A  projection  for  statistical  uncertainty  on  higher  order  cumulants of  net-­proton  multiplicity  

distributions

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Nirbhay Kumar  BeheraInha University

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Motivation

• Determination of freeze-out parameters from the ratio of cumulants of conserved charges.

• Possible search for chiral critical point.

1] S. Borsanyi et. al. Phys. Rev. Lett. 111, 062005 (2013)2] B. Friman et. al. Eur. Phys. J. C (2011) 71:1694

Freeze-out in hardonicphase

Freeze-out in the vicinity of chiral crossover temperature

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Centrality (%)0 10 20 30 40 50 60 70 80

p)

∆ ( 2/C 4C

0

0.2

0.4

0.6

0.8

1

1.2

1.4 ALICE Preliminary

c < 1.0 GeV/Tp0.4 <

| < 0.8η|

p p = p - ∆

Boxes: sys. errors

Pb-Pb = 5.02 TeVNNs = 2.76 TeVNNs

Skellam

ALI−PREL−159586

0

0.2

0.4

0.6

0.8

1

1.2

1.4

150 200 250 300 350T [MeV]

χB4/χB

2 HRGNt=6Nt=8

Nt=10Nt=12

WB continuum limit

Motivation: Recent measurementQuark  Matter  2018

• Net-proton C4/C2 result associated with large statistical uncertainties.• Large systematic in central events than the peripheral. Statistical fluctuations propagate to systematics.

Need precise determination of ratio of cumulants to constrain the freeze-out temperature

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The Model

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Model: a realistic approach• Model the proton and anti-proton multiplicity distributions using Pearson curve method.• Based on NKB, arXiv:1706.06558• Karl Pearson (1895): Probability density function of any frequency distribution can be derived, which satisfy

the following differential equation-

a, b0, b1, b2 are constant parameters and functions of first four moments (cumulants) of the distributions.Pearson curve method can be applied if the distribution satisfy following condition –Kurtosis – skewness2 – 1 > 0.

• There are mainly 7 family of Pearson curves, for example –• Normal distribution, Beta, Gamma, F-ratio distribution, StudentT distribution, etc.• In 22 types distributions are related to it at certain limits.• This method help to avoid arbitrariness of using different Probability density function to same frequency

data.• The PDF can exactly reproduce the first four cumulants of the distribution which is used to derive it

(Poisson distribution – only first cumulant, NBD – first and second cumulant)• A better tool to model the proton, anti-proton and net-proton distributions for MC study.

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• The efficiency corrected first four cumulants (C1, C2, C3 and C4 )of proton and anti-proton distributions in Pb-Pb collisions at 2.76 TeV data are used. (QM 2018 prel. Results)

• [ 0.4 < pT < 1.0 GeV/c, -0.8 < η < 0.8 , Centrality bin: 0-10%]

• The probability distributions of proton and anti-proton multiplicity distributions are obtained using Pearson curve method.

Model: a realistic approach

PDF of proton multiplicity distribution:

f(x) = 2.85855×10-149 (x + 4.9037)16.0641 (70.6873 – x)72.222

at -4.9037 < x < 70.6873

PDF of anti-proton multiplicity distribution:

f(x) = 2.92714×10-254 (x + 5.83732)19.4225 (96.9067 – x)118.259

at -5.83732 < x < 96.9067

(Rescaled and shifted) Beta distribution

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MC study: flow chart

Randomly get the proton and anti-proton numbers from the obtained pdfs.(Independent proton and anti-proton distribution)

Assignee the pT to each proton and anti-proton (pT spectra shape taken from ALICE published spectra)

Apply reconstruction efficiency to each particles

Store the event information of net-proton both at generation and reconstruction level.

• Use the pT dependent efficiency correction method* to get the efficiency corrected cumulants (up to 6th order).

• Statistical uncertainties are estimated using Subsample method with 30 numbers of subsample.

Repeat for N events

Event level

Track level

(GeV/c)T

p0.4 0.5 0.6 0.7 0.8 0.9 1

Effic

ienc

y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 2.76 TeVNNsPb-Pb

0-5% Centralitypp

* T. Nonaka et. al. PRC 95, 064912, (2017)

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At a given event statistics, the statistical uncertainties on cumulants depend on reconstruction efficiency

Why the efficiency corrected results?

X.  Luo,  N.  Xu  Nucl.  Sci.  Tech.  28,  112  (2017)   (arXiv:1701.02105)

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Sample0 1 2 3 4 5 6

2/C 4C

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.86 10×Event statistics: 5

Sample0 1 2 3 4 5 6

2/C 4C

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.86 10×Event statistics: 100

Sample0 1 2 3 4 5 6

2/C 4

Rel

ativ

e Er

ror o

f C

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.16 10×Event statistics: 100

Sample0 1 2 3 4 5 6

2/C 4

Rel

ativ

e Er

ror o

f C

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.56 10×Event statistics: 5

Result: C4/C2

Relative  error=  |  dx/x  |

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Result: C4/C2

• Event statistics used: 1, 5, 10, 20, 50, 100 and 200 Millions

• Study is done with five different samples by changing pseudorandom number seed.

)6 10×Events (1 10 210

2/C 4C

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Sample 1Sample 2Sample 3Sample 4Sample 5

)6 10×Events (1 10 210

2/C 4

Stat

istic

al u

ncer

tain

ty o

f C

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Sample 1Sample 2Sample 3Sample 4Sample 5

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Sample0 1 2 3 4 5 6

2/C 6

Rel

ativ

e Er

ror o

f C

0

1

2

3

4

5

6

7

8

9

106 10×Event statistics: 5

Sample0 1 2 3 4 5 6

2/C 6C

100−

50−

0

50

1006 10×Event statistics: 5

Sample0 1 2 3 4 5 6

2/C 6C

100−

50−

0

50

1006 10×Event statistics: 100

Sample0 1 2 3 4 5 6

2/C 6

Rela

tive

Erro

r of C

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

56 10×Event statistics: 100

Result: C6/C2

Relative  error=  |  dx/x  |

Not a good idea to see the relative uncertainties !

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Result: C6/C2

)6 10×Events (1 10 210

2/C 6C

100−

50−

0

50

100

Sample 1Sample 2Sample 3Sample 4Sample 5

)6 10×Events (1 10 210

2/C 6

Stat

istic

al u

ncer

tain

ty o

f C

0

10

20

30

40

50

60

Sample 1Sample 2Sample 3Sample 4Sample 5

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Scaled uncertainties with respect to variance

)6 10×Events (1 10 210

2/C 4

Scal

ed S

tatis

tical

unc

erta

inty

of C

0

0.005

0.01

0.015

0.02

0.025

Sample 1Sample 2Sample 3Sample 4Sample 5

)6 10×Events (1 10 210

2/C 6

Scal

ed s

tatis

tical

unc

erta

inty

of C

0

0.5

1

1.5

2

2.5

3

3.5

4

Sample 1Sample 2Sample 3Sample 4Sample 5

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Summary:

• For higher order cumulants measurement, high event statistics are needed.

• For C4/C2 at least 50 Millions of event and for C6/C2, more than 200 millions of events are required!

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