Vienna 11 Aug 05 1

Rapidity equilibration in d + Au and Au + Au systems

Rapidity equilibration in d + Au and Au + Au systems

Georg WolschinHeidelberg University

Theoretical Physics

Georg WolschinHeidelberg University

Theoretical Physics

Vienna 11 Aug 05 2

TopicsTopics

Relativistic Diffusion Model for R(y;t) with

three sources for d +Au, Au + Au at RHIC energies

Net protons dn/dy and produced charged particles dn/d at 200 A GeV

Importance of the equilibrated midrapidity

source in d+Au compared to Au+Au Conclusion

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Relativistic Diffusion ModelRelativistic Diffusion Model

GW, EPJ A5, 85(1999), 2 sources;

Phys. Lett. B 569, 67 (2003), 3 sources

Nonequilibrium-statistical description of the distribution function R(y,t) in rapidity space,

y = 0.5 · ln((E + p)/(E − p)).

k=1: target-likek=2: proj.-like

k=3: central distr.

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Relativistic Diffusion Model for R(y,t)Relativistic Diffusion Model for R(y,t)

• Nonequilibrium-statistical approach to relativistic many-body collisions

• Macroscopic distribution function R(y,t) for the rapidity y

-The drift function (y-yeq)/y

determines the shift of the mean rapidity towards the equilibrium value

- The diffusion coefficient D accounts for the broadening of the distributions due to interactions and particle creations. It is related

to y via a dissipation-fluct.

Theorem, but enhanced due to longitudinal

expansion.

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Dissipation-fluctuation relation in y- space

Dissipation-fluctuation relation in y- space

The rapidity diffusion coefficient Dy is calculated fromy and the equilibrium temperature T in the weak-coupling limit as

Note that D ~ T|y as in the Einstein relation of Brownian motion.

The diffusion coefficient as obtained from this statistical consideration

is further enhanced (Dyeff) due to collective expansion.

GW, Eur. Phys. Lett. 47, 30 (1999)

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Linear RDM in y- spaceLinear RDM in y- space

The rapidity relaxation time y determines the peak positionsThe rapidity diffusion coefficient Dy determines thevariances.

Note that D ~ T|y as in the Einstein relation of Brownian motion

In a moments expansion and for function initial conditions, the mean values become

and the variances are

<y3(t)>=yeq[1-exp(-t/y)]

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Linear RDM in y-spaceLinear RDM in y-space

The equilibrium value yeq that appears in the drift functionJ(y) = -(yeq-y)/y (and in the mean value of the rapidity) is

obtained from energy- and momentum conservation in thesubsystem of participants as

y eq.12ln

.m 1t ey b .m 2

t ey b

.m 2t ey b .m 1

t ey b

with the beam rapidity yb, and transverse masses

mkt mk

2 p kt 2

yeq = 0 for symmetric systems but ≠ 0 for asymmetric systems

mk = participant masses (k=1,2)

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Linear RDM in y-spaceLinear RDM in y-space

Analytical RDM-solutions with -function initial conditions at the beam rapidities ±yb in the two-sources model

are obtained as

R( ),y 0 = .12

y yb y yb

In the 3-sources model, a third (equilibrium) term appears.Is there evidence for a 3-sources-approach? Yes

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RDM: time evolution of the rapidity density for net protons

RDM: time evolution of the rapidity density for net protons

• Time evolution for 200 A GeV/c S+Au, linear model; function initial conditions

• Selected weighted solutions of the transport eq. at various values of ty

• The stationary solution (Gaussian) is approached for t/Y »1

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Central Collisions at AGS, SPSCentral Collisions at AGS, SPS

Rapidity density distributions evolve from bell-shape to double-hump as the energy increases from AGS (4.9 GeV) to SPS (17.3 GeV)

Diffusion-model solutions are shown for SPS energies:

The data clearly show the drift

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Central Au+Au @ RHIC vs. SPSCentral Au+Au @ RHIC vs. SPS

• BRAHMS data at SNN=200

GeV for net protons• Central 10% of the cross

section• Relativistic Diffusion Model

for the nonequilibrium contributions

• Discontinuous transition to local statistical equilibrium at midrapidity may indicate deconfinement.

GW, PLB 569, 67 (2003) and Phys. Rev. C 69 (2004)

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3-sources RDM for Au+Au3-sources RDM for Au+Au Incoherent superposition of nonequilibrium and equilibrium solutions of the transport equation

yields a very satisfactory representation of the data Disontinuous evolution of the distribution functions with time towards the local thermal equilibrium

distribution

N eq≈ 56 baryons (22 protons)

N1,2≈ 169 baryons (68 protons)

GW, Phys. Lett. B 569, 67 (2003); Phys. Rev. C 69, 024906 (2004)

dN ,y t intdy = .N 1 R 1 ,y t int

.N 2 R 2 ,y t int.N eq R eq( )y

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Au+Au at 62.4/200 A GeVAu+Au at 62.4/200 A GeV

