1. ICPAQGP 2010 Goa, Dec. 6-10, 2010 1 Percolation & Deconfinement Brijesh K Srivastava Department of Physics Purdue University USA

2. The general formulation of the percolation problem is concerned with elementary geometrical objects placed at random in a d-dimensional lattice. The objects have a well defined connectivity radius λ, and two objects are said to communicate if the distance between them is less than λ. One is interested in how many objects can form a cluster of communication and, especially, when and how the cluster become infinite. The control parameter is the density of the objects or the dimensionless filling factor ξ. The percolation threshold ξ = ξ c corresponding to the minimum concentration at which an infinite cluster spans the space. Thus the percolation model exhibits two essential features: Critical behavior Long range correlations Percolation : General 2

3. 3 Rev. Mod. Phys. 64, 961 (1992).

4. Parton distributions in the transverse plane of nucleus-nucleus collisions Parton Percolation De-confinement is expected when the density of quarks and gluons becomes so high that it no longer makes sense to partition them into color-neutral hadrons, since these would overlap strongly. We have clusters within which color is not confined -> De-confinement is thus related to cluster formation. This is the central topic of percolation theory, and hence a connection between percolation and de-confinement seems very likely. 4 H. Satz, Rep. Prog. Phys. 63, 1511(2000). H. Satz, hep-ph/0212046

5. Color Strings Multiparticle production at high energies is currently described in terms of color strings stretched between the projectile and target. Hadronizing these strings produce the observed hadrons. The no. of strings grow with energy and the no. of participating nuclei and one expects that interaction between them becomes essential. This problem acquires even more importance, considering the idea that at very high energy collisions of heavy nuclei (RHIC) may produce Quark-gluon Plasma (QGP). The interaction between strings then has to make the system evolve towards the QGP state. 5

6. Color Strings At low energies, valence quarks of nucleons form strings that then hadronize  wounded nucleon model. At high energies, contribution of sea quarks and gluons becomes dominant. Additional color strings formed. 1. Dual Parton Model (DPM): A. Capella et al., Phys. Rep. 236, 225 (1994). 2. A. Capella and A. Krzywicki, Phys. Rev. D184,120 (1978). 6

7. In two dimensions, for uniform string density, the percolation threshold for overlapping discs is: The fractional area covered by discs at the critical threshold is: = critical percolation density Satz, hep-ph/0007069 Parton Percolation 7

8. Multiplicity (  n ) Average Transverse Momentum Multiplicity and of particles produced by a cluster of n strings Color Strings + Percolation = CSPM 8 M. A. Braun and C. Pajares, Eur.Phys. J. C16,349 (2000) M. A. Braun et al, Phys. Rev. C65, 024907 (2002) = Color suppression factor (due to overlapping of discs). ξ is the percolation density parameter. N s = # of strings S 1 = disc area S N = total nuclear overlap area

9. To compute the p T distribution, a parameterization of the pp data is used: a, p 0 and n are parameters fit to the data. This parameterization can be used for nucleus-nucleus collisions, accounting for percolation by: Using the p T spectrum to calculate ξ M. A. Braun, et al. hep-ph/0208182. In pp at low energy, pp = 1 ± 0.1, due to low string overlap probability in pp collisions. CSPM 9

10. Percolation and Color Glass Condensate  Many of the results obtained in the framework of percolation of strings are very similar to the one obtained in the CGC.  In particular, very similar scaling laws are obtained for the product and the ratio of the multiplicities and transverse momentum. Percolation : Clustering of strings CGC : Gluon saturation Both are based on parton coherence phenomena.  Both provide explanation for multiplicity suppression and scaling w/ dN/dy. 10

11. Percolation and Color Glass Condensate Universal relation between dn/dy and Percolation: CGC: Phys. Lett. B 581, 156(2004) Phys. Lett. B 514, 29 ( 2001) Nucl. Phys. A 705, 494 (2002) Momentum Q s which establishes the scale in CGC with the corresponding one in percolation of strings hep-ph/1011.1099 11 For large value of ξ

