Recent Results on Cumulants of Net-Particle Distributions ... · See A. Chatterjee, Poster #534 The...

Recent Results on Cumulants of Net-Particle Distributions in Au+Au

Collisions at STAR

1

Toshihiro Nonaka for the STAR CollaborationUniversity of Tsukuba

Central China Normal University

Toshihiro Nonaka, QM2018, Venice, Italy

Outline

2

✓ Introduction- Observables

- Analysis methods

✓ Experimental results- Net-Λ cumulants

- Off-diagonal cumulants

- Sixth-order cumulants

✓Non-binomial efficiencies


QCD phase diagram

3

✓ Higher-order fluctuations of net-particle distributions.

- Crossover at μB=0

- 1st-order phase transition at large μB?

- Critical point?


Higher-order fluctuation

4

✓ Moments: mean (M), standard deviation (σ), skewness (S) and kurtosis (κ).✓ S and κ are non-gaussian fluctuations.

κ > 0

κ < 0

skewness→asymmetry kurtosis→sharpness

from wikipedia

✓ Cumulant ⇄ Moment ✓ Cumulant : additivity

proportional to volume

✦ Moments and cumulants are mathematical measures of “shape” of a distribution which probe the fluctuation of observables.


Existing analysis methods

5

- X.Luo, J. Xu, B. Mohanty and N. Xu. J. Phys. G40,105104(2013)

✓ Efficiency correction on cumulants have been done assuming the binomial efficiencies.

- M. Kitazawa : PRC.86.024904, M. Kitazawa and M. Asakawa : PRC.86.024904- A. Bzdak and V. Koch : PRC.86.044904, PRC.91.027901, X. Luo : PRC.91.034907- T. Nonaka et al : PRC.94.034909, T. Nonaka, M. Kitazawa, S. Esumi : PRC.95.064912

✓Centrality bin width averaging is done for the reduction of the initial volume fluctuation.

✓Calculate the cumulants at each value of the multiplicity used for centrality, then weighted-average these in each centrality bin.

N1, N2 : lowest and highest multiplicity bin in the centralitynr : # of events in rth multiplicity bin


Net-Λ cumulants

6

✓ Strange hadrons freeze-out earlier than light flavor hadrons?✓ Net-Λ cumulants might provide additional constraints on freeze-out

conditions.

J. Noronha-Hostler, C. Ratti, P. Parotto, R. Bellwied, arXiv:1805.00088

HRG fit to STAR net-K using PDG16+

HRG fit to STAR net-p,Q (1403.4903)

STAR thermal fits ,p,K (1701.07065)

STAR thermal fits ,p,K, , , , ,Ks0 (1701.07065)

0 50 100 150 200 250100

120

140

160

180

B [MeV]

T[M

eV]


Net-Λ cumulants

7

✓ Consistent with Poisson/NBD baselines.

✓ C1 and C2 are above UrQMD results.

✓ C3 shows better agreement with UrQMD.

>part<N50 100 150 200 250 300 350

01234 19.6 GeV

1C

>part<N50 100 150 200 250 300 350

02468

10

cumulantsΛnet -

Au + Au collisionsat STAR

2C

100 200 300

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1

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33C

>part<N50 100 150 200 250 300 350

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>part<N50 100 150 200 250 300 350

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2C

0.5−

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1

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>part<N50 100 150 200 250 300 350

)σ

(S 2/C 3C

0

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3.5

4 62.4 GeV

>part<N50 100 150 200 250 300 350

)σ

(S 2/C 3C

0

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100 200 300

)σ

(S 2/C 3C

0.5−

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>part<N50 100 150 200 250 300 350

)σ

(S 2/C 3C

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PoissonNBDUrQMD

200 GeV

>part<N50 100 150 200 250 300 350

)σ

(S 2/C 3C

0

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100 200 300

)σ

(S 2/C 3C

0.5−

0

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Efficiency corrected (GeV/c) < 2.0

T0.9 < p

|y| < 0.5

STAR Preliminary

See N. Kulathunga, Poster #528

Toshihiro Nonaka, QM2018, Venice, Italy 8

✓ C2/C1 is close to HRG results (same kinematic range) with kaon freeze-out condition, and far away from those of charge and proton.

