Collision energy dependence of moments of net-kaon multiplicity distributions at RHIC

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Collision

energy

dependence

of

moments

of

net-kaon

multiplicity

distributions

at

RHIC

STAR

Collaboration

L. Adamczyk

a

,

J.R. Adams

ac

,

J.K. Adkins

s

, G. Agakishiev

q

,

M.M. Aggarwal

ae

,

Z. Ahammed

ay

,

N.N. Ajitanand

an

,

I. Alekseev

o

,

z

,

D.M. Anderson

ap

,

R. Aoyama

at

,

A. Aparin

q

,

D. Arkhipkin

c

,

E.C. Aschenauer

c

,

M.U. Ashraf

as

,

A. Attri

ae

,

G.S. Averichev

q

,

X. Bai

g

,

V. Bairathi

aa

,

K. Barish

av

,

A. Behera

an

,

R. Bellwied

ar

,

A. Bhasin

p

,

A.K. Bhati

ae

,

P. Bhattarai

aq

,

J. Bielcik

j

,

J. Bielcikova

k

,

L.C. Bland

c

,

I.G. Bordyuzhin

o

,

J. Bouchet

r

,

J.D. Brandenburg

aj

,

A.V. Brandin

z

,

D. Brown

w

,

I. Bunzarov

q

,

J. Butterworth

aj

,

H. Caines

bc

,

M. Calderón de la Barca Sánchez

e

, J.M. Campbell

ac

,

D. Cebra

e

,

I. Chakaberia

c

,

P. Chaloupka

j

,

Z. Chang

ap

, N. Chankova-Bunzarova

q

,

A. Chatterjee

ay

,

S. Chattopadhyay

ay

,

J.H. Chen

am

,

X. Chen

u

,

X. Chen

ak

,

J. Cheng

as

, M. Cherney

i

,

W. Christie

c

,

G. Contin

v

,

H.J. Crawford

d

,

S. Das

g

, L.C. De Silva

i

,

R.R. Debbe

c

,

T.G. Dedovich

q

, J. Deng

al

,

A.A. Derevschikov

ag

, L. Didenko

c

, C. Dilks

af

, X. Dong

v

, J.L. Drachenberg

t

,

J.E. Draper

e

,

L.E. Dunkelberger

f

,

J.C. Dunlop

c

,

L.G. Efimov

q

,

N. Elsey

ba

,

J. Engelage

d

,

G. Eppley

aj

,

R. Esha

f

, S. Esumi

at

,

O. Evdokimov

h

, J. Ewigleben

w

,

O. Eyser

c

,

R. Fatemi

s

,

S. Fazio

c

,

P. Federic

k

,

P. Federicova

j

,

J. Fedorisin

q

, Z. Feng

g

,

P. Filip

q

,

E. Finch

au

,

Y. Fisyak

c

,

C.E. Flores

e

,

J. Fujita

i

,

L. Fulek

a

, C.A. Gagliardi

ap

, D. Garand

ah

, F. Geurts

aj

,

A. Gibson

ax

,

M. Girard

az

,

D. Grosnick

ax

,

D.S. Gunarathne

ao

,

Y. Guo

r

, S. Gupta

p

,

A. Gupta

p

, W. Guryn

c

,

A.I. Hamad

r

,

A. Hamed

ap

,

A. Harlenderova

j

,

J.W. Harris

bc

,

L. He

ah

,

S. Heppelmann

e

,

S. Heppelmann

af

,

A. Hirsch

ah

,

G.W. Hoffmann

aq

,

S. Horvat

bc

,

X. Huang

as

, H.Z. Huang

f

,

T. Huang

ab

,

B. Huang

h

,

T.J. Humanic

ac

,

P. Huo

an

,

G. Igo

f

,

W.W. Jacobs

n

, A. Jentsch

aq

,

J. Jia

c

,

an

, K. Jiang

ak

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S. Jowzaee

ba

,

E.G. Judd

d

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S. Kabana

r

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D. Kalinkin

n

, K. Kang

as

,

D. Kapukchyan

av

,

K. Kauder

ba

, H.W. Ke

c

,

D. Keane

r

,

A. Kechechyan

q

,

Z. Khan

h

,

D.P. Kikoła

az

, C. Kim

av

, I. Kisel

l

, A. Kisiel

az

,

L. Kochenda

z

,

M. Kocmanek

k

,

T. Kollegger

l

,

L.K. Kosarzewski

az

,

A.F. Kraishan

ao

,

L. Krauth

av

,

P. Kravtsov

z

,

K. Krueger

b

,

N. Kulathunga

ar

,

L. Kumar

ae

,

J. Kvapil

j

, J.H. Kwasizur

n

,

R. Lacey

an

,

J.M. Landgraf

c

,

K.D. Landry

f

,

J. Lauret

c

,

A. Lebedev

c

,

R. Lednicky

q

,

J.H. Lee

c

,

C. Li

ak

, W. Li

am

, Y. Li

as

,

X. Li

ak

,

J. Lidrych

j

,

T. Lin

n

,

M.A. Lisa

ac

, P. Liu

an

,

F. Liu

g

,

H. Liu

n

, Y. Liu

ap

,

T. Ljubicic

c

,

W.J. Llope

ba

,

M. Lomnitz

v

,

R.S. Longacre

c

, X. Luo

g

,

