# Presentation: 406692

## Systematic studies of global observables by PHENIX

**Systematic studies of global observables by PHENIX**

**Longitudinal density fluctuations**
**Meson-meson and baryon-meson correlation**
**Kensuke Homma **
**for the PHENIX collaboration**
**Hiroshima University**

**Feb 9, 2008 at QM2008 in Jaipur, India**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
I would like to present systematic studies of global observable by PHENIX. My name is Kensuke Homma from Hiroshima Univ. The focus of my talk is search for critical behavior via global observables.

## Understanding of QCD phase structure

- 1st order ?

-  CEP ?

- T

- B

**What RHIC achieved**
**Dense medium**
**Deconfined phase**
**   ****with partonic d.o.f**

**Quark number scaling of elliptic flow**

- Tc

**Is accessible region by RHIC**
**really crossover?**
**Crossover for any kinds**
**of order parameters?**

- _[Phys. Rev. ](http://link.aps.org/abstract/PRL/v98/e162301)__[Lett](http://link.aps.org/abstract/PRL/v98/e162301)__[. 98, 162301 (2007)](http://link.aps.org/abstract/PRL/v98/e162301)__[ ](http://link.aps.org/abstract/PRL/v98/e162301)_

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Understanding of QCD phase structure is one of crucial subjects to be intensively studied. What RHIC achieved so far is the formation of dense medium and the observation that the medium carry the partonic dof. Although it is believed that RHIC can across the crossover region, the current theoretical understanding is not conclusive in finite small density. And even if crossover region, critical temperature seems to depend on the order parameter we use. Therefore we would like to introduce an order parameter as general as possible in the following analysis.

## Intuitive observable: blob intensity  x blob size 

- GL with higher order terms

- Many length scales appear
- (a typical ***k* disappears)

- T=Tc

- T<Tc

**Order parameter**
********
- ******<<1 in T>>Tc**,
- Ginzburg-Landau(GL)
- free energy up to
- 2nd order term

**Two point correlation <********************)>**
**in 1-D longitudinal space**

**Non monotonic increase**
**of**** ******** ****indicates**** T~Tc**
**w.r.t. monotonically**
**decreasing baseline**
**as mean density <********>**
**increases.**

- At RHIC

- Kensuke Homma / Hiroshima Univ.

**Notes:**
The order parameter phi is defined as fluctuation size from the mean density as a function of spatial coordinate. What is shown here is the phase transition of water in a kettle in my kitchen. At RHIC, the initial temperature is much higher than Tc. In such a case, this order parameter would be very small, since the medium in higher temperature would be dense and homogenous. Therefore, as a free energy of the thermalized system, we can apply GL potential ie polynomical expansion up to the lowest 2nd order term of phi. Then we focus on the two point correlation of this order parameter. If the dimension of the system is effectively 1d, we can derive the correlation function as a simple damping form as a function of the relative distance with a typical correlation length xi. As a general signature of the phase transition, we usually look for the size of the blob as seen in this picture. However, the intensity of blob with large correlation length is also good indicator of T=Tc. Then the increase of the product between them can be sensitive quantity to determine Tc. Actually we can derive a relation that axi corresponds to chi*T based on the GL free potential form. It monotonically decreases as system density increases. However, if the T~Tc, the product jumps up. So the ........

## Density measurement: inclusive dNch/d

**Au+Au@62.4GeV**

**Au+Au@200GeV**

**Cu+Cu@200GeV**

**Cu+Cu@62.4GeV**

**N****ch****/< N****ch**** >**

**p+p@200GeV**
**Cu+Cu@22.5GeV**

**Negative Binomial Distribution**
**(NBD) perfectly describes**
**multiplicities in all collision**
**systems and centralities**
**at RHIC.**

**P(N****ch****)**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
As a density measurement, we naturally use the number of charged particles.. What is shown here is the multiplicity distribution divided by the mean from pp to AuAu taken by PHENIX so far. The fit curve is NBD. Multiplicity in all collision systems and all centralities are well described by NBD curves.

