Importance of separated efficiencies between
positively and negatively charged particles
for cumulant calculations
Toshihiro Nonaka, University of Tsukuba
ATHIC2016 @New Delhi, India
Feb.17, 20161

Outline•

Introduction•

Cumulants of conserved quantities•

Efficiency correction•

MC toy model•

Analytical calculation•

Beam energy dependence•

Summary

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Search for the critical point✓ Beam energy scan phase Ⅰ✓ √sNN = 7.7, 11.5, 19.6, 27, 39, 62.4,200GeVPRL112 032302 (2014)→ net-proton✓ Observables are measured as afunction of beam energy✓ Fluctuation of conserved quantities

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Cumulants of conserved quantities✓ Extensive variable✓ Proportional to the power of correlation length✓ Directly connected to susceptibilities

arXiv:1512.05038“Skewness”“Kurtosis”
Poisson baseline
Poisson - Poisson = “Skellam” distribution

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

4Efficiency correction✓ Effect of tracking efficiency must be corrected for measured cumulants.
✓ Averaged efficiency (ε++ε-)/2 between positively and negatively charged
particles were used in published net-charge and net-proton results.
PRC 86(2012) 044904
p : tracking efficiency
f : factorial momentc : measured cumulant
K: corrected cumulant✓ Actually there is finite difference of tracking efficiency between positively andnegatively charged particles.
✓ How will the published results be changed if the separated efficiencies are used？
✓ Difference between two correction methods are studied by➡ MC toy model assuming net-proton distribution.
➡ Analytical calculation from the correction formula.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

5Example
N-μ+= 10μ- = 8

1. Generate two Poisson distributions.
2. Random sampling according to binomial efficiency.
3. Apply correction using averaged or separated efficiencies.
1M events
ε+=0.65
ε- =0.66

Input

Reducedεave=(ε++ε-)/2=0.655

N+
efficiency correction doesn’t work
efficiency correction works well✓ From next page, toy model assuming net-proton will be shown. Parameters weretaken from published results.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

6Order dependence• Relative difference from input value (true-corr)/true as a function of eachorder of cumulants.
• Assume net-proton distribution in the most central collisions at 200GeV.
assume net-proton, 200GeV, 0-5%

Odd order cumulants systematically deviate from input value??

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

7Analytical calculation, C1
N± : # of produced particles
M± : # of observed particles
ε± : efficiency for charged or anti-charged particles
ε : averaged efficiency
Δε : efficiency difference

Taylor expansion around Δε=0
Δε=0Difference ΔK1 is proportional to the sum of multiplicity.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

8Analytical calculation, C2•

Difference of C2 can be calculated by similar approach

represented by “net-charge” terms

Difference ΔK2 is proportional to the difference of multiplicity.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

9Analytical calculation, C3•

Difference of C3 can be also calculated by similar approach

represented by “multiplicity” terms

Difference ΔK3 is proportional to the sum of multiplicity.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

10net-proton, 200GeV, 0-5%

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

12net-proton, 200GeV, 0-5%

How about low beam energies？

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

13Order dependence (7.7GeV)
assume net-proton, 7.7GeV, 0-5%

Difference of odd order is as large as even order→All the cumulant ratios don’t deviate from input value.

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Beam energy dependence✓ Relative difference is calculated for each BES energy.
✓ There is ~5% difference for Sσ at experimental interest region(~39GeV).
✓ Conclusions in net-proton published paper won’t be changed.

~5% difference

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Weighted average?
N : # of events
M : # of particles
ε : efficiency✓ At lower energies, weighted averaged efficiency gives a better results(C1,C3)
than averaged efficiency.
✓ At higher energies, however, the difference is as large as averaged efficiency.

➡ Separated efficiencies should be used.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

16Summary•

The results of MC toy model calculations indicate that
➡ odd order cumulants and C3/C2 systematically deviate from input
value in case of averaged efficiency @200GeV.
➡ deviation of even order cumulants is as large as odd order
cumulants @7.7GeV.

•According to analytical calculations, this is because➡ deviation of odd order cumulants is proportional to the sum ofmultiplicity.
➡ deviation of even order cumulants is proportional to the differenceof multiplicity.

•Beam energy dependence indicates that➡ conclusions in published net-proton paper won’t be changed.

•Please note that current fluctuation analysis at STAR is being
done by using separated efficiencies.

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Back up18

Skellam baseline of Sσ
efficiency vanishes!!

➡ Baseline is also changed

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Weighted average @7.7GeV
N : # of events
M : # of particles
ε : efficiency

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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Weighted average @200GeV
N : # of events
M : # of particles
ε : efficiency

Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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All energies and centralities
0-5%

5-10% 10-20% 20-30% 30-40% 40-50% 50-60% 60-70% 70-80%

7.7
11.5
19.6
2739
62.4
200
Each bin : C1, C2, C3, C4, Sσ, Sσ/Skellam and κσ2 from left to right.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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All energies and centralities
0-5%5-10% 10-20% 20-30% 30-40% 40-50% 50-60% 60-70% 70-80%

7.7
11.5
19.6
2739
62.4
200
Each bin : C1, C2, C3, C4, Sσ, Sσ/Skellam and κσ2 from left to right.
Feb.17, 2016

Toshihiro Nonaka, ATHIC2016@New Delhi

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