1. Search for the QCD Critical Point -
Fluctuations of Conserved Quantities in High Energy Nuclear Collisions at RHIC Xiaofeng Luo (罗晓峰) Central China Normal University (CCNU) Oct. 8, 2015
Search for the QCD Critical Point -

2. QCD Phase Diagram (Conjectured)
Very rich phase structure in the QCD phase diagram.
K. Fukushima and T. Hatsuda, Rept. Prog. Phys. 74, (2011); arXiv:
Large Uncertainties in determining phase structure by theory at finite μB.

3. I would rather fishing than thinking about the Critical Point…..
Fishing Diagram: Theorist’s thinking
I would rather fishing than thinking about the Critical Point…..
Fish, fish, fish…..

4. The QCD Critical Point Need proper observables in HIC.
Singularity of EOS: Diverges of the thermodynamics quantities, such as correlation length (ξ), Susceptibilities (χ), heat capacity (CV).
Long wavelength fluctuations of order parameter.
Critical Opalescence in liquid gas PT.
Water: H2O
T. Andrews.
Phil. Trans. Royal Soc., 159:575, 1869
Need proper observables in HIC.
1. finite size/time.
2. Non-CP physics effects.
3. Need CFO closer enough to CP.
4. Signal didn’t wash out after evolution.
ξ=2 ~ 3 fm
Very Challenging !!!
武汉·汤逊湖，
工欲善其事，必先利其器
Sharp tools make good work

5. Location of QCD Critical Point: Theory
Lattice QCD:
DES:
Lattice QCD：
1): Reweighting:Fodor&Katz,2004:
μEB/TE ~ 2.2  √sNN ~ 9.5 GeV
2): Tylor Expansion: Gavai&Gupta 2013
μEB/TE ~ 1.7  √sNN ~ 14.5 GeV
DSE:
1): Y. X. Liu, et al., PRD90, (2014).
μEB/TE ~ 2.88  √sNN ~ 8 GeV
2): C. S. Fischer et al., PRD90, (2014).
μEB/TE ~ 4.4  √sNN ~ 6 GeV
In my talk, I will focus on how we search for the QCD critical point by using fluctuation observable in heavy-ion experiment. Due lattice QCD
Proposed Colliding Energy Range for CP Search:√sNN = 6 ~ 14.5 GeV

6. Observables: Higher Moments (fluctuations)
“Shape” of the fluctuations can be measured: non-Gaussian moments.
C2,x~ ξ2
C3,x ~ ξ4.5
C4,x~ ξ7
M. A. Stephanov, PRL102, (2009); PRL107, (2011);
M. Akasawa, et al., PRL103, (2009).
Lattice
Cheng et al, PRD (2009)
B. Friman et al., EPJC 71 (2011) 1694.
H.T. Ding et al,
Susceptibility ratios  Moments of Conserved Charges  Cumulant Ratios
(q=B, Q, S)

7. Expectations from Skellam Distributions
If proton and anti-proton are independent Poissonian distributions, the distributions of net-protons is Skellam distributions, which is the case in Hadron Resonance Gas Model.
Npbar : Mean number of anti-protons
Np : Mean number of protons
Denote statistical/thermal fluctuations.
Then we have the skellam expectations for various moments/cumulants :

8. RHIC Beam Energy Scan-Phase I
√s (GeV)
Statistics(Millions)
(0-80%)
Year
μB (MeV)
T (MeV)
μB /T
7.7 ~4
2010
420
140
3.020
11.5
~12
315
152
2.084
14.5
~ 20
2014
266
156
1.705
19.6
~36
2011
205
160
1.287 27
~70
155
163
0.961 39
~130
115
164
0.684
62.4
~67 70
165
0.439
200
~350 20
166
0.142
μB , T : J. Cleymans et al., Phys. Rev. C 73, (2006). 