S. Luo

h

,

G.L. Ma

am

,

L. Ma

am

,

Y.G. Ma

am

,

R. Ma

c

,

N. Magdy

an

,

R. Majka

bc

,

D. Mallick

aa

,

S. Margetis

r

,

C. Markert

aq

,

H.S. Matis

v

,

K. Meehan

e

,

J.C. Mei

al

,

Z.W. Miller

h

,

N.G. Minaev

ag

, S. Mioduszewski

ap

,

D. Mishra

aa

, S. Mizuno

v

,

B. Mohanty

aa

, M.M. Mondal

m

,

D.A. Morozov

ag

,

M.K. Mustafa

v

,

Md. Nasim

f

,

T.K. Nayak

ay

,

J.M. Nelson

d

,

M. Nie

am

,

G. Nigmatkulov

z

,

T. Niida

ba

,

L.V. Nogach

ag

,

T. Nonaka

at

,

S.B. Nurushev

ag

,

G. Odyniec

v

,

A. Ogawa

c

,

K. Oh

ai

,

V.A. Okorokov

z

,

D. Olvitt Jr.

ao

,

B.S. Page

c

,

R. Pak

c

,

Y. Pandit

h

, Y. Panebratsev

q

,

B. Pawlik

ad

,

H. Pei

g

,

C. Perkins

d

,

P. Pile

c

,

J. Pluta

az

,

K. Poniatowska

az

,

J. Porter

v

, M. Posik

ao

,

E-mailaddress:xfluo@mail.ccnu.edu.cn(X. Luo).

https://doi.org/10.1016/j.physletb.2018.07.066

N.K. Pruthi

ae

,

M. Przybycien

a

,

J. Putschke

ba

, H. Qiu

ah

,

A. Quintero

ao

, S. Ramachandran

s

,

R.L. Ray

aq

, R. Reed

w

,

M.J. Rehbein

i

, H.G. Ritter

v

,

J.B. Roberts

aj

,

O.V. Rogachevskiy

q

,

J.L. Romero

e

,

J.D. Roth

i

, L. Ruan

c

, J. Rusnak

k

,

O. Rusnakova

j

,

N.R. Sahoo

ap

,

P.K. Sahu

m

,

S. Salur

v

, J. Sandweiss

bc

, A. Sarkar

bd

,

M. Saur

k

,

J. Schambach

aq

, A.M. Schmah

v

,

W.B. Schmidke

c

,

N. Schmitz

x

,

B.R. Schweid

an

, J. Seger

i

,

M. Sergeeva

f

,

R. Seto

av

,

P. Seyboth

x

,

N. Shah

am

,

E. Shahaliev

q

,

P.V. Shanmuganathan

w

,

M. Shao

ak

,

M.K. Sharma

p

,

A. Sharma

p

,

W.Q. Shen

am

,

Z. Shi

v

, S.S. Shi

g

, Q.Y. Shou

am

,

E.P. Sichtermann

v

,

R. Sikora

a

,

M. Simko

k

,

S. Singha

r

,

M.J. Skoby

n

, D. Smirnov

c

,

N. Smirnov

bc

,

W. Solyst

n

,

L. Song

ar

,

P. Sorensen

c

,

H.M. Spinka

b

,

B. Srivastava

ah

,

T.D.S. Stanislaus

ax

,

M. Strikhanov

z

,

B. Stringfellow

ah

,

T. Sugiura

at

,

M. Sumbera

k

, B. Summa

af

,

X.M. Sun

g

, Y. Sun

ak

,

X. Sun

g

,

B. Surrow

ao

,

D.N. Svirida

o

,

A.H. Tang

c

,

Z. Tang

ak

,

A. Taranenko

z

, T. Tarnowsky

y

,

A. Tawfik

bb

,

J. Thäder

v

,

J.H. Thomas

v

, A.R. Timmins

ar

, D. Tlusty

aj

,

T. Todoroki

c

,

M. Tokarev

q

,

S. Trentalange

f

,

R.E. Tribble

ap

,

P. Tribedy

c

,

S.K. Tripathy

m

,

B.A. Trzeciak

j

,

O.D. Tsai

f

,

T. Ullrich

c

,

D.G. Underwood

b

,

I. Upsal

ac

,

G. Van Buren

c

,

G. van Nieuwenhuizen

c

,

A.N. Vasiliev

ag

,

F. Videbæk

c

,

S. Vokal

q

,

S.A. Voloshin

ba

,

A. Vossen

n

,

F. Wang

ah

,

Y. Wang

g

, G. Wang

f

,

Y. Wang

as

,

J.C. Webb

c

, G. Webb

c

,

L. Wen

f

,

G.D. Westfall

y

,

H. Wieman

v

, S.W. Wissink

n

,

R. Witt

aw

,

Y. Wu

r

,

Z.G. Xiao

as

, G. Xie

ak

,

W. Xie

ah

,

Z. Xu

c

,

N. Xu

v

,

Y.F. Xu

am

,

Q.H. Xu

al

, J. Xu

g

,

Q. Yang

ak

, C. Yang

al

,

S. Yang

c

,

Y. Yang

ab

,

Z. Ye

h

,

Z. Ye

h

,

L. Yi

bc

,

K. Yip

c

,

I.-K. Yoo

ai

,

N. Yu

g

, H. Zbroszczyk

az

,

W. Zha

ak

,

X.P. Zhang

as

, S. Zhang

am

, J.B. Zhang

g

,

J. Zhang

v

,

Z. Zhang

am

,

S. Zhang

ak

, J. Zhang

u

,

Y. Zhang

ak

,

J. Zhao

ah

,

C. Zhong

am

,

L. Zhou

ak

, C. Zhou

am

,

Z. Zhu

al

,

X. Zhu

as

, M. Zyzak

l

a_{AGHUniversityofScienceandTechnology,FPACS,Cracow30-059,Poland} b_{ArgonneNationalLaboratory,Argonne,IL 60439}

