dynamic modeling of rhic collisionsdynamic modeling of...

Steffen A. Bass CTEQ 2004 Summer School #1

Dynamic Modeling of RHIC CollisionsDynamic Modeling of RHIC Collisions

Steffen A. BassDuke 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


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 natureis responsible for most of the mass of ordinary matterholds protons and neutrons together in atomic nucleibasic constituents of matter: quarks and gluons

• The QCD vacuum: ground-state of QCD has a complicated structurecontains scalar and vector condensates

explore vacuum-structure by heating/melting QCD matter

Quark-Gluon-Plasma

0 and G 0uu dd Gµνµν+ ≠ ≠


Phases of Normal MatterPhases of Normal Matter

electromagnetic interactions determine phase structure of normal matter

solid liquid gas


Phases of QCD MatterPhases 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

new state of matter!

Quark-GluonPlasma

HadronGas Solid


QGP and the Early UniverseQGP 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 MatterCompressing 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 ModelingThe Purpose of Dynamic Modeling

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

• rigorous calculation of QCD quantities• works in the infinite size / equilibrium limit

Lattice-Gauge Theory:

• full description of collision dynamics• connects intermediate state to observables• provides link between LGT and data

Transport-Theory:

• only observe the final state• rely on QGP signatures predicted by Theory

Experiments:


Microscopic Transport ModelsMicroscopic 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 -

classical approach:

, 0N

Nf f Ht

∂+ =

∂Liouville’s Equation:

use BBKGY hierarchy and cut off at 1-body level

1( ) 0r r pp U f

t m∂ + ∇ − ∇ ∇ = ∂

a) interaction based only on potentials: Vlasov Equation

1collr

p f It 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 pσ ′ ′= Ω − −∫ ∫

b) interaction based only on scattering: Boltzmann Equation

with


Kinetic Theory IIKinetic Theory II

quantum approach:

start with Dyson Equation on contour C (or Kadanoff-Baym eqns):

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

G G G G′ ′ ′′ ′′′ ′′′ ′′′ ′′ ′′= + Σ∫ ∫with G: path ordered non-equilibrium Green’s function

use approximation scheme for self-energy Σ (e.g. T-Matrix approx.)

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

i T T G + ′ ′ ′ ′ ′′ ′ ′′Σ = − − ∫ ∫

Perform Wigner-Transformation of two-point functions A(1,1’) to obtain classical quantities (smooth phase-space functions)

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

WA R p ye A R y R y⋅= + −∫


The Vlasov-Uehling-Uhlenbeck EquationThe 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 pp f r

g dd

f f f

p d p d p p p p pm d

p t

ff f

t m

f f

σδπ

+ +

′

∂ + +∇ ℜΣ ⋅∇ −∇ ℜΣ ⋅

′ ′ ′+ − −Ω

′ ′× −

∇ ∂

′′− −

=

−

−

∫ ∫ ∫

perform Wigner-transform• Connect Σ to scattering rates and potential• identify correlation functions with f• use quasi-particle approximation

quantum approach:

• combine Vlasov- and Boltzmann-equationsclassical approach:

•the Uehling-Uhlenbeck terms are added to ensure the Pauli-Principle


Collision Integral: Monte-Carlo TreatmentCollision 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 σ σ σπ

= =

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 rescatteramong each other

•initial state: 416 nucleons (p,n)•reaction time: 30 fm/c•final state: > 1000 hadrons


Initial Particle Production in UrQMDInitial Particle Production in UrQMD


Meson Baryon Cross Section in UrQMDMeson Baryon Cross Section in UrQMDmodel degrees of freedom determine the interaction to be used

300 MeVN*1990300 MeV∆1950

200 MeVN*1720350 MeV∆1930

110 MeVN*1710200 MeV∆1920

100 MeVN*1700250 MeV∆1910

130 MeVN*1680350 MeV∆1905

150 MeVN*1675200 MeV∆1900

150 MeVN*1650300 MeV∆1700

150 MeVN*1535120 MeV∆1620

125 MeVN*1520350 MeV∆1600

200 MeVN*1440120 MeV∆1232

widthN*width∆*

422

*,2

)()12)(12(12

totsMpIII

R

totMBR

cmsNR MB

RMBtot Γ

→

∆= +−

ΓΓ++

+= ∑ πσcalculate cross section according to:


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


Basic Principles of the PCMBasic 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 nucleonstructure 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


Initial State: Parton MomentaInitial 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 pβ −= =


Binary Processes in the PCMBinary Processes in the PCM

( ) ( ),

ˆ ˆˆ ˆ ab cdabc d

ssσ σ →=∑

( )max

min

ˆ

ˆ

ˆˆ ˆ ˆ( , , ) ˆˆ ˆˆ

t

ab cdt ab cd

d s t us dtdt

σ→

→

′ ′ ′= ′ ∫

• the total cross section for a binary collision is given by:

σwith partial cross sections:

• now the probability of a particular channel is:

( ) ( )( )

