastic and inelastic processes. (inelastic scattering is factor 5 more effective for isotropization in momentum space, than elastic : Z. Xu and C. Greiner,hep-ph/ )

4 Hadrons at High Q²: higher resolution Low Q² High Q² L.McLerran, hep-ph/

5 Saturation Scenario: Small x partons are produced in cascades via QCD-Bremsstrahlung At low values of x, QCD evolution predicts a fast increase of the parton densities which violates unitarity constraints (N*A parton < A Nucleon ). But: QCD evolution equations neglect interactions between partons in the cascades. If parton density is high, partons from different cascades may interact and merge. These interactions will lead to saturation of the parton densities.

6 Saturation Result of competition between gluon bremsstrahlung and recombination effects. when the density of produced partons is very high the recombination processes (~α s n²) compete with the emission of new partons (~n) and the gluon number saturates. Saturation condition: α s n² ~ n i.e. n ~ 1/α s

7 Saturation Momentum Saturation momentum scale: cross section parton-probe(photon): σ~α s /Q² Unitarity limit: N parton σ= N parton α s /Q²= πr A2 ~A 2/3 with N~A follows Q S2 ~α s A 1/3 larger for heavy nuclei. Saturation momentum is dependent on x and A. <Q s2 (x = 0.01)> = 2.0 GeV² Au+Au@RHIC At small enough x, all hadrons become the same: specific properties of the hadrons (like their size or atomic number A) enter only via the saturation scale Q s ²(x,A).

8 QCD at high energies can be described as a many-body theory of partons which are weakly coupled albeit non-perturbative due to the large number of partons. This system is called Color Glass Condensate

9 Color Glass Condensate Color, since the gluons are colored. Glass because of the strong analogy to actual glasses. Gluonic system is disordered and evolves very slowly relative to natural time scales: it is like a solid on short time scales and like a liquid on much longer time scales. Condensate because of very high density of massless gluons whose momenta are peaked about some characteristic momentum. n~1/α s (Bose Condesation)

10 Theory of Color Glass Condensate CGC is an effective theory to describe nucleons and nuclei at high energies within the saturation scenario Dynamical Wee modes Valence modes-are static (Gaussian random) sources for wee modes

11 Color Glass Condensate A simple, idealistic initial parton distribution (Current work) f x, p x=0 = c s N c 1 0 p z Q s 2 p t 2 Boost-invariant initial condition. Saturation for p t <Q S, gluons with higher momenta do not contribute Kharzeev, Levin, Nardi approach (Future work): x ;k t2 ;r t ~ 1 dn d 2 r dy ~ d 2 p t p t 2 2 Q S s Q 2 S max Q 2 S,k 2 t d 2 k t s A x 1,p 2 t B x 2, p t k t 2 with x 1,2 =p t exp ±y / s Saturation for p t <Q S. ~1/p t 4 for p t >Q S

12 Bottom-Up Scenario of thermalization v 2 measured at RHIC can be understood if the expanding quark-gluon matter is described by ideal hydrodynamics. This suggests that fast thermalization of QGP. The initial situation of the QGP is far from thermal equilibrium => it is important to understand how and which microscopic partonic interactions can thermalize the system within a short timescale.

13 Bottom-Up Scenario of thermalization Baier, Mueller, Schiff, Son Phys.Lett.B 502, 51(2001) Q 1 S t 3/2 1 Q S Initial system is dominated by hard gluons with p t ~Q S. 3/2 Q S 1 t 5/2 Q S 1 Hard gluons still outnumber soft ones. In inelastic collisions soft gluons are produced. Production of hard supressed by the Landau-Pomeranchuk-Effekt. Number of soft gluons increases. Soft gluons thermalize among themselves and build up a thermal bath. 5/2 Q 1 S t 13/5 1 Q S Hard gluons lose their entire energy to the thermal bath, built up by soft gluons. The system thermalizes. Emphasizes the role of inelastic collisions, though 3->2 collisions not considered Predicts enhancement of total particle number due to soft gluon production Estimated timescale for thermalization 13/5 Q S 1

14 BAMPS Z.Xu, C.Greiner, O.Fochler, A.El Boltzmann Approach for Multi- Parton Scatterings Solving the Boltzmann equation within partonic cascade modell (Monte Carlo simulation) in 1-Dim Bjorken expanding geometry 2<->3 processes included (detailed balance) pqcd calculated crossections

15 Construction 1-Dim expansion: v z = z Boost-invariant tube geometry Within a time step collisions only with particles from the same cell In transversal direction: reflection on cylindrical wall. One wall scattering process per time step Test particle method applied, at least 11 particles/cell Bins are constructed equidistant in longitudinal η-space: particles with similar η come into same bin Stochastical Method applied to calculate collision probabilities