12. Parametrization of pp UA1 data at 130 GeV from 200, 500 and 900 GeV ISR 53 and 23 GeV QM 2001 PHENIX p 0 = 1.71 and n = 12.42 Ref: Nucl. Phys. A698, 331 (2002). STAR has also extrapolated UA1 data from 200-900 GeV to 130 GeV p 0 = 1.90 and n = 12.98 Ref: Phys. Rev. C 70, 044901( 2004). UA1 results at 200 GeV p 0 = 1.80 and n = 12.14 Ref: Nucl. Phys. B335, 261 ( 1990) 12 pp fit parameter

13. 13 STAR Collaboration, Nukleonika 51, S109 (2006) Npart Percolation density parameter ξ Au+Au 200 GeV 62 GeV STAR Preliminary ξ c =1.2 Now the aim is to connect F(ξ) with Temperature and Energy density

14. Relation between Temperature & Color Suppression factor F(ξ) Ref : 1. Fluctuations of the string and transverse mass distribution A. Bialas, Phys. Lett. B 466 (1999) 301. 2.Percolation of color sources and critical temperature J. Dias de Deus and C. Pajares, Phys.Lett B 642 (2006) 455 14

15. Temperature It is shown that quantum fluctuations of the string tension can account for the ‘thermal” distributions of hadrons created in the decay of color string. 15 The tension of the macroscopic cluster fluctuates around its mean value because the chromoelectric field is not constant. Assuming a Gaussian form for these fluctuations one arrives at the probability distribution of transverse momentum: which gives rise to thermal distribution With temperature Average transverse momentum of a single string Average string tension

16. 16 Temperature At the critical percolation density = 167 MeV For Au+Au@ 200 GeV 0-10% centrality ξ = 2.88 T = 193 MeV 1. Chemical freeze out temperature P. Braun-Munzinger et al., Phys. Lett. B596, 61 (2006) 2. Universal chemical freeze out temperature F. Becattini et al, Eur. Phys. J. C66, 377 (2010).

17. Energy Density Bjorken Phys. Rev. D 27, 140 (1983) Transverse overlap area Proper Time is the QED production time for a boson which can be scaled from QED to QCD and is given by 17 Introduction to high energy heavy ion collisions C. Y. Wong

18. 18 Energy Density STAR Collaboration has published charged particle multiplicities and transverse momentum of particles in for Au+Au collisions at various energies. It is found that 1.D. Cebra, STAR Collaboration, QM08 2.B. I. Abelev et al, STAR Collaboration, Phys. Rev.C79, 34909 (2009) 3.B. I. Abelev et al, STAR Collaboration, Phys. Rev.C81, 24911(2010)

19. Equation of state 19 vs

20. Energy Density 0-10% centrality Lattice QCD : Bazavov et al, Phys. Rev. D 80, 014504(2009). 20

21. Thermodynamic Relations 21

22. Finally, we have energy density, temperature and sound velocity as a function of color suppression factor F(ξ) or percolation density parameter ξ. 22 : an analytic function of ξ for the EOS of QGP for T≥ T c Sound Velocity

23. Velocity of Sound expressed as Lattice QCD : Bazavov et al, Phys. Rev. D 80, 014504(2009). Physical hadron gas: Castorina et al, arXiv:0906.2289/hep-ph 23

24. Entropy Density 0-10% centrality 24 Lattice QCD : Bazavov et al, Phys. Rev. D 80, 014504(2009).

25. Color Suppression Factor F(ξ) Energy density ε/T 4 Entropy density s/T 3 Temperature Sound velocity CGC Shear viscosity η ? Summary H. Satz : Quantum Field Theory in Extreme Environments, Paris, April 2009 Clustering and percolation can provide a conceptual basis for the QCD phase diagram which is more general than symmetry breaking.