Dynamic Modeling of RHIC Collisions

1. Dynamic Modeling of RHIC Collisions Steffen A. Bass Duke University & RIKEN BNL Research Center • Motivation: why do heavy-ion collisions? • Introduction: the basics of kinetic theory • Examples of transport models and their application: • the hadronic world: UrQMD • the parton world: PCM • macroscopic point of view: hydrodynamics • the future: hybrid approaches Steffen A. Bass

2. Why do Heavy-Ion Physics? • QCD Vacuum • Bulk Properties of Nuclear Matter • Early Universe Steffen A. Bass

3. QCD and it’s Ground State (Vacuum) • Quantum-Chromo-Dynamics (QCD) • one of the four basic forces of nature • is responsible for most of the mass of ordinary matter • holds protons and neutrons together in atomic nuclei • basic constituents of matter: quarks and gluons • The QCD vacuum: ground-state of QCD • has a complicated structure • contains scalar and vector condensates • explore vacuum-structure by heating/melting QCD matter • Quark-Gluon-Plasma Steffen A. Bass

4. electromagnetic interactions determine phase structure of normal matter Phases of Normal Matter solid liquid gas Steffen A. Bass

5. strong interaction analogues of the familiar phases: Nuclei behave like a liquid Nucleons are like molecules Quark Gluon Plasma: “ionize” nucleons with heat “compress” them with density new state of matter! Phases of QCD Matter Quark-GluonPlasma HadronGas Solid Steffen A. Bass

6. QGP and the Early Universe • few microseconds after the Big Bang the entire Universe was in a QGP state • Compressing & heating nuclear matter allows to investigate the history of the Universe Steffen A. Bass

7. Compressing and Heating Nuclear Matter • accelerate and collide two heavy atomic nuclei The Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory Steffen A. Bass

8. Dynamic Modeling • purpose • fundamentals • current status Steffen A. Bass

9. hadronic phase and freeze-out QGP and hydrodynamic expansion initial state pre-equilibrium hadronization The Purpose of Dynamic Modeling Steffen A. Bass

10. Microscopic Transport Models microscopic transport models describe the time-evolution of a system of (microscopic) particles by solving a transport equation derived from kinetic theory key features: • describe the dynamics of a many-body system • connect to thermodynamic quantities • take multiple (re-)interactions among the dof’s into account key challenges: • quantum-mechanics: no exact solution for the many-body problem • covariance: no exact solution for interacting system of relativistic particles • QCD: limited range of applicability for perturbation theory Steffen A. Bass

11. Kinetic Theory:- formal language of transport models - classical approach: Liouville’s Equation: • use BBKGY hierarchy and cut off at 1-body level a) interaction based only on potentials: Vlasov Equation b) interaction based only on scattering: Boltzmann Equation with Steffen A. Bass

12. Kinetic Theory II quantum approach: start with Dyson Equation on contour C (or Kadanoff-Baym eqns): with G: path ordered non-equilibrium Green’s function • use approximation scheme for self-energy Σ (e.g. T-Matrix approx.) • Perform Wigner-Transformation of two-point functions A(1,1’) to obtain classical quantities (smooth phase-space functions) Steffen A. Bass

13. The Vlasov-Uehling-Uhlenbeck Equation the Uehling-Uhlenbeck terms are added to ensure the Pauli-Principle Steffen A. Bass

14. Collision Integral: Monte-Carlo Treatment • f1 is discretized into a sample of microscopic particles • particles move classical trajectories in phase-space • an interaction takes place if at the time of closes approach dminof two hadrons the following condition is fulfilled: • main parameter: • cross section: probability for an interaction to take place, which is interpreted geometrically dmin Steffen A. Bass

15. Example #1: the hadronic world • the UrQMD model Steffen A. Bass

16. Applying Transport Theory to Heavy-Ion Collisions Pb + Pb @ 160 GeV/nucleon (CERN/SPS) calculation done with the UrQMD (Ultra-relativistic Quantum Molecular Dynamics) model initial nucleon-nucleon collisions excite color-flux-tubes (chromo-electric fields) which decay into new particles all particles many rescatter among each other initial state: 416 nucleons (p,n) reaction time: 30 fm/c final state: > 1000 hadrons Steffen A. Bass

17. Initial Particle Production in UrQMD Steffen A. Bass

18. Meson Baryon Cross Section in UrQMD • model degrees of freedom determine the interaction to be used • calculate cross section according to: Steffen A. Bass

