# Presentation: 654038

## Search for critical temperature by measuring spatial correlation length via multiplicity density fluctuations

**Kensuke Homma / Hiroshima Univ.**
** ****from PHENIX collaboration**
**July 3, 2006  in**** ****Florence (Italy) at the Galileo Galilei Institute**
**Outline**
- What is the critical behavior ?
- Order parameter and phase transition
- Free energy coefficients and correlation function
- How can we define initial temperature ?
- PHENIX preliminary results
- Summary

- Kensuke Homma / Hiroshima Univ.

## What is the critical behavior ?

- A simulation based on two dimensional Ising model
- from ISBN4-563-02435-X C3342l

- Black

- Black & White

- Gray

- Various sizes
- from small to large

- Scale transformation

- Spatial pattern of
- ordered state

-  Large fluctuations of correlation sizes on order parameters:
-     **critical temperature (focus of this talk)**
-  Universality (power law behavior) around Tc caused
-     by basic symmetries and dimensions of an underlying system:
-     **critical exponent**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Spatial pattern of spin correlation. Black is the aligned state of spins.

## Order parameter and phase transition 

- In Ginzburg-Landau theory with Ornstein-Zernike picture,
- free energy density *g*  is given as

- external field *h* causes deviation of free energy
- from the equilibrium value *g**0*. Accordingly an order
- parameter ** fluctuates spatially.

- 1-D spatial multiplicity density fluctuation from the mean density is introduced as an order parameter in the following.

**spatial correlation**

**a>0**
**       ****a=0**
**             ****a<0**

**disappears**
**at Tc**

- In the vicinity of Tc, ** must vanish, hence

**b>0 for 2****nd**** order**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Interpretation. 

## Two point correlation function & Fourier transformation

**Two point correlation**

**Fourier transformation**

**Relative distance between two points**

- Kensuke Homma / Hiroshima Univ.

## Expectation value of |k|2 from free energy deviation

- Statistical weight can be obtained from free energy

- Fourier expression of order parameter

- Kensuke Homma / Hiroshima Univ.

## Function form of two point correlation function 

- A function form of correlation function is obtained by inverse Fourier transformation.

**From ********g (up to 2****nd**** order)**
**due to spatial fluctuation**

**Fourier transformation**
**of two point correlation**
**of order parameter**

- Kensuke Homma / Hiroshima Univ.

## From two point correlation to two particle correlation

**Rapidity independent**
**term is added.**

- Two point correlation function in 1-D case at fixed *T*

- Two particle correlation function

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Summarize

## Number of participants, Np and Centrality 

**Number of participants, Np and Centrality **

- Multiplicity distribution

## Can Np be related with initial temperature?

**Np scan may be a fine scan on the initial temperature T, while collision energy is a coarse scan (?). Tc should be rather investigated with fine scan. **
**Let’s suppose that Np can be a monotonic function of T.**

**Transverse energy E****T**

- Kensuke Homma / Hiroshima Univ.

## Multiplicity density measurements in PHENIX

**Δη****<0.7 integrated over ****Δφ****<****π****/2**
**PHENIX: Au+Au  √sNN=200GeV**

**PHENIX Preliminary**

**Negative Binomial Distribution**
**can describe data very well. **

- n/<n>

**small ******

**large ******

- Probability (A.U.)

**Zero magnetic field to **
**enhance low pt statistics**
**per collision event**

- Kensuke Homma / Hiroshima Univ.

## Relations betweenN.B.D k and integrated correlation function

- Integrated correlation function can be related with 1/k

- Negative Binomial Distribution
- (Distribution from *k *Bose-Einstein emission sources)

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Summarize

## N.B.D. k vs. d

**PHENIX Preliminary**

**PHENIX Preliminary**

**10 % centrality**
**bin width**

**5% centrality**
**bin width**

****

**k(********)**

**Function can fit the data**
**remarkably well !**

- Kensuke Homma / Hiroshima Univ.

## Correlation length  and static susceptibility 

**PHENIX Preliminary**

**PHENIX Preliminary **

**Np**

**Np**

- k=0 * T

**Divergence of susceptibility is the**
**indication of 2****nd**** order phase transition. **

**Divergence of correlation length is the**
**indication of a critical temperature. **

**Correlation length ******

**T~Tc?**

**Au+Au  √sNN=200GeV**

**Au+Au  √sNN=200GeV**

- Kensuke Homma / Hiroshima Univ.

## What  parameter represents ?

** ****can absorb all rapidity **
**independent fluctuations **
**caused by;**
- 1. finite centrality bin width
- (initial temperature fluctuations)
- 2. azimuthal correlations
- (under investigation)
- 3. Whatever you want.

**Np**

- Our parametrization
- can produce stable results
- on  and .

## Summary

**Multiplicity density distribution in Au+Au collisions at √S****NN****=200 GeV can be well approximated by N.B.D..**
**Two point correlation lengths have been extracted based on the function form by relating pseudo rapidity density fluctuations to the GL theory up to the second order term in the free energy. The lengths as a function of Np indicates non monotonic increase at Np~100.**
**The product of the static susceptibility and the corresponding temperature shows no obvious discontinuity within the large systematic errors at the same Np where the correlation length is increased.**

- Kensuke Homma / Hiroshima Univ.