[PPT] - Extraction of moments of net-particle event-by-event fluctuations in PowerPoint Presentation

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Extraction of moments of net-particle event-by-event fluctuations in the CBM experiment

V. Vovchenko, I. Kisel

for the CBM collaboration DPG Spring Meeting Darmstadt, Germany 15 March 2016

HGS-HIRe

Helmholtz Graduate School for Hadron and Ion Research

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Outline

Higher-order fluctuations on phase diagram
Rate of statistical convergence of different moments
Efficiency corrections
GEANT simulation and reconstruction of fluctuations
Summary

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SLIDE 3

Introduction

A future fixed target experiment at FAIR facility. Up to 107 Au+Au collisions per second at 4-11A GeV (SIS100) and 11-35A GeV (SIS300). Measurement of bulk and rare probes. Physics programme Equation of state at high baryonic densities Phase transitions at high µB QCD critical point, probed by e-by-e fluctuations Subthreshold production of hadrons Hypernuclei production

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Higher-order moments of fluctuations

Let N be a random variable and P(N) its probability distribution. k-th moment: Nk =

N

Nk P(N) Variance: σ2 = (∆N)2 = (N − N)2 Scaled variance: σ2 M = κ2 κ1 = σ2 N width Skewness: Sσ = κ3 κ2 = (∆N)3 σ2 asymmetry Kurtosis: κσ2 = κ4 κ2 = (∆N)4 − 3 (∆N)22 σ2 peakedness and so on... In heavy-ion collisions N can be conserved charge (baryon, electric, strangeness) or some particle number in a specific phase-space region

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SLIDE 5

Fluctuations in thermodynamics

Why are fluctuations interesting? In thermodynamics fluctuations are related to susceptibilities χ(n) χ(n) = ∂n(p/T 4) ∂(µ/T)n σ2 M = χ(2) χ(1) , Sσ = χ(3) χ(2) , κσ2 = χ(4) χ(2) , Fluctuations are very sensitive to QCD equation of state and can be used to study QCD phase transitions

1 2 3 1 2 3 10 2 1 0.5

mixed phase liquid T/Tc n/nc gas

0.1

σ

2/M

0.01 0.10 1.00 10.00

Near CP ∼ increasing powers of ξ χ(2) ∼ ξ2 χ(3) ∼ ξ4.5 χ(4) ∼ ξ7 Infinite system: ξ → ∞ at CP In HIC ξ 2 − 3 fm

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Fluctuations in T − µ plane: VDW nuclear matter

Nuclear matter as van der Waals system of nucleons

880 890 900 910 920 930 5 10 15 20 25 30 35 liquid T (MeV) (MeV) gas n (fm

3

) 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 880 890 900 910 920 930 5 10 15 20 25 30 35 10 2 1 0.5 liquid T (MeV) (MeV) gas 0.1 0.01 0.10 1.00 10.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

10 1

1

liquid T (MeV) (MeV) gas

10

S

40.00
10.00
1.00

0.00 1.00 10.00 40.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

100 100 10 1 10 1

1

liquid T (MeV) (MeV) gas

10
40.00
10.00
1.00

0.00 1.00 10.00 40.00

Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015)

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Fluctuations in T − µ plane: VDW nuclear matter

Nuclear matter as van der Waals system of nucleons

880 890 900 910 920 930 5 10 15 20 25 30 35 liquid T (MeV) (MeV) gas n (fm

3

) 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 880 890 900 910 920 930 5 10 15 20 25 30 35 10 2 1 0.5 liquid T (MeV) (MeV) gas 0.1 0.01 0.10 1.00 10.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

10 1

1

liquid T (MeV) (MeV) gas

10

S

40.00
10.00
1.00

0.00 1.00 10.00 40.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

100 100 10 1 10 1

1

liquid T (MeV) (MeV) gas

10
40.00
10.00
1.00

0.00 1.00 10.00 40.00

Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015)

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SLIDE 8

Fluctuations in T − µ plane: VDW nuclear matter

Nuclear matter as van der Waals system of nucleons

880 890 900 910 920 930 5 10 15 20 25 30 35 liquid T (MeV) (MeV) gas n (fm

3

) 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 880 890 900 910 920 930 5 10 15 20 25 30 35 10 2 1 0.5 liquid T (MeV) (MeV) gas 0.1 0.01 0.10 1.00 10.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

10 1

1

liquid T (MeV) (MeV) gas

10

S

40.00
10.00
1.00

0.00 1.00 10.00 40.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

100 100 10 1 10 1

1

liquid T (MeV) (MeV) gas

10
40.00
10.00
1.00

0.00 1.00 10.00 40.00

Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015)

