Identical pion intensity interferometry in central Au + Au collisions at 1.23A GeV

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Physics Letters B

www.elsevier.com/locate/physletb

Identical pion intensity interferometry in central Au + Au collisions at 1.23 A GeV

HADES Collaboration

J. Adamczewski-Musch

^d

, O. Arnold

^j,i

, C. Behnke

^h

, A. Belounnas

^p

, A. Belyaev

^g

, J.C. Berger-Chen

^j,i

, J. Biernat

^c

, A. Blanco

^b

, C. Blume

^h

, M. Böhmer

^j

, P. Bordalo

^b

,

S. Chernenko

^g,1

, L. Chlad

^q

, C. Deveaux

^k

, J. Dreyer

^f

, A. Dybczak

^c

, E. Epple

^j,i

, L. Fabbietti

^j,i

, O. Fateev

^g

, P. Filip

^a

, P. Fonte

^b,2

, C. Franco

^b

, J. Friese

^j

, I. Fröhlich

^h

, T. Galatyuk

^e,d

,

J.A. Garzón

^r

, R. Gernhäuser

^j

, M. Golubeva

^l

, R. Greifenhagen

^f,3

, F. Guber

^l

,

M. Gumberidze

^d,4

, S. Harabasz

^e,c

, T. Heinz

^d

, T. Hennino

^p

, S. Hlavac

^a

, C. Höhne

^k,d

, R. Holzmann

^d

, A. Ierusalimov

^g

, A. Ivashkin

^l

, B. Kämpfer

^f,3

, T. Karavicheva

^l

, B. Kardan

^h

, I. Koenig

^d

, W. Koenig

^d

, B.W. Kolb

^d

, G. Korcyl

^c

, G. Kornakov

^e

, R. Kotte

^f

, A. Kugler

^q

, T. Kunz

^j

, A. Kurepin

^l

, A. Kurilkin

^g

, P. Kurilkin

^g

, V. Ladygin

^g

, R. Lalik

^c

, K. Lapidus

^j,i

, A. Lebedev

^m

, L. Lopes

^b

, M. Lorenz

^h

, T. Mahmoud

^k

, L. Maier

^j

, A. Mangiarotti

^b

, J. Markert

^d

, T. Matulewicz

^s

, S. Maurus

^j

, V. Metag

^k

, J. Michel

^h

, D.M. Mihaylov

^j,i

, S. Morozov

^l,n

, C. Müntz

^h

, R. Münzer

^j,i

, L. Naumann

^f

, K. Nowakowski

^c

, M. Palka

^c

, Y. Parpottas

^o,5

, V. Pechenov

^d

, O. Pechenova

^d

, O. Petukhov

^l

, K. Piasecki

^s

, J. Pietraszko

^d

, W. Przygoda

^c

, S. Ramos

^b

, B. Ramstein

^p

, A. Reshetin

^l

, P. Rodriguez-Ramos

^q

, P. Rosier

^p

, A. Rost

^e

, A. Sadovsky

^l

, P. Salabura

^c

, T. Scheib

^h

, H. Schuldes

^h

, E. Schwab

^d

, F. Scozzi

^e,p

, F. Seck

^e

, P. Sellheim

^h

, I. Selyuzhenkov

^d,n

, J. Siebenson

^j

, L. Silva

^b

, Yu.G. Sobolev

^q

, S. Spataro

^t

, S. Spies

^h

, H. Ströbele

^h

, J. Stroth

^h,d

, P. Strzempek

^c

, C. Sturm

^d

, O. Svoboda

^q

, M. Szala

^h

, P. Tlusty

^q

, M. Traxler

^d

, H. Tsertos

^o

, E. Usenko

^l

, V. Wagner

^q

, C. Wendisch

^d

, M.G. Wiebusch

^h

, J. Wirth

^j,i

, D. Wójcik

^s

, Y. Zanevsky

^g,1

, P. Zumbruch

^d

aInstituteofPhysics,SlovakAcademyofSciences,84228Bratislava,Slovakia

bLIP-LaboratóriodeInstrumentaçãoeFísicaExperimentaldePartículas,3004-516Coimbra,Portugal cSmoluchowskiInstituteofPhysics,JagiellonianUniversityofCracow,30-059Kraków,Poland dGSIHelmholtzzentrumfürSchwerionenforschungGmbH,64291Darmstadt,Germany eTechnischeUniversitätDarmstadt,64289Darmstadt,Germany