✓ The error on C3/C2 is presently too large to provide a meaningful constraint on the HRG model.

Net-Λ cumulants

(GeV)NNs20 30 40 50 60 100 200

/M)

2σ ( 1

/C 2C

0

2

4

6

8

10 Λ0 - 5% net -

FO at kaon FO (HRG)ΛAssuming

FO at charge/proton FO (HRG)ΛAssuming

Au + Au collisions

STAR Preliminary

(GeV)NNs20 30 40 50 100 200

)σ

(S 2/C 3

C

0.5−

0

0.5

1

(GeV) < 2.0T

0.9 < p|y| < 0.5

Efficiency corrected.STAR Preliminary


Off-diagonal cumulants of conserved charges will provide additional constraints on the freeze-out conditions.

9

• A. Majumder and B. Muller, Phy. Rev. C 74 (2006) • A. Bazavov et al. Phys. Rev. D86 (2012) 034509 • A. Chatterjee et al. J. Phys. G: Nucl. Part. Phys.

43 (2016) 125103• Z. Yang et al. Phys. Rev. C 95 014914 (2017)

✓ Covariance shows linear dependence with respect to the centrality and η acceptance.

✓ The correlation between net-protons and net-kaons changes from positive to negative with increasing beam energy.

See A. Chatterjee, Poster #534

Centrality dependence, |η|<0.5 η acceptance dependence, 0-5%

Efficiency corrected-0.5

0

0.5net-proton, net-kaon (p-k)

Co-

varia

nce

(σ1,

1 )

0

20

40

60

0 100 200 300

net-charge, net-kaon (Q-k)

⟨Npart⟩

0

10

20

30

40 net-charge, net-proton (Q-p)

0 100 200 300

Beam energy

0.4 < pT < 1.6 GeV/c|η| < 0.5

bars: syst. errorstat. error within the marker

α1x

⟨Npart⟩

7.7 GeV11.5 GeV14.5 GeV19.6 GeV

27 GeV

39 GeV62.4 GeV200 GeV

CLT

-0.5

0


Co-

varia

nce

(σ1,

1 )

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0.4 < pT < 1.6 GeV/c|η| < 0.5


α1x

⟨Npart⟩

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27 GeV


CLT-0.5

0


Co-

varia

nce

(σ1,

1 )

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60

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0.4 < pT < 1.6 GeV/c|η| < 0.5


α1x

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27 GeV


CLTSTAR Preliminary

-0.5

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Co-

varia

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1 )

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0.4 < pT < 1.6 GeV/c|η| < 0.5


α1x

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7.7 GeV11.5 GeV14.5 GeV19.6 GeV

27 GeV


CLT

-0.5

0


Co-

varia

nce

(σ1,

1 )

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0.4 < pT < 1.6 GeV/c|η| < 0.5


α1x

⟨Npart⟩

7.7 GeV11.5 GeV14.5 GeV19.6 GeV

27 GeV


CLT

Efficiency correctedSTAR Preliminary

The 2nd-order off-diagonal cumulants

0.4<pT<1.6 (GeV/c)


✓ Normalized p-k correlation is positive at low energies and negative at high energies, which are also consistent with UrQMD.

✓ Significant excess is observed in Q-k and Q-p with respect to the Poisson baseline and UrQMD.

✓ This excess increases with the beam energy in central collisions compared to peripheral collisions.

See A. Chatterjee, Poster #534

The 2nd-order off-diagonal cumulants

STAR Preliminary

1

1.4

1.8

10 20 60 200(GeV)

CQ

,p=(σ

1,1

Q,p

/σ2 p)

√sNN

1

1.4

1.8 0.4<pT<1.6 GeV/c, |η|<0.5

CQ

,k=(σ

1,1

Q,k

/σ2 k)

0

0.04

0.08

Cp,

k=(σ

1,1

p,k/σ

2 k)

data 0-5%data 70-80%PoissonUrQMD 0-5%UrQMD 70-80%


C.Schmidt,Prog.Theor.Phys.Suppl.186,563–566(2010) Cheng et al, Phys. Rev. D 79, 074505 (2009) Friman et al, Eur. Phys. J. C (2011) 71:1694

✓ There isn’t yet any direct experimental evidence for the smooth crossover at μB~0.✓ Sixth-order cumulants of net-charge and net-baryon distributions are predicted to

be negative if the chemical freeze-out is close enough to the phase transition.

