1. Search for the critical point of strongly interacting matter
M. Gazdzicki
Frankfurt,Kielce
THE PROBLEM
Strongly interacting matter:
phases and transitions
AND ATTEMPTS TO SOLVE IT:
Observation of the onset of deconfinement
Search for the critical point

2. Observation of the onset of deconfinement implies that:
there are two phases of strongly interacting matter
(popular nicknames are hadron gas and quark-gluon plasma)
and thus there is a transition line/region between them
The next goals are to study properties of
- QGP (LHC),
- hadron gas (SIS, Nuclotron) and
- the transition line (SPS, RHIC, NICA)
In particular, a possibility to discover a hypothetical
critical end point of strong interacting matter attracts
a significant attention.

3. Phase diagram of water Cross-over (continuous change) End point
transition)
1st order PT
(discontinuous
transition)

4. Phase diagrams of strongly interacting matter
The most popular one
and an example of
a less popular one
Asakawa, Yazaki, NP A504, 668 (89)
within the Nambu, Jona-Lasinio model
Gorenstein, M.G., Greiner,
PR C72, (05)
within the quark-gluon bag model

5. The SPS and RHIC experimental programs are
mostly based on the most popular phase diagram
cross-over
1st order PT
End point

6. Properties of the end point of the 1st order line are believed
to resemble the properties of the 2nd order phase transition
1st order phase transition:
discontinuity of energy
and entropy density at TC
The two phases are very
different. For small
fluctuations of energy
density there are
only small fluctuations
of the phase composition.
2nd order phase transition:
energy and entropy densities
are continuous at TC
The two phases are similar
For small fluctuations of
energy density
there are large fluctuations
of the phase composition.

7. in the vicinity of the End point large volume
fluctuations of matter droplets are expected.
Fluctuations in the vicinity of the critical point:
micro-state 1
micro-state 2
micro-state 3
Fluctuations in the mixed phase (1st order PT):
micro-state 1
micro-state 2
micro-state 3

8. specific heat capacity: c ≡ dε /dT
Infinite c for T → TC
is typical for the 2nd order PT
Maximum of c at T < TC
appears for the 3rd and
higher order PT
as well as the cross-over

9. Temperature fluctuations in A+A collisions:
Stodolsky, PRL 75, 1044 (95),
Shuryak, PL B423, 9 (98)
(MCE with fixed E)
Particle number fluctuations in A+A collisions:
Mrowczynski, PL B430, 9 (98)
(GCE with fixed T)
Energy fluctuations in A+A collisions:
Stephanov, Rajagopal, Shuryak
PR D60, (99)
(GCE with fixed T)

10. Anomaly in fluctuations in a narrow domain
of the phase diagram is considered as
the main signal of the critical end point
but predictions what and how should
strongly fluctuate are model-dependent
Search for the critical end point in heavy ion
collisions implies a scan in the parameters
controlled in laboratory (collision energy and
centrality, system size). By changing them
we change freeze-out conditions (T, µB).
In the case they are close to the critical end
point anomaly in fluctuations should be observed.

11. E Schematic dependance of freeze-out
and early stage (T, µB) on collision energy
Early stage E
freeze-out

12. Freeze-out (T, µB) dependence on the laboratory controlled parameters
Collision energy dependence for central Pb+Pb (Au+Au)

13. NA49: System size dependence at 158A GeV
p+p
C+C
Si+Si
central Pb+Pb
Becattini, Manninen, MG, PR C73, (06)
Kraus, JP G31, S147 (05)

14. STAR: Centrality dependence for Au+Au
CPOD 2011
consistent with the NA49 findings

15. STAR: Centrality dependence for Au+Au
CPOD 2011
Seems to contradict the NA49 findings: possible reasons:
-STAR limited acceptance, limited set of yields,
-strangeness enhancement not taken into account in the STAR fits (neither CE nor γS)

16. T µB Summary on freeze-out (T, µB) dependence on
the laboratory controlled parameters T
energy A µB
Search for the critical point makes
sense only at energies larger than
the onset one (30A GeV)

