1. Event-by-event Fluctuation & Phase Transition
OUTLINE
Motivation
Fluctuation measures:
<pT> fluctuation
Multiplicity fluctuation
Particle ratio, strangeness
Balance functions
Net charge fluctuation
Moments of net charge
DCC
Long range correlations
Near term activities
at RHIC
at LHC
Summary
critical
point
Tapan K. Nayak
CERN & VECC
Strangeness in Quark Matter
UCLA
March 28, 2006
Event-by-event fluctuation and phase transition

2. QCD phase diagram critical point ? Tc hadron gas nucleon gas nuclei r0
Stephanov, Rajagopal & Shuryak PRL 81 (1998)
Phase transition/Latent heat
Supercooling
QGP droplet formation
<pT>, Multiplicity fluctuations
Baryon-strangeness correlations
Moments of strangeness, baryon number and net charge distributions
- (recent calculations by Ejiri-Karsch-Redlich, Gavai-Gupta and Koch-Majumdar-Randrup)
Location of the critical point
detailed study of particle ratio and fluctuations
Chiral symmetry restoration
formation of DCC
charge-neutral fluctuations Tc r0
baryon density
Temperature
Neutron stars
Early universe
nuclei
nucleon gas
hadron gas
colour
superconductor
quark-gluon plasma
critical point ?
vacuum
CFL
At the CRITICAL POINT:
singularities in thermodynamical observables
=> LARGE EbyE FLUCTUATIONS
Event-by-event fluctuation and phase transition

3. Event-by-event fluctuation and phase transition
Lattice predictions
Karsch et al.
Gavai, Gupta hep-lat/
Fodor, Katz JHEP 0404 (2004) 050
Lattice calculations have not yet converged on the location of Critical Point. The best guess so far: around c.m. energy of 5-20 GeV/nucleon.
CRITICAL END POINT
From lattice: TC ~ 170 15 MeV
eC ~ GeV/fm3
Location of the Critical point
Theoretical expectations
Fluctuation measures
Fluctuation sources (statistical+dynamic)
geometrical:
impact parameter
number of participants
detector Acceptance (y, pT)
energy, momentum, charge conservation
anisotropic flow
Bose-Einstein correlation
resonance decays
jets and mini-jets
formation of DCC
color collective phenomena ….
Role of strangeness
Dedicated measurements?
Points for discussion:
Event-by-event fluctuation and phase transition

4. <pT> fluctuations
NA49, Phys Lett B459 (1999) 679
Central Pb+Pb
√s = 17.2 GeV
data
mixed
events
charged hadrons
y>4.0
<pT> fluctuations
<pT> of emitted particles is related to the temperature of the system. EbyE fluctuations of <pT> is sensitive to temperature fluctuations predicted for QCD phase transition.
non-statistical (dynamical) part of the <pT> fluctuation provides valuable information regarding:
critical point of phase transition
droplet formation
Formation of DCC
Can be measured experimentally with high precision.
Event-by-event <pT> compared to
stochastic reference (mixed events)
STAR: Phys. Rev. C 72 (2005)
The following are used to construct various fluctuation measures:
pT of particle
Mean pT of the event (<pT>)
Mean of the <pT> distribution
Event-by-event fluctuation and phase transition

5. <pT> fluctuations: centrality dependence
CERES
K. Perl
PRC 70 (2004)
H. Sako QM04
NA49
M. Tannenbaum
J. Mitchell
nucl-ex/
Different observables are sensitive to different processes.
STAR sees a smooth dependence on collision centrality whereas NA49 and PHENIX see larger fluctuations in mid-central collisions. STAR attributes this difference due to effects of acceptance and elliptic flow (Pruneau QM05, Voloshin Bergen05)
Phys. Rev. C 72 (2005)
PHENIX
STAR
PRL 93 (04)
Event-by-event fluctuation and phase transition

6. <pT> fluctuations: energy dependence
C. Pruneau QM05
Adamova et al., Nucl. Phys. A727, 97 (2003)
<pT> fluctuations in (h-f) bins
STAR: nucl-ex/
fluctuations
correlations
200 GeV
No Energy dependence of <pT> fluctuations is seen from CERES & STAR data.
This study is also useful for studying
contributions from (mini)jets to fluctuations.
Event-by-event fluctuation and phase transition

7. Multiplicity fluctuations
NA49: M. Rybczynski, QM2004
PRC 65 (2002)
Charged particles
Photons
Gaussians for narrow bins in centrality
WA98: Fine bins in centrality so that fluctuation from Npart is minimal.
Centrality dependence of multiplicity fluctuations do not show evidence of non-statistical contribution.
However recent NA49 analysis of scaled variance show non-statistical fluctuations at mid-central collisions.
Charged
Particles
Photons
Photons
w = s2/ < N >
Fine bins in centrality
Event-by-event fluctuation and phase transition

