Update  on  the  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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The Model

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Model: a realistic approach• Model the proton and anti-proton multiplicity distributions using Pearson curve method.• Probability density function of any frequency distribution can be derived, which satisfy the following

differential equation [Karl Pearson (1895)]

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.

(arXiv:1706.06558) with mean (µ) at 0……..(1)

With µ ≠ 0,

1𝑓(𝑥)

𝑑𝑓(𝑥)𝑑𝑥 = −  

𝑎+𝑥 +  𝑎-𝑏- +  𝑏+𝑥 +  𝑏/𝑥/

……..(2)

• The values of constant parameters in Eq(2) are different than of Eq(1)• Mathematica (Inbuilt function – PearsonDistribution) is used to get the form of f(x).

(https://reference.wolfram.com/language/ref/PearsonDistribution.html)

Used for this 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

1) 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

2) 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

• Normalization constant very very small !! How to get the normalization constant ( see some example given in J. H. Pollard, ISBN 0 52129750, 1977 )

• Are they make sense? Can we plot them? – Next slide.

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Proton and anti-proton distribution from Pearson curve methodUsing ROOT

Proton

Anti-proton

0 5 10 15 20 25 300.00

0.05

0.10

0.15

x0 5 10 15 20 25 30

f(x)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.1872.222(70.6873 - p)16.0641(p + 4.9037)-14910×f(p) = 2.85855

118.259)p(96.9067 - 19.4225 + 5.83732)p(-25410×) = 2.92714pf(pp

Using Mathematica

TF1 *funcProton = new TF1("funcProton","2.85855e-149*TMath::Power(x + 4.9037, 16.0641)*TMath::Power(70.6873 - x,72.222 )" , 0., 30.);

TF1 *funcAntiProton = new TF1("funcAntiProton","2.92714e-254*TMath::Power(x + 5.83732, 19.4225 )*TMath::Power(96.9067 - x, 118.259 )" , 0., 30.);

x

f(x)

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MC study:

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)

For each sample:

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Xiaofeng Luo,  Phy.  Rev C  91,  034907  (2015)

MC study:

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Result: 𝜿4/𝜿2

)6 10×Events (1 10 210

2κ/ 4

κ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8(a) Sample 1 Sample 2

Sample 3 Sample 4Sample 5 Analytic

ALICE preliminary

)6 10×Number of events (1 10 210

Stat

istic

al u

ncer

tain

ty

00.05

0.10.15

0.20.250.3 -0.5x 3.55

2κ-610×6.15

OLD

)6 10×Events (1 10 210

2κ/ 4

κ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 (a) Sample 1 Sample 2Sample 3 Sample 4Sample 5 Analytic

ALICE preliminary

)6 10×Number of events (1 10 210

Stat

istic

al u

ncer

tain

ty

00.05

0.10.15

0.20.25

0.3 -0.353x 4.292κ

-710×4.0

The statistical uncertainty is estimated by taking the rms of the 𝜿4/ 𝜿2

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)6 10×Events (1 10 210

2κ/ 6

κ

150−

100−

50−

0

50

100

150 (b) Sample 1 Sample 2Sample 3 Sample 4Sample 5 Analytic

)6 10×Number of events (1 10 210

Stat

istic

al u

ncer

tain

ty

010203040506070

-0.5x 3.552κ

-310×1.26

Result: 𝜿6/ 𝜿2

)6 10×Events (1 10 210

2κ/ 6

κ

150−

100−

50−

0

50

100

150 (b) Sample 1 Sample 2Sample 3 Sample 4Sample 5 Analytic

)6 10×Number of events (1 10 210

Stat

istic

al u

ncer

tain

ty

010203040506070

-0.55x 3.562κ

-310×1.35OLD

• The statistical uncertainty is estimated by taking the rms of the 𝜿6/ 𝜿2.• To get 1% statistical uncertainty on 𝜿6/ 𝜿2, we need ~ 200x109 events (estimated using the parameterized function).

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Result: 𝜿6/ 𝜿2

• Should the x-axis start from 10M events?• Present figure represent the statistical uncertainty for

the Run1 event statistics.• Consistent with the Figure (a).

)6 10×Events (1 10 210

2κ/ 6

κ

150−

100−

50−

0

50

100

150 (b) Sample 1 Sample 2Sample 3 Sample 4Sample 5 Analytic

)6 10×Number of events (1 10 210

Stat

istic

al u

ncer

tain

ty

010203040506070

-0.55x 3.562κ

-310×1.35

R. Esha, Quark Matter 2017 [NPA 967 (2017) 457–460]

(~200M events for 0-10% centrality)

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Thank  You

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Backups

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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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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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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)