# Presentation: 573992

## Introduction: 

**Introduction:**** **
** ****RHIC Scientists serve up “Perfect Liquid”, April 18, 2005**
** ****BRAHMS, PHENIX, PHOBOS, STAR White Papers, NPA, 2005**
** ****AIP Top Physics Story 2005**
**Hydrodynamics and scaling of soft observables**
** ****Exact (i.e. not numerical) integrals of fluid dynamics **
** ****non-relativistic and relativistic solutions**
** ****Evidence for hydrodynamic scaling in RHIC data**
**  ****Scaling of slope parameters**
**  ****Scaling of Bose-Einstein /HBT radii**
**  ****Universal scaling of elliptic and higher order flows**
** ****Intermediate p****t**** region: breaking of the hydro scaling**

## Location of experiments: BNL

**RHIC: ****Au+Au @ E****cms ****= ****200**** ****A****Ge****V**
**Au+Au****, Cu+Cu,**** ****p****p+p****p****,**** ****d+Au**** ****collisions**
**4 experiments:**** **
**	****BRAHMS, PHENIX, PHOBOS, STAR**
**Brazilian participation: PHENIX**** and STAR:**** **

## PHENIX

**photons, electrons muons and hadrons**
**Investigates all stages of the reaction**
**Penetrating probes: early state of the reaction**
**Hadrons**

## 1st milestone: new phenomena

**1st milestone: new phenomena**

**Suppression of high pt particle production in Au+Au collisions at RHIC**

## 2nd milestone: new form of matter

**d+Au: no suppression**
**Its not the nuclear effect**
**on the structure functions**
**Au+Au:**
** ****new form of matter !**

## 3rd milestone: Top Physics Story 2005

**3****rd**** milestone: Top Physics Story 2005**

_**http://arxiv.org/abs/nucl-ex/0410003**_

## Discovering New Laws

- First we guess it.
- Then we compare the consequences of the guess to see
- what would be implied if this law that we guessed is right.
- Then we compare the result of the computation to nature,
- with  experiment or experience, compare it directly with observation,
- to  see if it works.
- If it disagrees with experiment it is wrong.
- In that simple  statement  is the key to science.
- It does  not  make  any difference  how  beautiful your guess is.
- It does  not  make  any difference  how  smart you are,
- who made the guess, or  what  his name is —
- if it disagrees with experiment it is wrong.”
- 						/R.P. Feynman/

**Discovering New Laws**

##  An observation: 

- increase linearly with mass:
- T = T0 + m<ut>2
- Increase is stronger in more head-on collisions.
- Suggests collective radial flow, local thermalization and hydrodynamics

** ****An observation: **

- PHENIX, Phys. Rev. C69, 034909 (2004)

## Notation for fluid dynamics

**nonrelativistic hydro:**
**t: ****	****time,**
**r:****	****	****coordinate 3-vector, r = (r****x****, r****y****, r****z****),**
**m: ****	****mass,**
**field i.e. (t,r) dependent variables:**
**n:****	****	****number density,**
**p:****	****	****pressure,**
******: ****	****energy density,**
**T: ****	****temperature,**
**v:****	****	****velocity 3-vector, v = (v****x****, v****y****, v****z****),**
**relativistic hydro:**
**x********:****	****coordinate 4-vector, x******** = (t, ****r****x****, r****y****, r****z****),**** **
**k********:****	****momentum 4-vector, k******** = (E, ****k****x****, k****y****, k****z****),   k******** k********= m****2 ****,**
**additional fields in relativistic hydro:**
**u********:****	****velocity 4-vector, u******** = ********(1, ****v****x****, v****y****, v****z****), ****	****    u******** u********= 1,******
**g********:****	****metric tensor, g********= diag(1,-1,-1,-1),**
**T********:  ****	****energy-momentum tensor .**

## Nonrelativistic dynamics of perfect fluids

**Equations of nonrelativistic hydro:**
**local conservation of **
**charge: continuity**
**momentum: Euler**
**energy**
**Not closed, EoS needed:**
**Perfect fluid****: definitions are equivalent, term used by PDG**
** ****# 1: no bulk and shear viscosities, and no heat conduction.**
** ****# 2: T******** = diag(e,-p,-p,-p) in the local rest frame.**
**ideal fluid: ambiguously defined term, discouraged**
** ****	****	****	****	** #1: keeps its volume, but conforms to the outline of its container
- 				 #2: an inviscid fluid

