1. Christoph Blume University of Heidelberg
International School on: Quark-Gluon Plasma and Heavy Ion Collisions: Past, Present, Future Villa Gualino, Turino, Italy Soft Probes III
Christoph Blume
University of Heidelberg

2. Observables Temperature Strangeness Resonances Femtoscopy Fluctuations
Chemical Freeze-Out
Kinetic Freeze-Out
Temperature
Femtoscopy
Fluctuations
Flow
Jets +
Heavy Flavor
Photons
Strangeness
Resonances

3. Fluctuations

4. The Early Universe ...

5. Fluctuations in Cosmology
WMAP
Only 1 Event
Fluctuations on the level of < 10-4

6. Fluctuations in Heavy Ion Physics
Probe the medium response (susceptibilities)
Study hadronization properties
Might be sensitive to QGP phase
Hadron gas reacts differently than QGP
Different number of degrees of freedom
Nature of the phase transition
Order of the transition (cross over ⇔ 1st order)
Existence of critical point
⇒ sudden increase of fluctuations
Quark number susceptibility
from lattice QCD (Bielefeld group)

7. Fluctuations Measures (I): Basics
Basic event-by-event observables:
Multiplicities
Average transverse momenta 〈pT〉
Particle ratios (e.g. K/π)
Conserved quantities (charge Q, strangeness S, baryon number B )
Fluctuations usually characterized by second moments ⇒ variance
Higher moments (kurtosis) recently investigated
Two averages: inside a given event and over all events
Large and uniform detector acceptance is helpful
Need to separate simple statistical fluctuations from dynamical ones
Large effect in heavy ion physics: volume (impact parameter) fluctuations

8. Example: 〈pT〉 Fluctuations

9. Fluctuation Measures (II): Means and Variances
Observable x (e.g. pT ) for a single particle i
⇒ mean in a given event of multiplicity Nj :
Mean over all events of a quantity Xj,
which characterizes each event :
The weighting factor is wj = 1 for quantities
such as the event-wise multiplicity (i.e ).
In the case (e.g. average pT) we have
The variance of Xj is :
see also: NPA727, 97 (2003)

10. Fluctuation Measures (III): Means and Variances
Mean over all particles i and events j
of the single particle observable xi :
Corresponding variance :
Mean over all events j of the event-wise
mean Mx (e.g. average pT):
Variance of Mx :

11. Example: Multiplicity Fluctuations
NA49: PRC75,
(2007)

12. Fluctuation Measures (IV): Φx
Properties:
Φx = 0 for independent
particle emission
(no interparticle correlations)
Φx(A+A) = Φx(p+p) if A+A was
a simple superposition of p+p
M. Gazdzicki and S. Mrowczynski,
ZPC54, 127 (1992)
Not a dimensionless quantity
〈...〉 : averaging over events

13. Example: 〈pT〉 Fluctuations
central √sNN = 17.3 GeV
NA49: PLB459, 679 (1999)

14. Example: 〈pT〉 Fluctuations
NA49: PRC70, (2004)

15. Fluctuation Measures (V): σdyn
Definition :
S. Voloshin, V. Koch, H.G. Ritter,
PRC60, (1999)
If only statistical fluctuations
are present ⇒
Normalized dynamical fluctuation:
NA45: NPA727, 97 (2003)
Normalization removes energy dependencies,
e.g. due to increase of 〈pT〉

16. Example: 〈pT〉 Fluctuations
NA45: NPA727, 97 (2003)

17. Fluctuation Measures (VI): Particle Ratios A/B
Mixed events as reference
PRC79, (2009)
Poisson statistics as reference:
C. Pruneau, S. Gavin, and S. Voloshin, PRC66, (2002)
Negative values imply dominating correlations between A and B