17. NA61 and NA49 search for the critical point

18. Central collisions of light and medium size nuclei
are required for the proposed fluctuation studies
The fluctuations of the number of projectile participants are suppressed by selecting collisions with fixed number of projectile spectators
(in NA61 measured by PSD)
The fluctuations of the number of target participants can be suppressed only by selection of very central collisions
peripheral
central
Number of projectile participants
Fluctuations of target participants

19. and expected signals: hills of fluctuations
Critical Point:
freeze-out close to critical point,
and system large enough,
expected signal: a hill in fluctuations
and local power-law fluctuations
Pb+Pb
Be+Be 13

20. NA49 pT and multiplicity fluctuations
p+p
Pb+Pb
Pb+Pb
p+p
Pb+Pb
system size at 158A GeV
energy for central Pb+Pb
First hint of the
fluctuation hill?
PR C78:034914
PR D60:114028

21. NA49 di-pion and proton intermittency
Diakonos at CPOD 2011

22. CPOD 2011

23. CPOD 2011

24. Future belongs to NA61

25. Additional slides

26. x measure (ZP C54, 127 (1992)) of fluctuations (x=pT, f, Q)
Fluctuation measures (NA49 and/or NA61):
sdyn measure of dynamical particle
ration fluctuations (K/p, p/p, K/p)
E-by-e fit of particle multiplicities required in NA49
Mixed events used as reference
s2dyn ∝ 1/NW (PR C81, (2010), arXiv: )
Scaled variance w of multiplicity distribution
Intensive measure
For Poissonian multiplicity distribution w=1
In wounded nucleon model (superposition) w(A+A) = w(N+N) + <n>wW
<n> - mean multiplicity of hadrons from a single N+N; wW - fluctuations in NW w is strongly dependent on NW fluctuations
relative width (of K/π, p/π, K/p) σ= 𝑅𝑀𝑆 𝑀𝑒𝑎𝑛 ⋅100 % σ 𝑑𝑦𝑛 =𝑠𝑖𝑔𝑛 σ 𝑑𝑎𝑡𝑎 2 − σ 𝑚𝑖𝑥𝑒𝑑 ∣ σ 𝑑𝑎𝑡𝑎 2 − σ 𝑚𝑖𝑥𝑒𝑑 2 ∣ σ 𝑑𝑦𝑛 2 ≈∣ ν 𝑑𝑦𝑛 ∣
ω= 𝑁 2 − 𝑁 2 𝑁
x measure (ZP C54, 127 (1992)) of fluctuations (x=pT, f, Q)
In superposition model x(A+A) = x(N+N)
For independent particle emission x=0
In superposition model x is independent of NW and NW
fluctuations (strongly intensive)
𝑧 𝑥 =𝑥− 𝑥 ˉ ; 𝑥 ˉ − inclusive average event variable 𝑍 𝑥 = 𝑖=1 𝑁 𝑥 𝑖 − 𝑥 ˉ Φ 𝑥 = 𝑍 𝑥 2 𝑁 − 𝑧 𝑥 2 ˉ

27. 3rd moment of pT measure (pT(3))
Strongly intensive (PL B465, 8 (1999))
Intermittency in low mass p+p- pair
density fluctuations in pT space
Proper mass window and multiplicity required
Mixed events used as reference
Power-law behavior from s mode expected:
Critical QCD prediction f2 = 2/3
Φ 𝑝 𝑇 3 =  𝑍 𝑝 𝑇 𝑁  −  𝑧 𝑝 𝑇 3 ˉ  1 3
2D transv. momentum factorial moments: 𝐹 𝑝 𝑀 = 1 𝑀 2 𝑖=1 𝑀 2 𝑛 𝑖 𝑛 𝑖 − 𝑛 𝑖 −𝑝 𝑀 2 𝑖=1 𝑀 2 𝑛 𝑖 𝑝 𝑀 2 − number of cells in 𝑝 𝑇 space of di−pion 𝑝 𝑇,ππ = 𝑝 𝑇, π 𝑝 𝑇, π − 𝑛 𝑖 − number of reconstruc. di−pions in𝑖−th cell Δ 𝐹 2 𝑀 − combinatorial background subtracted (by use of mixed events) second factorial moment
Δ 𝐹 2 ~ 𝑀 2 φ 2