1. Katarzyna Grebieszkow Warsaw University of Technology
News from pT fluctuations
October 7-11, Wrocław NA61 and NA49 Collaboration Meeting
K. Grebieszkow for the NA61 Collaboration
"Status and plans of the ion program of NA61 at the CERN SPS"
The NA61/SHINE experiment at the CERN SPS is a new program to study hadron
production in hadron+hadron, hadron+nucleus and nucleus+nucleus interactions.
The main goal of the NA61 ion program is to explore the most
interesting region of the phase diagram of strongly interacting matter.
Within the expected (T - mu_B) interval we plan to study the
properties of the onset of deconfinement and to search for the signatures
of the critical point. Such a 2D scan of the phase diagram will be performed by
varying the energy (13A-158A GeV) and system size (p+p, Be+Be, Ar+Ca, Xe+La)
of the collision. The first data samples of the p+p energy scan were recorded
in 2009 and 2011.
This presentation will summarize the status and plans of the NA61/SHINE
ion program. In particular the detector upgrades, data taking schedule and
the first results on spectra and correlations will be discussed.

2. May serve as a signature of the onset of deconfinement
Fluctuations and correlations:
May serve as a signature of the onset of deconfinement
Close to the phase transition Equation of State changes rapidly which can impact energy dependence of fluctuations
Can help to locate the critical point of strongly interacting matter
Analogy to critical opalescence – enlarged fluctuations close to the critical point. For strongly interacting matter maximum of CP signal expected when freeze-out happens near CP

3. Measures of transverse momentum fluctuations used in NA49
Methods applied:
1. Average pT per event M(pT) distributions (data and mixed events)
2. pT measure (ZPC 54, 127 (1992))
In superposition model pT(A+A) = pT(N+N)
For independent particle emission pT=0
In superposition model pT is independent of NW and
NW fluctuations (strongly intensive)
FpT measure determines magnitude but no information about its origin (event-by-event fluctuations?, two-particle/multi-particle correlations?, etc.) Þ
3. Two-particle correlations in cumulative variable x (PRC 70, (2004))
4. 3rd moment of pT fluctuations F(3)pT
Strongly intensive (PL B465, 8 (1999)).
For independent particle emission (3)pT=0
→ see back-up slides for results
5. This presentation: New strongly intensive measures of pT fluctuations (DXN, SXN )
𝑧 𝑝 𝑇 = 𝑝 𝑇 − 𝑝 𝑇 ; 𝑝 𝑇 − inclusive average event variable 𝑍 𝑝 𝑇 = 𝑖=1 𝑁 𝑝 𝑇,𝑖 − 𝑝 𝑇 Φ 𝑝 𝑇 = 𝑍 𝑝 𝑇 𝑁 − 𝑧 𝑝 𝑇 2
𝑧 𝑥 =𝑥− 𝑥 ; 𝑥 − inclusive average event variable 𝑍 𝑥 = 𝑖=1 𝑁 𝑥 𝑖 − 𝑥 Φ 𝑥 = 𝑍 𝑥 2 𝑁 − 𝑧 𝑥 2
Φ 𝑝 𝑇 3 = 𝑍 𝑝 𝑇 𝑁 − 𝑧 𝑝 𝑇

4. System size dependence Energy dependence of average
(p+p, C+C, Si+Si, and Pb+Pb)
of average pT and multiplicity
fluctuations at 158A GeV
Energy dependence of average
pT and multiplicity fluctuations
for central Pb+Pb
all charged
Maximum of pT and w for C+C and Si+Si at 158A GeV
No significant energy dependence
all charged
7.2% most central
(semi)central
1% most central
1% most central
Forward-rapidity, limited azimuthal acceptance
upper left: PRC 70, (2004)
lower left: p+p – PRC 75, (2007); Pb+Pb – PRC 78, (2008); C+C, Si+Si - B. Lungwitz, PhD thesis
upper right: PRC 79, (2009)
lower right: PRC 78, (2008)
For energy dependence of FpT important cut on y*p to get rid of artificial effect of event-by-event centrality fluctuations while studying only forward-rapidity → for details see separate paper KG, PRC 76, (2007)

