Higher-Order
Moments of ProtonNumber Fluctuations
in Au+Au Collisions at1.23A GeV with
HADES
Melanie Szala

Melanie Szala

1

Outline
Motivation
HADES spectrometer
Higher order moments of proton number fluctuations- Efficiency corrections- Volume fluctuations- Results
Conclusions

Melanie Szala

2

Motivation•

Cross-over at low µB and1st-order phase transition at high µB
à Critical point•

Event-by-event fluctuations of conserved quantities
are expected to be a signature of discontinuity in
the QCD phase diagram- Charge Q / baryon number B / strangeness S•

Experimental observable:- Cumulants of event-by-event net-particle
multiplicity distributions- (Net-)proton (proxy for net-baryon)

Melanie Szala

3

Experimental Observables
Cumulants of event-by-event net-particle multiplicity distributions- (Net-)proton (proxy for net-baryon)

MeanVariance
Skewness
Kurtosis

0-10%, 10-20%, 20-30%, 30-40%
Cumulant ratios to cancel volume effects:
à Cumulant rations cancel only mean volume effects!
Melanie Szala

S ⋅σ =K3
K2

κ ⋅σ 2 =K4
K2
4

HADES Spectrometer
High Acceptance DiElectron Spectrometer
Located at SIS18, GSI
Large acceptance• Full azimuthal and polar angle
coverage of Θ = 18 – 85°
Fast detector• Trigger rate up to 8kHz
(large statistics)+ plastic scint. FW
Θ = 0.5 - 7.5°

Au + Au @1.23 AGeV, √sNN=2.41 GeV•
15 fold segmented Au target•
7.4 x 109 events recorded•
Trigger on 47% most central collisions

Melanie Szala

50.90.8

Exp. Daten Protonen

π-π+

106

K+-K

pt

1105

3He

0.7×103

2000

1000
1800
1600

800

1400

0.6d/4He0.5t

104

1200

600

1000

103
800

0.4
2100.3400600
4000.2100.10−1000

p [MeV/c]βProton Analysis−50005001000

1500

2000

2002001mom / Z [MeV/c]

000.2

0.4

0.6

0.8

11.21.4ycm = 0.74

1.6

1.820yEvent selection• 0-10%, 10-20%, 20-30%, 30-40% central events
selected with the Forward Wall
Track selection• y = ycm ± 0.2• pt = 400-1600 MeV/c
If Np= 0 à Poisson distribution: S ⋅ σ = κ ⋅ σ 2 = 1
Melanie Szala

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Efficiency correction
Efficiency = acc x det. eff x rec. eff
Investigations of efficiency correction in UrQMD simulations:à Correct the moments
Bzdak & Koch, PRC 86 (2012);
Xiaofeng Luo, arXiv:1410.3914 (2014)
à Unfolding
G. D‘Agostino, Nucl. Instr. Meth. A 362 (1995) 487.
J. Albert et al. (MAGIC), Nucl. Instr. Meth. A 583 (2007) 494.
S. Schmitt, J. Instr. 7 (2012) T10003.

Melanie Szala

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Correct the moments

Reconstructed proton
distribution

Calculatemoments

Meanmeasured
Variancemeasured

Correctmoments

Skewnessmeasured

Meantrue
Variancetrue
Skewnesstrue
Kurtosistrue

Kurtosismeasured
efficiencycorrection

Skew*Sigmeasured

Skew*Sigtrue
Kurt*Sig2true

Kurt*Sig2measured

Self-consistency check• UrQMD protons à reference• UrQMD + Geant reconstructed protons à efficiency correction
Bzdak & Koch, PRC 86 (2012);
Xiaofeng Luo, arXiv:1410.3914 (2014)
Melanie Szala

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Correct the moments

Melanie Szala

UrQMD

9

Efficiency correction

Melanie Szala

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Efficiency correction
Efficiency: strong effect on
centrality
Efficiency depends on number oftracks per event

à Efficiency correction• Event-by-event• Particle number in sector• y and pt dependent

efficiency

Detector divided into 6 sectors• Efficiency is sector dependent

30-40 %
0-10 %

Ntracks/sector
Melanie Szala

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UrQMD
Correct moments – event-by-event -Melanie Szala

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UrQMD
Correct moments – event-by-event -Melanie Szala

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Unfolding

measured distribution

true distribution
UnfoldingMatrix inversion

Folding
UrQMD + GEANT(recon. protons)

Melanie Szala

UrQMD protons

14

Unfolding procedure
Unfolding via matrix inversion

y=Ax
à finding x !

y:x:
A:

measured histogram
true histogram
response matrix

Unfortunately, A is often quasi-singular and can not be inverted (ill-conditioned problem!)
ROOT package by S. Schmitt à TUnfold, TUnfoldSys, TUnfoldDensity
Minimize in a least-squares procedure the „Lagrangian“:
L1: least square minimization
L2: describes regularisation
L3: area constraint
Melanie Szala