19. Example #2: the partonic world • The Parton Cascade Model • applications Steffen A. Bass

20. Basic Principles of the PCM provide a microscopic space-time description of relativistic heavy-ion collisions based on perturbative QCD degrees of freedom: quarks and gluons classical trajectories in phase space (with relativistic kinematics) initial state constructed from experimentally measured nucleon structure functions and elastic form factors system evolves through a sequence of binary (22) elastic and inelastic scatterings of partons and initial and final state radiations within a leading-logarithmic approximation (2N) binary cross sections are calculated in leading order pQCD with either a momentum cut-off or Debye screening to regularize IR behaviour guiding scales: initialization scale Q0, pT cut-off p0 / Debye-mass μD, intrinsic kT / saturation momentum QS, virtuality > μ0 Steffen A. Bass

21. Initial State: Parton Momenta flavour and x are sampled from PDFs at an initial scale Q0 and low x cut-off xmin initial kt is sampled from a Gaussian of width Q0 in case of no initial state radiation virtualities are determined by: Steffen A. Bass

22. Binary Processes in the PCM the total cross section for a binary collision is given by: with partial cross sections: now the probability of a particular channel is: finally, the momentum transfer & scattering angle are sampled via Steffen A. Bass

23. Parton-Parton Scattering Cross-Sections a common factor of παs2(Q2)/s2 etc. further decomposition according to color flow Steffen A. Bass

24. Initial and final state radiation Probability for a branching is given in terms of the Sudakov form factors: space-like branchings: time-like branchings: Altarelli-Parisi splitting functions included: Pqqg , Pggg , Pgqqbar & Pqqγ Steffen A. Bass

25. Higher Order Corrections and Microcausality higher order corrections to the cross section are taken into account by multiplying the lo pQCD cross section with a (constant) factor: K-factor corrections include initial and final state gluon radiation numerical problem: the hard, binary, collision has to be performed in order to determine the momentum scale for the space-like radiation space-like radiation may alter the incoming momenta (i.e. the sampled parton distribution function) and affect the scale of the hard collision Steffen A. Bass

26. Parton Fusion (21) Processes • in order to account for detailed balance and study equilibration, one needs to account for the reverse processes of parton splittings: • explicit treatment of 32 processes (D. Molnar, C. Greiner) • glue fusion: qg  q* gg  g* work in progress Steffen A. Bass

27. Hadronization • requires modeling & parameters beyond the PCM pQCD framework • microscopic theory of hadronization needs yet to be established • phenomenological recombination + fragmentation approach may provide insight into hadronization dynamics • avoid hadronization by focusing on: • net-baryons • direct photons Steffen A. Bass

28. Testing the PCM Kernel: Collisions in leading order pQCD, the hard cross section σQCD is given by: number of hard collisions Nhard (b) is related to σQCD by: • equivalence to PCM implies: • keeping factorization scale Q2 = Q02 with αs evaluated at Q2 • restricting PCM to eikonal mode Steffen A. Bass

29. Testing the PCM Kernel: pt distribution the minijet cross section is given by: • equivalence to PCM implies: • keeping the factorization scale Q2 = Q02 with αs evaluated at Q2 • restricting PCM to eikonal mode, without initial & final state radiation • results shown are for b=0 fm Steffen A. Bass

30. Debye Screening in the PCM • the Debye screening mass μD can be calculated in the one-loop approximation [Biro, Mueller & Wang: PLB 283 (1992) 171]: • PCM input are the (time-dependent) parton phase-space distributions F(p) • Note: ideally a local and time-dependent μD should be used to self-consistently calculate the parton scattering cross sections • currently beyond the scope of the numerical implementation of the PCM Steffen A. Bass

31. Choice of pTmin: Screening Mass as Indicator • screening mass μD is calculated in one-loop approximation • time-evolution of μD reflects dynamics of collision: varies by factor of 2! • model self-consistency demands pTmin> μD : • lower boundary for pTmin : approx. 0.8 GeV Steffen A. Bass

32. Photon Production in the PCM relevant processes: • Compton: q g q γ • annihilation: q qbar  g γ • bremsstrahlung: q*q γ • photon yield very sensitive to parton-parton rescattering Steffen A. Bass

33. What can we learn from photons? primary-primary collision contribution to yield is < 10% emission duration of pre-equilibrium phase: ~ 0.5 fm/c • photon yield directly proportional to the # of hard collisions • photon yield scales with Npart4/3 Steffen A. Bass

34. Stopping at RHIC: Initial or Final State Effect? • net-baryon contribution from initial state (structure functions) is non-zero, even at mid-rapidity! • initial state alone accounts for dNnet-baryon/dy5 • is the PCM capable of filling up mid-rapidity region? • is the baryon number transported or released at similar x? Steffen A. Bass

35. Stopping at RHIC: PCM Results • primary-primary scattering releases baryon-number at corresponding y • multiple rescattering & fragmentation fill up mid-rapidity domain • initial state & parton cascading can fully account for data! Steffen A. Bass