5. CP2 CP1 Data are consistent with the CP2 predictions
Average pT and multiplicity fluctuations: dependence on phase diagram coordinates
all charged
CP2
two locations of CP considered
all charged
CP1
A or
system size
sNN
(Pb+Pb)=6 fm ↘ (p+p)=2 fm (dashed), and 3 fm ↘ 1 fm (solid)
 = 6 fm (dashed)
 = 3 fm (solid)
Maximum of pT and w observed for C+C and Si+Si
Data are consistent with the CP2 predictions
Grebieszkow, Nucl. Phys. A830,
547C-550C (2009)

6. important relation: Φ 𝑝 𝑇 = 𝑝 𝑇 ω 𝑝 𝑇 Σ 𝑋𝑁 −1
New strongly intensive measures of fluctuations D and S (here applied to pT fluctuations) where suggested in Gaździcki, Gorenstein, PRC 84, (2011).
Their normalization was proposed in Gaździcki et al. PRC 88, (2013)
important relation: Φ 𝑝 𝑇 = 𝑝 𝑇 ω 𝑝 𝑇 Σ 𝑋𝑁 −1
DXN uses only first two moments: <N>, <X>, <X2>, <N2>
SXN uses also correlation term: <XN>-<X><N>
thus D and S can be sensitive to several physics effects in different ways

7. Independent Particle Model (IPM) Model of Independent Sources (MIS)
unit
No fluctuations;
N = const.
X = const.
Independent Particle Model (IPM)
Model of Independent Sources (MIS)
FpT
MeV/c
FpT = 0
FpT in MIS not dependent on NS and its fluct. thus
FpT (A+A) = FpT (N+N)
DXN
dimensionless
DXN = 0
DXN = 1
DXN (A+A) = DXN (N+N)
SXN
SXN = 0
SXN = 1
SXN (A+A) = SXN (N+N)
Φ 𝑝 𝑇 =− 𝑝 𝑇 ω 𝑝 𝑇
D and S are dimensionless and have scale which allows for a quantitative comparison of fluctuations of different, in general dimensional, extensive quantities
Several effects were studied:
1. IPM, MIS, source-by-source T fluctuations (example of MIS), event-by-event (global) T fluctuations, M(pT) vs N correlation → see M.I. Gorenstein, K. Grebieszkow, arXiv:
2. quantum (BE & FD) effects → see M.I. Gorenstein, M. Rybczyński, arXiv:
3. system size and energy dependence using UrQMD → see M. Gaździcki, M.I. Gorenstein,
M. Maćkowak-Pawłowska, PRC 88, (2013) [arXiv: ]
One of conclusions (supported by UrQMD tests): D, S, and F quantities measure deviations from the superposition model in different ways. Therefore, in the analysis of experimental data a simultaneous measurement of all three quantities would be highly desirable.

8. “Know your reference”
- What does the elliptic flow coefficient v2=0.1 means?
- It means that 50% more particles are emitted
“in plane” than “out of plane”. Huge effect!
- What does the FpT = 10 MeV/c mean ?
- Nothing! We do not know whether it is a large or a small effect.
Especially when the magnitudes of FpT from several “trivial” effects
(BE statistics, resonance decays, etc.) are not estimated.
- What does the SXN = 1.1 mean?
- It means that (for this specific combination of moments → S quantity) we measure 10% deviation from Independent Particle Model (fluctuations
are 10% larger than in IPM)
Similar advantage for w measure of N fluctuations → here Poisson N distribution
(instead of IPM) used as the reference model:
wN = 0 for N = const. and wN = 1 for Poisson N distribution.
Thus for any P(N) distribution:
wN > 1 (or wN >> 1) corresponds to “large” (or “very large”) fluctuations of N,
wN < 1 (or wN << 1) corresponds to “small” (or “very small”) fluctuations of N

9. Notation: D/S[PT,N] º D/SXN
M.I. Gorenstein, K. Grebieszkow, arXiv:
Model of Independent Sources (MIS) reduced to Independent Particle Model (IPM)
Each event composed by a given number of identical single sources.
For each source the number of particles generated from the Poisson distribution with a mean value of 5.
Particle pT generated from exp. mT spectrum with inverse slope T=150 MeV.
Number of sources composing an event was either constant (circles) or selected from Poisson (triangles) or from Negative Binomial distribution (squares). For Negative Binomial distribution its dispersion
sqrt(Var(NS)) was large and taken to be equal <NS>/2.
Confirmation that these measures are intensive (circles) and strongly intensive (triangles, squares). For these simulations FpT = 0.