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Response function
UrQMD + HGEANT reconstr. protons

detector response function50

pt = 400-1600 MeV/c|y|± 0.240106105

1043010320102
10100

Melanie Szala

01020304050607080
UrQMD protons

116Regularization
τ is a free parameter controlling the strength of regularization
parameter is typically difficult to choose using only a priori information• But its value usually has a major impact on the unfolded spectrum
several methods to find τin TUnfold implemented methods• L-curve scan
• Minimising global correlation coefficients

τ= 0Melanie Szala

redblue= original distribution= unfolded distribution

τ= 117

Methods to find best regularization
Minimising globalcorrelation coefficientsρ

L-curve Scan

kink = optimum
optimum

log10(τ)à τ ≈ 10-4 - 10-5
Melanie Szala

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Unfolded distributions
red = original distribution
blue = unfolded distribution

Melanie Szala

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Comparing methods– Correct moments vs. Unfolding -Melanie Szala

UrQMD

20

RealData

Mean

20

Correct moments

6Unfolding

155105Sigma

7σMResults (eff. Corr.)

430-10 %

10-20 %

20-30 %

30-40 %

κSSkewness

0-10 %

0.5

10-20 %

20-30 %

30-40 %

Kurtosis

0.6
0.4
0.2

0010-20 %20-30 %

30-40 %

Sσ

0-10 %

κ σ²

Sσ0-10 %

2

10-20 %

20-30 %30-40 %κ σ²51000-10 %Melanie Szala

10-20 %

20-30 %

30-40 %

0-10 %

10-20 %

20-30 %

30-40 %
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Poissonizer

Melanie Szala∆ytotal≫∆yaccept≫∆ycorr

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Experimental Observables
Cumulants of event-by-event net-particle multiplicity distributions- (Net-)proton (proxy for net-baryon)
Mean
Variance
Skewness
Kurtosis

Cumulant ratios to cancel volume effects

S ⋅σ =K3
K2

κ ⋅σ 2 =K4
K2

à Cumulant ratios cancel only mean volume effects!

Melanie Szala

23

Volume Fluctuations
V. Skokov, B. Friman, and K. Redlich, Physical Review C 88, 034911 (2013)

à The effect of volume fluctuations on cumulants of the net baryon number

kn baryon number cumulants
cn volume affected cumulants
vn volume fluctuations cumulants

à Npart ~ volume
à Modeled (Glauber and iQMD) Npart distribution tocalculate the volume fluctuation cumulants

Melanie Szala

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Npart from IQMD
Tracks in MDC

Forward Wall

• For each IQMD event b is translated into Npart using Gosset et al. PRC 16• FW sum of charge has worse resolution
Melanie Szala

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Results with volume corrections (IQMD FW)

Au+Au data

Au+Au data

y0 ± 0.2
pt = 400- 1600 MeV/c

Melanie Szala

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Poisson limit
Poisson limit: S ⋅ σ →1 andκ ⋅ σ 2 →1 for Δy → 0

Efficiency corrected protons with volume corrections:Sk ⋅ σκ ⋅ σ2

100400 < p < 1600 MeV

2T

0-10 % central

8030-40 % central060

κ ⋅ σ2Sk ⋅ σ−240−4400 < p < 1600 MeV−6T0-10 % central

20

30-40 % central−80y ± 0.5
0y ± 0.40y ± 0.30y ± 0.20y ± 0.1
0y ± 0.50y ± 0.40

y ± 0.30y ± 0.2
0y ± 0.10! Strong dependence on Δy for most central collisions!
Melanie Szala27

pt and y dependance of κσ2 =K4/K2
Bzdak, Koch arXiv:1707.02640

s

Melanie Szala

NN= 7.7 GeV, 0-5 % centrals

NN= 2.41 GeV, 0-10 % central

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Cumulants & multi-particle correlators
Bzdak, Koch & Strodthoff, PRC 95, 054906 (2017) based on STAR data (X. Luo et al., CPOD2014)•Cumulants mix correlations of different order,e.g. K4 contains of two-, three- and fourparticle correlations•Measuring couplings of the multi-particle correlation functions could provide cleaner
information on possible non-trivial dynamics in heavy-ion collisions
à extract true correlation functions from the measured cumulants••

Melanie Szala

andReduced correlation functions

29

Npart dependance of proton correlations• Ck vs. Npart as a better approach to isolate critical fluctuations:• Factors to reflect their relative contribution to the fourth order cumulant

κ 4 = N + 7C2 + 6C3 + C4

y0 ± 0.2
pt = 400- 1600 MeV/c

𝜅𝑘:𝐶k :
Melanie Szala

cumulants
multi-particle correlators
30

Proton correlations Cn/Np vs. Beam Energy• At energies larger that 19.6 GeV anti-protons become non-negligible, and thus thephysical interpretation is less clear.
• K4/K2 is dominated by C4 at low energies and by C2 at 19.6 GeV .
à should not forget that there are sources of correlations other that critical dynamics. need
to be removed and understood