RDM-prediction for 62.4 A GeV

(the lower RHIC energy measured by BRAHMS; data analysis is underway)

Dashed areas are thermal equilibrium results with longitudinal expansion

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

GW, hep-ph/0502123, submitted to Phys. Lett. B

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Produced particles in the 3 sources RDM:

Charged-hadron (pseudo-) rapidity distributions

Produced particles in the 3 sources RDM:

Charged-hadron (pseudo-) rapidity distributions

PHOBOS data at SNN=130, 200 GeV for charged hadronsCentral collisions (0-6%)

Total number of particles in the 3 sources: 448:3134:448 @ 130 GeV 551:3858:551 @ 200 GeV

Most of the produced charged hadrons at RHIC are in the equilibrated midrapidity region(78%)

M. Biyajima et al., Prog. Theor. Phys.Suppl. 153, 344 (2004)

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d+Au 200 GeV charged hadrons:

PHOBOS data d+Au 200 GeV charged hadrons:

PHOBOS data PHOBOS data in

centrality bins

Use 3-sources-RDM with initial particle creation at beam- and equilibrium (yeq)values of the rapidity in the analysis.

10

20

30

40

0B.B. Back et al., PHOBOS collab. 、Phys. Rev. C, in press

(nucl-ex/0409021)

= -ln(tan(/2))

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d+Au 200 GeV charged hadrons:

RDM-analysis d+Au 200 GeV charged hadrons:

RDM-analysis

bk

2.5334.8486.1777.2588.1958.494

1: 0-20 %

2: 20-40%

3: 40-60 %

4: 60-80 %

5: 80-100 %

bmax=8.49 fm

Nk

12.7410.2376.7444.3272.0087.372

Min.bias

y eqj

0.9440.760.5640.3470.1690.664

NGlk

13.58.95.42.91.66.6

B.B.Back et al.

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d+Au 200 GeV charged hadrons:

RDM-analysis d+Au 200 GeV charged hadrons:

RDM-analysis In the RDM-analysis,determine in particularthe importance of theMoving equilibrium(gluonic) source

Data: B.B. Back et al., PHOBOS coll. 、Phys. Rev. C

Data are for= -ln(tan(/2)) ≈

y GW, M.Biyajima,T.Mizoguchi, N.Suzuki, hep-ph/0503212

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d+Au 200 GeV charged hadrons:

RDM-analysis d+Au 200 GeV charged hadrons:

RDM-analysis Mean value and variance ofthe 3 sources for particle production:

Dashed: d- and Au-likeSolid: equilibrated midrapidity sourceTop: total no. of particles

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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Convert to pseudorapidity spaceConvert to pseudorapidity space

The conversion from rapidity (y-) to pseudorapidity space requires the knowledge of the Jacobian

The mean transverse momentum is taken to be 0.4 GeV/c.

The average mass is approximated by the pion mass (0.14 GeV) in the central region. In the Au-like region, it is ≈ 0.17 GeV.

[ < m >≈ mp · Z1/N1ch +m · (N1ch − Z1)/N1ch ≈ 0.17GeV ]

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d+Au 200 GeV RDM-analysis d+Au 200 GeV RDM-analysis

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Data: PHOBOS coll.Phys. Rev. C, in press

GW, MB,TM,NS, hep-ph/0503212

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d+Au indiv. Distributions in the RDM

d+Au indiv. Distributions in the RDM

Data: PHOBOS coll. Phys. Rev. C, in press

GW, MB,TM,NS

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d+Au 200 GeV RDM-analysis d+Au 200 GeV RDM-analysis

Full lines:RDM-results with2-minimization

Data: PHOBOS coll.Phys. Rev. C, in press

GW, MB,TM, NS

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d+Au 200 GeV time developmentd+Au 200 GeV time development

• The mean values of all three -distributions approach eq for large

times

• The actual collision stops before full equilibrium is reached

Num. Calc. By T. Mizoguchi

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Compare d+Au/Au+Au 200 GeV

central source Compare d+Au/Au+Au 200 GeV

central source

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Heavy Relativistic SystemsHeavy Relativistic Systems

Parameters for heavy relativistic systems at AGS, SPS and RHIC energies. The beam rapidity is expressed in the c.m. system. The ratio int/y determines how fast the net-baryon system equilibrates in rapidity space. The effective rapidity width coefficient is y

eff, the longitudinal expansion velocity vcoll.

*At 62.4 GeV, yeff will need adjustment to

forthcoming data.

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ConclusionConclusion

The Relativistic Diffusion Model with 3 sources describes rapidity distributions for net protons in central Au + Au at RHIC.

The centrality dependence of pseudorapidity distributions for produced charged hadrons in d+Au and Au+Au collisions at RHIC energies (200A GeV) is also accurately described in the RDM.

For produced particles, the equilibrated central source carries 78% of the charged-particle content in Au+Au central collisions, but only 17% in d+Au collisions.

This thermalized central source may be interpreted as indirect evidence for a locally equilibrated quark-gluon plasma.