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
CTEQ 2004 Summer School #1
Why do Heavy-Ion Physics?
•QCD Vacuum
•Bulk Properties of Nuclear Matter
•Early Universe
Steffen A. Bass
CTEQ 2004 Summer School #2
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
uu  dd  0 and G  G  0
 explore vacuum-structure by
heating/melting QCD matter
 Quark-Gluon-Plasma
Steffen A. Bass
CTEQ 2004 Summer School #3
Phases of Normal Matter
solid
liquid
gas
 electromagnetic interactions
determine phase structure of
normal matter
Steffen A. Bass
CTEQ 2004 Summer School #4
Phases of QCD Matter
• strong interaction analogues of
the familiar phases:
Quark-Gluon
Plasma
• Nuclei behave like a liquid
– Nucleons are like molecules
• Quark Gluon Plasma:
– “ionize” nucleons with heat
– “compress” them with density
 new state of matter!
Steffen A. Bass
Hadron
Gas
Solid
CTEQ 2004 Summer School #5
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
CTEQ 2004 Summer School #6
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
CTEQ 2004 Summer School #7
Dynamic Modeling
• purpose
• fundamentals
• current status
Steffen A. Bass
CTEQ 2004 Summer School #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
CTEQ 2004 Summer School #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
CTEQ 2004 Summer School #10
Kinetic Theory:
- formal language of transport models classical approach:
Liouville’s Equation:
f N
 f N , H  0
t
use BBKGY hierarchy and cut off at 1-body level
a) interaction based only on potentials: Vlasov Equation
 p
 1



(

U
)

r
p f  0
 t m r

b) interaction based only on scattering: Boltzmann Equation
with
 p  1
 t  m  r  f  I coll
I coll  N   d  dp2 v1  v2  f1 ( p1) f1 ( p2 )  f1 ( p1 ) f1 ( p2 ) 
Steffen A. Bass
CTEQ 2004 Summer School #11
Kinetic Theory II
quantum approach:
start with Dyson Equation on contour C (or Kadanoff-Baym eqns):
G(1,1)  G0 (1,1)   d1 d1G0 (1,1)(1,1)G(1,1)
C
C
with G: path ordered non-equilibrium Green’s function
use approximation scheme for self-energy Σ (e.g. T-Matrix approx.)
(1,1)  i









d2
d2
12
T
1
2

12
T
2
1
G
(2
,
2
)
  

C
C
Perform Wigner-Transformation of two-point functions A(1,1’) to
obtain classical quantities (smooth phase-space functions)
AW ( R, p)   d 4 yeip y / A( R  12 y, R  12 y )
Steffen A. Bass
CTEQ 2004 Summer School #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
   p1



 t   m   p1    r1   r1    p1  f1 (r1 , p1 , t )




2g
d
3
3
3
 2
d p2  d p1  d p2  ( p1  p2  p1  p2 )
3 
m (2 )
d
  f1f 2(1  f1 )(1  f 2 )  f1 f 2 (1  f1)(1  f 2 ) 


•the Uehling-Uhlenbeck terms are added to ensure the Pauli-Principle
Steffen A. Bass
CTEQ 2004 Summer School #13
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:
d min 
 tot
with  tot   tot


s , h1 , h2

• main parameter:
– cross section: probability for an interaction to take place,
which is interpreted geometrically
dmin
Steffen A. Bass
CTEQ 2004 Summer School #14
Example #1: the hadronic world
• the UrQMD model
Steffen A. Bass
CTEQ 2004 Summer School #15
Applying Transport Theory to HeavyIon 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 (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
CTEQ 2004 Summer School #16
Initial Particle Production in UrQMD
Steffen A. Bass
CTEQ 2004 Summer School #17
Meson Baryon Cross Section in UrQMD
 model degrees of freedom determine the interaction to be used
MB

calculate cross section according to:  tot
Steffen A. Bass
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
RMBtot
2I R  1

2

2
tot
2
R   , N * ( 2 I B  1)( 2 I M  1) pcms ( M R  s )  4
CTEQ 2004 Summer School #18
Example #2: the partonic world
• The Parton Cascade Model
• applications
Steffen A. Bass
CTEQ 2004 Summer School #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 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
CTEQ 2004 Summer School #20
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:
2
2
2
2

i 
i  
i  
i 



M
E
p
p
p





 
x 
y 
z
i  i
 i
 i

with pzi  xi PzN and E i   N1 pzi
Steffen A. Bass
2
N
CTEQ 2004 Summer School #21
Binary Processes in the PCM
• the total cross section for a binary collision is given by:
ˆ ab  sˆ    ˆ abcd  sˆ 
c ,d
tˆmax
with partial cross sections:
 dˆ ( sˆ, tˆ, uˆ) 
ˆ abcd  sˆ    
dtˆ

dtˆ
abcd
tˆmin 
• now the probability of a particular channel is:
ˆ abcd  sˆ 
Pabcd  sˆ  
ˆ ab  sˆ 
• finally, the momentum transfer & scattering angle are sampled via
tˆ
ˆ ( sˆ, tˆ, uˆ ) 

d

  tˆ  
dtˆ



ˆ abcd  sˆ  tˆmin 
dtˆ
abcd
1
Steffen A. Bass
CTEQ 2004 Summer School #22
Parton-Parton Scattering Cross-Sections
gggg
q g q g
g g  q qbar
qqqq
q qbar  q qbar
q qbar  g g
9
tu su st 
3