c_{BrookhavenNationalLaboratory,Upton,NY 11973} d_{UniversityofCalifornia,Berkeley,CA 94720} e_{UniversityofCalifornia,Davis,CA 95616} f_{UniversityofCalifornia,LosAngeles,CA 90095} g_{CentralChinaNormalUniversity,Wuhan,Hubei430079} h_{UniversityofIllinoisatChicago,Chicago,IL 60607} i_{CreightonUniversity,Omaha,NE 68178}

j_{CzechTechnicalUniversityinPrague,FNSPE,Prague,11519,CzechRepublic} k_{NuclearPhysicsInstituteASCR,25068Prague,CzechRepublic}

l_{FrankfurtInstituteforAdvancedStudiesFIAS,Frankfurt60438,Germany} m_{InstituteofPhysics,Bhubaneswar751005,India}

n_{IndianaUniversity,Bloomington,IN 47408}

o_{AlikhanovInstituteforTheoreticalandExperimentalPhysics,Moscow117218,Russia} p_{UniversityofJammu,Jammu180001,India}

q_{JointInstituteforNuclearResearch,Dubna,141980,Russia} r_{KentStateUniversity,Kent,OH 44242}

s_{UniversityofKentucky,Lexington,KY 40506-0055} t_{LamarUniversity,PhysicsDepartment,Beaumont,TX 77710}

u_{InstituteofModernPhysics,ChineseAcademyofSciences,Lanzhou,Gansu730000} v_{LawrenceBerkeleyNationalLaboratory,Berkeley,CA 94720}

w_{LehighUniversity,Bethlehem,PA 18015}

x_{Max-Planck-InstitutfürPhysik,Munich80805,Germany} y_{MichiganStateUniversity,EastLansing,MI 48824}

z_{NationalResearchNuclearUniversityMEPhI,Moscow115409,Russia}

aa_{NationalInstituteofScienceEducationandResearch,Bhubaneswar751005,India} ab_{NationalChengKungUniversity,Tainan70101}

ac_{OhioStateUniversity,Columbus,OH 43210}

ad_{InstituteofNuclearPhysicsPAN,Cracow31-342,Poland} ae_{PanjabUniversity,Chandigarh160014,India} af_{PennsylvaniaStateUniversity,UniversityPark,PA 16802} ag_{InstituteofHighEnergyPhysics,Protvino142281,Russia} ah_{PurdueUniversity,WestLafayette,IN 47907}

ai_{PusanNationalUniversity,Pusan46241,RepublicofKorea} aj_{RiceUniversity,Houston,TX 77251}

ak_{UniversityofScienceandTechnologyofChina,Hefei,Anhui230026} al_{ShandongUniversity,Jinan,Shandong250100}

am_{ShanghaiInstituteofAppliedPhysics,ChineseAcademyofSciences,Shanghai201800} an_{StateUniversityofNewYork,StonyBrook,NY 11794}

ao_{TempleUniversity,Philadelphia,PA 19122} ap_{TexasA&MUniversity,CollegeStation,TX 77843} aq_{UniversityofTexas,Austin,TX 78712} ar_{UniversityofHouston,Houston,TX 77204} as_{TsinghuaUniversity,Beijing100084}

au_{SouthernConnecticutStateUniversity,NewHaven,CT 06515} av_{UniversityofCalifornia,Riverside,CA 92521}

aw_{UnitedStatesNavalAcademy,Annapolis,MD 21402} ax_{ValparaisoUniversity,Valparaiso,IN 46383}

ay_{VariableEnergyCyclotronCentre,Kolkata700064,India} az_{WarsawUniversityofTechnology,Warsaw00-661,Poland} ba_{WayneStateUniversity,Detroit,MI 48201}

bb_{WorldLaboratoryforCosmologyandParticlePhysics(WLCAPP),Cairo11571,Egypt} bc_{YaleUniversity,NewHaven,CT 06520}

bd_{IndianInstituteofTechnology,Mumbai400076,India}

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Articlehistory:

Received2September2017 Receivedinrevisedform10July2018 Accepted10July2018

Availableonline23August2018 Editor:V.Metag

Fluctuations ofconserved quantitiessuch as baryonnumber, charge, and strangeness are sensitive to the correlationlengthofthehot and densemattercreatedinrelativisticheavy-ioncollisions and can be used to search for the QCD critical point. We report the first measurements of the moments of net-kaonmultiplicitydistributionsinAu+Aucollisions at√sNN=7.7,11.5,14.5,19.6,27,39,62.4,and 200 GeV.Thecollisioncentralityandenergydependenceofthemean(M),variance(

σ

2_{),skewness(S),} andkurtosis(

κ

)fornet-kaonmultiplicitydistributionsaswellastheratio

σ

2_{/M andtheproductsS}

_σ

and

κσ

2_{arepresented.ComparisonsaremadewithPoissonandnegativebinomialbaselinecalculations} aswellaswithUrQMD,atransportmodel(UrQMD)thatdoesnotincludeeffectsfromtheQCDcritical point.Withincurrentuncertainties,thenet-kaoncumulantratiosappeartobemonotonicasafunction ofcollisionenergy.