Melanie Szala

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What we did...
1) Systematic checks in simulations (UrQMD + Geant)• Centrality selection• Efficiency correction- (Pt-y dependend eff correction)- Pt-y and track multiplicity dependend eff correction- Unfolding2) Au+Au data
• Centrality with EvtCharaClass: FWSumCharge• Two different eff correction methods➔ Higher order moments of proton distribution➔ Higher order moments of proton and deuteron distributions
3) Volume fluctuations• Glauber model• IQMD
Melanie Szala32

What to do...
1) Systematic checks in simulations (IQMD + Geant)• Centrality selection- FW- Resolution vs autocorrelation (Hits/Tracks/FW)• Include fragments (efficiency, ...)

2) Possible contamination of moments• Resolution centrality estimation• Au+C reactions• Pile up effects• ...

Melanie Szala

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Conclusions
Study of centrality selection and efficiency correction in simulation
Introduction of two different eff. correction methods• Within error bars, both methods deliver compatible results

Choice of phase-space bin ? (pt, y)βFirst look to volume fluctuation correction• Volume correction influences higher order moments
10.9

Fluctuations on baryon number• Bound protons?!
Possible contamination of moments

0.8π-π+106+K-K

p105

3He0.70.6

d/4He0.5

t1041030.4102

0.30.2100.1
0−1000−5000500

1000

1500

2000

1

mom / Z [MeV/c]
Melanie Szala

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The HADES Collaboration

Melanie Szala

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Backup

Melanie Szala

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Comparison with STAR - (Net-) Proton
STAR analysis: Xiaofeng Luo et al., PoS (CPOD2014) 019

HADES:
pt = 400 – 1600 MeV/cy = y0 +- 0.2
Forward Wall
Volume Corrected

arXiv:1503.02558v2

HADES
preliminary

HADES
preliminarySTAR:
pt = 400 – 2000 MeV/cy = y0 +- 0.5
NOT Volume Corrected

red/black = unfolding (preferred method) + vol. flucs. corr.
green = evt-by-evt eff correction of factorial moments + vol. flucs. corr.
Melanie Szala

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Theoretical Expectation
Misha Stephanov, CPOD 2014S ⋅σ

κ ⋅σ 2κ ⋅σ 2S ⋅σ

STAR

STAR

HADES?

Melanie Szala

HADES?

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Correct the moments

Bzdak & Koch, PRC 86 (2012);
Xiaofeng Luo, arXiv:1410.3914 (2014)
Melanie Szala

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Volume Corrections
M

Evt-by-Evt Eff. ∈(N)
Evt-by-Evt Eff. ∈(N) - Volume corr.
Unfolding
Unfolding - Volume corr.

15Sigma

7σMean2065410350-10 %

10-20 %

20-30 %

0-10 %

10-20 %

20-30 %

30-40 %

Kurtosisκ

Skewness

S230-40 %

0.5

0.5

00−0.510-20 %

20-30 %

30-40 %

Sσ

0-10 %

κ σ²

Sσ0-10 %

2

10-20 %

20-30 %

30-40 %

κσ²50

0−50-10 %Melanie Szala

10-20 %

20-30 %

30-40 %

0-10 %

10-20 %

20-30 %

30-40 %
40

Centrality determination• Number of tracks in MDCs
• Number of hits in time-of-flight detectors(TOF+RPC)• Sum of charge in Forward Wall(spectators)

Forward Wall•••

Melanie Szala

Plastic Scintilator
Θ = 0.5 - 7.5°
3 different sizes

41Volume Correction: Glauber vs IQMD
E-by-e eff. corr.
+ Volume Glauber SelectTracks+ Volume iQMD SelectTracks+ Volume iQMD FW

E-by-e eff. corr.

8+ Volume Glauber SelectTracks+ Volume iQMD SelectTracks

26+ Volume iQMD FW

1κ ⋅ σ2Sk ⋅ σ4200−2−1y0 ± 0.2
pt = 400- 1600 MeV/c50

Melanie Szala

100

150200−4250300<Npart>−650100

150200250300<Npart>42

Poisson Limit: Δyà0
HADES
preliminaryce30-40%

HADES
preliminary

ntralityc30-40%Melanie Szala

entrality43

Proton n-particle correlations: Cn vs Npart
à Going from cumulants 𝜅𝑛 to correlations C𝑛:y0 ± 0.2
pt = 400- 1600 MeV/cà C4 is clearly the dominant contribution to the fourth order cumulant
Melanie Szala

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Proton correlations Cn/Np vs. Npart

y0 ± 0.2
pt = 400- 1600 MeV/c

Melanie Szala

y0 ± 0.2
pt = 400- 1600 MeV/c

y0 ± 0.2
pt = 400- 1600 MeV/c45