## Two point correlation via NBD

**k=1   Bose-Einstein**
**k=∞ Poisson**

- source 1

- source 2

- source 1+2

**NBD**

**Uncorrelated**
**sources**

**Correlated**
**sources**

- k=k1+k2

- k=k1

- k=k2

- k=k1

- k=k2

- k!=k1+k2

**1/k corresponds to integral**
**of two point correlation**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Let me explain how to discuss two point correlation with the multiplicity distributions. First of all, NBD consists of mean charged particle multiplicity mu and the width parameter 1/k. If we think about a convolution between two uncorrelated emission sources, the k value of convolution can be represented by the sum of k in each source. In this case the correlation of multiplicity between two sources is round shape. On the other hand, if the emission sources are correlated, the k parameter of the convolution deviates from the sum of k in each emission sources. In such a case, the correlation of multiplicity between two sources looks like this. By measuring NBD k by fits, we can describe any kind of two point correlation. If k=1, it corresponds to BE and if k is infinite, it corresponds to Poisson distribution. Mathematically it is known that 1/k corresponds to the integral of two point correlation function.

## Differential multiplicity measurements

**Differential multiplicity measurements**

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

**NBD can well describe**
**differential distribution too. **

**n/******

**small ******

**large ******

- Probability (A.U.)

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

****

- _[Phys. Rev. C 76, 034903 (2007)](http://link.aps.org/abstract/PRC/v76/e034903)__ _

- Kensuke Homma / Hiroshima Univ.

**Notes:**
As a measurement of the two point correlation, we perform the differential analysis on the multiplicity. We especially focus on the longitudinal space coordinate, since the system expansion speed is expected high and the intially embedded fluctuation might be able to survive even if we look at soft pions. In order to enhance the low pt statistics per collision which would be relevant for the discussion of phase transition, we used B-off data. What is shown here is the multiplicity distribution for different pseudo rapidity intervals from small to large. Again, distributions are well described by NBD.

## Extraction of  product

**Parametrization of **
**two particle correlation**

**Exact relation with NBD *****k***

**Fit with approximated functional form**

*** *****absorbs rapidity independent**
**bias such as centrality bin width**

**Look at**
**slopes**

- 10%

- 5%

**Approximated**
**functional form**

******

***k*********

- _[Phys. Rev. C 76, 034903 (2007)](http://link.aps.org/abstract/PRC/v76/e034903)__ _

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Let me explain how to extract alpha*xi from the NBD. Firstly two particle correlation function can be parameterized by the functional form based on the GL free energy. However, we found that a constant parameter is really necessary to absorb the experimental biases like centrailitiy bin width which is independent of the rapidity interval size. The inverse of NBD k is mathematically related with 2nd order factorial moment over window size. If the xi is smaller than the window size we use, we can approximate the relation like this. If beta is zero, we notice that we can extract a*xi from the slope parameter in k vs dEta.  

## αξ, β vs. Npart

**β**** is systematically shift to lower values as the centrality bin width becomes smaller from 10% to 5%. This is understood as fluctuations of Npart for given bin widths**
**αξ**** product, which is monotonically related with ****χ*****k=0***** indicates the non-monotonic behavior around Npart ~ 90.**
**Significance with Power + Gaussian:**
**3.98 ****σ**** (5%), 3.21 ****σ**** (10%)**
**Significance with Line + Gaussian:**
**1.24 ****σ**** (5%), 1.69 ****σ**** (10%)**

- β

- αξ

- ●5%
- ○10%

- ●5%
- ○10%

- Npart

**Au+Au@200GeV**

**Dominantly Npart fluctuations**
**and possibly correlation in azimuth**

- _[Phys. Rev. C 76, 034903 (2007)](http://link.aps.org/abstract/PRC/v76/e034903)__ _

- Kensuke Homma / Hiroshima Univ.

**Notes:**
What is shown here is the beta and axi as a function of Npart in AuAU@200GeV system. As you expect, beta is systematic shift to lower value from 10% case to 5% case, while the a*x is rather stable. There seems to be non monotonic behavior in a*xi at around Npar=90 and it might be related with T~Tc. The significance depends on how we assume the base line monotonic shape. If we assume power function which is naturally expected since rho increases monotonically as a function Npart, the significance levels are 4 sigma for 5% and 3 sigma for 10% bin width cases, respectively.

## Analysis in smaller system: Cu+Cu@200GeV

**Cu+Cu@200GeV**

- 5% bin width

**Analysis in smaller system: Cu+Cu@200GeV**

**Cu+Cu@200GeV**

- 5% bin width

- Kensuke Homma / Hiroshima Univ.