10. the measures are strongly intensive
M.I. Gorenstein, K. Grebieszkow, arXiv:
For each source the number of particles from Poisson with a mean value of 5.
Particle pT generated from exp. mT spectrum with average inv. slope <T>=150 MeV. T generated separately for each single source (source-by-source T fluctuations → MIS) from Gaussian shape with dispersion sT=25 MeV. Number of sources composing an event generated from the Poisson distribution.
Lines → analytical calculations for mT exponential shape (see the paper);
solid line for pion mass and dashed line for massless particles
Positive signal FpT > 0 (»24 MeV/c, not shown) , DXN and SXN > 1;
the measures are strongly intensive

11. Strong dependence of DXN and SXN on the number of sources
M.I. Gorenstein, K. Grebieszkow, arXiv:
The same as previous page, but:
source-by-source T fluctuations replaced by event-by-event (global) T fluctuations. For each event T generated from Gaussian shape with dispersion sT=25 MeV.
Lines → analytical calculations for mT exponential shape (see the paper);
solid line for pion mass and dashed line for massless particles
Strong dependence of DXN and SXN on the number of sources
for event-by-event T fluctuations (the same observation for FpT – not shown)

12. solid line for pion mass and dashed line for massless particles
M.I. Gorenstein, K. Grebieszkow, arXiv:
The same as previous page.
Event-by-event T fluctuations T varied from event to event following Gaussian distribution with dispersion sT. In order to avoid negative T values only events within T=150 ±3sT MeV were accepted.
The number of sources composing an event was generated from the Poisson distribution with a mean value of 100.
Lines → analytical calculations for mT exponential shape (see the paper);
solid line for pion mass and dashed line for massless particles
The values of all fluctuation measures (also for FpT which is not shown)
increase when event-by-event ”temperature” fluctuations are stronger (higher sT )

13. Previous slides → the same behaviour and magnitudes of DXN and SXN
The example that those two measures can be different → see calculations within UrQMD 3.3 model
M. Gaździcki, M.I. Gorenstein, M. Maćkowak-Pawłowska, PRC 88, (2013) [arXiv: ]

14. M(pT) – average transverse momentum per event
M.I. Gorenstein, K. Grebieszkow, arXiv:
M(pT) – average transverse momentum per event
Known from years correlation between M(pT) and N in elementary interactions.
Here such a correlation taken from p+p at 158 GeV/c (forward-rapidty): NA49, PRC 70, (2004). <M(pT)> versus N values from NA49 (red triangles in right panel) used as 2T values in fast generator where dn/dmT = C mT exp (-mT/T)
DXN = ±
SXN = ±
FpT = 0.82 ± 0.19 MeV/c

15. Acceptance used by NA49 for the analysis of FpT and Ff
NA49 azimuthal acceptance is limited. Detector is left-right symmetric. Acceptance for positive and negative particles is the same, provided the azimuthal angle for one charge is reflected
To allow quantitative comparison of Ff (azimuthal fluct.) for pos. and neg. charged particles we rotated particles of one charge
System size dependence
p+p, C+C(15.3%), Si+Si(12.2%), Pb+Pb(5% or MB) at 158A GeV
2.0 <y*p<2.2
Energy scan
7.2% Pb+Pb
See PRC 70, (2004) and PRC 79, (2009) for detailed parametrization of
acceptance regions (particles inside black lines) in each rapidity bin

16. Note: for pT analysis (and now also for DXN and SXN) I use:
y*p < y*beam – 0.5
(additional cut for
the energy scan)
Note: for pT analysis (and now also for DXN and SXN) I use:
Forward rapidity only (1.1 < y* < 2.6), < pT < 1.5 GeV/c
Limited azimuthal acceptance (much more limited in case of energy scan)

17. Raw data (not corrected for two-track resolution, statistical errors only)

18. Raw data (not corrected for two-track resolution, statistical errors only)

19. Two-track resolution correction (based on mixed events)
1. D (TTR effect) = a given fluctuation measure, DXN or SXN (after Geant + reconstruction) minus this fluctuation measure for mixed events. D (TTR effect) is expected to be negative.
2. Fluctuation measure corrected = Fluctuation measure raw – corresponding D (TTR effect)
For 20A and 30A GeV – no correction applied; instead the upper systematic error will be increased by (for all charge combinations)
For 20A and 30A GeV – no correction applied; instead the upper systematic error will be
increased by (for all charge comb.)
Remark: corrections for DXN and SXN are relatively small !! In case of FpT the magnitudes
of TTR corrections and the magnitudes of the final values were of the same order :(

