Thermalization of gluon matter including gg-ggg interactions

a r X i v :h e p -p h /0609278v 1 27 S e p 2006

Thermalization of gluon matter including gg ?ggg interactions

A.El a ?,C.Greiner a ?and Z.Xu a ?

a

Institut f¨u r Theoretische Physik,Johann Wolfgang Goethe Universit¨a t Frankfurt,Max-von-Laue Str.1,D-60438Frankfurt,Germany

Within a pQCD inspired kinetic parton cascade we simulate the space time evolution of gluons which are produced initially in a heavy ion collision at RHIC energy.The inelastic gluonic interactions gg ?ggg do play an important role:For various initial conditions it is found that thermalization and the close to ideal ?uid dynamical behaviour sets in at very early times.Special emphasis is put on color glass condensate initial conditions and the ‘bottom up thermalization’scenario.O?-equilibrium 3→2processes make up the very beginning of the evolution leading to an initial decrease in gluon number and a temporary avalanche of the gluon momentum distribution to higher transversal momenta.1.INTRODUCTION

It had been demonstrated that the measured momentum anisotropy parameter v 2at RHIC energy can be well understood if the expanding quark-gluon matter is described by ideal hydrodynamics.This important ?nding suggests that a strongly interacting and locally thermalized state of matter has been created which behaves almost like a perfect ?uid.On the other hand,the initial situation of the quark-gluon system is far from thermal equilibrium.It is thus important to understand how and which microscopic partonic interactions can thermalize the system within a short timescale and can be responsible as well for its (nearly)ideal hydrodynamical behaviour.

A traditional way to study thermalization of particles is to carry out microscopic trans-port simulations.A standard parton cascade analysis incorporating only elastic (and forward directed)2?2collisions described via one-gluon exchange,shows that thermal-ization and early (quasi-)hydrodynamical behaviour (for achieving su?cient elliptic ?ow)can not be built up or maintained,but only if a much higher,constant and isotropic cross section σeff ≈45mb is being employed [1].The gluons being treated as semi-classical degrees of freedom would then aquire a collional width Γ≈nσeff v rel being much larger than the typical energy E ≈3T .The gluons would then resemble very broad excitations and can not be considered at all as quasi-particles,if such large and constant cross sections are employed.

In contrast,a kinetic parton cascade algorithm [2]has beeb deneloped with pertur-bative QCD inspired processes including for the ?rst time inelastic (‘Bremsstrahlung’)

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A.El,C.Greiner and Z.Xu

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.6

0.7

0.8

0.9

1.0

d E

/d y |

(t ) / d E

/d y |

(t =0.5f m /c )

t [fm/c] m inijets p =2.0 GeV

m inijets p =1.5 GeV

m inijets p

=1.4 GeV

m inijets p =1.3 GeV

color glass condensate

hy dro. ~ t Figure 1.Time evolution of nomalized transverse energy at midrapidity.Results are obtained from simulations with di?er-ent initial conditions of gluons for a cen-tral Au Au collision.

1

2

3

4

5

v

2

t [fm/c]

Figure 2.Time evolution of the elliptic

?ow v 2at midrapidity for various impact parameter b .collisions gg ?ggg .The multiparticle back reation channel is treated fully consistently by respecting detailed balance within the same algorithm.The three-body gluonic inter-actions are described by the matrix element |M gg →ggg |2

=

9g

4

(q 2⊥+m 2D )

2

12g 2q 2⊥

Thermalization of gluon matter3

p

[

G

e

V]

p

[

G

e

V]

p

[

G

e

V]

p [GeV]

p

[

G

e

V]

p [GeV]

Figure3.The evolution of the local mo-

mentum occupation within a typical color

glass condensate initial condition taken at

τ0=0.4fm/c is shown for four subsequent

times.

1/

N

d

N/

p

d

p

d

|

[

G

e

V

]

p [GeV]p [GeV]

Figure4.Respective transverse momen-

tum spectrum in the central region.

energies at t=0.5fm/c.One sees a rather unique behavior of a decreasing,normalized dE T/dy|y=0,especially for initial conditions employing minijets with p0=1.3?1.5GeV and CGC with Q s=https://www.sodocs.net/doc/2c7454458.html,parisons to the normalized transverse energy per unit rapidity for an ideal hydrodynamical expansion show that the collective expansion due to the pQCD interactions is quasi ideal at0.5?1.5fm/c.At later times the decrease of dE T/dy|y=0slows down,since the collision rates become smaller,especially in the outer, transversally expanding region.