©2018TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Oneprimary goalofhighenergyheavy-ioncollisions isto ex-plorethephasestructureofstronglyinteractinghot,densenuclear matter.Itcanbedisplayedinthequantumchromodynamics(QCD) phasediagram,whichischaracterizedbythetemperature(T )and thebaryonchemical potential (

μ

B).LatticeQCDcalculations sug-gestthatthephasetransitionbetweenthehadronicphaseandthe quark–gluonplasma (QGP)phaseatlarge

μ

B andlow T isofthe firstorder[1,2],whileinthelow

μ

B andhighT region,thephase transitionisasmoothcrossover[3].Theendpoint ofthefirst or-derphase boundarytowardsthe crossoverregionisthe so-called critical point [4,5]. Experimental search for the critical point is one ofthe central goals ofthe beamenergy scan (BES) program attheRelativisticHeavy-IonCollider(RHIC)facilityatBrookhaven NationalLaboratory.

Fluctuations of conserved quantities, such as baryon number (B), charge ( Q ), and strangeness (S) are sensitive to the QCD phase transition and QCD critical point [6–8]. Experimentally, one can measure the moments (mean (M), variance (

σ

2_),

skew-ness (S),andkurtosis(

κ

))oftheevent-by-eventnet-particle distri-butions(particlemultiplicity minusantiparticlemultiplicity),such asnet-proton,net-kaonandnet-chargemultiplicitydistributionsin heavy-ioncollisions.Thesemomentsareconnectedtothe thermo-dynamic susceptibilities that can be computed in lattice QCD [5, 9–15] andinthehadronresonancegasmodel(HRG)[16–19].They areexpectedtobesensitivetothecorrelationlength(ξ)ofthehot anddense medium created in the heavy-ion collisions [6]. Non-monotonic variation of fluctuations in conserved quantities with thecollidingbeamenergyisconsidered to beone ofthe charac-teristicsignatureoftheQCDcriticalpoint.

Themoments

σ

2_{, S,and}

_κ

_{havebeenshowntobe relatedto}

powers of the correlation length as

ξ

2,

ξ

4.5 and

ξ

7 [6], respec-tively.Thenth order susceptibilities

χ

(n) _{are relatedto cumulant}

as

χ

(n)

₌

_C

n

/

V T3 [8], where V

,

T are the volume and tempera-tureof the system, Cn is thenth order cumulant of multiplicity distributions. The moment products S

σ

and

κσ

2 _{and the ratio}

σ

2

_/

_{M are constructed to cancel the volume term. The moment}

products arerelatedtotheratiosofvarious ordersof susceptibil-ities according to

κσ

2

₌

_χ

(4) i

/

χ

(2) i and S

σ

=

χ

(3) i

/

χ

(2) i , where i indicatestheconservedquantity.Duetothesensitivitytothe cor-relation length andthe connection with the susceptibilities, one canusethemomentsofconserved-quantitydistributionstoaidin thesearchfortheQCDcriticalpointandtheQCDphasetransition [6–8,16,20–31]. In addition, the moments of net-particle fluctua-tionscanbeusedtodeterminefreeze-outpointsontheQCDphase diagrambycomparingdirectlytofirst-principlelatticeQCD calcu-lations [12]. Specifically, by comparingthe lattice QCD resultsto the measured

σ

2

_/

_{M for netkaons, one can infer the}

hadroniza-tiontemperatureofstrangequarks [32].

As apartofthe BES,Au+Au collisionswere run by RHICwith energies rangingfrom

√

sNN

=

200 GeVdownto 7.7GeV [33–35]

corresponding to

μ

B from20 to 420MeV. Inthis paper, we re-port the firstmeasurements for themoments ofnet-kaon multi-plicitydistributions inAu+Aucollisions at

√

sNN

=

7.7, 11.5,14.5,

19.6, 27,39,62.4,and200GeV. Theseresultsare comparedwith baseline calculations(Poissonandnegative binomial) andthe Ul-trarelativisticQuantum MolecularDynamics (UrQMD,version2.3) modelcalculations[36].

The manuscript is organized asfollows. In section 2, we de-finetheobservablesusedintheanalysis.Insection3,wedescribe theSTAR(SolenoidalTrackerAtRHIC)experimentatBNLandthe analysistechniques.Insection 4,we presenttheexperimental re-sultsforthemomentsofthenet-kaonmultiplicitydistributionsin Au+AucollisionsatRHICBESenergies.Asummaryisgivenin sec-tion5.