20. p+p, C+C, Si+Si, peripheral Pb+Pb (bin 6), mid-peripheral Pb+Pb (bin 3) and central Pb+Pb (7.2%) data used to prepare these plots
For negatively charged – no correction applied; instead the symmetric systematic error will be increased by (all systems)
For positively charged and all charged – correction applied accordingly to the parametrization of correction vs. mean multiplicity
For all 3 charge combinations – correction applied accordingly to the parametrization of correction vs. mean multiplicity

21. Components of the systematic error for DXN (half of the difference between the highest and the lowest value – within “reasonable” range of the control parameter)
negatively charged
positively charged
ntf – number of tracks used to fit main vertex
nto – number of tracks registered in TPC's
np – number of measured points for a track
nmp – maximal number of points (from geometry of the track)

22. Components of the systematic error for SXN

23. Components of the systematic error for DXN Not measured bins of Pb+Pb:
For centrality bin 5 – mean value between bins 6 and 3
For centrality bin 4 – mean value between bins 6 and 3
For centrality bin 2 – mean value between bins 3 and 1

24. Components of the systematic error for SXN Not measured bins of Pb+Pb:
For centrality bin 5 – mean value between bins 6 and 3
For centrality bin 4 – mean value between bins 6 and 3
For centrality bin 2 – mean value between bins 3 and 1

25. Energy dependence of DXN and SXN measures of pT fluctuations
Lines – UrQMD 1.3
NA49 shows tendency similar to UrQMD predictions
Local minimum (all charged and pos. charged) of DXN for 80A GeV (data and UrQMD)?

26. NA49 published
(FpT)

28. System size dependence of DXN and SXN measures of pT fluctuations
Lines – UrQMD 3.3
NA49 data: maximum for peripheral Pb+Pb
UrQMD (6 cent. of Pb+Pb): no significant system size
dependence, only a small maximum for DXN in Pb+Pb(6)

29. NA49 published
(FpT)

30. Please note: mean multiplicities are sometimes
different for data and for UrQMD 3.3,
examples for all charged:
p+p: 1.41 (data) and 1.21 (UrQMD)
C+C: 10.2 (data) and 8.2 (UrQMD)
Si+Si: 27.4 (data) and 24.2 (UrQMD)
Pb+Pb (6): 40.6 (data) and 11.9 (UrQMD)
Pb+Pb (5): 75.7 (data) and 50.0 (UrQMD)
Pb+Pb (4): (data) and 84.3 (UrQMD)
Pb+Pb (3): (data) and (UrQMD)
Pb+Pb (2): (data) and (UrQMD)
Pb+Pb (1): (data) and (UrQMD)

31. For NA61 only stat. errors shown
Comparisons of NA49 A+A with NA61 p+p data (Tobiasz's results)
in the same (NA49) acceptance
For NA61 only stat. errors shown
Forward rapidity
1.1 < y* < 2.6;
y*p < y*beam – 0.5
Common (for all energies) narrow azimuthal angle
Similar behaviour for Pb+Pb and p+p; difference only for negatively charged particles

32. Common (for all energies) narrow azimuthal angle
Forward rapidity
1.1 < y* < 2.6;
y*p < y*beam – 0.5
Common (for all energies) narrow azimuthal angle
Forward rapidity
1.0 < y* < ybeam
Energy dependent azimuthal angle acceptance → as available in NA49 detector
Interesting result for w (difference between Pb+Pb and p+p)
→ violation of the Wounded Nucleon Model

33. Forward rapidity
1.1 < y* < 2.6
Wide azimuthal angle – nearly as available at 158 GeV/c
Similar behaviour of FpT and SXN (the same “family” of strongly intensive measures)
Increase of FpT  two times larger for all charged than for negatively charged particles (as predicted for CP)

34. Wide azimuthal angle – nearly as available at 158 GeV/c
Forward rapidity
1.1 < y* < 2.6
Wide azimuthal angle – nearly as available at 158 GeV/c
Forward rapidity
1.0 < y* < ybeam
Wide az. angle – as available at 158 GeV/c
For p+p points → acceptance used for pT fluct. (above)
this difference is
under investigations
Increase of w  two times larger for all charged than for negatively charged particles (as predicted for CP)