ˆ ˆˆ

ˆ ˆab cd

ab cdab

sP s

sσσ

→→ =

• finally, the momentum transfer & scattering angle are sampled via

( ) ( )min

ˆ

ˆ

ˆˆ ˆ ˆ1 ( , , )ˆ ˆˆˆ ˆ

t

ab cd t ab cd

d s t ut dts dt

σσ → →

′ ′ ′Ξ = ′ ∫


Parton-Parton Scattering Cross-SectionsParton-Parton Scattering Cross-Sections

q qbar → g g

q qbar → γ γq qbar → q qbar

q qbar → g γq q → q q

q g →q γg g → q qbar

q qbar→ q’ qbar’q g→ q g

q q’ → q q’g g → g g 2 2 2

9 32

tu su sts t u

− − −

2 2

2

49s u s uu s t

+ − + +

2 2

2

1 36 8t u t uu t s

+ + − 2 2 2 2 2

2 2

4 89 27s u s t st u tu

+ ++ −

2 2

2

49s ut+

2 2

2

49t us+

2

3qe u ss u

− +

289 q

u tet u

+

423 q

u tet u

+

2 2 2 2 2

2 2

4 89 27s u u t ut s st

+ ++ −

2 2

2

32 827 3

t u t uu t s

+ + −

• a common factor of παs2(Q2)/s2 etc.

• further decomposition according to color flow


Initial and final state radiationInitial and final state radiationProbability for a branching is given in terms of the Sudakov form factors:

( ) ( ) ( ) ( )( )

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

xα

π′ ′ ′

′→′

′ ′ ′= − ′ ∑∫ ∫

space-like branchings:

( ) ( ) ( )max

max, , exp2

t

d dat

d d es P zt

T x t t dt dzα

π ′′

→

′ ′= −

∑∫ ∫

• Altarelli-Parisi splitting functions included: Pq→qg , Pg→gg , Pg→qqbar & Pq→qγ

time-like branchings:


Higher Order Corrections and MicrocausalityHigher 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 (2→1) ProcessesParton Fusion (2→1) Processes

work i

n pro

gress

• 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 3→2 processes (D. Molnar, C. Greiner)• glue fusion:


HadronizationHadronization

•requires modeling & parameters beyond the PCM pQCD framework•microscopic theory of hadronizationneeds yet to be established•phenomenological recombination + fragmentation approach may provide insight into hadronization dynamics•avoid hadronization by focusing on:

net-baryonsdirect photons


Testing the PCM Kernel: CollisionsTesting 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

dtσ

σ = −∑ ∫ ∫ ∫• 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 bν ν ν νπ ω

σ= ×

= −

= =

∫

• equivalence to PCM implies:keeping factorization scale Q2 =

Q02 with αs evaluated at Q2

restricting PCM to eikonal mode


Testing the PCM Kernel: pt distributionTesting the PCM Kernel: pt distribution• the minijet cross section is given by:

( ) ,2 21 2 1 22

,1 2

ˆ( , ) ( , ) 1 2 1ˆ 2

jet ij i ji j

i jt

d dx x f x Q f x Q

dp dy dy dtσ σ δ

= + ↔ −

∑

• 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 PCMDebye Screening in the PCM

( ) ( ) ( ) 2 32 0

3 1lim6g q q

sD pq q

pd p F p F p F pq

q pαµπ →

= ⋅∇ + + ⋅

∑∫

•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


Choice of pTmin: Screening Mass as IndicatorChoice 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 pT

min> µD : lower boundary for pT

min : approx. 0.8 GeV


Photon Production in the PCMPhoton 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?

•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


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/dy≈5

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


Stopping at RHIC: PCM ResultsStopping at RHIC: PCM Results

•primary-primary scattering releases baryon-number at corresponding y•multiple rescattering & fragmentation fill up mid-rapidity domain

initial state & partoncascading can fully account for data!


Example #3: hydrodynamics


Nuclear Fluid DynamicsNuclear 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: OverviewCollective 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)


Elliptic flow: early creationElliptic flow: early creation

initial energy density distribution: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 rescatteringElliptic 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 EoSAnisotropies: sensitive to the QCD EoSTeaney, 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 HydrodynamicsLimits 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 ModelA combined Macro/Micro Transport Model

Hydrodynamics + micro. transport (UrQMD)

• ideally suited for dense systemsmodel early QGP reaction stage

• well defined Equation of StateIncorporate 1st order p.t.

• parameters:– initial conditions (fit to

experiment)– Equation of State

• no equilibrium assumptionsmodel break-up stagecalculate freeze-out

• parameters:– (total/partial) cross sections– resonance parameters

(full/partial widths)

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 FlowFlavor 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 ratesnearly 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

Hydro+Jet modelT.HiranoT.Hirano. & . & Y.NaraY.Nara: : Phys.Rev.Phys.Rev.C66C66 041901, 2002041901, 2002

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

x

yuse GLV 1use GLV 1stst order formula for order formula for partonpartonenergy loss (energy loss (M.GyulassyM.Gyulassy et al. et al. ’’00)00)

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

•• Jet quenching analysis takingJet quenching analysis takingaccount of (2+1)D hydro resultsaccount of (2+1)D hydro results

((M.GyulassyM.Gyulassy et al. et al. ’’02)02)


Transport Theory at RHICTransport Theory at RHIC

PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

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 developedexciting area of research with lots of challenges and opportunities!

dynamic modeling of rhic collisionsdynamic modeling of...

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