2. Observables

Distributions canbe characterizedby themoments M,

σ

2_{, S,}

and

κ

aswellasintermsofcumulantsC1,C2,C3,andC4 [37].

writ-Fig. 1. (Coloronline.) RawNK distributionsinAu+Aucollisionsfrom √sNN=7.7 to200GeVfor 0–5%,30–40%,and70–80%collisioncentralitiesatmidrapidity.The distributionsarenotcorrectedforthefinitecentralitybinwidtheffectnorthereconstructionefficiency.

ten as

δ

N

=

N

− 

N



. The various order cumulants of event-by-eventdistributionsofN aredefinedas:

C1

= 

N



(1)

C2

= (δ

N

)

2



(2)

C3

= (δ

N

)

3



(3)

C4

= (δ

N

)

4

 −

3

(δ

N

)

2



2 (4)

Themomentscanbewrittenintermsofthecumulantsas: M

=

C1

,

σ

2

=

C2

,

S

=

C3

(

C2

)

3 2

,

κ

=

C4

(

C2

)

2 (5)

Inaddition,theproducts ofmoments

κσ

2 _{and S}

_σ

_canbe

ex-pressedintermsofcumulantratios:

κσ

2

=

C4 C2

,

S

σ

=

C3 C2 (6) 3. Dataanalysis

Theresultspresentedinthispaperarebasedonthedatataken atSTAR[38] for Au+Au collisions at

√

sNN

=

7.7,11.5, 14.5, 19.6,

27,39,62.4and200GeV.The7.7,11.5,39,62.4,and200GeVdata were collected inthe year2010, the 19.6and27GeV datawere collected intheyear2011, andthe 14.5GeVdatawere collected intheyear2014.

The STAR detector has a large uniform acceptance at midra-pidity (

|

η

|

<

1) with excellent particle identification capabilities, i.e., allowing to identify kaons from other charged particles for 0.2

<

pT

<

1.6 GeV/c. Energy loss(dE/dx) in thetime projection chamber(TPC)[39] andmass-squared(m2)fromthetime-of-flight detector(TOF)[40] areusedtoidentifyK+ andK−.Toutilizethe energylossmeasuredintheTPC,aquantityn

σ

X isdefinedas:

n

σ

X

=

ln

[(

dE

/

dx

)

measured

/(

dE

/

dx

)

theory

]

σ

X

(7)

where

(

dE

/

dx

)

measured istheionization energylossfromTPC, and

(

dE

/

dx

)

theoryistheBethe–Bloch[41] expectationforthegiven

par-ticle type (e.g.

π

,

K

,

p).

σ

X is the dE

/

dx resolution of TPC. We select K+ and K− particles by using a cut

|

n

σ

K aon

|

<

2 within transverse momentum range 0.2

<

pT

<

1.6 GeV/c and rapidity

|

y

|

<

0.5. The TOF detector measures the time of flight

(

t

)

of a particle fromthe primary vertexof the collision.Combined with thepathlength(L)measuredintheTPC,onecandirectlycalculate thevelocity

(

v

)

oftheparticlesandtheirrestmass

(

m

)

using:

β

=

v c

=

L ct (8) m2c2

=

p2



1

β

2

−

1



=

p2



c2t2 L2

−

1



(9)

In this analysis, we use mass-squared cut 0.15

<

m2

<

0.4 GeV2

_/

_c4 _toselectK+_andK−_{withinthe pT range0.4}

_<

_pT

_<

1.6 GeV/c togethighpurityofkaonsample(betterthan99%).For the pT range0.2

<

pT

<

0.4 GeV/c,weuseonlytheTPCto iden-tify K+ and K−. The kaon purity between 0.2 and0.4 GeV/c is about95%,whereonlyTPCisused.

The collision centrality is determined using the efficiency-uncorrectedchargedparticlemultiplicityexcludingidentifiedkaons within pseudorapidity

|

η

|

<

1.0 measuredwiththeTPC. This def-inition maximizesthe numberofparticles usedtodetermine the collisioncentralityandavoidsself-correlationsbetweenthekaons used to calculatethe moments andkaons in the reference mul-tiplicity [42]. Using the distribution of thisreferencemultiplicity along withthe Glaubermodel [43] simulations,thecollision cen-tralityisdetermined.Thisreferencemultiplicity issimilarin con-cept tothereferencemultiplicityusedbySTARtostudymoments of net-proton distributions [29], where the referencemultiplicity was calculated using all charged particles within

|

η

|

<

1.0 ex-cluding identified protons and antiprotons. Using this definition, collision centralitybinsof0–5%,5–10%,10–20%,20–30%,30–40%, 40–50%,50–60%,60–70%,and70–80%ofthemultiplicity distribu-tionswereusedwith0–5%representingthemostcentralcollisions. Fig. 1 shows the raw event-by-event net-kaon multiplicity (NK

=

NK+

−

NK−) distributions inAu+Au collisionsat

√

sNN

=

Fig. 2. (Coloronline.) CollisioncentralitydependenceofthepT-averagedefficienciesinAu+Aucollisions.ForthelowerpTrange(0.2<pT<0.4 GeV/c),onlytheTPCisused. ForthehigherpT range(0.4<pT<1.6 GeV/c),boththeTPCandTOFareusedforparticleidentification(PID).

centrality bin width correction [42]. Those distributions are not correctedfor the finitecentrality binwidth effectand alsotrack reconstructionefficiency.However,allthecumulantsandtheir ra-tiospresentedin thispaperarecorrected forthefinitecentrality binwidtheffectandefficiencyofK+andK−.