35. Summary of new pT fluct. measures in NA49 forward-rapidity
Energy dependence for 7.2% central Pb+Pb:
DXN values < 1 (smaller than IPM), no significant energy dependence. The same behaviour for NA61 p+p in the same (NA49) acceptance.
SXN values (for data and UrQMD) close to 1 (all charged, pos. charged) or slightly above 1 (neg. charged). No significant energy dependence. Similar behaviour for NA61 p+p but there SXN for neg. charged also close to 1.
System size dependence at 158A GeV:
Maximum of DXN and SXN for periph. Pb+Pb. For DXN this max. is more like a rapid jump.
Maximum of DXN in peripheral Pb+Pb seems to be seen also in UrQMD but the magnitude is much smaller.
When only central collisions are considered (Tchem dependence) all DXN values < 1 and SXN values for all charged and neg. charged > 1 (1 = IPM).
In SXN versus Tchem dependence for central A+A, the maximum for C+C and Si+Si (all charged) is about 5% higher than the base line defined by Independent Particle Model.
The behaviour of SXN is similar to the behaviour of FpT→ they belong to the same S-class of strongly intensive measures (the same moments used).
Predictions within CMC model [NP A693, 799 (2001); NP A761, 149 (2005)] (mixture of their sigmas
and indep. produced pions) for Si+Si at 158A GeV in NA49 acc.: DXN ~ 0.76 and SXN ~ 1.35 (back-up) → should not be directly comp. with Stephanov et al. (different correlation length).
 All the measured values (DXN and SXN with new normalization) are annoyingly close to 1 (1 = Independent Particle Model).
 Two-track resolution corrections for DXN are small when compared to “raw” values (in contrary to what was seen for FpT).

36. Back-up slides

37. Increase  two times larger for all charged than for negatively charged particles
(as predicted for CP)
←Here only (semi)central A+A shown; see PRC 70, , 2004 (and back-up) for several centralities of Pb+Pb
CP2 location:
B(CP2)  250 MeV = B (A+A at 158A GeV)
T (CP2) = 178 MeV = Tchem (p+p)
CP2 predictions (curves) normalized to reproduce pT and w value for central Pb+Pb collisions
Grebieszkow, Nucl. Phys. A830,
547C-550C (2009)

38. 5 remaining centralities of Pb+Pb also shown
NA49 values of Tchem for p+p, C+C, Si+Si, and Pb+Pb (5%) - PRC 73, (2006), and for remaining centralities of Pb+Pb (lighter colors) – linear interpolation between two points for Si+Si and Pb+Pb (5%)

39. Data systematic errors are large (still not final, will be enlarged)
Energy and system size dependence of 3rd moment of average pT fluctuations
F(3)pT has strongly intensive property like FpT (PL B465, 8 (1999))
NA49 preliminary
Higher moments are expected to be more sensitive to fluctuations but no quantitative predictions for fluctuations at CP yet
Data systematic errors are large (still not final, will be enlarged)
Melkumov, …, Grebieszkow,
et al. (NA49), Phys. Atom.
Nucl. 75, 556 (2012)
forward-rapidity region,
limited azimuthal acceptance (for details see PRC 70, (2004) and PRC 79, (2009))

40. Forward rapidity
1.1 < y* < 2.6
Wide azimuthal angle – nearly as available at 158 GeV/c

41. Forward rapidity
1.1 < y* < 2.6
Wide azimuthal angle – nearly as available at 158 GeV/c
Forward rapidity
1.0 < y* < ybeam
Wide az. angle – as available at 158 GeV/c
For p+p points → acceptance used for pT fluct. (above)

42. Wide az. angle – as available at 158 GeV/c
Lower panel only:
C+C, Si+Si, Pb+Pb
Forward rapidity
1.0 < y* < ybeam
Wide az. angle – as available at 158 GeV/c
p+p
Forward rapidity
1.1 < y* < 2.6
Wide azimuthal angle – nearly as available at 158 GeV/c

43. Energy dependent azimuthal angle acceptance → as available in NA49 detector
Lower panel:
1.0 < y* < ybeam

44. Motivation: to predict magnitude of fluctuations (pT, azimuthal angle, higher moments of pT fluct., etc. ) at the critical point using CMC model [N. G. Antoniou et al., Nucl. Phys. A693, 799 (2001); N. G. Antoniou et al., Nucl. Phys. A761, 149 (2005)] - model with sigmas decaying into two pions. Default parameters used (fixed by Nikos to fit Si+Si at 158A GeV)
For each event approx. 18 sigmas are generated – the number for default parameters.
Nikos's slide from NA49/NA61 Analysis Meeting, February 2011, WUT

