# Presentation: 115629

## Tomoaki Nakamura 
for the PHENIX collaboration
Hiroshima University

- for the PHENIX collaboration
- Hiroshima University

**Measurement of event-by-event fluctuations ****and order parameters ****in PHENIX**

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Phase transitions

- According to the classical
-   classification of the phase
-      transition, the order of phase
-      transition is defined by
-      discontinuities in derivatives
-      of free energy.
- In this aspect of bulk property, discontinuity in thermodynamic variables or order parameters as a function of the temperature or time evolution are available to search for the critical point of phase.
- In particular, the second order phase transitions are often accompanied by the divergence with respect to thermodynamic variables as a results of critical phenomena.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Some thermodynamic variables as order parameters of phase can be obtained from event-by-event fluctuations.

Particles correlation length 
scale dependence of multiplicity fluctuations
Specific heat
temperature fluctuations from average pT fluctuations.
    PRL. 93 (2004) 092301
    M. J. Tannenbaum : poster #120

We have performed measurements of several fluctuations to explore the QCD phase transition using the PHENIX detector at RHIC.

- Particles correlation length
- scale dependence of multiplicity fluctuations
- Specific heat
- temperature fluctuations from average *p****T*** fluctuations.
-     PRL. 93 (2004) 092301
-     **M. J. Tannenbaum : poster #120**
- We have performed measurements of several fluctuations to explore the QCD phase transition using the PHENIX detector at RHIC.

_**Event-by-event fluctuations in heavy-ion collisions measured by PHENIX**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Multiplicity fluctuations by variance

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Normalized variance vs. participants

_**Normalized variance vs. participants**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Charged particle multiplicity distributions and negative binomial distribution (NBD)

_**Charged particle multiplicity distributions and negative binomial distribution (NBD)**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Negative binomial distribution (NBD)

_**Negative binomial distribution (NBD)**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Charged particle multiplicity distributions in PHENIX

_**Charged particle multiplicity distributions in PHENIX**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k parametersas a function of average multiplicity

_**NBD k parameters**__**as a function of average multiplicity**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k parametersas a function of number of participants

_**NBD k parameters**__**as a function of number of participants**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k parameters in Cu+Cu

_**NBD k parameters in Cu+Cu**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k parametersas functions of δpT  (pT>0.2 GeV/c)

_**NBD k parameters**__**as functions of **__**δ**__***p***__***T***__**  (**__***p***__***T***__**>0.2 GeV/c)**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Extraction of two particle correlation 

_**Extraction of two particle correlation **_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k and pseudo rapidity gap

- Two component model well agree with data.
- Correlation function dose not go to 0 at δηequal 0. It dose not suggest the intermittency effect.

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

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Participants dependence of ξ and b

- Smaller value of two particle correlation length have been observed at RHIC energy as compared to the past experiments.
- Low density p+p collisions (UA5, p+p √s = 540 GeV : **ξ**= 2.9)
- Low energy N+N collisions (E802 O+Cu 14.6AGev/c : **ξ**= 0.18±0.05)
- Both **ξ** and b decrease with increasing the number of participants.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Linearity in the log-log plot ξ vs. number of participants

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Conclusions

- A systematic study on charged particle multiplicity fluctuations have been performed in Au+Au, Cu+Cu, d+Au and p+p collisions with respect to two collision energy of 200GeV and 62GeV.
- Multiplicity distributions measured by PHENIX also agree with the negative binomial distributions at the RHIC energy.
- Multiplicity fluctuations by the NBD k parameters are not scaled by the average multiplicity but scaled by the number of participants or system size in Au+Au collisions.
- Scale dependence of NBD k parameters are presented with respect to pseudo rapidity gap and transverse momentum range.
- Two particle correlation length have been observed by the two component model from the multiplicity fluctuations.
- Extracted correlation length have a linearity as a function of the number of participants in the logarithmic scale (log-log plot).