8. Particle ratio & fluctuations
<K+>/<p+>
Particle Ratio:
<K/p> has an increasing trend with energy, whereas a horn structure seen in <K+/ p+>.
<K->/<p->
s2data - s2mix = s2dynamic
Fluctuation in Ratio:
K/p fluctuations are large at low beam energy & decrease with increasing energy.
p/p fluctuations are negative, indicating a strong contribution from resonance decays.
J. Phys. G30 (2004) S1381
M. Gazdzicki QM04
C. Roland (NA49)
SQM2004
sdyn
Event-by-event fluctuation and phase transition

9. K/p fluctuation in STAR
Supriya Das: SQM’06 Symposium
s = rms/mean
sdyn = sqrt(sdata2 – smixed2)
Fluctuation in K/p decreases with increasing energy till the top SPS energy and remains flat above it. The amount of fluctuation decreases with increasing centrality and is similar for 62 GeV as well as 200GeV AuAu collisions.
Event-by-event fluctuation and phase transition

10. Event-by-event fluctuation and phase transition
Balance functions
Z=0
Bass-Danielewicz-Pratt, PRL 85, 2000
D. Drijard et al, NP B(155), 1979
Early Hadronization
 Large 
Opposite charged particles are created at the same location of space–time.
Charge–anticharge particles created earlier (early stage hadronization) get further separated in rapidity.
Particle pairs that were created later (late stage hadronization) are correlated at small Δy.
The Balance Function quantifies the degree of this separation and relates it with the time of hadronization.
Late Hadronization
 Small 
Event-by-event fluctuation and phase transition

11. Balance functions: centrality & energy dependence
Gary Westfall: STAR
Panos Christakoglou: NA49
Panos Christakoglou
STAR: √sNN = 130 GeV PRL 90 (2003)
NA49: √sNN = 17.2 GeV PRC 71 (2005)
STAR data
NA49 data
NA49 shuffling
STAR shuffling
simulation
NA49 data
STAR data
W is a normalized measure of the time of hadronization with respect to uncorrelated data sample.
This is consistent with delayed hadronization at RHIC compared to SPS energies.
peripheral
central
DATA show a strong centrality dependence of balance function width.
Event-by-event fluctuation and phase transition

12. Balance functions for identified particles
Bass-Danielewicz-Pratt, PRL 85, 2000
and Gary Westfall, J.Phys.G30, S345-S349 (2004)
Heavier particles are characterized by narrower bf distributions:
The balance function width for pions get narrower with increasing centrality, remains constant for kaons.
HIJING reproduces results for kaons, but not for pions.
The ratio of widths of pions to kaons is consistent with delayed hadronization picture.
STAR Preliminary
pions
Panos Christakoglou in ALICE PPR p
kaons K
ALICE simulation showing BF widths of p,K,p p
Mass (GeV)
Event-by-event fluctuation and phase transition

13. Net charge fluctuations
confined:
few d.o.f.
deconfined:
many d.o.f.
Charged multiplicity: nch = n+ + n–
Net charge: Q = n+ - n–
Charge ratio: R = n+ / n-
(1) v(Q)  Var(Q)/<nch>
(for stochastic emission, v(Q) = 1)
(2) v(R)  Var(R) * <nch>
(for stochastic emission, v(R) = 4)
(3) F(Q)
ndynamic
Moments of Net charge distributions
Prediction: A drastic decrease in the EbyE fluctuations of net charge in local phase space regions in the deconfined QGP phase compared to that of the confined case hadronic gas.
QGP:4 and pion gas: 1-2
Jeon, Koch: PRL (2000) 2076
Asakawa, Heinz & Muller: PRL (2000) 2072
Evolution of fluctuation
Shuryak & Stephanov: PR C63 (2001)
Heiselberg & Jackson: PR C63 (2001)
Mohanty, Alam & TN: PR C67 (2003)
Event-by-event fluctuation and phase transition

14. Event-by-event fluctuation and phase transition
Net charge fluctuation: energy dependence
J. Mitchell, QM’04
STAR: Au+Au
Preliminary
nucl-ex/
peripheral
central
PHENIX ||<0.35, =/2
CERES 2.0<  <2.9
STAR: 5% Central Au+Au
C. Pruneau
QM05
Net charge fluctuations measured by PHENIX & NA49 are consistent with independent emission.
Net charge fluctuations measured by STAR are close to the quark coalescence model of Bialas.
Fluctuations are larger at SPS compared to RHIC, but remain constant over a large range of energy.
Event-by-event fluctuation and phase transition