45. Obtained for 10k events
Here:
angle = atan2(py,px)* (180/3.14)
Later, in the analysis of FpT and Ff it will be defined as atan2(py,px) with a range -p, +p

46. Only sigmas (approx. 18 sigmas in one event):
pT (all charged) = MeV/c <Nall> =
pT (neg. charged) = MeV/c <Nneg> =
pT (pos. charged) = MeV/c <Npos> =
For a comparison: our Si+Si mean multiplicities (PR C70, (2004)) at forward-rapidity, azimuthal angle slightly restricted (acceptance for 158A GeV): 27 (all charged), 12 (neg. charged), and 15 (pos. charged).
Adding independently produced particles to generated sigmas:
Fotios's suggestion of mixture: 1 sigma per 4 independently produced pions
18 sigmas/event => 72 indep. prod. pions/event (as a first step 50% positive and 50% negative)
N_negative= Poisson(36)
N_positive=Poisson(36)
In such simulation 1/3 of produced particles come from sigma decays!
mT of particles generated from exponential shape (range ,1.5065) with slope T = mean_pt_inclusive/2.0 mean_pt_inclusive taken from PRC 70, (2004) (values at forward-rapidity): GeV/c (for negative) and GeV/c (for positive)
rapidity generated from Gauss with <y*>=0 and sy = 1.7 (range of generation: -4 to +4)
azimuthal angle – flat distribution (range of generation: to 3.14)

47. Full phase space:
pT (all charged) = MeV/c <Nall> =
pT (neg. charged) = MeV/c <Nneg> =
pT (pos. charged) = MeV/c <Npos> =
Forward-rapidity (1.1 < y* < 2.6) – for both sigmas and indep. prod. pions
pT (all charged) = MeV/c <Nall> =
pT (neg. charged) = MeV/c <Nneg> =
pT (pos. charged) = MeV/c <Npos> =
Forward-rapidity (1.1 < y* < 2.6, precisely 4.0 < yLAB < 5.5) and azimuthal angle restricted (as in NA49 system size dep.) – for both sigmas and indep. prod. pions
pT (all charged) = MeV/c <Nall> =
pT (neg. charged) = MeV/c <Nneg> =
pT (pos. charged) = MeV/c <Npos> =

48. Predictions used by us so far:
Magnitude of fluctuations at CP from Stephanov, Rajagopal, Shuryak PR D60, (1999)
pT (additive correction to uncorrelated particle production):
Remark: uncorrelated particle production results in pT = 0
Assuming correlation length  = 6 fm
4p acceptance: 40 MeV/c (all charged)
20 MeV/c (one charge only)
acceptance correction in rapidity: 0.6 (forward-rapidity)
acceptance correction in azimuthal angle: close to 1.0 for system size dependence
(0.4 for so-called common acceptance in energy scan)
Finally (sys. size dep.): 40*0.6*1.0 = 24 MeV/c (all charged)
12 MeV/c (like-sign particles)
However, the correlation length may not exceed 3 fm...
Assuming correlation length  = 3 fm
4p acceptance: 10 MeV/c (all charged)
5 MeV/c (one charge only)
acceptance correction in rapidity: 0.6
acceptance correction in azimuthal angle: 1.0
Finally (sys. size dep.): 6 MeV/c (all charged)
3 MeV/c (like-sign particles)

49. min (limit due to finite system size, limit due to finite life time)
Predictions used by us so far:
Magnitude of fluctuations at CP from Stephanov, Rajagopal, Shuryak PR D60, (1999)
with correlation length  = min ( c1 A1/3, c2 A1/9 ) =
min (limit due to finite system size, limit due to finite life time)
(M. Stephanov, private communication)
where c1 and c2 are fixed such that
(Pb+Pb) = 6 fm and (p+p) = 2 fm (c1 = 2, c2 = 3.32)
(Pb+Pb) = 3 fm and (p+p) = 1 fm (c1 = 1, c2 = 1.66)
Width of CP region in (T, mB) plane based on Hatta, Ikeda PR D67, (2003)
s(mB)  30 MeV and s(T)  10 MeV
Chemical freeze-out parameters, T(A,sNN) and mB(A,sNN) from Beccatini, Manninen, Gaździcki PRC 73, (2006)
Location of the Critical Point:
two examples considered
mB(CP1) = 360 MeV (Fodor, Katz JHEP 0404, 050 (2004))
T(CP1)  147 (chemical freeze-out temperature Tchem
for central Pb+Pb at mB = 360 MeV)
mB(CP2)  250 MeV (mB for A+A collisions at 158A GeV)
T(CP2) = 178 MeV (Tchem for p+p collisions at 158 GeV)