9. STAR Detector System Large, Uniform Acceptance at Mid-y
EEMC
Magnet
MTD
BEMC
TPC
TOF
BBC
Large, Uniform Acceptance at Mid-y
Excellent Particle Identification
Full TOF became available in 2010

10. First Order Phase Transition ?
Courtesy of Nu Xu
Indication of a First Order Phase Transition.
D.H. Rischke et al. HIP1, 309(1995) H. Stoecker, NPA750, 121(2005)
J. Steinheimer et al., arXiv: P. Konchakovski et al., arXiv:

11. How to ? Raw net-p prob. distribution Moments/Cumulants
1. Finite particle detection efficiency.
Need efficiency correction. It is not
straight forward for higher moments.
2. Initial volume fluctuations.
Improve centrality resolution and
apply centrality bin width correction.
3. Remove auto-correlation.
Particles used in the analysis
are excluded in centrality definition.
4. Proper error calculations.
STAR: PRL112, 32302(14)
STAR: PRL113,092301(14)
X. Luo, J. Phys.: Conf. Ser (2011).
X. Luo, JPG 39, (2012).
X. Luo,et al., JPG 40, (2013).
X. Luo, PRC 91, (2015).
Delta theorem and Bootstrap

12. Higher Moments Results
Net-proton results:
All data show deviations below Poisson for κσ2 at all energies. Larger deviation at √sNN ~ 20 GeV
Net-charge results:
Need more statistics.
Poisson: κσ2=1
STAR: PRL112, 32302(14) / arXiv:
STAR: PRL113,092301(14) / arXiv:

13. New Net-proton results: Larger pT Acceptance
Published net-proton results: Only TPC used for proton/anti-proton PID.
TOF PID extends the phase space coverage.
Au+Au  protons
STAR Preliminary
TOF
TOF
TPC
TPC
Acceptance: |y| ≤ 0.5, 0.4 ≤ pT ≤ 2 GeV/c
Efficiency corrections:
TPC (0.4 ≤ pT ≤ 0.8 GeV/c): εTPC ~ 0.8
TPC+TOF (0.8 ≤ pT ≤ 2 GeV/c): εTPC*εTOF ~ 0.5

14. Efficiencies for Protons and Anti-protons
TPC
TPC+TOF
STAR Preliminary
Systematic errors: 1) Uncertainties on efficiency, 2) PID, 3) Track Cuts.

15. Efficiency Correlation and Error Estimation
We provide a unified description of efficiency correction and error estimation for higher moments analysis in heavy-ion collisions.
X. Luo, PRC91, (2015).
Fitting formula:
We can express the moments and cumulants in terms of the factorial moments, which can be easily efficiency corrected.
One can also see:
A. Bzdak and V. Koch,
PRC91,027901(2015),
PRC86，044904(2012).

16. Statistical Errors Comparison Between for Net-Q, Net-P and Net-K
X. Luo, JPG 39, (2012).
14.5 GeV: 0-5%
Net-Charge (Q)
Net-proton (B)
Net-Kaon (K)
Typical Width (σ)
12.2
4.2
3.4
Aver. Efficiency (ε)
65%
75%
38%
355 32 82
With the same # of events: error(Net-Q) > error(Net-K) > error (Net-P)

17. Cumulants: C1~C4
In general, cumulants are increasing with <Npart>.
Significant increase for C4 at most two central bin at 7.7 GeV after eff. corr. .

18. Centrality Dependence:Cumulants Ratios
Large increase for Kσ2 are observed in the most central 7.7 GeV
Oscillations are observed at 0-5% and 5-10% for 14.5 and 19.6 GeV.

19. Energy Dependence for Net-proton κσ2
Net-proton as proxy for net-baryon.
Non-monotonic trend is observed
for the 0-5% most central Au+Au collisions. Dip structure is observed
around 19.6 GeV.
Separation and flipping for the
results of 0-5% and 5-10% centrality are observed at 14.5 and 19.6 GeV. ( Oscillation Pattern observed !)
UrQMD (no CP) results show
suppression at low energies.
Consistent with the effects of baryon number conservation.

20. Sign of Kurtosis :Model and Theoretical Calculations
M.A. Stephanov,
PRL107, (2011).
PQM
Schaefer&Wanger,
PRD 85, (2012)
PQM
V. Skokov, QM2012
NJL
VDW
Memory Effects
JW Chen et al., arXiv:
Vovchenko et al., arXiv:
Swagato,et al, PRC92, (2015).

21. Lattice Calculation μEB/TE~1.7.
Large uncertainty will appear when approaching critical point.
S. Gupta et al., Lattice 2013

22. Oscillation Pattern: Signature of Critical Region ?
Blue: Positive Contribution
Red: Negative Contribution
Depending on relative position between reaction trajectories/freeze out position and critical region.
0-5%
5-10%
14.5 GeV
1+Pos.
1+Neg.
19.6 GeV
“Oscillation pattern” around baseline for Kurtosis may indicate a signature of critical region.
Propose to scan 16.5 GeV (μB =238 MeV) or even finer step between 14.5 and 19.6 GeV,expect to see bigger dip and no separation for the results of the 0-5% and 5-10%.