**Notes:**
We extend this analysis to another collision systems. As the different size with the same energy, we measure a*xi in Cu+Cu@200. NBD k vs dEta. Fit good with the same functional form..

## Analysis in lower energy: Au+Au@62.4GeV

**Au+Au@62.4GeV**

**Analysis in lower energy: Au+Au@62.4GeV**

**Au+Au@62.4GeV**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
As the lower energy system keeping the same system size, we analyzed AuAu@62.4 data. Again, fit to k vs dEeta is enough accurate.

## Comparison of three collision systems

**Cu+Cu@200GeV**

**Au+Au@62.4GeV**

**Comparison of three collision systems**

**Au+Au@200GeV**

**Au+Au@200GeV**

**αξ**

**<********c****>/<********c****>****@AuAu200**

**Normalized mean**
**multiplicity to that**
**of top 5% in**
**Au+Au@200GeV**

**Npart~90 in **
**AuAu@200GeV**
******BJ********~2.4GeV/fm****2****/c**

- _[Phys. Rev. C 76, 034903 (2007)](http://link.aps.org/abstract/PRC/v76/e034903)__ _

- _[Phys. Rev. C 76, 034903 (2007)](http://link.aps.org/abstract/PRC/v76/e034903)__ _

- Kensuke Homma / Hiroshima Univ.

**Notes:**
This is a comparison of a*xi as a function of normalized mean multiplicity to that of the top 5% centrality in AuAu@200GeV. So the maximum horizontal axis value is set at 1.0. As you see, three collision systems quantitatively agree in low and higher multiplicity. However, the  possible non monotonicity is prominent only in AuAu@200. We still need careful study on other systems to make a conclusive statement on the non monotonicity. If we put a biased line based on AuAu@200, the corresponding Npart is around 90. Bease upon the measurement of dET/dEta, we can deduce the corresponding Bjorken energy density and Npart~90 corresponds to 2.4GeV/fm2/c..

## Are there symptoms in other observables at around the same Npart?

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Given this bias, let us look for other observable which show peculiar behavior as a function of Npart in AuAu@200GeV.

## Meson-meson and baryon-meson fluctuations

**Meson-meson and baryon-meson fluctuations**

**Npart ~90**

**Au+Au@200GeV**

**Au+Au@200GeV**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
PHENIX is analyzing K/pi fluctuaitons and also p pi fluctuations. What is shown here are Nu_dyn as a function of Npart, which is defined as a combination of the 2nd order factorial moments of each species and the mixed term. As you see, the quantity in p to pi has a qualitative different behavior from that of K to pi.

## Deviation from scaling at low KET region ?

**In lower KE****T****, there seems to be different behaviors between baryon and mesons. The transition is at Npart~90.**

**Deviation from scaling at low KE****T**** region ?**

**Npart  ~90**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Next one is Ratio of measured v2 to fit scaling curves as a function of KET/nq in each centrality class. They indicate good NCQ scalings at around KET ~ 1 GeV. However, if we focus on the deviation from the scaling curve, the trend of proton curves is qualitatively different from that of pi and K as you seen in blue points. The relation flips at around Npart~90.

## Conclusion

**Correlation function derived from GL free energy density up to 2****nd**** order term in the high temperature limit is consistent with what was observed in NBD k vs ******** in three collision systems. ****This provides a way to directly determine  transition points without tunable model parameters with relatively fewer event statistics.**
**The product of susceptibility and temperature, ******** as a function of Npart indicates a possible non monotonic increase at Npart~90. The corresponding ****Bjorken energy density is 2.4GeV/fm****3**** with ********=1.0 fm/c and the transverse area=60fm****2**
**The trends of ******** in smaller system in the same collision energy (Cu+Cu  200GeV) and in the same system size in lower collision energy (Au+Au 62.4) as a function of mean multiplicity are similar to that of Au+Au at 200GeV except the region where the possible non monotonicity is seen. We need careful error estimates and increase of statistics for smaller size and lower energy systems to obtain the conclusive result.**
**Combining other symptoms in the same multiplicity region, we hope to understand possibly interesting behaviors.**
**         **

- Kensuke Homma / Hiroshima Univ.

## Backup

- Kensuke Homma / Hiroshima Univ.