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SLIDE 9

Fluctuations in T − µ plane: VDW nuclear matter

Nuclear matter as van der Waals system of nucleons

880 890 900 910 920 930 5 10 15 20 25 30 35 liquid T (MeV) (MeV) gas n (fm

3

) 0.00 0.02 0.04 0.07 0.09 0.11 0.13 0.16 0.18 880 890 900 910 920 930 5 10 15 20 25 30 35 10 2 1 0.5 liquid T (MeV) (MeV) gas 0.1 0.01 0.10 1.00 10.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

10 1

1

liquid T (MeV) (MeV) gas

10

S

40.00
10.00
1.00

0.00 1.00 10.00 40.00 880 890 900 910 920 930 5 10 15 20 25 30 35

1

100 100 10 1 10 1

1

liquid T (MeV) (MeV) gas

10
40.00
10.00
1.00

0.00 1.00 10.00 40.00

Vovchenko et al., PRC 91, 064314 (2015) and PRC 92, 054901 (2015)

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SLIDE 10

Beam energy dependence

Can be measured in different acceptance windows at different energies For small window fluctuations approach ideal gas For large window global charge conservation plays role Measurements should be performed in different windows Peculiarities in energy dependence may signal criticality

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SLIDE 11

Needed statistics to measure higher moments

How much statistics are needed for accurate estimation of higher moments? For a large sample of Gaussian distributed variables ∆(Sσ) =

6σ2

n , ∆(κσ2) =

24σ4

n , ∆(κ6/κ2) =

720σ8

n . More rigorously: Delta theorem, X. Luo, JPG 39, 025008 (2012) Simulation result

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1%

Dash-dotted: σ

2 = 40

Solid: σ

2 = 15

∆(Sσ) ∆(κσ

2)

∆(κ6/κ2)

Events

Poisson

Dashed: σ

2 = 5

10%

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SLIDE 12

Monte Carlo simulation: Poisson statistics

1012 Poisson-distributed numbers with ¯ N = 5 Expected values: σ2/M = Sσ = κσ2 = κ6/κ2 = 1

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σ

2/M

Events

σ

2 = 5

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SLIDE 13

Monte Carlo simulation: Poisson statistics

1012 Poisson-distributed numbers with ¯ N = 5 Expected values: σ2/M = Sσ = κσ2 = κ6/κ2 = 1

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σ

2 = 5

σ

2/M

Sσ

Events

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SLIDE 14

Monte Carlo simulation: Poisson statistics

1012 Poisson-distributed numbers with ¯ N = 5 Expected values: σ2/M = Sσ = κσ2 = κ6/κ2 = 1

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σ

2 = 5

σ

2/M

Sσ κσ

2

Events

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SLIDE 15

Monte Carlo simulation: Poisson statistics

1012 Poisson-distributed numbers with ¯ N = 5 Expected values: σ2/M = Sσ = κσ2 = κ6/κ2 = 1

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σ

2 = 5

σ

2/M

Sσ κσ

2

κ6/κ2

Events Higher fluctuation moments require higher statistics.

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Monte Carlo simulation: Poisson statistics

1012 Poisson-distributed numbers with ¯ N = 15 Expected values: σ2/M = Sσ = κσ2 = κ6/κ2 = 1

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σ

2 = 15

σ

2/M

Sσ κσ

2

κ6/κ2

Events Statistical error grows with ¯ N. Convergence rate will depend on kinematic window.

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SLIDE 17

Monte Carlo simulation: Poisson statistics

1012 Poisson-distributed numbers with ¯ N = 40 Expected values: σ2/M = Sσ = κσ2 = κ6/κ2 = 1

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σ

2 = 40

σ

2/M

Sσ κσ

2

κ6/κ2

Events Statistical error grows with ¯ N. Convergence rate will depend on kinematic window.

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SLIDE 18

Efficiency corrections

Since not all particles are reconstructed and identified, the efficiency corrections are needed The simplest one is the binomial correction Binomial correction assumptions Detection of all particles is independent of each other Probability to register particle is binomial Only a single efficiency parameter ε is needed Original cumulants Ki reconstructed from measured ki K1 = k1 ε K2 = k2 + (ε − 1) k1 ε2 K3 = k3 + 3(ε − 1)k2 + (ε − 1)(ε − 2) k1 ε3 · · · For net-particle numbers (N = N+ − N−) more complicated correction involving factorial moments exists.