## Old idea: Quark Gluon Plasma
“Ionize” nucleons with heat
“Compress” them with density
New state(s?) of matter

_**Old**_** idea: Quark Gluon Plasma**
**“****Ionize” nucleons with heat**
**“****Compress” them with density**
**New state(s?) of matter**

**Input from lattice: EoS of QCD Matter**

**General input for hydro: p(********,T)**
**LQCD for RHIC region: p~p(T),**
**c****s****2**** = ********p/********e = c****s****2****(T) = 1/********(T)**
**It’s in the family analytic hydro solutions!**** **

## New parametric, ellipsoidal hydro solutions 

**Ansatz: the density n (and T and ********) depend on coordinates **
**only through a scale parameter s  **
- T. Cs. Acta Phys. Polonica B37 (2006) 1001, hep-ph/0111139
**Scale parameters:**
**	****(X,Y,Z) = (X(t), Y(t), Z(t))**
**    ****Density=const on ellipsoids.     Directional Hubble flow.**
** ****g(s): arbitrary scaling function.   Notation: n ~ ********(s), T ~ ********(s) etc.**

## New exact, ellipsoidal hydro solutions 

**A new family of PARAMETRIC, exact, scale-invariant solutions **
- T. Cs. Acta Phys. Polonica B37 (2006) 1001, hep-ph/0111139
- Volume is introduced as V = XYZ
- 												_For ____ = ____(T) exact solutions_, see
- 												T. Cs, S.V. Akkelin, Y. Hama,
- 												B. Lukács, Yu. Sinyukov,
- 												hep-ph/0108067, Phys.Rev.C67:034904,2003
- The dynamics is reduced to coupled, nonlinear but ordinary differential equations for the scales X,Y,Z
- All hydro problems (initial conditions, role of EoS, freeze-out conditions)
- can be easily illustrated and understood on the equivalent problem:
-  		a classical potential motion of a mass-point in a conservative potential (a shot)!
- Note: temperature scaling function (s) remains arbitrary! (s) depends on (s). -> FAMILY of solutions.

## i

**Canonical coordinates, **
**canonical momenta:**
**Hamiltonian of the motion **
**(for EoS c****s****2**** = 1/******** = 2/3):**
**The role of initial boundary conditions, EoS and freeze-out **
**in hydro can be understood from potential motion!**

*i*

## From the new family of exact solutions, the initial conditions:

**	****Initial coordinates:**
**     ****(nuclear geometry + **
**     ****time of thermalization)**
**    ****Initial velocities:   **
**	****(pre-equilibrium+ time of thermalization)**
**    ****Initial temperature: ****	**
**	****Initial  density:  ****	**
**    ****Initial profile function:  **
**	**** ****(energy deposition **
**       ****and pre-equilibrium process)**

## Role of initial temperature profile 

**Initial temperature profile = arbitrary positive function**
**Infinitly rich class of solutions**
**Matching initial conditions for the density profile **
- T. Cs. Acta Phys. Polonica B37 (2006) 1001, hep-ph/0111139
**Homogeneous temperature  ******** Gaussian density**
**Buda-Lund profile:****	****	****	****           Zimányi-Bondorf-Garpman profile:**

## Illustrated initial T-> density profiles

- Determines density profile!
- Examples of density profiles
- - Fireball
- - Ring of fire
- - Embedded shells of fire
- Exact integrals of hydro
- Scales expand in time

- Time evolution of the scales (X,Y,Z)
- follows a classic potential motion.
- Scales at freeze out  -> observables.
- info on history _LOST_!
- No go theorem  -  constraints
- on  initial conditions
- (penetrating probels) indispensable.

## From the new exact hydro solutions,

**the conditions to stop the evolution:**
**    **_**Freeze-out temperature**_**:**** **
**	**
**	****Final coordinates:**
**     ****(cancel from measurables, diverge)**
**    ****Final velocities:   **
**	****(determine observables, tend to constants)**
**	**
**	****Final density:  ****	**
**	****(cancels from measurables, tends to 0)**
**    ****Final profile function:  **
**	**** ****(= initial profile function! from solution)**

## g

**The potential depends **
**on  ********p:**
** **

- Time evolution of the scales (X,Y,Z) follows a classic potential motion.
- Scales at freeze out  determine the observables. Info on history _LOST_!
- No go theorem  -  constraints on  initial conditions
- (information on spectra, elliptic flow of penetrating probels) indispensable.
- The arrow hits the target, but can one determine g from this information??