Friman et al, Eur. Phys. J. C (2011) 71:1694 : O(4) scaling functions

Cheng et al, Phys. Rev. D 79, 074505 (2009) : Lattice QCD

Net-charge sixth-order cumulant

Predicted scenario for this measurement


✓ The first result of C6/C2 of net-charge.✓ Net-charge has much larger errors than net-proton because the error strongly depends

on the standard deviation.✓ Results of net-charge C6/C2 are consistent with zero within large statistical errors.✓ Negative values are observed in net-proton C6/C2 systematically from peripheral to

central collisions.partN

0 100 200 300600−

400−

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Binomialzero

QM2017Au+Au, √sNN = 200 GeVSTAR PreliminaryNet-Proton, |y|<0.5

<Npart>

error(Cr) / �rpNeve

partN0 100 200 300

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Run10+Run11Run10, Central triggerBinomialzero

See T. Sugiura, Poster #532

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4000Au+Au, √sNN = 200 GeVSTAR PreliminaryNet-Charge, |η|<0.5

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400−

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200

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4000

<Npart>

Net-charge sixth order cumulant


✓ The detector efficiency may not be binomial, which would be due to the particle mis-identification, track splitting/merging effects, and many other reasons.

✓ Residual dependence of efficiency inside one multiplicity bin (for centrality) needs to be taken into account.

A. Bzdak, R. Holzmann, V. Koch : PRC.94.064907

A. Bzdak, R. Holzmann, V. Koch : PRC.94.064907

1. Experimental effects

√sNN = 200 GeV

"0lowp

= �0.00032

"0lowpbar

= �0.00033

"0highp

= �0.00025

"0highpbar

= �0.00023

STAR Preliminary

T. Nonaka, WPCF2017 Multiplicity

2. Multiplicity dependent efficiency

➡ One example of non-binomial distribution, Beta-binomial, is wider distribution than binomial

Non-binomial efficiencies


✓ We performed MC simulations by embedding protons and antiprotons, e.g., Np=60 and Npbar=15 (which would be an extreme number), and see whether those particles can be reconstructed or not.

✓ The response matrix is close to the beta-binomial distribution, which is wider than binomial.

See T. Nonaka, Poster #453

➡ “Urn model” for beta-binomial distribution, where the parameter α controls the deviation from binomial.

20 30 40 50 60

5−10

4−10

3−10

2−10

1−10

20 30 40 50 600.80.91

1.11.21.3

np : # of reconstructed protonsda

ta/fi

t

Binomial : χ2/NDF ~ 3.3Beta-binomial (α=15) :χ2/NDF ~ 1.0

STAR Preliminary

√sNN = 19.6 GeV, 0-2.5% centralityR(np;Np, Npbar)

Prob

abilit

y

Nw : white balls, Nb: black balls, ε: efficiencyNw = ↵Np " = Nw/(Nw +Nb)

Non-binomial efficiencies


√sNN = 19.6 GeV, 0-5% centrality, |y|<0.5, 0.4 < pT < 2.0, embedding simulation

Response matrices✓ The deviation from binomial would depend on the # of embedded protons and antiprotons.✓ 4-D response matrices are determined by embedding simulation, which can be directly

used for unfolding in order to reconstruct the distribution itself.


5 10 15 20 25 30 350

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10

=3pbar=10, NpN

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Forward response matrix

n pba

r : #

of r

econ

stru

cted

an

tipro

tons

STAR Preliminary

np : # of reconstructed protons

R(np, npbar;Np, Npbar) Reversed response matrix

5 10 15 20 25 30 350

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=4pbar=18, npn

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0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

=4pbar=21, npn

Np : # of embedded protons

Npb

ar :

# of

em

bedd

ed a

ntip

roto

ns

R(Np, Npbar;np, npbar)


✓ For unfolding, 2.5% centrality width averaging has been done.

✓ Systematic suppression is observed for C2 and C3 with respect to the results of efficiency correction assuming binomial efficiencies.

✓ C4, C3/C2 and C4/C2 are consistent within large systematic uncertainties limited by embedding samples.


