Dynamic Modeling of RHIC Collisions

Steffen A. Bass CTEQ 2004 Summer School #1

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

Dynamic Modeling of RHIC Collisions

Dynamic Modeling of RHIC Collisions


Why do Heavy-Ion Physics?•QCD Vacuum•Bulk Properties of Nuclear Matter•Early Universe


QCD and it’s Ground State (Vacuum)

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

0 and G 0uu dd G


Phases of Normal Matter

Phases of Normal Matter

electromagnetic interactions determine phase structure of normal matter

solid liquid gas


• 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

Phases of QCD Matter

Quark-GluonPlasma

HadronGas Solid


QGP and the Early Universe

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


Compressing and Heating Nuclear Matter

Compressing and Heating Nuclear Matter

accelerate and collide two heavy atomic nucleiThe Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory


Dynamic Modeling• purpose• fundamentals• current status


The Purpose of Dynamic Modeling

The Purpose of Dynamic Modeling

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

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


Microscopic Transport Models

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


Kinetic Theory:- formal language of transport

models -

Kinetic Theory:- formal language of transport

models -

, 0N

Nff H

t

1( ) 0r r p

pU f

t m

1collr

pf I

t m

coll 2 1 2 1 1 1 2 1 1 1 2d d ( ) ( ) ( ) ( )I N p v v f p f p f p f p

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


Kinetic Theory IIKinetic Theory II

0 0(1,1 ) (1,1 ) d1 d1 (1,1 ) (1 ,1 ) (1 ,1)C C

G G G G

(1,1 ) d2 d2 12 1 2 12 2 1 (2 ,2 )C C

i T T G

4 / 1 12 2( , ) d ( , )ip y

WA R p ye A R y R y

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)


The Vlasov-Uehling-Uhlenbeck Equation

The Vlasov-Uehling-Uhlenbeck Equation

1 1 1 1

3 3 32 1 2 1 2 1 22 3

1

11 1

1 2

1

2 2 12 1(

2( )

(

1 )(1 ) (1

2

( , , )

)( )

)

1

p r r p

pf r

g dd

f f f

p d p d p p p p pm d

p t

ff f

t m

f f

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


Collision Integral: Monte-Carlo Treatment

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 dmin of two hadrons the following condition is fulfilled:

• main parameter: – cross section: probability for an interaction to take

place, which is interpreted geometrically

1 2min with , ,tot

tot tot s h hd

dmin


Example #1: the hadronic world• the UrQMD model


Applying Transport Theory to Heavy-Ion Collisions

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


Initial Particle Production in UrQMD

Initial Particle Production in UrQMD


Meson Baryon Cross Section in UrQMD

Meson Baryon Cross Section in UrQMD

model degrees of freedom determine the interaction to be used

Δ* width N* width

Δ1232 120 MeV

N*1440 200 MeV

Δ1600 350 MeV

N*1520 125 MeV

Δ1620 120 MeV

N*1535 150 MeV

Δ1700 300 MeV

N*1650 150 MeV

Δ1900 200 MeV

N*1675 150 MeV

Δ1905 350 MeV

N*1680 130 MeV

Δ1910 250 MeV

N*1700 100 MeV

Δ1920 200 MeV

N*1710 110 MeV

Δ1930 350 MeV

N*1720 200 MeV

Δ1950 300 MeV

N*1990 300 MeV

422

*,2

)()12)(12(

12totsMpII

I

R

totMBR

cmsNR MB

RMBtot

calculate cross section according to:


Example #2: the partonic world• The Parton Cascade Model• applications


Basic Principles of the PCM

Basic Principles of the PCM

• 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 (22) elastic and inelastic scatterings of partons and initial and final state radiations within a leading-logarithmic approximation (2N)• 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

provide a microscopic space-time description of relativistic heavy-ion collisions based on perturbative QCD


Initial State: Parton Momenta

Initial State: Parton Momenta

• virtualities are determined by:

• 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

2 2 2 2

2N

i i i ix y z

i i i iME p p p

1with and i i N i iz z N zp x P E p


Binary Processes in the PCM

Binary Processes in the PCM

,

ˆ ˆˆ ˆ ab cdabc d

ss

max

min

ˆ

ˆ

ˆˆ ˆ ˆ( , , ) ˆˆ ˆˆ

t

ab cd

t ab cd

d s t us dt

dt

min

ˆ

ˆ

ˆˆ ˆ ˆ1 ( , , )ˆ ˆˆˆ ˆ

t

ab cd t ab cd

d s t ut dt

s dt

ˆ ˆˆ

ˆ ˆab cd

ab cdab

sP s

s

• 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


Parton-Parton Scattering Cross-Sections

Parton-Parton Scattering Cross-Sections

g g g g q q’ q q’

q g q gq qbar q’ qbar’

g g q qbar q g q γ

q q q q q qbar g γ

q qbar q qbar

q qbar γ γ

q qbar g g

2 2 2

93

2

tu su st

s t u

2 2

2

4

9

s u

t

2 2

2

4

9

t u

s

2

3qe u s

s u

28

9 q

u te

t u

42

3 q

u te

t u

2 2

2

4

9

s u s u

u s t

2 2

2

1 3

6 8

t u t u

u t s

2 2

2

32 8

27 3

t u t u

u t s

2 2 2 2 2

2 2

4 8

9 27

s u s t s

t u tu

2 2 2 2 2

2 2

4 8

9 27

s u u t u

t s st

• a common factor of παs2(Q2)/s2 etc.

