Dynamic Modeling of RHIC Collisions Steffen A. Bass Duke University & RIKEN BNL Research Center • Motivation: why do heavy-ion collisions? • Introduction: the basics.
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Transcript Dynamic Modeling of RHIC Collisions Steffen A. Bass Duke University & RIKEN BNL Research Center • Motivation: why do heavy-ion collisions? • Introduction: the basics.
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
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CTEQ 2004 Summer School #3
Phases of Normal Matter
solid
liquid
gas
electromagnetic interactions
determine phase structure of
normal matter
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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
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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
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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
I coll
Steffen A. Bass
p 1
t m r f I coll
N d dp2 v1 v2 f1 ( p1) f1 ( p2 ) f1 ( p1 ) f1 ( p2 )
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 d1G0 (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 )
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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
f1f 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
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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
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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
RMBtot
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 (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
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 Ei N1 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 abcd sˆ
c,d
tˆmax
dˆ ( sˆ, tˆ, uˆ)
with partial cross sections: ˆ abcd sˆ
dtˆ
dtˆ
abcd
tˆmin
• now the probability of a particular channel is:
ˆ abcd sˆ
Pabcd sˆ
ˆ ab sˆ
• finally, the momentum transfer & scattering angle are sampled via
tˆ
ˆ ( sˆ, tˆ, uˆ )
d
tˆ
dtˆ
ˆ abcd sˆ tˆmin
dtˆ
abcd
1
Steffen A. Bass
CTEQ 2004 Summer School #22
Parton-Parton Scattering Cross-Sections
gggg
q g q g
g g q qbar
qqqq
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
9u 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 2u t
eq
9 t u
2 4u 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
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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 Paae 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:
Pqqg , Pggg , Pgqqbar & Pqqγ
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 (21) 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 32 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
dˆ 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
dˆ 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/dy5
•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