18. Example: K/π Fluctuations
STAR: arXiv:

19. Example: K/π Fluctuations
Comparison of energy and system size dependence of νdyn
STAR: arXiv:

20. Example: K/p and p/π Fluctuations
S/B fluctuation as QGP signal
V. Koch, A. Majumder, and J. Randrup,
PRL95, (2005)
T < Tc: S and B can be unrelated
(Kaons: S = -1, B = 0)
T > Tc: S and B are correlated
(s-Quark: S = -1, B = 1/3)
K/p
p/π
Dominated by resonance decays

21. Fluctuation Measures (VII): pT Correlations
Covariance of
transverse momenta
of different particles
STAR: PRC72, (2005)
with
Independent of detection
efficiencies
Influence of other effects
(e.g. Coulomb interaction
or Bose-Einstein corr.)
can more easily be studied

22. Example: pT Correlations
STAR: PRC72, (2005)

23. Example: Net-Charge Fluctuations
Hadron Gas:
Charge unit = 1
Quark Gluon Plasma:
Charge unit = 1/3
⇒ Charge fluctuations should be reduced
in QGP relative to hadron gas
S. Jeon and V. Koch, PRL85, 2076 (2000)
M. Asakawa, U. Heinz and B. Müller, PRL85, 2072 (2000)

24. Example: Net-Charge Fluctuations
Charge Conservation Limit
HIJING
QGP
Au+Au, √sNN = 130 GeV
Signal obscured by resonance decays
Strongly acceptance dependent
STAR: PRC (2003)

25. Balance Function
⇒ Sensitive to hadronization time in an expanding system

26. Balance Function With, e.g., being the density of pairs inside a
given relative pseudo-rapidity range
Analysis done as a function of
S. Bass, P. Danielewicz, and S. Pratt, PRL85, 2689 (2000)

27. Balance Function Possible evidence for delayed hadronization
Shuffled: randomly shuffle
charges inside a given event
⇒ largest possible BF-width
Possible evidence
for delayed hadronization
STAR: PRC82, (2010)

28. Fluctuations Fluctuations observed on the level of 1 - 10%
Many “trivial” effects
Volume fluctuations
Resonance decays
Acceptance effects
Short range correlations (Bose-Einstein)
Conservation laws
(Mini-)jets
Elliptic flow
...
But clear evidence for dynamical fluctuations
with non-trivial energy or system size dependencies

29. QCD Critical Point

30. The QCD Phase Diagram

31. Analogy: Phase Diagram of Water
Cross over
Critical point
1st order
phase boundary

32. The QCD Phase Diagram
K. Rajagopal, CPOD Conference 09

33. Critical Point Predictions
Lattice QCD calculation
at finite μB
Z. Fodor and S. Katz
JHEP 0404, 050 (2004)
But current predictions scatter quite a lot
The CP might even not exist at all ...
P. de Forcrand and O. Philipsen, JHEP01, 077 (2007)
M. Stephanov,
CPOD conference 09

34. Critical Point Predictions
Larger critical area possible
Y. Hatta and T. Ikeda,
PRD67, (2003)
Focusing effect
Proximity of critical point might
influence isentropic trajectories
M. Askawa et al.,
PRL101, (2008)

35. First Attempts Multiplicity fluctuations as a function of B
NA49 data:
Phys. Rev. C79,
(2009)
B from stat.
model fit:
F. Becattini et al.,
Phys. Rev. C73,
(2006)
Amplitude of
Fluctuations:
M. Stephanov et al.
Phys. Rev. D60,
(1999)
Width of
crit. region:
Y. Hatta and T. Ikeda,
Phys. Rev. D67,
(2003)
Position of
crit. point:
Z. Fodor and S. Katz
JHEP 0404, 050 (2004)

36. Strategy: Energy Scan STAR at RHIC NA61 at the SPS CBM at FAIR
Observables:
Fluctuations
Flow
Spectra
Overview: arXiv:

38. The QCD Phase Diagram

39. Critical Endpoint from Lattice QCD

40. Order of the Phase Transition