## How about <cc> suppression?

**How about <cc> suppression?**

**Npart~90 in **
**AuAu@200GeV**
******BJ********~2.4GeV/fm2/c**

**Cu+Cu@200GeV**

**Au+Au@200GeV**

**If we put a biased line ...**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
If the possible non monotinicity is really related with dynamical phase transtion, we should see the symptoms in other observable at around same energy density region. 

## Other symptoms?: baryon-meson correlation

**Other symptoms?: baryon-meson correlation**

**Npart  ~90**

**KE****T****/n****q**

**Npart ~90**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
V2 divided by scaled curve of v2 by Nq as a function KET/nq.

## KET + Number of constituent Quarks (NCQ) scaling

**KE****T ****+ Number of constituent Quarks (NCQ) scaling**

-  Scaling holds well for different centralities
-  Deviations at low KET may be due to radial flow

- Kensuke Homma / Hiroshima Univ.

## Future prospect

**It would be important to see coherent behaviors on other observables like; J/******** suppression pattern, breaking point of quark number scaling of V2, fluctuation on baryon-meson production, low pt photon yield and so on to investigate what kind of phase transition is associated with ********measurement********if ******** is really the indication of a phase transition.**
**Quick finer energy and species scan with 100M events for each system would provide enough information on the structure of the possible non monotonicity. This would reveal the relation between initial temperature and speed of blob evolution and speed of medium expansion.**
**         **

- Kensuke Homma / Hiroshima Univ.

## What is the energy density at Npart~90?

**Npart~90 corresponds to ********BJ****~2.4GeV/fm****2****/c**

**Measurement of transverse energy E****T**

**Preliminary**

- Kensuke Homma / Hiroshima Univ.

## Density correlation in longitudinal space

**In narrow midrapidity region like PHENIX, cosh(y)~1 and y~******

**Longitudinal space coordinate z can be transformed into rapidity**
**coordinate in each proper frame of sub element characterized by **
**a formation time ********where dominant density fluctuations are embedded.**

**Due to relatively rapid expansion in y, analysis in y would**
**have an advantage to extract initial fluctuations**
**compared to analysis in transverse plane.**

**Longitudinal multiplicity density fluctuation from the mean density can be an order parameter:**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Interpretation. 

## Direct observable for Tc determination

**GL free energy density *****g *****with ********** ~ 0 from high temperature side is insensitive to transition order, but it can be sensitive to Tc**

**spatial correlation**

******disappears at Tc →**

**Susceptibility in long wavelength limit **

**Correlation length**

**1-D two point correlation function**

**Fourier analysis**

**Product between correlation **
**length and amplitude can also**
**be a good indicator for T~Tc**

**Susceptibility**

- Kensuke Homma / Hiroshima Univ.

**Notes:**
Interpretation. 

## NBD fitsin CuCu@200

- 16 fit examples in most left edge in top 10% events
- out of 28/2*(1+28) times NBD fits

**Level (window size)**
**L=2****8****(1-********PHENIX****)**

- L=0

- L=240

- Kensuke Homma / Hiroshima Univ.

## Hit and dead map of East arm

- 3-sigma cut as the central cut
- to define dead map.
- Today only 3-sigma result will be shown.

- Number of hits (counts*events per minimum bin size)

- bins

- 256  bins

- Hit map

- Dead map

- Number of bins

- Kensuke Homma / Hiroshima Univ.

## Position dependent NBD corrections

- Require geometrical correction factor on NBD k below 2.0
-  -window size  is defined as:
-      L=28(1-/**PHENIX**)

**Correction factor on NBD k**

**Corrected NBD k**

**L=0**

**L=72**

**L=152**

**L=224**

**L=79**

**L=7**

**L=159**

******>0.06**

******=0.7**

**Example of 5% most central sample**

**L=231**

- Kensuke Homma / Hiroshima Univ.

## Cu+Cu@200GeV with only statistical errors

**10% bin width**

**5% bin width**
**with 2.5% shift**

**Confirmation of absorption**
**of bin width bias**

**Fit with only statistical errors**
**in k vs. ******

- Kensuke Homma / Hiroshima Univ.

## Corrected mean multiplicity <c>

**Cu+Cu@200GeV**
**5% bin width**

**Cu+Cu@200GeV**
**10% bin width**

**Au+Au@62.4GeV**
**10% bin width**

- Kensuke Homma / Hiroshima Univ.