Dynamic Modeling of RHIC Collisions Steffen A Bass
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: Ur. QMD the parton world: PCM macroscopic point of view: hydrodynamics the future: hybrid approaches Steffen A. Bass 1
Why do Heavy-Ion Physics? • QCD Vacuum • Bulk Properties of Nuclear Matter • Early Universe Steffen A. Bass 2
QCD and it’s Ground State (Vacuum) • Quantum-Chromo-Dynamics (QCD) Ø Ø 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 Ø has a complicated structure Ø contains scalar and vector condensates Ø explore vacuum-structure by heating/melting QCD matter Ø Quark-Gluon-Plasma Steffen A. Bass 3
Phases of Normal Matter solid liquid gas Ø electromagnetic interactions determine phase structure of normal matter Steffen A. Bass 4
Phases of QCD Matter • 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 Ø new state of matter! Steffen A. Bass Quark-Gluon Plasma Hadron Gas Solid 5
QGP and the Early Universe • few microseconds after the Big Bang the entire Universe was in a QGP state ØCompressing & heating nuclear matter allows to investigate the history of the Universe Steffen A. Bass 6
Compressing and Heating Nuclear Matter Ø accelerate and collide two heavy atomic nuclei The Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory Steffen A. Bass 7
Dynamic Modeling • purpose • fundamentals • current status Steffen A. Bass 8
The Purpose of Dynamic Modeling hadronic phase and freeze-out QGP and hydrodynamic expansion initial state pre-equilibrium hadronization Lattice-Gauge Theory: • rigorous calculation of QCD quantities • works in the infinite size / equilibrium limit Experiments: • only observe the final state • rely on QGP signatures predicted by Theory Transport-Theory: • full description of collision dynamics • connects intermediate state to observables • provides link between LGT and data Steffen A. Bass 9
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 10
Kinetic Theory: - formal language of transport models classical approach: Liouville’s Equation: Øuse 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 11
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 Øuse approximation scheme for self-energy Σ (e. g. T-Matrix approx. ) ØPerform Wigner-Transformation of two-point functions A(1, 1’) to obtain classical quantities (smooth phase-space functions) Steffen A. Bass 12
The Vlasov-Uehling-Uhlenbeck Equation classical approach: • combine Vlasov- and Boltzmann-equations quantum approach: • • perform Wigner-transform Connect Σ to scattering rates and potential identify correlation functions with f use quasi-particle approximation • the Uehling-Uhlenbeck terms are added to ensure the Pauli-Principle Steffen A. Bass 13
Collision Integral: Monte-Carlo Treatment • f 1 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 dmin of 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 14
Example #1: the hadronic world • the Ur. QMD model Steffen A. Bass 15
Applying Transport Theory to Heavy. Ion Collisions Pb + Pb @ 160 Ge. V/nucleon (CERN/SPS) • calculation done with the Ur. QMD (Ultra-relativistic Quantum Molecular Dynamics) model • initial nucleon-nucleon collisions excite color-flux-tubes (chromoelectric 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 16
Initial Particle Production in Ur. QMD Steffen A. Bass 17
Meson Baryon Cross Section in Ur. QMD Ø model degrees of freedom determine the interaction to be used Δ* width N* width Δ 1232 120 Me. V N*1440 200 Me. V Δ 1600 350 Me. V N*1520 125 Me. V Δ 1620 120 Me. V N*1535 150 Me. V Δ 1700 300 Me. V N*1650 150 Me. V Δ 1900 200 Me. V N*1675 150 Me. V Δ 1905 350 Me. V N*1680 130 Me. V Δ 1910 250 Me. V N*1700 100 Me. V Δ 1920 200 Me. V N*1710 110 Me. V Δ 1930 350 Me. V N*1720 200 Me. V Δ 1950 300 Me. V N*1990 300 Me. V Øcalculate cross section according to: Steffen A. Bass 18
Example #2: the partonic world • The Parton Cascade Model • applications Steffen A. Bass 19
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 p. QCD with either a momentum cut-off or Debye screening to regularize IR behaviour • guiding scales: initialization scale Q 0, p. T cut-off p 0 / Debye-mass μD, intrinsic k. T / saturation momentum QS, virtuality > μ 0 Steffen A. Bass 20
Initial State: Parton Momenta • flavour and x are sampled from PDFs at an initial scale Q 0 and low x cut-off xmin • initial kt is sampled from a Gaussian of width Q 0 in case of no initial state radiation • virtualities are determined by: Steffen A. Bass 21
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 22
Parton-Parton Scattering Cross-Sections gg gg q q’ q g q qbar q’ qbar’ g g q qbar q g q γ qq qq q qbar g γ q qbar γ γ q qbar g g • a common factor of παs 2(Q 2)/s 2 etc. • further decomposition according to color flow Steffen A. Bass 23
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 24
Higher Order Corrections and Microcausality • higher order corrections to the cross section are taken into account by multiplying the lo p. QCD 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 25
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: re ss • explicit treatment of 3 2 processes (D. Molnar, C. Greiner) • glue fusion: w or k in pr og • qg q* • gg g* Steffen A. Bass 26
Hadronization • requires modeling & parameters beyond the PCM p. QCD 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: Ø net-baryons Ø direct photons Steffen A. Bass 27
