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Rapidity equilibration in d + Au and Au + Au systems

Rapidity equilibration in d + Au and Au + Au systems. Georg Wolschin Heidelberg University Theoretical Physics. Topics. 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

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Rapidity equilibration in d + Au and Au + Au systems

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  1. Rapidity equilibration in d + Au and Au + Au systems Georg Wolschin Heidelberg University Theoretical Physics 11 Aug 05

  2. Topics • 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 11 Aug 05

  3. Relativistic Diffusion Model Nonequilibrium-statistical description of the distribution function R(y,t) in rapidity space, y = 0.5 · ln((E + p)/(E − p)). k=1: target-like k=2: proj.-like k=3: central distr. GW, EPJ A5, 85(1999), 2 sources; Phys. Lett. B 569, 67 (2003), 3 sources 11 Aug 05

  4. 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)/ydetermines 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. 11 Aug 05

  5. Dissipation-fluctuation relation in y- space The rapidity diffusion coefficient Dy is calculated from y and the equilibrium temperature T in the weak-coupling limit as GW, Eur. Phys. Lett. 47, 30 (1999) Note that D ~ T|yas in the Einstein relation of Brownian motion. The diffusion coefficient as obtained from this statistical consideration is further enhanced (Dyeff) due to collective expansion. 11 Aug 05

  6. Linear RDM in y- space In a moments expansion and for function initial conditions, the mean values become The rapidity relaxation time y determines the peak positions The rapidity diffusion coefficient Dy determines the variances. Note that D ~ T|y as in the Einstein relation of Brownian motion <y3(t)>=yeq[1-exp(-t/y)] and the variances are 11 Aug 05

  7. Linear RDM in y-space The equilibrium value yeq that appears in the drift function J(y) = -(yeq-y)/y (and in the mean value of the rapidity) is obtained from energy- and momentum conservation in the subsystem of participants as with the beam rapidity yb, and transverse masses mk = participant masses (k=1,2) yeq = 0 for symmetric systems but ≠ 0 for asymmetric systems 11 Aug 05

  8. Linear RDM in y-space Analytical RDM-solutions with -function initial conditions at the beam rapidities ±yb in the two-sources model are obtained as = In the 3-sources model, a third (equilibrium) term appears. Is there evidence for a 3-sources-approach? Yes 11 Aug 05

  9. 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 t/y • The stationary solution (Gaussian) is approached for t/Y »1 11 Aug 05

  10. Central 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 11 Aug 05

  11. Central Au+Au @ RHIC vs. SPS • BRAHMS data at SNN=200GeV 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) 11 Aug 05

  12. 3-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) 11 Aug 05

  13. Au+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 GW, hep-ph/0502123, submitted to Phys. Lett. B 11 Aug 05

  14. Produced particles in the 3 sources RDM: Charged-hadron (pseudo-) rapidity distributions M. Biyajima et al., Prog. Theor. Phys.Suppl. 153, 344 (2004) PHOBOS data at SNN=130, 200GeV for charged hadrons Central 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%) 11 Aug 05

  15. 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. 40 30 20 10 B.B. Back et al., PHOBOS collab.、 Phys. Rev. C, in press (nucl-ex/0409021) 0 = -ln(tan(/2)) 11 Aug 05

  16. d+Au 200 GeV charged hadrons: RDM-analysis 1: 0-20 % 2: 20-40% 3: 40-60 % 4: 60-80 % 5: 80-100 % bmax=8.49 fm B.B.Back et al. Min.bias 11 Aug 05

  17. d+Au 200 GeV charged hadrons: RDM-analysis In the RDM-analysis, determine in particular the importance of the Movingequilibrium (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 11 Aug 05

  18. d+Au 200 GeV charged hadrons: RDM-analysis Mean value and variance of the 3 sources for particle production: Dashed: d- and Au-like Solid: equilibrated midrapidity source Top:total no. of particles 11 Aug 05

  19. Convert 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 ] 11 Aug 05

  20. d+Au 200 GeV RDM-analysis GW, MB,TM,NS, hep-ph/0503212 Data: PHOBOS coll. Phys. Rev. C, in press 11 Aug 05

  21. d+Au indiv. Distributions in the RDM GW, MB,TM,NS Data: PHOBOS coll. Phys. Rev. C, in press 11 Aug 05

  22. d+Au 200 GeV RDM-analysis Full lines: RDM-results with 2-minimization Data: PHOBOS coll. Phys. Rev. C, in press GW, MB,TM, NS 11 Aug 05

  23. d+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 11 Aug 05

  24. Compare d+Au/Au+Au 200 GeV central source 11 Aug 05

  25. Heavy 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 yeff, the longitudinal expansion velocity vcoll. *At 62.4 GeV, yeff will need adjustment to forthcoming data. 11 Aug 05

  26. Conclusion • 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. 11 Aug 05

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