Themomentsandcumulantscanbeexpressedintermsof fac-torial moments, which can be easily corrected for efficiency [45, 46]. The efficiency correction is done by assuming the response function of the efficiency is a binomial probability distribution. Fig.2showsthecollisioncentralitydependenceofthepT-averaged

efficiencies of tracking and PID combined for two pT ranges.

One can seethat atthe lower pT range(0.2

<

pT

<

0.4 GeV/c), kaonshavealowerefficiencycomparedwiththehigher pT range (0.4

<

pT

<

1.6 GeV/c). The efficiencies increase monotonically withthe centrality changing frommost central (0

∼

5%) to pe-ripheral(70

∼

80%). K+andK−havesimilarefficiencies.

By calculating the covariance between the various order fac-torial moments, one can obtain the statistical uncertainties for theefficiencycorrected momentsbasedon theerrorpropagation derived from the Deltatheorem [42,46,47]. The statistical uncer-tainties ofvarious order cumulants andcumulant ratios strongly depend on the width(

σ

) of the measured multiplicity distribu-tionsaswellastheefficiencies(

ε

).Onecanroughly estimatethe statistical uncertainties of S

σ

and

κσ

2 _{as error}

₍

_S

_σ

₎

_∝

σ

ε3/2 and

error

(

κσ

2

₎

_∝

σ2

ε2.That explainswhy we observe largerstatistical uncertaintiesforcentralthanperipheralcollisions,asonthewidth ofthenet-kaondistributionsgrowsfromperipheraltocentral. Fur-thermore,duetothesmallerdetectionefficiencyofkaonsthanthe protons, we observe larger statistical uncertainties of cumulants andcumulantratiosthanthoseofthenet-protonfluctuations [29]. Systematic uncertainties are estimated by varying the following trackquality cuts:distanceofclosestapproach,thenumberoffit pointsusedintrackreconstruction,thedE

/

dx selectioncriteriafor identification,andadditional5%uncertaintiesinthereconstruction efficiency.Thetypicalsystematicuncertainties areoftheorderof 15%forC1 andC2,21%forC3,and65%forC4.Thestatisticaland

systematic(caps)errorsarepresentedseparatelyinthefigures. 4. HIJING+GEANTsimulations

Toevaluateourmethodsofdataanalysis,wehavedonea stan-dard STARGEANTsimulation withrealistic detector environment

Fig. 3. (Coloronline.) Transversemomentumdependenceoftheefficiency(tracking+ acceptance)forK+and K− inAu+Aucollisionsat19.6GeVfromHIJING+GEANT simulation.

andinput fromHIJINGmodel.We generated4.8millionmini-bias HIJINGeventsforAu+Aucollisionsat

√

sNN

=

19.6 GeVandpassed

theparticlesfromHIJINGeventsthroughSTARdetectorsimulated by usingGEANTframework. The trackreconstruction inthe sim-ulation is done withthe sametracking algorithm asused in the STARexperiment.Wealso appliedthesametrackselection crite-ria, kinematiccuts(0.2

<

pT

<

1.6 GeV,

|

y

|

<

0.5)for the recon-structedkaons(K+and K−)asweusedintherealdataanalysis. Toavoidauto-correlation,themultiplicitiesofchargedprotonsand pionswithin pseudo-rapidityrange

|

η

|

<

1 areusedtodefine the collisioncentralities.Thecentralitybinwidthcorrectionisalso ap-plied to suppressthe volume fluctuationswithin wide centrality bins. As thecharged particleionization energy lossdE

/

dx in the TPCgasisnotsimulatedwithsufficientdetailsinthecurrentSTAR simulations andthe time-of-flight (TOF) detectorsare not imple-mentedintheSTARGeant,we thusdidn’tincludedE/dxandTOF identification efficiencies inthe simulations. The matching of re-constructed and simulated tracks is determined by requiring at least 10shared commonhit points. By doing the simulation,we cantest theanalysismethodsaswell asthe detectorresponse in termsofthetrackingperformancewiththerealistictracking envi-ronmentandalgorithm.

Fig. 3 shows the transverse momentum (pT) dependence of

Fig. 4. (Coloronline.) Event-by-eventnet-kaonmultiplicity distributionsfor0–5% Au+Aucollisionsat√sNN=19.6 GeVfromHIJINGinputandHIJING+GEANT, respec-tively.

in HIJING+GEANT simulation for Au+Au collisions at

√

sNN

=

19.6 GeV. These efficiencies are obtainedby takingthe ratio be-tweenthekaonpT spectrafromHIJING+GEANTandHIJINGinput, viatheformula:



ε

 =

 

_dN dpT



RCdpT

 

_dN dpT



MCdpT (10)

where the RC and MC represent the tracks reconstructed from

HIJING+GEANT and from HIJING input, respectively. Those aver-ageefficiencies for K+ and K− arecalculated within 0.2

<

pT

<

1.6 GeV/c,

|

y

|

<

0.5 at nine centralities in Au+Au collisions at

√

sNN

=

19.6 GeVandarewiththesimilar(

∼

50%–55%)valuesas

theTPCtrackingefficienciesobtainedfromSTARembedding simu-lation.

