# Presentation: 934925

## Modeling of Ultra-Relativistic Heavy-Ion Collisions at RHIC

- Duke University

-  Overview:
-  Relativistic Fluid Dynamics
-  Microscopic Transport Models
-  Hybrid Approaches
-  Comprehensive Data Analysis

## Transport Theory at RHIC:

- Motivation
- Survey of available concepts
- Discussion

## The Purpose of Dynamic Modeling

| 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 |

## Transport Theory at RHIC

**hadronization**

**initial state**

**pre-equilibrium**

**QGP and**
**hydrodynamic expansion**

**hadronic phase**
**and freeze-out**

## RHIC Transport Theory: Discussion

- collisions at RHIC cover a sequence of vastly different dynamical regimes
- standard transport approaches are usually only applicable to a subset of the phases of the reaction or restricted to a particular regime
- in particular the dynamical modeling of the very early reaction stage and thermalization mechanisms remain a challenge
- hybrid models can extend the range of applicability of conventional approaches

## Relativistic 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*

## Collision Geometry: Elliptic Flow

**elliptic flow (v****2****):**
-  gradients of almond-shape surface will lead to preferential emission in the reaction plane
-  asymmetry out- vs. in-plane emission is quantified by 2nd Fourier coefficient of angular distribution: v2
-  calculable with fluid-dynamics

- The application of fluid-dynamics implies that the medium is in local thermal equilibrium!
- Note that fluid-dynamics cannot make any statements how the medium reached the equilibrium stage...

## Elliptic flow: early creation

-  P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000) 054909

- Most model calculations suggest that flow anisotropies are generated at the earliest stages of the expansion, on a **timescale of ~ 5 fm/c** if a QGP EoS is assumed.

**Notes:**
Anisotropic observables are of special interest as they reflect the dynamics of the earliest stages of the reaction, and thus the dynamics of the hottest phases, eventually the plasma phase that we hope to achieve in heavy ion collisions at RHIC. The overlap region of two nuclei at finite impact parameter is strongly deformed, which you can see here characterized by contours of constant wounded nucleon density in the plane orthogonal to the beam axis. 
If the system is driven apart by the pressure in the system, then you see that it will expand faster into the direction of the strongest pressure gradients, and that is in the direction of the reaction plane.  Upon building up an anisotropic flow field, the system will therefore get rounder with time, as you can see in those subsequent shots of the transverse plane at different times.
This reduction of the spatial eccentricity can be quantified by averaging the difference of the width in x direction and the width in y direction over this transverse distribution function. You see how this leads to a rapid decrease of the initial large geometric anisotropy over timescales relevant for heavy ion dynamics, once displayed here for a system with a very hard equation of state of a ideal gas, where the transition is very rapid, and once for an equation of state that involves a phase transition from hadronic to partonic degrees of freedom.
While the system becomes isotropic in coordinate space, it builds up anisotropies in momentum space, which you can characterize by the difference of the diagonal components of the energy momentum tensor. And here you see nicely how, as the system gets round in coordinate space, no further momentum anisotropy is built up. Note that for a system with a large mixed phase, the generation of further anisotropies is stalled significantly earlier. The momentum anisotropy you observe is really built up in the plasma stage!
The timescale suggested from all model calculations, no matter if macroscopic or microscopic is generally below 5 fm/c. Anisotropies thus probe the earliest stages of the collision. 

## Elliptic Flow: ultra-cold Fermi-Gas

- Li-atoms released from an optical trap exhibit elliptic flow analogous to what is observed in ultra-relativistic heavy-ion collisions
- Elliptic flow is a general feature of strongly interacting systems!

## Survey of Hydrodynamic Models

- PT  spectra

- Elliptic Flow

- PHENIX:Nucl.Phys. A757 (2005) 184

- Au + Au                   GeV

- 

- 

| Ref |  (GeV/fm3)or  s(fm-3) | fm/c | Latent heat
(GeV/fm3) | Hadronic stage | Tf (MeV) |
| --- | --- | --- | --- | --- | --- |
| [1] | 23()(eWN) | 0.6 | 1.15 | Full equil. | 120 |
| [2] | 110(s)(0.75sWN+0.25sBC) | 0.6 | 1.15 | Partial chemical equil. | 100 |
| [3] | 35()(eBC) | 0.6  | 1.7 | Partial chemical equil. | 100,120,140 |
| [4] | 16.7()(sWN) | 1.0 | 0.8 | RQMD | 100 |

- [1]Huovinen et.al., PLB(130) [2]Kolb et.al. PRC, [3]Hirano et.al.PRC(130), [4]Teaney et al.

