T. WPCF’06, Sao Paulo, 2006/9/10 1 T. Csörgő, M. Csanád and M. Nagy MTA KFKI RMKI, Budapest, Hungary Anomalous diffusion of pions at RHIC Introduction: Normal diffusion Anomalous diffusion Model simulation of anomalous diffusion of pions at RHIC PHENIX nucl-ex/ : evidence for a heavy tail of S(r) Selection of a model Rescattering effects Powerlaw tails Particle id dependence Anomalous diffusion and a second order QCD phase transition: future measurements: how to falsify these scenarios?

T. WPCF’06, Sao Paulo, 2006/9/10 2 "In general we look for a new law by the following process. First we guess it. Then we compare the consequences of the guess to see what would be implied if this law that we guessed is right. Then we compare the result of the computation to nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is — if it disagrees with experiment it is wrong.” /R.P. Feynman/ Discovering New Laws

T. WPCF’06, Sao Paulo, 2006/9/10 3 Normal diffusion and Gaussian sources W j (t) ~ S(x,t) Jumps nearest neighbouring cells j-1 -> j, j+1 -> j Master equation ~ Boltzmann equation Diffusion equation Scattering: time independent mean free path -> corresponds to Gaussian random walk normal (Gaussian) diffusion of pions in homogeneous medium

T. WPCF’06, Sao Paulo, 2006/9/10 4 Anomalous diffusion and Levy stable laws Continuous random walk models jump length has distribution (x) waiting time as a distribution w(t) Rescattering in a time dependent mean free path system -> corresponds to random Levy walk with Poissonian waiting (decay) time distributions anomalous diffusion of pions in an expanding fireball

T. WPCF’06, Sao Paulo, 2006/9/10 5 Anomalous diffusion and Levy stable laws Relevant mathematical tools: Generalized Boltzmann equation Rescattering in a time dependent mean free path system -> corresponds to random Levy walk but let us look for a realistic model first

T. WPCF’06, Sao Paulo, 2006/9/10 6 Selection of the MC model - Conventional hadronic cascade model - Describes single particle spectra - Describes elliptic flow data - Describes HBT data [ w/o any puzzle] hence yields a good description of the hadronic final state i.e. an acceptable model of S(r) ! - Well documented and easy to use - Works at CERN SPS as well as at RHIC energies - Contains the most important short and long lived resonances e.g. ’ (halo of long-lived resonances) AND includes their rescattering in the code

T. WPCF’06, Sao Paulo, 2006/9/10 7 HRC and exact Buda-Lund hydro solutions Hadronic Rescattering Model (HRM): first use of the word „shooting” when predicting spectra. HRM and Buda-Lund hydrodynamical calculations: self-quenching effect.

T. WPCF’06, Sao Paulo, 2006/9/10 8 HRC and exact Buda-Lund hydro solutions: HBT HRM and Buda-Lund hydrodynamical calculations: self-quenching effect. HBT radii stop to evolve in time, although the system keeps on expanding (rescatterings or hydrodynamical evolution).

T. WPCF’06, Sao Paulo, 2006/9/10 9 HRC and exact Buda-Lund hydro solutions: v2 HRM and Buda-Lund hydrodynamical calculations: self-quenching effect. Elliptic flow freezes out at the same time when the spectra (slopes) stop to evolve in time, although the system keeps on expanding (rescatterings or hydrodynamical evolution).

T. WPCF’06, Sao Paulo, 2006/9/10 10 Solution of the “HBT puzzle” HBT volume Full volume Geometrical sizes keep on increasing. Expansion velocities tend to constants. HBT radii R x, R y, R z approach a direction independent constant. Slope parameters tend to direction dependent constants. General property, independent of initial conditions - a beautiful exact result.

T. WPCF’06, Sao Paulo, 2006/9/10 11 Understanding hydro & HRC New exact solutions of 3d nonrelativistic hydrodynamics+ HRC calculations: time evolution similar to a shot! Hydro:Shot of an arrow: Desription of dataHitting the target Initial conditions(IC)Initial position and velocity Equations of stateStrength of the potential Freeze-out (FC)Position of the target Data constrain EOSHitting the target tells about the potential (?) Different IC lead toDifferent archers can exactly the same FChit target simultaneously (!) EoS and IC can co-varyInitial conditions can be co-varied with the potential Universal scaling of v 2 In a perfect shot, the shape of trajectory is a parabola Viscosity effectsDrag force of air

T. WPCF’06, Sao Paulo, 2006/9/10 12 More than hydro: tails of particle production What about the rescattering after hydrodynamics freezes out? How to distinguish hydro + ideal freeze-out (Therminator) from cascades and hydro+cascade scenario?