Fig.4showstheevent-by-eventnet-kaonmultiplicity distribu-tions for0–5% Au+Aucollisions at

√

sNN

=

19.6 GeV fromHIJING

input andHIJING+GEANT, respectively. Due to the reconstruction efficiency loss of the kaons, the mean values and the width of net-kaon distributions becomesmaller than thatof HIJING input. Basedonthesimulations,wehavedoneabinomialtestforthe re-sponsefunctionofkaonefficiencies.Byfixingtheinputnumberof

K+,wefittedtheevent-by-eventnumberofreconstructedK+ dis-tributions withbinomial distribution function (reddashed lines). ThefittingresultsareshowninFig.5andresultsindifferent pan-els representdifferentinput numberof K+,whichisvariedfrom 30to 22andcorresponds tothe0–10% collisioncentrality. Fig.6 showsthe centralitydependence ofthe cumulantsandcumulant ratios ofnet-kaonmultiplicity distributionsinAu+Aucollisions at

√

sNN

=

19.6 GeVfromtheHIJING+GEANTsimulations.Thered

cir-clesrepresenttheresultsfromHIJINGinput.Thebluecrossesand black starsrepresenttheefficiencycorrected anduncorrected re-sults, respectively. The efficiency correction is done by using the analytical formula derived from factorial moment assuming bi-nomial response function of the kaon efficiencies [45,46]. With this simplified simulation approach, the efficiency corrected cu-mulant andcumulant ratiosare consistent with the resultsfrom the originalHIJING INPUTfornet-kaons.The efficiencycorrection proceduretends tosignificantlyincreasethestatisticalerrors.The STARcollaborationiscurrentlyexploringotherfunctionsincluding abeta-binomialdistributiontodescribetheefficiencies.Unfolding

Fig. 6. (Coloronline.) Cumulants(1–4th order)ofnet-kaondistributionsinAu+Aucollisionsat√sNN=19.6 GeVfromHIJING+Geantsimulations.Theredcirclerepresents theresultsfromHIJINGINPUT.ThebluecrossesandblackstarsaretheresultsofefficiencycorrectedanduncorrectedresultsafterHIJINGeventspassingthroughtheGEANT simulationwithrealisticSTARdetectorenvironment.

Fig. 7. (Coloronline.) Collisioncentralitydependenceofcumulants(C1,C2,C3,andC4)ofNK distributionsforAu+Aucollisionsat√sNN=7.7–200 GeV.Theerrorbarsare statisticaluncertaintiesandthecapsrepresentsystematicuncertainties.ThePoissonandNBDexpectationsareshownasdashedandbluesolidlines,respectively.

methodsarebeingevaluated ascorrectionmethodforefficiencies includingthose from PID cuts andtracking [48]. The PID effects from the TPC and the TOF are not currently simulated through GEANT.These improvements will be necessary forthe beam en-ergyscan phaseIIprogramwhereanorderofmagnitudeincrease inthedatasampleisexpected.

5. Results

Fig. 7 showsthe centrality dependence of cumulants (C1–C4)

of net-kaon (NK) multiplicity distributions in Au+Au collisions at

√

sNN

=

7.7–200 GeV.Thecollisionscentralitiesarerepresented

Fig. 8. Collision centrality dependence of M/σ2_forN

K distributions in Au+Au collisions at√sNN=7.7–200 GeV. The Poisson expectations are shown as dashed lines.

Fig. 9. (Coloronline.) CollisioncentralitydependenceoftheSσforNKdistributionsfromAu+Aucollisionsat√sNN=7.7–200 GeV.Theerrorbarsarestatistical uncertain-tiesandthecapsrepresentsystematicuncertainties.ThePoisson(dashedline)andNBD(bluesolidline)expectationsarealsoshown.

are obtainedby Glauber model simulation.The efficiency correc-tionshavebeendoneusingthevaluesshowninFig.2.Ingeneral, the various order cumulants show a nearly linear variation with



Npart



,whichcanbeunderstoodastheadditivitypropertyofthe

cumulantsbyincreasingthevolumeofthesystem.Thisreflectsthe fact that the cumulants are extensivequantities that are propor-tionaltothesystemvolume.ThedecreaseoftheC1andC3 values

with increasing collision energy indicates that the ratio K+

/

K−

approachesunityforthehighercollisionenergies.Fig.7alsoshows the Poissonandnegative binomial distribution (NBD) [49,50] ex-pectations.ThePoissonbaselineisconstructedusingthemeasured meanvaluesofthemultiplicitydistributionsofK+andK−,while theNBDbaseline isconstructed usingbothmeans andvariances. Assumingthattheevent-by-eventmultiplicitiesofK+andK−are independentrandom variables,thePoissonandNBDassumptions provide references for the moments of the net-kaon multiplicity distributions. Within uncertainties, the measured cumulants val-uesof C3 andC4 are consistent withboth the Poissonand NBD

baselinesformostcentralities.