## 3D-Hydro: Results

- separate chemical f.o.
- simulated by rescaling p,K

- good agreement to data
- 3D-Hydro gives a reliable description of bulk dynamics at RHIC

## Transport Theory at RHIC

**hadronization**

**initial state**

**pre-equilibrium**

**QGP and**
**hydrodynamic expansion**

**hadronic phase**
**and freeze-out**

## Microscopic Transport Models

- Parton Cascade Models
- Hadronic Transport Models

## Microscopic Transport Models

- the state of all particles of the system is defined by the N-body distribution function *f**N **, *which can be reduced to the 1-body distribution function *f**1* via the BBGKY hierarchy
- In the low density limit and assuming that *f**1* only changes via 2-body scattering, the time-evolution of *f**1* can be described by a Boltzmann-Equation:

-     microscopic transport models describe the time-evolution of a system of (microscopic) particles by solving a transport equation derived from kinetic theory

## Basic Principles of the PCM

-  degrees of freedom: quarks and gluons
-  classical trajectories in phase space (with relativistic kinematics)
-  initial state constructed from structure functions and elastic form factors
-  interaction criterion based on geometric interpretation of cross section:
-  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
-  guiding scales: initialization scale Q0, pT cut-off p0 / Debye-mass μD,     		              intrinsic kT / saturation momentum QS, virtuality > μ0

## Parton-Parton Cross-Sections

| g g  g g |  | q q’  q q’ |  |
| --- | --- | --- | --- |
| q g q g |  | q 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 |  |  |  |

-  a common factor of παs2(Q2)/s2  etc.
-  divergences in the matrix elements require special integration schemes

## Initial and final state radiation

**space-like branchings:**

**time-like branchings:**

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

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

- consistent treatment of splittings requires inclusion of gluon-fusion (or multi-particle collision) processes and the LPM effect!

## Applications: Photons & BN-Transport

- w/o LPM: overprediction of photon yield
- w/ LPM: photon yield for pt < 6 GeV strongly reduced
- strong pt dependence of suppression
- good agreement with data

- multiple scattering & fragmentation fill up mid-rapidity domain
- Initial state + parton cascading can fully account for data!

- Bass et al. PRL 91 052302 (2005)

- Renk et al. PLB 632, 632 (2006)

## PCM: Discussion

**future improvements:**
-  treatment detailed balance: (N  2 processes)
-  selfconsistent medium corrections
-  need to augment the PCM with dynamical description of soft modes

- the PCM is an important tool to study high-pt phenomena at RHIC, but in its current incarnation cannot provide the full dynamics of the created (bulk) matter

## String/Hadron Transport Models

- Example: UrQMD
- elementary degrees of freedom: hadrons, const. (di)quarks
- classical trajectories in phase-space (relativistic kinematics)
- initial high energy phase of the reaction is modeled via the excitation and fragmentation of strings
- 55 baryon- and 32 meson species, among those 25 N*, Δ* resonances and 29 hyperon/hyperon resonance species
- full baryon-antibaryon and isospin symmetry
- main physics input and parameters:
**cross sections**: total and partial cross sections, angular distributions
**resonance parameters**: total and partial decay widths
**string fragmentation scheme**: fragmentation functions, formation time
- UrQMD is ideal for the description of excited **hadronic** matter

## Visualization in UrQMD

- Pb + Pb @ 160 GeV/nucleon (CERN/SPS)

-  visualization performed with POVRAY on basis of one UrQMD event
- 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

-  educational benefits & useful for visual debugging

## Hadronic Transport: Discussion

- string/hadron models have a proven track-record for the description of heavy-ion reactions at BEVALAC, SIS, AGS up to low SPS energies
- conditions at RHIC exclude a purely hadronic evolution
-  string fragmentation as dominant source of initial particle production  creates insufficient transverse pressure to account for the observed collective flow phenomena
- highly effective for describing the late, viscous hadronic reaction phase
-  flavor-dependence of hadronic cross sections not treated in hydro
-  finite mean free path of hadrons in break-up stage implies viscosity
- use as part of a hybrid model!

