# Presentation: 400527

## HBT based on perfect fluid hydrodynamics

- T. Csörgő1, M. Csanád2
- 1Eötvös University Budapest, 2KFKI RMKI

## Contents

- Equations of hydrodynamics, solutions
- The Buda-Lund model
- Succesful predictions based on scaling
- Succesfully described data
- Further predictions
- Summary

- M. Csanád, ISMD 2007, Berkeley

## Motto

*Make everything as simple as possible, but not simpler.*
*Any f**l can make things bigger, more complex, and more violent. It takes a touch of genius – and a lot of courage – to move in the opposite direction*.
- A. Einstein

- M. Csanád, ISMD 2007, Berkeley

## Introduction

- History:
- _Fermi, 1950_: basic idea: thermal behavior, based on mean free path
- _Landau, 1953-57_: formulation of relativistic hydrodynamics, exact, but parametric solution, realistic rapidity profile calculation
- _Hwa, 1974 & Bjorken, 1983_: (very) simple relativistic solution, (unrealistic) rapidity distribution, initial energy density estimation
- Nowadays:
- CERN and RHIC: _scaling laws_ observed in data
- Hydrodynamics provides good frame of _successful models_
- Buda-Lund model: successful description of data, based on exact (nonrelativistic and relativistic) analytic hydro solutions
- Important (and not so easy) task: to find new, relativistic, analytic, accelerating solutions

- M. Csanád, ISMD 2007, Berkeley

## Equations of relativistic hydro

- Assuming local thermal equilibrium
- For a perfect fluid:
- Equations in four-vector form and nonrelativistic notation
- Euler equation:
- Energy conservation:
- Charge conservation:

- comoving proper-time derivative

- comoving derivative

- M. Csanád, ISMD 2007, Berkeley

## The most famous (1D) solutions

- Landau’s solution:
- Accelerating, implicit, complicated, 1D
- L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51
- I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529
- L.D.Landau and S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309
- Hwa-Bjorken solution:
- Non-accelerating, explicit, simple, 1D, boost-invariant
- R.C. Hwa, Phys. Rev. D10, 2260 (1974)
- J.D. Bjorken, Phys. Rev. D27, 40(1983)

- M. Csanád, ISMD 2007, Berkeley

## 3D solutions

- Nonrelativistic, spherically symmetric solution
- P. Csizmadia, T. Csörgő, B. Lukács, nucl-th/9805006
- Relativistic, spherically symmetric solution
- T. Csörgő, L. Csernai, Y. Hama, T. Kodama, nucl-th/0306004
- Accelerationless
- Hubble flow profile (flow proportional to distance)
- Relativistic, spherically symmetric solution
- T. Csörgő, M. Nagy, M. Csanád, nucl-th/0605070
- Accelerating (the only besides Landau’s)
- Realistic rapidity distributions (data described by it)
- Advanced energy and lifetime estimate
- All describe expanding fireballs
- Sometimes: rings/shells of fire

- M. Csanád, ISMD 2007, Berkeley

## Where we are

- Other accelerationless solutions:
- T. S. Biró, Phys. Lett. B 474, 21 (2000)
- Yu. M. Sinyukov and I. A. Karpenko, nucl-th/0505041
- Solutions by coordinate transformations:
- S. Pratt, nucl-th/0612010
- There are some exotic solutions as well
- Need for solutions that are:
- explicit
- simple
- accelerating
- relativistic
- realistic / compatible with the data

- M. Csanád, ISMD 2007, Berkeley

## How analytic hydro works

- Take hydro equations and EoS
- Local conservation laws
- Find a solution
- That will contain parameters (like in GR)
- A lot of examples shown before
- Use a freeze-out condition
- At a given proper time
- On a given hyper-surface
- Calculate the hadron source function
- Probability of generating a particle at a given spacetime point and at a given momentum
- Calculate observables
- E.g. spectra, flow, correlations
- Straightforward calculation