• further decomposition according to color flow


Initial and final state radiation

Initial and final state radiation

space-like branchings:

time-like branchings:

max

max

,, , exp

2 ,

ts a a a

a aa a a at

a ae

t x f x tS x t t dt dz

x f tP z

x

Probability for a branching is given in terms of the Sudakov form factors:

max

max, , exp2

t

d dat

d d es P zt

T x t t dt dz

• Altarelli-Parisi splitting functions included: Pqqg , Pggg , Pgqqbar & Pqqγ


Higher Order Corrections and Microcausality

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


Parton Fusion (21) Processes

Parton Fusion (21) Processes

wor

k in

pro

gres

s• qg q*• gg g*

•in order to account for detailed balance and study equilibration, one needs to account for the reverse processes of parton splittings:

• explicit treatment of 32 processes (D. Molnar, C. Greiner)• glue fusion:


HadronizationHadronization

•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


Testing the PCM Kernel: Collisions

Testing the PCM Kernel: Collisions

• in leading order pQCD, the hard cross section σQCD is given by:

min min

1 12 2 2 min 2

1 2 1 2,

ˆˆ( ) ( , ) ( , ) θ ( )

ˆij

QCD i j Ti j x x

ds dx dx dt f x Q f x Q Q p

dt

• number of hard collisions Nhard (b) is related to σQCD by:

2

23

3

( ) ( )

( ) ' ' ( ')

1 K ;

96

har QCDdN b A b

A b d b h b b h b

b b

• equivalence to PCM implies: keeping factorization scale Q2 = Q0

2 with αs evaluated at Q2

restricting PCM to eikonal mode


Testing the PCM Kernel: pt distribution

Testing the PCM Kernel: pt distribution

,2 21 2 1 22

,1 2

ˆ( , ) ( , ) 1 2 1

ˆ 2jet ij i j

i ji jt

d dx x f x Q f x Q

dp dy dy dt

• the minijet cross section is given by:

• equivalence to PCM implies:

keeping the factorization scale Q2 = Q0

2 with αs evaluated at Q2

restricting PCM to eikonal mode, without initial & final state radiation

• results shown are for b=0 fm


Debye Screening in the PCM

Debye Screening in the PCM

2 32 0

3 1lim

6g q qs

D pqq

pd p F p F p F pq

q p

•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 sectionscurrently beyond the scope of the numerical implementation of the PCM


Choice of pTmin: Screening Mass as Indicator

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


Photon Production in the PCM

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


What can we learn from photons?

What can we learn from photons?

•photon yield directly proportional to the # of hard collisions photon yield scales with Npart

4/3

•primary-primary collision contribution to yield is < 10%•emission duration of pre-equilibrium phase: ~ 0.5 fm/c


Stopping at RHIC: Initial or Final State

Effect?

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/dy5

•is the PCM capable of filling up mid-rapidity region?•is the baryon number transported or released at similar x?


Stopping at RHIC: PCM Results

Stopping at RHIC: PCM Results

•primary-primary scattering releases baryon-number at corresponding y•multiple rescattering & fragmentation fill up mid-rapidity domaininitial state & parton cascading can fully account for data!


Example #3: hydrodynamics


Nuclear Fluid Dynamics

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


Collective Flow: Overview

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)


initial energy density distribution:

Elliptic flow: early creation

Elliptic flow: early creation

time evolution of the energy density:

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.

spatial eccentricity

momentumanisotropy


Elliptic flow: strong rescattering

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, indicating insufficient thermalization of 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


Anisotropies: sensitive to the QCD EoS

Anisotropies: sensitive to the QCD EoS

Teaney, Lauret, Shuryak, nucl-th/0110037P. Kolb and U. Heinz, hep-ph/0204061

the data favor an equation of state with a soft phase and a latent heat e between 0.8 and 1.6 GeV/fm3


Example #4: hybrid approaches• motivation• applications• outlook


Limits of Hydrodynamics

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)


A combined Macro/Micro Transport Model

A combined Macro/Micro Transport Model

• 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)

Hydrodynamics + micro. transport (UrQMD)

matching conditions:• 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


Flavor Dynamics: Radial Flow

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


Connecting high-pt partons with the dynamics of an expanding

QGP

Connecting high-pt partons with the dynamics of an expanding

QGP

color: QGP fluid densitysymbols: mini-jets

Au+Au 200AGeV, b=8 fmtransverse plane@midrapidityFragmentation switched off

hydro+jet model• Jet quenching analysis takingJet quenching analysis takingaccount of (2+1)D hydro resultsaccount of (2+1)D hydro results (M.Gyulassy et al. ’02)(M.Gyulassy et al. ’02)

Hydro+Jet model T.Hirano. & Y.Nara: T.Hirano. & Y.Nara: Phys.Rev.Phys.Rev.C66C66 041901, 2002 041901, 2002

take Parton density take Parton density ρρ((xx) ) from full 3D hydrodynamic from full 3D hydrodynamic calculationcalculation

x

y use GLV 1use GLV 1stst order formula for order formula for parton parton energy loss (M.Gyulassy et al. energy loss (M.Gyulassy et al. ’00)’00)

Movie and data of Movie and data of ρρ((xx) are available at) are available athttp://quark.phy.bnl.gov/~hirano/http://quark.phy.bnl.gov/~hirano/


PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

Transport Theory at RHIC

Transport Theory at RHIC

hadronization

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronic phaseand freeze-out


Last words…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!

Dynamic Modeling of RHIC Collisions

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