## Hybrid Hydro+Micro Approaches

-  Sequential Models
-  Lockstep Models
-  Diffusion Models

## 3D-Hydro + UrQMD Model

**Full 3-d Hydrodynamics**
**QGP evolution**

- Cooper-Frye
- formula

**UrQMD**

- t fm/c

- hadronic rescattering

- Monte Carlo

**Hadronization**

*T**C*

*T**SW*

- Bass & Dumitru, PRC61,064909(2000)
- Teaney et al, nucl-th/0110037
- Nonaka & Bass, nucl-th/0510038
- Hirano et al. nucl-th/0511046

**ideally suited for dense systems**
- model early QGP reaction stage
**well defined Equation of State**
**parameters:**
- initial conditions
- Equation of State

**Hydrodynamics****   ****     ****+****  ****micro. transport (UrQMD)**

- no equilibrium assumptions
- model break-up stage
- calculate freeze-out
- includes viscosity in hadronic phase
- parameters:
- (total/partial) cross sections

- matching condition:
-  use same set of hadronic states for EoS as in UrQMD
-  generate hadrons in each cell using local T and μB

## 3D-Hydro+UrQMD: Results

- significantly improved agreement to data
- good description of cross section dependent features

## 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 density**
**symbols: mini-jets**

- Au+Au 200*A*GeV, *b*=8 fm
- transverse plane@midrapidity
- Fragmentation switched off

**hydro+jet model**

** **Jet quenching analysis taking
- account of (2+1)D hydro results
-                      (M.Gyulassy et al. ’02)

**Hydro+Jet model**
** **T.Hirano. & Y.Nara: Phys.Rev.**C66** 041901, 2002

- take Parton density *ρ*(*x*) from full 3D hydrodynamic calculation

***x***

***y***

** **use GLV 1st order formula for parton
- energy loss (M.Gyulassy et al. ’00)

- next step:
- treat dissipated energy in hydro

## Diffusion of Heavy Quarks in a Hydrodynamic Medium

- propagation of heavy quarks in a QGP medium resembles a **diffusion process**
- use evolution of 3D-Hydro as medium
- T, μ and vflow are known as function of r& τ
- describe propagation with a **Langevin Eqn**:
- drag coefficient κ/2T can be locally calculated from 3D hydro evolution

- Teaney & Moore: PRC 71, 064904 (2005)
- Van Hees, Greco & Rapp: nucl-th/0508055
- Bass, Asakawa & Mueller in preparation

## Comprehensive Data Analysis 

- utilizing Transport Models

## Scientific Goals

- extraction of the Equation of State of strongly interacting matter under RHIC conditions & comparison to Lattice predictions
- determination of equilibration mechanisms and initial conditions
- extraction of transport coefficients
- test of Lattice predictions for spectral functions

## Strategy: from data to QCD

- develop a comprehensive transport model (or set of consistent interlocking transport approaches), capable of describing the full time-evolution of a heavy-ion collision at RHIC, starting from the coherent glue-field dominated initial state up to the hadronic final state
- identify the relevant physics parameters (EoS, QCD transport coefficients, matrix elements etc.) which are sensitive to the observables measured at RHIC
- conduct a systematic study in that multi-dimensional parameter-space and via comparison to data to determine the properties of the QCD medium created at RHIC

## The next steps

- improved treatment of multi-particle interactions and detailed balance in cascade-type models
- treatment of coherence effects (LPM) in PCM-type models
- development of a fully 3D viscous hydro
- lockstep-models: develop hydro+PCM type model including the dynamical treatment of dissipated energy from hard partons

- Molnar; Greiner & Xu

- Renk & Bass; Greiner & Xu

- Teaney; U. Heinz et al.

- Nonaka & Bass; Hirano & Nara

## Required Resources

- manpower:
-  lack of programmatic funds for translating basic advances into applicable & available tools
-  missed opportunities for personnel training
- computing resources:
-  hardware resources for model development
-  high performance computing facilities for production
-  large memory unit(s) for hydrodynamic calculations
-  multiple processors for hybrid monte-carlos
-  large amount of disk storage for event database
-  event database needs to be publicly accessible for analysis

## Summary and Outlook

- 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 reaction remains to be developed
- field has matured to a point where it will yield significant and quantitative results in the near to medium term future

## The End

## Beyond the PCM: Soft Degrees of Freedom

- Describing the full microscopic dynamics of the deconfined phase
- requires the consistent & simultaneous treatment of particles & fields

- combine a *hard* PCM-type collision term with *soft* field dynamics (described by Wong’s and Yang-Mills eqns.):

- with fq,g the phase-space distribution functions of quarks & *hard* gluons

- couple to Yang-Mills eqn:

- and

- H.T. Elze & U. Heinz, Phys. Rept. 183 (1989)
- Bass, Mueller & Poschl, J. Phys. G25, L109 (1999)