- M. Csanád, ISMD 2007, Berkeley

## Which solution to choose?

- Different solutions with different initial conditions → Same final state
- Possible even with viscous hydro solutions!! (see T. Csörgő, WPCF’07)

- M. Csanád, ISMD 2007, Berkeley

## The Buda-Lund model

- Interpolator of the above solutions
- Equals to each of them at a certain limit
- Analytic expressions for the observables
- Ellipsoidally symmetric, relativistic expansion
- Hence non-flat temperature profile _possible_
- Freeze-out surface pseudo-orthogonal to flow
- Treats resonances by the Core/Halo model
- Core: hydrodynamic evolution
- Halo: decay products of long-lived resonances
- Has parameters (like the Friedmann-solution of GR)

- M. Csanád, ISMD 2007, Berkeley

## What does the data tell us 

- 		data

- What does the data tell us

- M. Csanád, ISMD 2007, Berkeley

## Predictions of the Buda-Lund model

- Hydro predicts scaling (even viscous)
- What does a scaling mean?
- See Hubble’s law – or Newtonian gravity:
- Cannot predict acceleration or height
- Collective, thermal behavior →
- 	Loss of information
- Spectra slopes:
- Elliptic flow:
- HBT radii:

- M. Csanád, ISMD 2007, Berkeley

## BudaLund fits to 130 GeV RHIC data

- M. Csanád, T. Csörgő, B. Lörstad, A. Ster, nucl-th/0311102, ISMD03

- M. Csanád, ISMD 2007, Berkeley

## BudaLund fits to 200 GeV RHIC data

- M. Csanád, T. Csörgő, B. Lörstad, A. Ster, nucl-th/0403074, QM04

- M. Csanád, ISMD 2007, Berkeley

## The HBT test

- Models with acceptable results:
- nucl-th/0204054	Multiphase Trasport model (AMPT)
- 					Z. Lin, C. M. Ko, S. Pal
- nucl-th/0205053	Hadron cascade model
- 					T. Humanic
- hep-ph/9509213 	Buda-Lund hydro model
- 	nucl-th/0305059	T. Csörgő, B. Lörstad, A. Ster
- hep-ph/0209054	Cracow (single freeze-out, thermal)
- 					W. Broniowski, W. Florkowski
- nucl-ex/0307026	Blast wave model
- 					F. Retiére for STAR

- M. Csanád, ISMD 2007, Berkeley

## Buda-Lund fits to flow data

- Prediction of 2003: scaling variable
- If plotted against ‘w’, data collapse:
- From 20 to 200 GeV
- From central to peripheral
- Pion, kaon, proton
- pt and  dependence
- Prediction:

**Csanád, Csörgő, Lörstad, Ster et al. nucl-th/0512078**

- M. Csanád, ISMD 2007, Berkeley

## Predictions on HBT(mt,φ,)

**Dependence only on m****t****, not on PID → m****t****-m scaling! **
**Dramatic change with y at low m****t**
** ****No change at high m****t**
** ****Radii decreasing with increasing m****t**** or y**

**R****out****	****	****	****  R****side****	****	****	****R****long**

- M. Csanád, ISMD 2007, Berkeley

## Hydro scaling in HBT

- Radii depend on mt and y through mt
- Same for all particles

- M. Csanád, ISMD 2007, Berkeley

## Prediction for kaon HBT

- Transverse mass scaling → same curve for pions and kaons if plotted versus mt

**K**

- M. Csanád, ISMD 2007, Berkeley

## Summary

- Solutions of relativistic hydrodynamics
- Old, famous, 1D solutions
- New 3D solutions
- Need for accelerating, realistic 3D solutions
- Successful predictions of the Buda-Lund
- Hydrodynamic scaling seen in the data
- RHIC data well described by the model
- HBT test rules out ~40 models
- Further predictions on HBT
- „mtbar” scaling

- M. Csanád, ISMD 2007, Berkeley

## Thank you for your attention!

- "In general we look for a new law by the following process.
- 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/