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## PHENIX posterson fluctuations and correlations

_**PHENIX posters**__**on fluctuations and correlations**_

- #109 : J. T. Mitchell
-            The low- to high-pT evolution of
-            charged hadron azimuthal
-            correlation functions: from HBT to jets
- #110 : J. T. Mitchell
-            A survey of multiplicity fluctuations
-            in PHENIX
- #120 : M. J. Tannenbaum
-            How to measure specific heat using
-            event-by-event average pT fluctuations

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Slide 19

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Backup Slide

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Variables of statistical mechanics as order parameters

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Multiplicity fluctuations by variance

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Multiplicity fluctuations in PHENIX

_**Multiplicity fluctuations in PHENIX**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Normalized variance as a function of average multiplicity

_**Normalized variance **__**as a function of average multiplicity**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k parametersas a function of average multiplicity

_**NBD k parameters**__**as a function of average multiplicity**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Multiplicity fluctuationsas a function of collision overlap geometry

- When plotted as a function of a measure of the collision overlap geometry (fractional impact parameter divided by the nuclear diameter - so a head-on collision = 1.0), the 62 GeV Cu+Cu fluctuations are less Poissonian.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## δpT (pT>0.2 GeV/c) dependence of NBD k

_**δ**__***p***__***T***__** (**__***p***__***T***__**>0.2 GeV/c) dependence of NBD k**_

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Comparison for Au+Au, d+Au and p+p

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Fit by the E802 type correlation function

_**Fit by the E802 type correlation function**_

- E802 type integrated two particle correlation function dose not agree with PHENIX data by taking all range of pseudo rapidity into account.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Two component model

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Zoom up for small δη

_**Zoom up for small **__**δη**_

- Lines are just extrapolated obtained fitting curve for small area (δη<0.01).
- K parameters converge finite value. Factorial moment/cumulant do not diverge.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Linearity in log-log plots

- α** **= -0.72 ± 0.032
- β = 0.097 ± 0.015

- α** **= -0.90 ± 0.027
- β = 4.02 ± 0.540

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Divergence...!?

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Divergence...!?could not find critical points by only the fitting

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## δηdependence of NBD k parameter

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Number of participants dependence of correlation length ξ

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Linear behavior of NBD k as a function of logarithmic δη

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Normalized factorial moment Fq

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## NBD k and factorial moment Fq

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Integral of correlation function and normalized factorial moments

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Specific heat from average pT fluctuation

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Random particle emission pattern based on NBD

_**Random particle emission pattern based on NBD**_

- Detector 1

- Detector 2

- Emission source

- In the case of there are no correlation about the particle emission, the value of NBD k parameters are summed up.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Correlated particle emission pattern into the phase-space

_**Correlated particle emission pattern into the phase-space**_

- Detector 1

- Detector 2

- Emission source

- If there are correlations, NBD k parameters do not increase according to the size of detector acceptance.

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Correlation functions and correlation length

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Average pT fluctuationPublished by Phys. Rev. Lett. 93, 092301 (2004)

**Already, famous!**
**It’s not a Gaussian...**
**it’s a Gamma distribution!**

**M.J. Tannenbaum, **
**Phys. Lett. B 498 (2001) 29**

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Contribution of Jet/Jet suppression to the average pT fluctuation

**This decrease due to jet suppression?**

- PYTHIA based simulation, which contains scaled hard-scattering probability factor (Sprob) by the nuclear modification factor (RAA), well agree with the measured FpT. It might be indicate that jet suppression might contribute to the average pT fluctuation.

- centrality 20-25%

- FpT is adjusted at the open circle for this simulation

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005

## Estimation of the magnitude of residual temperature fluctuations

| Experiment | √sNN | σT/<T> 
(most central) |
| --- | --- | --- |
| PHENIX | 200 | 1.8% |
| STAR | 130 | 1.7% |
| CERES | 17 | 1.3% |
| NA49 | 17 | 0.7% |

- Tomoaki Nakamura - Hiroshima Univ.

- 8/6/2005