Taking p0=1.4GeV for the initial minijets,the parton evolution for noncentral col-lisions at RHIC energy is simulated in order to calculate the elliptic?ow parameter v2. Figure2shows the time evolution of v2extracted at midrapidity for various impact pa-rameter b.These calculations are still preliminary.The results gives strong indication that an early pressure is being built up within that pQCD inspired description.The sym-bols in Fig.2mark the time from which the energy density in the central region decreases below1Gev/fm3.If we take the v2values at that marked times as the contribution from the partonic phase,they lie well in the region covered by the experimental data.

2.Thermalization of the color glass condensate

For the initial gluon distribution from the saturation picture we employ an idealized and boost-invariant form[5]and express it as

f(x,p)|z=0=

c

τf

δ(p z)Θ(Q2s?p2T),(2) which is described by Q s,the momentum scale at which gluon distribution saturates. The boost-invariance leads to the equality of momentum and space-time rapidity,i.e.,η=y,for the initial gluons.The value of parameters are taken from[5]:N c=3for

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A.El,C.Greiner and Z.Xu

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

2

4

6

8

10

d N z

e n t r a l /d y

t , fm/c

dN/dy ZENTRAL

1 GeV

2 GeV

3 GeV

Figure 5.Time evolution of particle num-ber per rapidity for various saturation mo-menta Q s .

0.1 1

0.1

1

10

100

T ,G e V

t , fm/c

Temperatur

Q s =1 GeV Q s =2 GeV Q s =3 GeV

Figure 6.Time evolution of the e?ective temperature T =E/3N .SU(3),c =1.3,αs =0.3,and the corresponding formation time,at which the gluon distribution becomes dilute enough,is given as τ0=c/(αs N c Q s ).Hence,the cascade operates from τ0=0.4fm/c for Q s =1GeV and τ0=0.18fm/c for Q s =2GeV and Q s =3GeV.For the calculation depicted in Figs.3and 4the gluons are produced within a transverse radius of 6fm (Au nucleus)and within |η|<3longitudinally.Fig.3states a ?rst dynamical realization of the so called ‘bottom up thermalization’scenario as advocated in [3].The evolution of the momentum occupation is shown for four subsequent times sampled within a space-time rapidity interval of ?η=0.1and a central transverse region of R ≤1.5fm.One recognizes the population of the ‘soft’gluons and a subsequent degradation of the ‘hard’initial gluons.In addition,also harder gluons are produced.All this happens roughly within the ?rst 1fm/c.In the last picture we also see that all particles are clearly more centered around the origin,demonstrating the ongoing cooling and quasi hydrodynamical behaviour from 2to 4fm/c.In Fig.4the transverse momentum spectrum is given for various https://www.sodocs.net/doc/2c7454458.html,paring the p T spectra between 2.0and 4.0fm/c,the local system at the central region clearly is at equilibrium at that later times,and the spectrum steepens continuously,since the thermodynamical work in outward direction cools down the system.

Thermalization somehow resembles the idealistic ‘bottom up scenario’[3]with low momentum gluons being produced and populated.On the other hand,a couple of im-portant di?erences do show up.In Fig.5the time evolution of particle number per rapidity for various saturation momenta Q s is shown.In these simulations strict Bjorken geometry in longitudinal direction is preserved by re?ection at cylindrical boundaries [4].The number of gluons is initially slightly decreasing,although a strong parametric enhancement has been advocated in [3]due to Bremsstrahlung production,which is not observed in the present calculations.This is true for all chosen saturation scales.The initial gluons are seemingly oversaturated,if the system thermalizes rather immediately [4].Only if thermalization would happen on a much longer time scale,the gluon number would become undersaturated at a subsequent time,at which net gluon production would

Thermalization of gluon matter

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0

1

2

3

4 5 6 7 8 9

T *t 1/3,G e V *f m 1/3

t,fm/c

Tt

1/3

, Q s =2 GeV

0 0.2 0.4 0.6 0.8 1

1

2

3

4 5

6

7

8

2

/

t , fm/c

Impulsanisotropie 2

>/

> Q s =2 GeV

aus der Simulation exponentielles Gesetz

Figure 7.Time evolution of the rescaled temperature T ·t

1

6 A.El,C.Greiner and Z.Xu

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