Ultra-peripheral Collisions at RHIC Spencer Klein, LBNL for the STAR Collaboration
Dynamic Modeling of RHIC Collisions
description
Transcript of 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
Steffen A. Bass CTEQ 2004 Summer School #2
Why do Heavy-Ion Physics?•QCD Vacuum•Bulk Properties of Nuclear Matter•Early Universe
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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
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Phases of Normal Matter
Phases of Normal Matter
electromagnetic interactions determine phase structure of normal matter
solid liquid gas
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• 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
Steffen A. Bass CTEQ 2004 Summer School #6
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
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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
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Dynamic Modeling• purpose• fundamentals• current status
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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
Steffen A. Bass CTEQ 2004 Summer School #10
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
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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
Steffen A. Bass CTEQ 2004 Summer School #12
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)
Steffen A. Bass CTEQ 2004 Summer School #13
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
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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
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Example #1: the hadronic world• the UrQMD model
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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
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Initial Particle Production in UrQMD
Initial Particle Production in UrQMD
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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:
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Example #2: the partonic world• The Parton Cascade Model• applications
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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
Steffen A. Bass CTEQ 2004 Summer School #21
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
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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
Steffen A. Bass CTEQ 2004 Summer School #23
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
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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
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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
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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:
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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
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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
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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
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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
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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
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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
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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
Steffen A. Bass CTEQ 2004 Summer School #34
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?
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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!
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Example #3: hydrodynamics
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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
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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)
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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
Steffen A. Bass CTEQ 2004 Summer School #40
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
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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
Steffen A. Bass CTEQ 2004 Summer School #42
Example #4: hybrid approaches• motivation• applications• outlook
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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)
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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
Steffen A. Bass CTEQ 2004 Summer School #45
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
Steffen A. Bass CTEQ 2004 Summer School #46
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/
Steffen A. Bass CTEQ 2004 Summer School #47
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
Steffen A. Bass CTEQ 2004 Summer School #48
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!