Theratiosbetweendifferentordercumulantsaretakento can-cel the volume dependence.Figs. 8, 9, and 10 show the



Npart



dependence of



NK distributions for cumulant ratios C1

/

C2

(

=

M

/

σ

2_{), C}

3

/

C2 (

=

S

σ

), and C4

/

C2 (

=

κσ

2), respectively. The

valuesofC1

/

C2,showninFig.8,systematicallydecreasewith

in-creasing collision energyfor all centralities. The Poissonbaseline forC1

/

C2 slightlyunderestimatesthedata,indicatingpossible

cor-relations between K+ and K− production. For C3

/

C2 (

=

S

σ

) in

Fig.9,thePoissonandNBDexpectationsareobservedtobelower than themeasured S

σ

valuesatlowcollision energies.The mea-suredvaluesforC4

/

C2(

=

κσ

2)inFig.10areconsistentwithboth

thePoissonandNBDbaselineswithinuncertainties.

The collision energy dependence of the cumulant ratios for



NK distributions in Au+Au collisions are presented in Fig. 11. The results are shown in two collision centrality bins, one cor-responding to most central (0–5%) and the other to peripheral (70–80%)collisions.ExpectationsfromthePoissonandNBD base-linesarederivedforcentral(0–5%)collisions.ThevaluesofM

/

σ

2

decreaseasthecollision energyincreases,andare largerfor cen-tral collisions compared with the peripheral collisions. For most centralcollisions, thePoissonbaselineforC1

/

C2 slightly

underes-timates thedata.Withinuncertainties,thevaluesof S

σ

and

κσ

2

areconsistentwithboththePoissonandNBDbaselinesincentral collisions.Thebluebandsgive theresultsfromtheUrQMDmodel calculationsforcentral (0–5%)Au+Aucollisions. The widthofthe bandsrepresentsthestatisticaluncertainties.TheUrQMD calcula-tions for S

σ

, and

κσ

2 _{are consistent withthe measured values}

Fig. 10. (Coloronline.) Collisioncentrality dependenceoftheκσ2 _forN

K distributionsfrom Au+Aucollisionsat√sNN=7.7–200 GeV.Theerrorbarsarestatistical uncertaintiesandthecapsrepresentsystematicuncertainties.ThePoisson(dashed-line)andNBD(blue-solid-line)expectationsarealsoshown.

Fig. 11. (Coloronline.) Collisionenergy dependenceofthevaluesof M/σ2_{, Sσ,}

κσ2 _for

NK multiplicitydistributionsfrom 0–5%mostcentraland70–80% pe-ripheralcollisionsinAu+Aucollisionsat√sNN=7.7,11.5,14.5,19.6,27,39,62.4 and200 GeV.Theerrorbarsarestatisticaluncertaintiesandthecapsrepresent sys-tematicuncertainties.TheexpectationsfromPoissonandNBDandtheresultsofthe UrQMDmodelcalculationsareallfromthe0–5%centrality.

ofthe net-kaon (NK) fluctuationsis less than that of the net-proton (Np) [53]. A much highstatistics dataset is neededfor thesearchoftheQCDcriticalpointwithstrangeness.

6. Summary

In heavy-ion collisions, fluctuations of conserved quantities, such asnet-baryon,net-charge andnet-strangeness numbers, are sensitiveobservablestosearchfortheQCDcriticalpoint.Nearthe QCDcriticalpoint,thosefluctuationsareexpectedtohavesimilar

energydependencebehavior.Experimentally,theSTARexperiment haspublishedtheenergydependenceofthenet-proton(proxyfor net-baryon) [29] andnet-charge [30] fluctuationsin Au+Au colli-sions from the first phase of the beam energy scan at RHIC. In

this paper, we present the first measurements of the moments

of net-kaon (proxy for net-strangeness) multiplicity distributions in Au+Au collisions from

√

sNN

=

7.7 to 200 GeV. The measured

M

/

σ

2 _{valuesdecreasemonotonicallywithincreasingcollision}

en-ergy. The Poisson baseline for C1

/

C2 slightly underestimates the

data.Nosignificantcollisioncentralitydependenceisobservedfor both S

σ

and

κσ

2 _{atall energies. ForC}

3

/

C2 (

=

S

σ

),the Poisson

andNBDexpectationsarelower thanthemeasured S

σ

valuesat lowcollisionenergies.ThemeasuredvaluesforC4

/

C2(

=

κσ

2)are

consistentwithboththePoissonandNBDbaselineswithin uncer-tainties.UrQMDcalculationsfor S

σ

and

κσ

2 _{areconsistent with}

data for the most central 0–5% Au+Au collisions. Within current uncertainties, the net-kaon cumulant ratios appear to be mono-tonicasa function ofcollision energy.The moments ofnet-kaon multiplicity distributions presented here can be used to extract freeze-outconditionsinheavy-ioncollisions bycomparingto Lat-tice QCD calculations. Future high statistics measurements with improved efficiency correction method will be made for fluctua-tion studies in the second phase ofthe RHIC Beam Energy Scan during2019–2020.

Acknowledgements

We thank the RHIC Operations Group and RCF at BNL, the

NERSC Center at LBNL, and the Open Science Grid consortium

SportsoftheRepublicofCroatia,ROSATOM ofRussiaandGerman BundesministeriumfurBildung,Wissenschaft,Forschungand Tech-nologie(BMBF)andtheHelmholtzAssociation.

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