** **

**Different**
**initial **
**conditions**
**but**
**same **
**freeze-out**
**conditions**
**ambiguity!**
**Penetrating**
**probes**
**radiate**
**through**
**the time **
**evolution!**

## Illustrations of exact hydro results

**Propagate the hydro solution in time numerically:**

## Solution of the “HBT puzzle”

- Geometrical sizes keep on increasing. Expansion velocities tend to constants. HBT radii Rx, Ry, Rz approach a direction independent constant.
- 		        Slope parameters tend to direction dependent constants.
- General property, independent of initial conditions - a beautiful exact result.

## Geometrical  & thermal & HBT radii

**   ****3d analytic hydro: exact time evolution **
**geometrical size (fugacity ~ const)**
**Thermal sizes (velocity ~ const)**
**HBT sizes (phase-space density ~ const)**
**HBT**** dominated by the smaller of the ****geometrical**** and ****thermal**** scales**
**nucl-th/9408022, hep-ph/9409327 **
**hep-ph/9509213, hep-ph/9503494**
**HBT radii**** approach a constant of time **
**HBT volume****  becomes  spherical**
**HBT radii**** -> ****thermal ~ constant sizes**
** **
**hep-ph/0108067, nucl-th/0206051**
**	****          ****animation by Máté Csanád**

## Scaling predictions of fluid dynamics

- - Slope parameters increase
-   linearly with mass
- - Elliptic flow is a universal function
-   its variable w is proportional to
-   transverse kinetic energy
-   and depends on slope differences.

- Inverse of the HBT radii
- increase linearly with mass
- analysis shows that they are
- asymptotically the same
- Relativistic correction: m -> mt
- hep-ph/0108067,
- nucl-th/0206051

## Relativistic Perfect Fluids 

**Rel. hydrodynamics of perfect fluids is defined by:**
**A recent family of exact solutions: ****nucl-th/0306004**
**Overcomes two shortcomings of Bjorken’s solution:**
**Yields finite rapidity distribution, includes transverse flow**
**Hubble flow ******** lack of acceleration: **
**Accelerating, new rel. hydro solutions: ****nucl-th/0605070**

##  Solutions of Relativistic Perfect Fluids 

**A new family of exact solutions: **
**T. Cs, M. I. Nagy, M. Csanád: nucl-th/0605070**
**Overcomes two shortcomings of Bjorken’s solution:**
**Finite Rapidity distribution ~ Landau’s solution**
**Includes relativistic acceleration**
**in 1+1 and 1+3 spherically symmetric**** **

##  Animation of the new exact solution 

**nucl-th/0605070**
**dimensionless**
****** ****= 2**
**1+1 d**
**both internal**
**and external**
**looks like A+A**

##  nucl-th/0605070: advanced estimate of 

**Width of dn/dy distribution is due to acceleration:**
**acceleration yields longitudinal explosion, thus**
**Bjorken estimate underestimates initial energy density by 50 %:**

## M. Csanád->  fits to  BRAHMS dn/d data
dn/d widths yields  correction factors of ~ 2.0 - 2.2
Yields inital energy density of    10- 30 GeV/fm^3 
a correction of 0/Bj ~ 2 as compared to PHENIX White Paper! 

**dn/d******** widths yields  correction factors of ~ 2.0 - 2.2**
**Yields inital energy density of   **************** 10- 30 GeV/fm^3**** **
**a correction of ********0****/********Bj**** ~ 2 as compared to PHENIX White Paper! **

** ****nucl-th/0605070: advanced estimate of **********

## Principles for Buda-Lund hydro model

**Analytic expressions for all the observables**
**3d expansion, local thermal equilibrium, symmetry**
**Goes back to known exact hydro solutions: **
**nonrel, Bjorken, and Hubble limits,  1+3 d ellipsoids  **
**but phenomenology,  extrapolation  for unsolved cases**
**Separation of the Core and the Halo****	**
**Core: perfect fluid dynamical evolution**
**Halo: decay products of long-lived resonances**
**Missing links: phenomenology needed **
**search for accelerating ellipsoidal rel. solutions**
**first accelerating rel. solution: nucl-th/0605070**** **