Au+Au, √sNN = 19.6 GeV

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STAR Preliminary

Eff.corrUnfolding

<Npart>0 100 200 3000.9

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STAR Preliminary

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STAR Preliminary

C4/C2

Results of unfolding


Summary

17

1. Net-Λ cumulants up to 3rd-order• Consistent with Poisson/NBD baselines.• The result of C2/C1 is closer to those of HRG with kaon freeze-out condition rather than light flavor hadrons.

2. Second-order off-diagonal cumulants• Q-k and Q-p correlations are in excess of the UrQMD results.

3. Sixth-order cumulant of net-charge and net-proton• Negative value is observed (although with extremely large uncertainties) in consistency with expectations

from O(4) scaling functions.4. Influence of non-binomial efficiencies

• One example of the response matrix is tested by embedding simulation, which is closer to beta-binomial than binomial.

• Unfolding has been applied at √sNN = 19.6 GeV in central collisions, where results show systematic suppression for C2 and C3 compared to the efficiency correction assuming binomial efficiencies, while C4, C3/C2 and C4/C2 are consistent within large systematic uncertainties limited by embedding samples.

Thank you for your attention

18

Back up

19


Centrality definition

20

✓ In order to avoid the auto-correlation, the centrality is defined by using different paricle species or acceptance.

• Net-proton• Net-Λ

• Net-charge• Off-diagonal (π, K, p)

Charged paricles excluding (anti)protons in |η|<1.0

Charged paricles excluding (anti)protons in |η|>0.5

See : Phys. Rev. Lett. 113, 092301(2014) See : Phys. Rev. Lett. 112, 032302(2014)


Lambda reconstruction

21

✓ Lambda is reconstructed event-by-event based on the decay topology.

2) GeV/c-π (P, invM1.08 1.1 1.12 1.14 1.16 1.18

Cou

nts

0

5

10

15

20

25

610×h_InvMass_UnFltd_0

Entries 5.788835e+08

Mean 1.128

Std Dev 0.02935

Signal

Background

39 GeV Au+Au Collisionsat STAR

)-πInv. mass (P,

STAR Preliminary

(p, π—)

(p, π—)Minv(p,⇡�) (GeV/c2)


Lambda reconstruction

22

✓ The geometrical cuts were varied in order to obtain high pure (>90%) sample.

Minv(p̄,⇡+) (GeV/c2)


Net-lambda distribution

23


See A. Chatterjee, Poster #534Centrality dependence

-0.5

0


Co-

varia

nce

(σ1,

1 )

0

20

40

60

0 100 200 300


⟨Npart⟩

0

10

20

30


0 100 200 300

Beam energy

0.4 < pT < 1.6 GeV/c|η| < 0.5


α1x

⟨Npart⟩

7.7 GeV11.5 GeV14.5 GeV19.6 GeV

27 GeV


CLT

STAR Preliminary

2nd-order off-diagonal cumulants


2nd-order off-diagonal cumulantsη acceptance dependence See A. Chatterjee, Poster #534

STAR Preliminary


Unfolding methodology

26

✓ Difference between exp.meas and sim.meas is applied to sim.true to get the corrected distribution, which is repeated until cumulants converge.

A) Exp. true C) Sim. true

D) Sim. measB) Exp. meas

RMrev

C.F_rec = B) - D)

C.F_gen = RMrev x C.F_rec

C’) = C) + C.F_gen

run MC iteratively

Np

Npbar

Generatedcoordinate

Reconstructedcoordinate

S. Esumi et al : in preparation


Test with non-binomial distributions

27

α1 2 3 4 5 6 7 8 9 10

Std.

Dev

1.3

1.4

1.5

1.6

1.7=0.6εN=10,

Beta-binomial

Hypergeometric

Binomial

n0 1 2 3 4 5 6 7 8 9 10

Hyp./Binom

ial

0.8

0.9

1

1.1

1.2

n0 1 2 3 4 5 6 7 8 9 10

Beta./Binom

ial

0.8

0.9

1

1.1

1.20 1 2 3 4 5 6 7 8 9 10

210

310

410

510

610

=1α

=3α

=5α

=10α

Binomial

N=10, ε=0.6

n0 1 2 3 4 5 6 7 8 9 10

Hyp./Binom

ial

0.8

0.9

1

1.1

1.2

n0 1 2 3 4 5 6 7 8 9 10

Beta./Binom

ial

0.8

0.9

1

1.1

1.2 0 1 2 3 4 5 6 7 8 9 10

410

510

610

=1α

=3α

=5α

=10α

Binomial

N=10, ε=0.6

Hypergeometric distributionDraw a ball from urn, if it is white, count particle. This is repeated without replacement.