Testing the PCM Kernel: Collisions • in leading order p. QCD, the hard cross section σQCD is given by: • number of hard collisions Nhard (b) is related to σQCD by: • equivalence to PCM implies: Ø keeping factorization scale Q 2 = Q 02 with αs evaluated at Q 2 Ø restricting PCM to eikonal mode Steffen A. Bass 28
Testing the PCM Kernel: pt distribution • the minijet cross section is given by: • equivalence to PCM implies: Ø keeping the factorization scale Q 2 = Q 02 with αs evaluated at Q 2 Ø restricting PCM to eikonal mode, without initial & final state radiation • results shown are for b=0 fm Steffen A. Bass 29
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 selfconsistently calculate the parton scattering cross sections Øcurrently beyond the scope of the numerical implementation of the PCM Steffen A. Bass 30
Choice of p. Tmin: 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 p. Tmin> μD : Ølower boundary for p. Tmin : approx. 0. 8 Ge. V Steffen A. Bass 31
Photon Production in the PCM relevant processes: • Compton: q g q γ • annihilation: q qbar g γ • bremsstrahlung: q* q γ Øphoton yield very sensitive to parton-parton rescattering Steffen A. Bass 32
What can we learn from photons? • primary-primary collision contribution to yield is < 10% • emission duration of preequilibrium phase: ~ 0. 5 fm/c Steffen A. Bass • photon yield directly proportional to the # of hard collisions Ø photon yield scales with Npart 4/3 33
Stopping at RHIC: Initial or Final State Effect? • net-baryon contribution from initial state (structure functions) is non-zero, even at midrapidity! Øinitial state alone accounts for d. Nnet-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 34
Stopping at RHIC: PCM Results • primary-primary scattering releases baryon-number at corresponding y • multiple rescattering & fragmentation fill up mid -rapidity domain Øinitial state & parton cascading can fully account for data! Steffen A. Bass 35
Example #3: hydrodynamics Steffen A. Bass 36
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) Ø connection to Lattice QCD calculation of Eo. S • initial conditions (i. e. thermalized QGP) required for calculation • assumes local thermal equilibrium, vanishing mean free path Ø 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 37
Collective Flow: Overview • directed flow (v 1, px, dir) – spectators deflected from dense reaction zone – sensitive to pressure • elliptic flow (v 2) – asymmetry out- vs. in-plane emission – emission mostly during early phase – strong sensitivity to Eo. S • radial flow (ßt) – isotropic expansion of participant zone – measurable via slope parameter of spectra (blue-shifted temperature) Steffen A. Bass 38
Elliptic flow: early creation P. Kolb, J. Sollfrank and U. Heinz, PRC 62 (2000) 054909 time evolution of the energy density: initial energy density distribution: spatial eccentricity momentum anisotropy 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 39
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 p. T ( > 2 Ge. V) the experimental results (as well as the parton cascade) saturate, indicating insufficient thermalization of the rapidly escaping particles to allow for a hydrodynamic description. • • D. Molnar and M. Gyulassy, NPA 698 (2002) 379 P. Kolb et al. , PLB 500 (2001) 232 Steffen A. Bass 40
Anisotropies: sensitive to the QCD Eo. S P. Kolb and U. Heinz, hep-ph/0204061 Teaney, Lauret, Shuryak, nucl-th/0110037 Ø the data favor an equation of state with a soft phase and a latent heat De between 0. 8 and 1. 6 Ge. V/fm 3 Steffen A. Bass 41
Example #4: hybrid approaches • motivation • applications • outlook Steffen A. Bass 42
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 • v 2 saturates for high pt vs. monotonic increase in hydro (onset of p. QCD physics) Steffen A. Bass 43
A combined Macro/Micro Transport Model Hydrodynamics • ideally suited for dense systems + micro. transport (Ur. QMD) • Ø model early QGP reaction stage • • Ø model break-up stage Ø calculate freeze-out well defined Equation of State Ø Incorporate 1 st order p. t. parameters: – initial conditions (fit to experiment) – Equation of State matching conditions: no equilibrium assumptions • parameters: – (total/partial) cross sections – resonance parameters (full/partial widths) • use same set of hadronic states for Eo. S as in Ur. QMD • perform transition at hadronization hypersurface: generate space-time distribution of hadrons for each cell according to local T and μB Ø use as initial configuration for Ur. QMD Steffen A. Bass 44
Flavor Dynamics: Radial Flow • • Ø Ø 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 45
Connecting high-pt partons with the dynamics of an expanding QGP • Jet quenching analysis taking hydro+jet model account of (2+1)D hydro results (M. Gyulassy et al. ’ 02) color: QGP fluid density symbols: mini-jets Hydro+Jet model Ø use GLV 1 st order formula for parton energy loss (M. Gyulassy et al. ’ 00) y T. Hirano. & Y. Nara: Phys. Rev. C 66 041901, 2002 Au+Au 200 AGe. V, b=8 fm transverse plane@midrapidity Fragmentation switched off Øtake Parton density ρ(x) from full 3 D hydrodynamic calculation Steffen A. Bass x Movie and data of ρ(x) are available at http: //quark. phy. bnl. gov/~hirano/ 46
Transport Theory at RHIC initial state pre-equilibrium Steffen A. Bass hadronic phase and freeze-out QGP and hydrodynamic expansion hadronization 47
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 Ø exciting area of research with lots of challenges and opportunities! Steffen A. Bass 48
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