50. Correlation length  = min ( c1 A1/3, c2 A1/9 )
= min (limit due to finite system size, limit due to finite life time)
Points for Pb+Pb (5%), Si+Si,
C+C and p+p at 158A GeV
(T(A)) for c1 = 2, c2 = 3.32
((T(A))/6 fm)^2
Assumed correlation length  for Pb+Pb = 6 fm and for smaller systems  decreased
Parameters c1 and c2 were fixed this way to obtain  for Pb+Pb = 6 fm and  for p+p = 2 fm. Precisely: I was starting with fixing c2 as equal 6fm/(208)^1/9 = 3.32 and then I tried to find a value of c1 to get  for Pb+Pb = 6 fm and  for p+p = 2 fm.
y= e-11*pow(x,5.0)
(T(A)) [fm]
1. Gaussian curve with maximum predicted by M. Stephanov and position at Tchem(p+p)
2. multiplied by ((T(A))/6fm)^2
3. Finally curve normalized for central Pb+Pb (it is curve shifted to cross Pb+Pb point)

51. Predictions used by us so far (final plot shown on QM 2009):
CP2 location:
B(CP2)  250 MeV = B (A+A at 158A GeV)
T (CP2) = 178 MeV = Tchem (p+p)
CP2 predictions (curves) normalized to reproduce pT for central Pb+Pb collisions
Increase  two times larger for all charged than for negatively charged particles
(as predicted for CP)

52. Full phase space:
DXN (all charged) = <Nall> =
DXN (neg. charged) = <Nneg> =
DXN (pos. charged) = <Npos> =
Forward-rapidity (1.1 < y* < 2.6) – for both sigmas and indep. prod. pions
DXN (all charged) = <Nall> =
DXN (neg. charged) = <Nneg> =
DXN (pos. charged) = <Npos> =
Forward-rapidity (1.1 < y* < 2.6, precisely 4.0 < yLAB < 5.5) and azimuthal angle restricted (as in NA49 system size dep.) – for both sigmas and indep. prod. pions
DXN (all charged) = <Nall> =
DXN (neg. charged) = <Nneg> =
DXN (pos. charged) = <Npos> =

53. Full phase space:
SXN (all charged) = <Nall> =
SXN (neg. charged) = <Nneg> =
SXN (pos. charged) = <Npos> =
Forward-rapidity (1.1 < y* < 2.6) – for both sigmas and indep. prod. pions
SXN (all charged) = <Nall> =
SXN (neg. charged) = <Nneg> =
SXN (pos. charged) = <Npos> =
Forward-rapidity (1.1 < y* < 2.6, precisely 4.0 < yLAB < 5.5) and azimuthal angle restricted (as in NA49 system size dep.) – for both sigmas and indep. prod. pions
SXN (all charged) = <Nall> =
SXN (neg. charged) = <Nneg> =
SXN (pos. charged) = <Npos> =

54. Multiplicity fluctuations in CMC model
Full phase space:
w (all charged) = <Nall> =
w (neg. charged) = <Nneg> =
w (pos. charged) = <Npos> =
Forward-rapidity (1.1 < y* < 2.6) – for both sigmas and indep. prod. pions
w (all charged) = <Nall> =
w (neg. charged) = <Nneg> =
w (pos. charged) = <Npos> =
Forward-rapidity (1.1 < y* < 2.6, precisely 4.0 < yLAB < 5.5) and azimuthal angle restricted (as in NA49 system size dep.) – for both sigmas and indep. prod. pions
w (all charged) = <Nall> =
w (neg. charged) = <Nneg> =
w (pos. charged) = <Npos> =

55. 3rd moment of pT fluctuations:
Only sigmas (approx. 18 sigmas in one event):
100k events generated
3pT (all charged) = MeV/c (<3pT>sub. = 626.1) <Nall> =
3pT (neg. charged) = MeV/c (<3pT>sub. = 370.9) <Nneg> =
3pT (pos. charged) = MeV/c (<3pT>sub. = 374.7) <Npos> =
Adding independently produced particles to generated sigmas:
mixture: 1 sigma per 4 independently produced pions
18 sigmas/event => 72 indep. prod. pions/event (as a first step 50% positive and 50% negative)
N_negative= Poisson(36)
N_positive=Poisson(36)
In such simulation 1/3 of produced particles come from sigma decays!
mT, rapidity, azimuthal angle – the same distributions as before