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SLIDE 19

Monte Carlo simulation: Binomial efficiency

Test of efficiency correction on non-trivial (non-Poisson) fluctuations Testing procedure

1 Take e-by-e proton yields from 5 million PHSD Au+Au events 2 Simulate detector response by performing Bernoulli trials on each

proton in each event with given efficiency ε

3 Compare efficiency corrected cumulants with original ones 0.85 0.90 0.95 1.00 PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

σ

2/M

5 million PHSD central Au+Au events at 10A GeV

σ

2/M

eff = 80%

0.85 0.90 0.95 1.00 PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

σ

2/M

5 million PHSD central Au+Au events at 10A GeV

σ

2/M

eff = 50%

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SLIDE 20

Monte Carlo simulation: Binomial efficiency

Test of efficiency correction on non-trivial (non-Poisson) fluctuations Skewness

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

Sσ

5 million PHSD central Au+Au events at 10A GeV

Sσ

eff = 50%

Kurtosis

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

κσ

2

5 million PHSD central Au+Au events at 10A GeV

κσ

2

eff = 50%

In ideal case correction works properly for non-trivial initial fluctuations

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SLIDE 21

Monte Carlo simulation: Fluctuating efficiency

In more realistic scenario efficiency is changes from event to event, e.g., due to fluctuations in number of tracks, momenta etc. Simulate efficiency fluctuations by Gaussian around ε with particular δε Skewness

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

δε = 0.01

PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

Sσ

5 million PHSD central Au+Au events at 10A GeV

Sσ

<ε> = 0.80

Kurtosis

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

κσ

2

5 million PHSD central Au+Au events at 10A GeV

κσ

2

<ε> = 0.80 δε = 0.01

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SLIDE 22

Monte Carlo simulation: Fluctuating efficiency

In more realistic scenario efficiency is changes from event to event, e.g., due to fluctuations in number of tracks, momenta etc. Simulate efficiency fluctuations by Gaussian around ε with particular δε Skewness

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

δε = 0.03

PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

Sσ

5 million PHSD central Au+Au events at 10A GeV

Sσ

<ε> = 0.80

Kurtosis

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 PHSD+Binomial, efficiency corrected PHSD+Binomial, efficiency uncorrected PHSD proton |y-ycm|<0.2 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

κσ

2

5 million PHSD central Au+Au events at 10A GeV

κσ

2

<ε> = 0.80 δε = 0.03

Small efficiency fluctuations may destroy agreement! Especially higher moments more strongly affected.

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SLIDE 23

First try with GEANT simulation

GEANT simulation Realistic CBM detector response UrQMD events put through GEANT Tracks and momenta rec. with STS Particle ID by TOF Implemented in KF Particle Finder

0.85 0.90 0.95 1.00 eff = 80% UrQMD+CBM, efficiency corrected UrQMD+CBM, efficiency uncorrected UrQMD net-proton |y|<0.5 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

σ

2/M

3 million UrQMD central Ni+Ni events at 15A GeV

σ

2/M

Problems Individual efficiency correlations e.g. track merging Particle misidentification Identification of primaries

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SLIDE 24

GEANT simulation: Efficiency corrections

GEANT simulation of CBM detector response

0.4 0.5 0.6 0.7 0.8 0.9 1.0 eff = 80% UrQMD+CBM, efficiency corrected UrQMD+CBM, efficiency uncorrected UrQMD net-proton |y|<0.5 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

Sσ

3 million UrQMD central Ni+Ni events at 15A GeV

Sσ

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 eff = 80% UrQMD+CBM, efficiency corrected UrQMD+CBM, efficiency uncorrected UrQMD net-proton |y|<0.5 0.4<pT<2.0 GeV/c 0.4<pT<0.8 GeV/c 0.4<pT<0.5 GeV/c

κσ

2

3 million UrQMD central Ni+Ni events at 15A GeV

κσ

2

The simple correction is not enough! Becomes worse with increasing moments and/or kinematic window Still preliminary, reconstruction can likely be improved

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SLIDE 25

Summary

1 Fluctuations of conserved charges carry information about finer details

f the equation of state and exhibit rich structures near critical point.

Both NN interactions and chiral criticality may play role at SIS100/300 energies.

2 Interaction rate at CBM should be enough to measure the efficiency

uncorrected moments up to sixth order.

3 The errors due to binomial correction increase with decreasing

efficiency and increasing cumulant order. The validity of the correction is very sensitive to fluctuations and correlations of efficiencies of individual particles.

4 Higher moments cannot be properly corrected by the binomial

correction in realistic situations. More elaborate procedure is likely needed.

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SLIDE 26

Summary

1 Fluctuations of conserved charges carry information about finer details

f the equation of state and exhibit rich structures near critical point.

Both NN interactions and chiral criticality may play role at SIS100/300 energies.

2 Interaction rate at CBM should be enough to measure the efficiency

uncorrected moments up to sixth order.

3 The errors due to binomial correction increase with decreasing

efficiency and increasing cumulant order. The validity of the correction is very sensitive to fluctuations and correlations of efficiencies of individual particles.

4 Higher moments cannot be properly corrected by the binomial

correction in realistic situations. More elaborate procedure is likely needed.

Thanks for your attention!

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