23. Energy Dependence for Net-proton Sσ
Non-monotonic trend is observed
for the 0-5% most central Au+Au collisions.
Peak structure is observed
around 14.5 GeV for both Sσ
and Sσ/Skellam
UrQMD (no CP) results show
suppression for (Sσ)/Skellam
at low energies.
STAR Preliminary

24. Κσ2 Vs. Sσ Taylor expansion in Lattice :
THERMO-meter :
BARYO-meter :
A structure is observed for 0-5% most central data while it is flat for peripheral collisions.
Can be directly compared with theoretical calculations.

25. Eff. Corrected and Eff. Uncorrected
Efficiency corrections are important not only for the values in the higher moments analysis, but also the statistical errors.

26. Acceptance Study: pT and Rapidity
Bjorken rapidity Δηcorr=ξ/τf~
<< Kinematic rapidity Δycorr~ 2 (Misha)
Significant pT and rapidity dependence are observed at low energies.
Large acceptance is crucial for the fluctuation measurement.

27. STAR: Moments of Net-Charge and Net-Kaon Distributions
STAR Preliminary
Within current errors, Net-Kaon
and Net-Charge κσ2 are consistent
with unity.
More statistics are needed to
make a conclusion.
UrQMD (no CP), show no energy
dependent.
σ： Measured width of distributions.
ε: Efficiency.
STAT, QM 2015.
In STAR, with the same # of events: error(Net-Q) > error(Net-K) > error (Net-P)

28. STAR Upgrades and BES Phase-II (2019-2020)
Larger rapidity acceptance crucial for
further critical point search with net-protons
iTPC proposal:
BES-II whitepaper:
Electron cooling upgrade will provide increased luminosity ~ 3-10 times.
Inner TPC(iTPC) upgrade : |η| < 1 to | η |< 1.5. Better dE/dx resolution.
Forward Event Plane Detector (EPD): Centrality and Event Plane Determination.
1.8 < |η| < 4.5

29. Summary Dip Discovery Potential at High Baryon Density:
Intriguing structures are observed at low energies.
Dip
Peak
Oscillation
Discovery Potential at High Baryon Density:
First order phase transition and QCD Critical Point etc.
The Race is on…..

30. It is time to discover the QCD Critical Point !
Thank you for your attention !

31. Exploring QCD Phase Structure3
RHIC, NICA, FAIR 1
LHC, RHIC 2
RHIC
RHIC BES-II
QCD phase structure and critical point
√sNN ≤ 20 GeV
RHIC
Quarkyonic matter?
For region μB > 500 MeV, √sNN ≤ 5 GeV, fixed-target experiments are much more efficient
RHIC
sQGP properties
√sNN = 200GeV

32. Observables: Higher Moments
Citation: 251
Cumulant Ratios: (q=B, Q, S)
Volume cancel.

33. Baryon Stopping
Phys. Rev. Lett. 93, (2004)
Phys.Rev. C74, (2006)
Phys. Rev. Lett. 93, (2004)
Baryon stopping strongly depends on the collision energy and become
more important at low energies.
Maximum freeze out baryon density is reached at √sNN ~ 8 GeV.
(higher when consider the finite size of the nucleon.)

34. Verification of the Eff. Correction with Model
AMPT model: Au+Au 39 GeV.
Set different efficiency for two pT range.
(0.4，0.8): 80%, (0.8, 2): 50% for p
and pbar.
X. Luo, PRC91, (2015).
Random Sampling
The eff. corrected results match the model inputs very well, which indicate the efficiency correction method works well.
2. The error estimation for eff. corrected results are based on the Delta theorem.

35. Centrality Determination
Defined by charged Kaon and Pion within |eta|<1, DCA<3, Nfits>10.
1. Avoid auto-correlation, 2. Suppress volume fluctuations.

36. Event-by-Event Net-proton Multiplicity Distributions

37. Transport Model Calculations

38. Resonance Decay Effect
Net-baryon
arXiv:
Resonance Decay Effect
Study by Hadron Resonance Gas Model
Net-charge
Net-proton is a good approximation of
Net-baryon.
Net-baryon has less resonance decay
effect compare to net-charge and
net-strangeness.
Net-strangeness

39. Trends change significantly from peripheral to central.
Collision Energy and Centrality Dependence
Trends change significantly from peripheral to central.