Beta-binomial distributionDraw a ball from urn, if it is white, count particle. And return two white balls to urn (similar for black balls).

✓ Smaller α for Hypergeometric distribution, becomes narrower than binomial distribution.

✓ Smaller α for Beta-binomial distribution, becomes wider than binomial distribution.

✓ Both non-binomial distributions become close to the binomial with large α.

✓ The beta-binomial and hypergeometric distributions can be defined by the urn model, in which the parameter α controls the devitation from the binomial distribution.

Nw : white balls, Nb: black balls, ε: efficiency


Response matrix from embedding simulation

28√sNN = 19.6 GeV, 0-2.5% centrality, 60 protons and 15 antiprotons are embedded2040608000.010.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.170.180.190.20.210.220.230.240.250.260.270.280.290.30.310.320.330.340.350.360.370.380.390.40.410.420.430.440.450.460.470.480.490.50.510.520.530.540.550.560.570.580.590.60.610.620.630.640.650.660.670.680.690.70.710.720.730.740.750.760.770.780.790.80.810.820.830.840.850.860.870.880.890.90.910.920.930.940.950.960.970.980.991Np=1, Npbar=42040608000.010.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.170.180.190.20.210.220.230.240.250.260.270.280.290.30.310.320.330.340.350.360.370.380.390.40.410.420.430.440.450.460.470.480.490.50.510.520.530.540.550.560.570.580.590.60.610.620.630.640.650.660.670.680.690.70.710.720.730.740.750.760.770.780.790.80.810.820.830.840.850.860.870.880.890.90.910.920.930.940.950.960.970.980.991Np=1, Npbar=82040608000.010.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.170.180.190.20.210.220.230.240.250.260.270.280.290.30.310.320.330.340.350.360.370.380.390.40.410.420.430.440.450.460.470.480.490.50.510.520.530.540.550.560.570.580.590.60.610.620.630.640.650.660.670.680.690.70.710.720.730.740.750.760.770.780.790.80.810.820.830.840.850.860.870.880.890.90.910.920.930.940.950.960.970.980.991Np=1, Npbar=15

20 40 60 800

2

4

6

Np=7, Npbar=2

20 40 60 800

2

4

6

Np=15, Npbar=1

20 40 60 800

2

4

6

Np=15, Npbar=4

20 40 60 800

2

4

6

Np=30, Npbar=1

20 40 60 800

2

4

6

Np=30, Npbar=8

20 40 60 800

2

4

6

Np=30, Npbar=15

20 40 60 800

2

4

6

Np=60, Npbar=1

20 40 60 800

2

4

6

Np=60, Npbar=8

20 40 60 800

2

4

6

Np=60, Npbar=15

Binomial

Hypergeometric

Beta-binomial

α : closeness to the binomial

χ2/N

DF

20 30 40 50 60

5−10

4−10

3−10

2−10

1−10

20 30 40 50 600.80.91

1.11.21.3

np : # of reconstructed protonsda

ta/fi

t

χ2/NDF ~ 3.3χ2/NDF ~ 1.0

✓ Embed 60 protons and 15 antiprotons into the real data (which would be the extreme number), and see whether those particles can be reconstructed or not.

✓ The response matrix is wider than the binomial, and it is close to the beta-binomial distribution.

✓ More detailes can be found in the poster #453.

STAR Preliminary

STAR Preliminary


Test with non-binomial distributions

29

✓ The beta-binomial distribution is wider than binomial, which can be defined by the urn model.

✓ The parameter α controls the devitation from the binomial distribution.

Nw : white balls, Nb: black balls, ε: efficiency

0 1 2 3 4 5 6 7 8 9 10

410

510

610

=1α

=3α

=5α

=10α

Binomial

N=10, ε=0.6

N

Recent Results on Cumulants of Net-Particle Distributions ... · See A. Chatterjee, Poster #534 The...

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