15. Event-by-event fluctuation and phase transition
Moments of net charge distributions
Lattice calculations
Ejiri, Karsch and Redlich: hep-ph/
Gavai, Gupta: hep-lat/
Calculation of Non-linear susceptibilities (higher order derivatives of pressure with respect to chemical potentials):
4th moment
Net charge
Isospin
Strangeness
2nd moment
6th moment
(similar to kurtosis)
=> Interesting structure close to T=TC
Is it possible to make precise measurement of higher moments of net charge?
bins in centrality
bins in pT
Event-by-event fluctuation and phase transition

16. MEAN of Q distributions
Q(net charge) distributions
Q distributions for AuAu 200GeV at 4 different centralities and 6 bins in pT
MEAN of Q distributions
<Q>
low pT
high pT
<Q>/Npart
Q (net charge)
<Q>/Npart is independent of centrality.
Moments of Q distributions have been analyzed.
Event-by-event fluctuation and phase transition

17. Variance and kurtosis of net charge distributions
n(Q) with pT binned
AuAu 200GeV
Kurtosis (4th moment)
Centrality & pT
n(Q) is low at low pT ad increases with increase of pT. Could be an effect of more resonance production at low pT.
First analysis of the 4th moment of net charge distribution is performed. Detailed comparison in terms of lattice calculations is expected soon.
Event-by-event fluctuation and phase transition

18. Event-by-event fluctuation and phase transition
Formation of DCC
Bjorken, Kowalski & Taylor SLAC-pub-6109 (1993)
Review: Mohanty & Serreau Phy Rep 414 (2005)
Methods of Analysis:
Gamma-Charge correlation
Discrete Wavelet analysis
Power spectrum analysis
‘Robust’ variables
Event shape analysis
Sliding window method (SWM)
=> WA98 and NA49 have put upper limit on DCC production at 3x10-3 level.
=> DCC production also shows up in other forms including strangeness correlations.
Large fluctuations in number of
photons and charged particles
Aggarwal, Sood, Viyogi
nucl-ex/
Recent simulation for RHIC show better sensitivity for DCC by using SWM with photon and charged multiplicity:
WA98
PMD & SPMD
PRC 67 (2003)
Event-by-event fluctuation and phase transition

19. Long-range multiplicity correlations
Correlation strength:
=> Study of correlations among particles produced in different rapidity regions.
=> The long-range correlations are expected to be much stronger in p-A and A-A, compared to p-p at the same energy.
Terence J Tarnowsky
Nuclear Dynamics,
San Diego March 2006
STAR Preliminary
STAR: forward region of 0.8<h<1.0 & backward of -1.0<h<-0.8.
Increase in correlation strength observed for central collisions compared to peripheral for AuAu collisions at 200GeV.

20. RHIC Search for critical point at RHIC Physics measure Energy Density
The QCD phase boundary is worthy of study, including that of the tri-critical point.
STAR experiment with the inclusion of TOF will be the ideal place for this study.
PHENIX will be able to carry out an extensive program for the search of critical point.
RHIC has an unique capability to scan the full range from the top AGS to top RHIC energy.
The idea is to have an energy scan from c.m. energy of 4.6GeV to 30GeV in suitable steps corresponding to baryon chemical potentials of 150MeV to 550MeV.
Fluctuation study especially with strangeness plays a major role in the search for critical point.
Physics measure
AGS SPS RHIC
RHIC
QCD Critical Point
Energy Density
Event-by-event fluctuation and phase transition

21. EbyE fluctuation in ALICE
EbyE measures in ALICE: simulation for Pb+Pb at 5.5TeV
Slope parameter
<pT> pions
<pT> kaons
<pT> protons
With the large multiplicity of several tens of thousands expected in each collision at LHC energies, EbyE analyses of several quantities become possible. This allows for a statistically significant global as well as detailed microscopic measures of various quantities.
ALICE-PPR
Event#1
Event#2
Event#3
EbyE HBT radii
K/p
p/p
Event-by-event fluctuation and phase transition

22. Event-by-event fluctuation and phase transition
Thermodynamic quantity / fluctuation in the quantity
Energy Density
Fluctuation behavior???
Critical point???
Summary
What’s done so far :
Fluctuations of thermodynamic quantities are fundamental to the study of phase transition – including quark-hadron phase transition.
Lattice calculations suggest fluctuation patterns in strangeness, baryon number & net charge even at small chemical potentials - increasing towards the critical point.
Exploratory study using many fluctuation measures continues - interpretation of results become complex because of several competing processes which contribute.
Indication of large fluctuation patterns around SPS energies.
What’s coming up:
Fluctuation study will play a major role in the search for the critical point at RHIC.
ALICE: detailed EbyE physics and fluctuation to understand the physics of bulk matter as well as high-pT particles and jets.
Future GSI facilities: CBM
Event-by-event fluctuation and phase transition