36. Example #3: hydrodynamics Steffen A. Bass

37. Nuclear Fluid Dynamics • transport of macroscopic degrees of freedom • based on conservation laws:μTμν=0 μjμ=0 • for ideal fluid:Tμν= (ε+p) uμ uν - p gμν and jiμ = ρi uμ • Equation of State needed to close system of PDE’s:p=p(T,ρi) • connection to Lattice QCD calculation of EoS • initial conditions (i.e. thermalized QGP) required for calculation • assumes local thermal equilibrium, vanishing mean free path • applicability of hydro is a strong signature for a thermalized system • simplest case: scaling hydrodynamics • assume longitudinal boost-invariance • cylindrically symmetric transverse expansion • no pressure between rapidity slices • conserved charge in each slice Steffen A. Bass

38. Collective Flow: Overview • directed flow (v1, px,dir) • spectators deflected from dense reaction zone • sensitive to pressure • elliptic flow (v2) • asymmetry out- vs. in-plane emission • emission mostly during early phase • strong sensitivity to EoS • radial flow (ßt) • isotropic expansion of participant zone • measurable via slope parameter of spectra (blue-shifted temperature) Steffen A. Bass

39. Elliptic flow: early creation momentum anisotropy spatial eccentricity time evolution of the energy density: initial energy density distribution: P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000) 054909 All model calculations suggest that flow anisotropies are generated at the earliest stages of the expansion, on a timescale of ~ 5 fm/c. Steffen A. Bass

40. Elliptic flow: strong rescattering cross-sections and/or gluon densities approx. 10 to 80 times the perturbative values are required to deliver sufficient anisotropies! at larger pT( > 2 GeV) the experimental results (as well as the parton cascade) saturate, indicatinginsufficient thermalizationof the rapidly escaping particles to allow for a hydrodynamic description. D. D. Molnar and M. Gyulassy, NPA 698 (2002) 379 P. Kolb et al., PLB 500 (2001) 232 Steffen A. Bass

41. Anisotropies: sensitive to the QCD EoS P. Kolb and U. Heinz, hep-ph/0204061 Teaney, Lauret, Shuryak, nucl-th/0110037 • the data favor an equation of state with a soft phase and a latent heat De between 0.8 and 1.6 GeV/fm3 Steffen A. Bass

42. Example #4: hybrid approaches • motivation • applications • outlook Steffen A. Bass

43. Limits of Hydrodynamics applicable only for high densities: i.e. vanishing mean free path λ local thermal equilibrium must be assumed, even in the dilute, break-up phase fixed freeze-out temperature: instantaneous transition from λ=0 to λ=  no flavor-dependent cross sections v2 saturates for high pt vs. monotonic increase in hydro (onset of pQCD physics) Steffen A. Bass

44. ideally suited for dense systems model early QGP reaction stage well defined Equation of State Incorporate 1st order p.t. parameters: initial conditions (fit to experiment) Equation of State no equilibrium assumptions model break-up stage calculate freeze-out parameters: (total/partial) cross sections resonance parameters (full/partial widths) A combined Macro/Micro Transport Model Hydrodynamics +micro. transport (UrQMD) • use same set of hadronic states for EoS as in UrQMD • perform transition at hadronization hypersurface: generate space-time distribution of hadrons for each cell according to local T and μB • use as initial configuration for UrQMD matching conditions: Steffen A. Bass

45. Flavor Dynamics: Radial Flow • Hydro: linear mass-dependence of slope parameter, strong radial flow • Hydro+Micro: softening of slopes for multistrange baryons • early decoupling due to low collision rates • nearly direct emission from the phase boundary Steffen A. Bass

46. Connecting high-pt partons with the dynamics of an expanding QGP • Jet quenching analysis taking account of (2+1)D hydro results (M.Gyulassy et al. ’02) hydro+jet model color: QGP fluid density symbols: mini-jets Hydro+Jet model T.Hirano. & Y.Nara: Phys.Rev.C66 041901, 2002 y • use GLV 1st order formula for parton energy loss (M.Gyulassy et al. ’00) Au+Au 200AGeV, b=8 fm transverse plane@midrapidity Fragmentation switched off x • take Parton density ρ(x) from full 3D hydrodynamic calculation Movie and data of ρ(x) are available at http://quark.phy.bnl.gov/~hirano/ Steffen A. Bass

47. Transport Theory at RHIC hadronic phase and freeze-out QGP and hydrodynamic expansion initial state pre-equilibrium hadronization Steffen A. Bass

48. Last words… • Dynamical Modeling provides insight into the microscopic reaction dynamics of a heavy-ion collision and connects the data to the properties of the deconfined phase and rigorous Lattice-Gauge calculations • a variety of different conceptual approaches exist, all tuned to different stages of the heavy-ion reaction • a “standard model” covering the entire time-evolution of a heavy-ion recation remains to be developed • exciting area of research with lots of challenges and opportunities! Steffen A. Bass