 2  2

2
2
s
t
u 
4  s u  s2  u2
   
9u s
t2
1  t u  3 t 2  u2
  
6  u t  8 s2
4  s2  u2 s2  t 2  8 s2



2
2
9 t
u  27 tu
4  s2  u2 u2  t 2  8 u2



9  t2
s 2  27 st
q q’  q q’
q qbar q’ qbar’
q g q γ
q qbar  g γ
q qbar  γ γ
4 s2  u2
9 t2
4 t 2  u2
9 s2
eq2  u s 
   
3  s u
8 2u t 
eq   
9  t u
2 4u t 
eq   
3  t u
32  t u  8 t 2  u 2
  
27  u t  3 s 2
• a common factor of παs2(Q2)/s2 etc.
• further decomposition according to color flow
Steffen A. Bass
CTEQ 2004 Summer School #23
Initial and final state radiation
Probability for a branching is given in terms of the Sudakov form factors:
space-like branchings:
 tmax  s  t  
xa f a  xa , t   

Sa  xa , tmax , t   exp   dt
dz Paae  z 




2

x
f
x
,
t
 
a
 t
a a a
time-like branchings:
 tmax  s  t  


Td  xd , tmax , t   exp   dt
dz Pd d e  z  


2

a
 t

• Altarelli-Parisi splitting functions included:
Pqqg , Pggg , Pgqqbar & Pqqγ
Steffen A. Bass
CTEQ 2004 Summer School #24
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
CTEQ 2004 Summer School #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:
• explicit treatment of 32 processes (D. Molnar, C. Greiner)
• glue fusion:
• qg  q*
• gg  g*
Steffen A. Bass
CTEQ 2004 Summer School #26
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
CTEQ 2004 Summer School #27
Testing the PCM Kernel: Collisions
• in leading order pQCD, the hard cross section σQCD is given by:
 QCD ( s)  
1

1
dx1
i , j xmin

dx2  dtˆ
xmin
dˆ ij
dtˆ

fi ( x1 , Q 2 ) f j ( x2 , Q 2 ) θ Q 2  ( pTmin )2
• number of hard collisions Nhard (b) is related to σQCD by:
N hard (b)   QCD  A(b)
A(b)   d 2b ' h

b b '
 h(b ')
2
1
3


b
K

b
;


  3 
96

• equivalence to PCM implies:
 keeping factorization scale Q2 =
Q02 with αs evaluated at Q2
 restricting PCM to eikonal mode
Steffen A. Bass
CTEQ 2004 Summer School #28

Testing the PCM Kernel: pt distribution
• the minijet cross section is given by:
d jet
dˆ ij

  i , j 
2
2
  x1 x2  fi ( x1 , Q ) f j ( x2 , Q )
 1  2  1 

2
ˆ
dpt dy1dy2 i , j
dt
2



• 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
CTEQ 2004 Summer School #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]:
 
2
D
3 s
2


1
lim  d p
q  p  Fg  p   Fq  p   Fq  p 
q 0
q p
6 q


3
p
•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
currently beyond the scope of the numerical implementation of
the PCM
Steffen A. Bass
CTEQ 2004 Summer School #30
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
CTEQ 2004 Summer School #31
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
CTEQ 2004 Summer School #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
 photon yield scales with Npart4/3
CTEQ 2004 Summer School #33
Stopping at RHIC:
Initial or Final State Effect?
•net-baryon contribution from
initial state (structure functions)
is non-zero, even at midrapidity!
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
CTEQ 2004 Summer School #34
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
CTEQ 2004 Summer School #35
Example #3: hydrodynamics
Steffen A. Bass
CTEQ 2004 Summer School #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)
 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
CTEQ 2004 Summer School #37
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
CTEQ 2004 Summer School #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
CTEQ 2004 Summer School #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 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
Steffen A. Bass
CTEQ 2004 Summer School #40
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 e between 0.8 and 1.6 GeV/fm3
Steffen A. Bass
CTEQ 2004 Summer School #41
Example #4: hybrid approaches
• motivation
• applications
• outlook
Steffen A. Bass
CTEQ 2004 Summer School #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
• v2 saturates for high pt vs.
monotonic increase in hydro
(onset of pQCD physics)
Steffen A. Bass
CTEQ 2004 Summer School #43
A combined Macro/Micro Transport Model
Hydrodynamics
•
ideally suited for dense systems
+ micro. transport (UrQMD)
•
 model early QGP reaction stage
•
•
 model break-up stage
 calculate freeze-out
well defined Equation of State
 Incorporate 1st 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 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
Steffen A. Bass
CTEQ 2004 Summer School #44
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
CTEQ 2004 Summer School #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
use GLV 1st order formula for parton
energy loss (M.Gyulassy et al. ’00)

y
T.Hirano. & Y.Nara: Phys.Rev.C66 041901, 2002
Au+Au 200AGeV, b=8 fm
transverse plane@midrapidity
Fragmentation switched off
take Parton density ρ(x) from
full 3D hydrodynamic calculation
Steffen A. Bass
x
Movie and data of ρ(x) are available at
http://quark.phy.bnl.gov/~hirano/
CTEQ 2004 Summer School #46
Transport Theory at RHIC
hadronic phase
and freeze-out
QGP and
hydrodynamic expansion
initial state
pre-equilibrium
hadronization
CYM & LGT
PCM & clust. hadronization
NFD
NFD & hadronic TM
string & hadronic TM
PCM & hadronic TM
Steffen A. Bass
CTEQ 2004 Summer School #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
 exciting area of research with lots of challenges and
opportunities!
Steffen A. Bass
CTEQ 2004 Summer School #48