## A useful analogy

**Core****	****	****	****	****	******** ****	****Sun**
**Halo****	****	****	****	****	******** ****	****Solar wind**
**T****0,RHIC **** ~ 210 MeV****	****	****	******** ****	****T****0,SUN ********~ 16 million K **
**T****surface,RHIC **** ~ 100 MeV****	****	******** ****	****T****surface,SUN ******** ~6000 K**

**Fireball at RHIC ******** our Sun**

## Buda-Lund hydro model

## The generalized Buda-Lund model

**The original model was for axial symmetry only, central coll.**
**In its  general hydrodynamical form:**
**Based on 3d relativistic and non-rel solutions of perfect fluid dynamics:**
**Have to assume special shapes:**
**Generalized Cooper-Frye prefactor:**
**Four-velocity distribution:**
**Temperature:**
**Fugacity:**

## Scaling predictions of non-rel hydro

- - Slope parameters increase
-   linearly with mass
- - Elliptic flow is a _universal function_
-   and variable w is proportional to
-   transverse kinetic energy
-   and depends on slope differences.

- Inverse of the HBT radii
- increase linearly with mass
- analysis shows that they are
- asymptotically the same
- Relativistic correction: m -> mt
- hep-ph/0108067,
- nucl-th/0206051

## Scaling predictions: Buda-Lund hydro

- - Slope parameters increase
-   linearly with transverse mass
- - Elliptic flow is same universal function.    - Scaling variable w is prop. to
-   generalized transv. kinetic energy
-   and depends on effective slope diffs.

- Inverse of the HBT radii
- increase linearly with mass
- analysis shows that they are
- asymptotically the same
- Relativistic correction: m -> mt
- hep-ph/0108067,
- nucl-th/0206051

## Some analytic Buda-Lund results

**HBT radii widths:**
**	****	****	****	****      **
**Slopes, effective temperatures:**
**Flow coefficients are **_**universal**_**:**
**	****	****	****	****	****	****	****	****	**
**	****	****	**

##   Buda-Lund hydro and Au+Au@RHIC

## Femptoscopy signal of sudden hadronization 

- Buda-Lund hydro
- 	fit indicates
- hydro predicted
- (1994-96)
- scaling of HBT radii
- T. Cs, L.P. Csernai
- 	hep-ph/9406365
- T. Cs, B. Lörstad
- 	hep-ph/9509213
- Hadrons with T>Tc :
- 	a hint  for
- 	cross-over
- M. Csanád, T. Cs, B. Lörstad and A. Ster,
- 	nucl-th/0403074

## Confirmation

- see nucl-th/0310040 and nucl-th/0403074,
- R. Lacey@QM2005/ISMD 2005
- A. Ster @ QM2005.

- Universal scaling
- PHOBOS v2(->w)

## Buda-Lund hydro prediction:		Exact non-rel. hydro solution:

PHENIX data: 

**PHENIX data:**** **

## Hydro scaling of Bose-Einstein/HBT radii

**R****side****/R****out**** ~ 1**

**R****side****/R****long**** ~ 1**

**R****out****/R****long**** ~ 1**

**1/R****2****side**** ~ m****t**

**1/R****2****out**** ~ m****t**

**1/R****2****long**** ~ m****t**

## G. Veres, PHOBOS data, proc QM2005

- Nucl. Phys. A774 (2006) in press

##  Hydro scaling of v2 and       dependence

## Universal scaling and v2(centrality,) 

## Universal v2 scaling and PID dependence

## Universal scaling and fine structure of v2

## PREDICTED

- PHENIX and PHOBOS
- data collapse to a
- Universal Scaling Law
- of perfect fluid hydro

## Summary on universal hydro scaling of v2

- Black line:
- Theoretically
- predicted, universal
- scaling function
- from analytic works
- on perfect fluid
- hydrodynamics:

- hep-ph/0108067,
- nucl-th/0310040

## Universal hydro 

- Universal hydro
- scaling breaks
- where scaling with number of
- 	VALENCE QUARKS
- sets in, pt ~ 1-2 GeV
- 	Fluid of QUARKS!!

- R. Lacey and M. Oldenburg, proc. QM’05
- A. Taranenko et al,
- PHENIX PPG062: nucl-ex/0608033

## Slide 50

## Slide 51