16 Collision Terms in the Boltzmann Equation t p 1 E 1 f 1 r, p,t =C 22 C 23 P 22,23 =v 22,23 t rel N test 3 x 23 = 1 1 d 3 p 1 d 3 p 2 d 3 p 3 M 2s 2 9 2E 1 2E 2 2E 1' 2' p' 1 p' 2 p 1 p 2 p 3 3 Matrix element 2->3: Gunion-Bertsch formula M g g ggg 2 = 9g4 2 s g 2 qt q 2 t m 2 D 2 k 2 t [ k t q t 2 m 2 D ] 1 I P 32 = E 1 E 2 E 3 N test t M 3 x '2 2 ' = 1 16 M 1' 2' I 32 = p 1 1 d d 3 p 2 M 2 6 2E 1 2E 1' 2' 3' p' 1 p' 2 p' 3 p 1 p 2 2

17 Implementation of LPM Effect In a medium the radiation of soft gluons i assumed to be supressed due to the LPM effect. The emission of a soft gluon should be completed before it scatters again: (formation time < mean free pass Λ g ) M g g ggg 2 = 9g4 2 s g 2 qt q 2 t m 2 D 2 k 2 t [ k k t q t 2 m 2 t g cosh y D ]

18 The initial conditions Color Glass Condensate f x,p = c 1 s N c t p Q 2 z s p 2 t dn 2 N 2 =c R c 1 2 Q d s s N c [ 3; 3] y= p t 2 [0;Q s 2 ] p z =p t sinh y RHIC: Q S =2 GeV (mean value of saturation momentum, small x) LHC: Q S =3-4 GeV dn log N p T dp T basically log d N N d 3 p p T

19 Results. Evolution towards thermal equilibrium Particle Number

20 Results Temperature >first slightly increasing due to particle annihilation >later falling with t 1/3 --->Hydro T* t 1/3 =const T~t 1/3 --->Ideal Hydro behavior System thermalized. Viscous hydro? thermalization times: Q s =4 GeV : ~0.75 fm/c Q s =3 GeV : ~1.0 fm/c 1 ~Q s Q s =2 GeV : ~1.5 fm/c

21 Momentum anisotropy Momentum isotropy 2 Q t = p z E 2 can be fitted via Q t = 1 3 p 2 z E t 1 3 exp t t rel t 0 with relaxation time θ rel Z. Xu and C. Greiner,hep-ph/

22 Transversal momentum spectra for Q S =3 GeV log 10 dn dp t p t d vs p t t=0.1 energy flows into high p t sector at the beginning. no increase of soft particle number t= high p t sector has thermal shape. particle number in soft sector increases.

23 Gluonic Socialism: take energy away from medium and put into the soft sector define 3 energy scales: p T < 1.5 GeV > soft 1.5 GeV < p T < Q S > medium p T > Q S > hard In which sector are gluons annihilated? In which produced and in what amount? What particles are produced in what processes (in middle)? 2->3: produce 1.5 soft particle per collision, from soft, medium, hard. 3->2: produce 0.5 hard particle per collision. annihilate 1 soft, less medium 2->2: produce very few soft and hard from medium

24 Thermalization process: Bottom-Up? Total particle number decreases first. As the sytem is almost thermal it is comparable to the initial number. Initial condition is oversaturated Increase of soft particle number is moderated by 3->2 processes in BAMPS Thermalization of soft and hard+medium sectors on comparable timescales, parallel Transversal spectra achieve a thermal shape in hard sector even earlier than in soft

25 Momentum spectra Transversal (p x ) and longitudinal(p z ) spectra. ~80% of particles isotropic at 1.0 fm/s

26 Quasi hydrodynamical behavior after 0.5 fm/c Transversal spectra have thermal shapes starting with 0.5 fm/c Slopes become steeper----> typical hydro behavior

27 Shear viscosity arxiv: manuscript in preparation Using the standard dissipative hydro: = 4 T xx T yy 2T zz with T ii = 1 V p ii 2 E s=4n n ln with = n n eq s 0.15 In BAMPS η/s proves to be a universal number, if going from RHIC to LHC energy (dependence of α S on Q S NOT considered in simulations) 1/4π from AdS/CFT

28 Summary If a simplified form of CGC applied, thermalization at RHIC energy achieved after ~0.75 fm/c Thermalization time is proportional to 1/Q S, in agreement with Bottom-Up Thermalization of soft and hard momentum scales proceeds on same timescale No enhancement of total particle number observed, in contradiction with Bottom-Up. 3->2 collisions moderate gluon production in soft sector. Quasi hydrodynamical behavior after energy. Shear viscosity/entropy density ratio small ~0.15, close to AdS/CFT limit. Shear Viscosity/entropy density ratio independent of saturation momentum

29 Outlook dn log p T dp T p T Consider the p T factorization (cooperation with HJ Drescher) What is the role of plasma instabilities? (modified Bottom-Up ) Can we understand the results using dissipative hydro? Can we find a transition between microscopic theory(boltzmann) and dissipative Hydro?

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