56. Full phase space: (predicted 3pT indeed higher than pT)
3pT (all charged) = MeV/c (<3pT>sub. = ) <Nall> =
3pT (neg. charged) = MeV/c (<3pT>sub. = ) <Nneg> =
3pT (pos. charged) = MeV/c (<3pT>sub. = ) <Npos> =
Forward-rapidity (1.1 < y* < 2.6) – for both sigmas and indep. prod. pions
3pT (all charged) = MeV/c (<3pT>sub. = -4.8) <Nall> =
3pT (neg. charged) = MeV/c (<3pT>sub. = 10.8) <Nneg> =
3pT (pos. charged) = MeV/c (<3pT>sub. = -1.2) <Npos> =
… but this higher sensitivity to CP of 3rd moment washed out by our acceptance
Forward-rapidity (1.1 < y* < 2.6, precisely 4.0 < yLAB < 5.5) and azimuthal angle restricted (as in NA49 system size dep.) – for both sigmas and indep. prod. pions
3pT (all charged) = MeV/c (<3pT>sub. = -9.0 ) <Nall> =
3pT (neg. charged) = MeV/c (<3pT>sub. = 7.0) <Nneg> =
3pT (pos. charged) = MeV/c (<3pT>sub. = 0.7) <Npos> =

57. Note: generally 3pT values are more unstable than FpT.
The value from the whole sample sometimes is very far from the mean value from 30 subsamples.
Additional test (perfect detector above mid-rapidity)
Forward-rapidity (y* > 0.0) – for both sigmas and indep. prod. pions
3rd moment
3pT (all charged) = MeV/c (<3pT>sub. = 105.0) <Nall> = 53.92
3pT (neg. charged) = MeV/c (<3pT>sub. = 50.2) <Nneg> = 26.96
3pT (pos. charged) = MeV/c (<3pT>sub. = 38.2) <Npos> = 26.97
2nd moment
pT (all charged) = MeV/c (<pT>sub. = 49.9) <Nall> = 53.92
pT (neg. charged) = MeV/c (<pT>sub. = 23.9) <Nneg> = 26.96
pT (pos. charged) = MeV/c (<pT>sub. = 21.5) <Npos> = 26.97

58. NA49 results for low and high pT regions:
Expectations: fluctuations due to the critical point originate mainly from low pT pions Stephanov, Rajagopal, Shuryak, PR D60, (1999)
NA49 results for low and high pT regions:
low pT region
pT < 0.5 GeV/c
signal observed
(similar for pT < 0.25 GeV/c)
NA49 preliminary
high pT region
pT > 0.5 GeV/c
fluctuations consistent with zero
Grebieszkow, PoS
EPS-HEP 2009, 030
 Correlations observed predominantly at low pT
 No more maximum of pT due to large correlations in Pb+Pb; their origin will be analyzed (short range correlations considered)

59. Energy dependence (7.2% most central Pb+Pb) of azimuthal fluctuations
Grebieszkow, APPB 43, 609 (2012)
NA49 preliminary
NA49 preliminary
Ff (negative) > 0; different than in UrQMD (1.3) (f in UrQMD rotated by FR)
Ff (positive) consistent with zero
This and next page: results for forward-rapidity region;
limited azimuthal acceptance (for details see PRC 70, (2004) and PRC 79, (2009))

60. System size dependence (at 158A GeV) of azimuthal fluctuations
Grebieszkow, APPB 43, 609 (2012)
NA49 preliminary
Ff > 0, maximum for peripheral Pb+Pb
UrQMD (3.3) does not reproduce data (f in UrQMD rotated by FR)
The magnitude of Ff reproduced by the effect of v1 and v2; the difference between positively and negatively charged particles also reproduced
(see Grebieszkow, Mrówczyński APPB, Proc. Suppl. 5, 727 (2012) for details)
In MC v1 and v2 (for pions and protons at forward-rapidity) taken from PRC 68, (2003); <N-> and <N+> from this analysis; effect of limited NA49 acceptance taken into account; percentage of protons in the studied kinematic rage estimated from UrQMD