T. WPCF’06, Sao Paulo, 2006/9/10 13 Demonstration: Model vs Au+Au 200GeV Hadronic Rescattering Model (HRM): Tom Humanic, reviewed in nucl-th/ Int.J.Mod.Phys.E15: ,2006 freeze-out OK at the freeze-out

T. WPCF’06, Sao Paulo, 2006/9/10 14 Important properties of Tom’s HRC - Conventional hadronic cascade model - Has an adaptive bin size in the time direction: hence no natural time scale in the code - Contains the cascading of the most abundant hadrons: direct  = 152 MeV = 120 MeV K * (K * )= 50 MeV = 8.4 MeV = MeV ’ ’= MeV ’= 4.4 MeV ’= 2.5x MeV - but neglect of electric charge core: h/  < 4 fm halo: h/  >40 fm  : h/  = 23.5 fm

T. WPCF’06, Sao Paulo, 2006/9/10 15 Tom’s HRC: cuts for Fig.1 of nucl-ex/ Au+Au 200 GeV 0-20 % centrality -0.5 < y < GeV < kT < 0.36 GeV pions (no charge selection) Gaussian to first few points fails at heavy tail core-core pairs reproduce heavy tail omega-core pairs: small dominate for r > 80 fm only all other pairs have negligible contribution

T. WPCF’06, Sao Paulo, 2006/9/10 16 nucl-ex/ Fig.1 b and HRM, log-log scale A power-law tail due to rescattering (in HRM) -> corresponds to Levy distributions Reason: adaptive scale in HRM, corresponding to a aa anomalous diffusion (in contrast to Gaussian diffusion)

T. WPCF’06, Sao Paulo, 2006/9/10 17 Same plot, on a larger, log-log scale Prediction: at a larger scale power-law tail has two components: core-core and omega-core -> corresponds to sum of Levy distributions but rescattering is dominant (core-core) for r < 80 fm in HRC

T. WPCF’06, Sao Paulo, 2006/9/10 18 More detailed decomposition Now all resonances are decomposed dominant component: pion-pion rescattering -> corresponds to random Levy walk anomalous diffusion of pions in an expanding fireball

T. WPCF’06, Sao Paulo, 2006/9/10 19 Detailed investigations: Investigation of the role of cuts -centrality -kT -PID Is the effect particular or general, valid for all cuts in nucl-ex/ Fig. 3 (a-c,d-f) Centrality k T1 (MeV)k T2 (MeV)k T3 (MeV) 0-10 % %sames %ss %ss % ss % %samesamesame

T. WPCF’06, Sao Paulo, 2006/9/10 20 Centrality dependence 0.2 GeV < k T < 0.4 GeV, pions After a Gaussian start, a heavy tail in all centrality class exponent (log-log slope) weakly depends on centrality

T. WPCF’06, Sao Paulo, 2006/9/10 21 Centrality dependence, 0.5 GeV < k T < 1.0 GeV, pions Same as at low k t : a heavy tail in all centrality class exponent (log-log slope) weakly depends on centrality

T. WPCF’06, Sao Paulo, 2006/9/10 22 Centrality dependence 0.2 GeV < k T < 0.4 GeV, kaons The Gaussian width decreases with decreasing overlap, a tail exists in all centrality class also for kaons exponent (log-log slope) weakly depends on centrality

T. WPCF’06, Sao Paulo, 2006/9/10 23 Centrality dependence 0.5 GeV < k T < 1.0 GeV, kaons The Gaussian width decreases with decreasing overlap, a tail exists in all centrality class also for kaons at higher k T exponent (log-log slope) weakly depends on centrality

T. WPCF’06, Sao Paulo, 2006/9/10 24 Centrality dependence 0.2 GeV < k T < 0.4 GeV, protons Problems with statistics in simulation for protons exponent (log-log slope) weakly depends on centrality

T. WPCF’06, Sao Paulo, 2006/9/10 25 Centrality dependence 0.5 GeV < k T < 1.0 GeV, protons The Gaussian width decreases with decreasing overlap, a tail exists in all centrality class also for protons and weakly depends on centrality in the MC simulation

T. WPCF’06, Sao Paulo, 2006/9/10 26 Momentum dependence 0-20 % and 40-80%, pions The Gaussian width decreases with decreasing overlap, a tail weakly depends on k T in the MC simulation

T. WPCF’06, Sao Paulo, 2006/9/10 27 Momentum dependence 0-20 % and 40-80%, kaons The Gaussian width decreases with decreasing overlap, a tail weakly depends on k T in the MC simulation even for kaons

T. WPCF’06, Sao Paulo, 2006/9/10 28 Momentum dependence 0-20 % and 40-80%, protons The Gaussian width decreases with decreasing overlap, a tail weakly depends on k T in the MC simulation even for protons - does it depend on anything at all??

T. WPCF’06, Sao Paulo, 2006/9/10 29 PID dependence 0-10 % and MeV The tail strongly depends on PID (particle type) in the MC simulation largest for kaons - that have the smallest cross sections

T. WPCF’06, Sao Paulo, 2006/9/10 30 PID dependence % and MeV The tail strongly depends on PID largest for kaons - that have the smallest cross sections

T. WPCF’06, Sao Paulo, 2006/9/10 31 PID dependence % and MeV The same, strong PID dependece

T. WPCF’06, Sao Paulo, 2006/9/10 32 Conclusions, summary In a hadronic rescattering MC simulation: - a heavy tail exists in all centrality, k T range of PHENIX nucl-ex/ it is of a power-law type (linear on log-log plot)  ~ exponent weakly depends on centrality and k T - exponent strongly depends on PID (particle cross section) in the MC simulation Consequence: a 2 nd order QCD phase transition  ~ 0.5 and rescattering lead to different Lévy exponents - check PID dependence of  first!