fInstitutfürStrahlenphysik,Helmholtz-ZentrumDresden-Rossendorf,01314Dresden,Germany gJointInstituteofNuclearResearch,141980Dubna,Russia

hInstitutfürKernphysik,Goethe-Universität,60438Frankfurt,Germany iExcellenceCluster‘OriginandStructureoftheUniverse’,85748Garching,Germany jPhysikDepartmentE62,TechnischeUniversitätMünchen,85748Garching,Germany kII.PhysikalischesInstitut,JustusLiebigUniversitätGiessen,35392Giessen,Germany lInstituteforNuclearResearch,RussianAcademyofScience,117312Moscow,Russia mInstituteofTheoreticalandExperimentalPhysics,117218Moscow,Russia

nNationalResearchNuclearUniversityMEPhI(MoscowEngineeringPhysicsInstitute),115409Moscow,Russia oDepartmentofPhysics,UniversityofCyprus,1678Nicosia,Cyprus

pInstitutdePhysiqueNucléaire,CNRS-IN2P3,Univ.Paris-Sud,UniversitéParis-Saclay,F-91406OrsayCedex,France

E-mailaddress:hades-info@gsi.de(J. Stroth).

1 Deceased.

2 AlsoatCoimbraPolytechnic- ISEC,Coimbra,Portugal.

3 AlsoatTechnischeUniversitätDresden,01062 Dresden,Germany.

4 AlsoatExtreMeMatterInstituteEMMI,64291 Darmstadt,Germany.

5 AlsoatFrederickUniversity,1036 Nicosia,Cyprus.

https://doi.org/10.1016/j.physletb.2019.06.047

0370-2693/©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

qNuclearPhysicsInstitute,TheCzechAcademyofSciences,25068Rez,CzechRepublic rLabCAF.F.Física,Univ.deSantiagodeCompostela,15706SantiagodeCompostela,Spain sUniwersytetWarszawski- InstytutFizykiDo´swiadczalnej,02-093Warszawa,Poland tDipartimentodiFisicaandINFN,UniversitàdiTorino,10125 Torino,Italy

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received15February2019

Receivedinrevisedform14June2019 Accepted24June2019

Availableonline27June2019 Editor:D.F.Geesaman

We investigate identicalpion HBTintensity interferometryin central Au + Aucollisions at 1.23 A GeV.

High-statistics π⁻π⁻ andπ⁺π⁺ data aremeasuredwithHADESatSIS18/GSI.Theradiusparameters, derivedfromthecorrelationfunctiondependingonrelativemomentainthelongitudinallycomovingsys- temandparametrizedasthree-dimensionalGaussiandistribution,arestudiedasfunctionoftransverse momentum.Asubstantialcharge-signdifferenceofthesourceradiiisfound,particularlypronouncedat lowtransversemomentum.Theextractedsourceparametersagreewellwithasmoothextrapolationof thecenter-of-massenergydependenceestablishedathigherenergies,extendingthecorrespondingexci- tationfunctionsdowntowardsaverylowenergy.

©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

Two-particleintensityinterferometryofhadronsiswidelyused to study the spatio-temporal size, shape and evolution of their sources created inheavy-ion collisions orother reactions involv- ing hadrons(for a review seeRef. [1]). The technique, pioneered byHanburyBrownandTwiss[2] tomeasureangularradiiofstars, later on named HBT interferometry, is based on the quantum- statistical interference of identical particles. Goldhaber et al. [3]

firstappliedintensityinterferometrytohadrons.Inheavy-ioncol- lisions, the intensity interferometry does not allow to measure directlythereaction volume,asthe emissionsource, changingin shapeand sizein the courseof the collision,is affectedby den- sityandtemperaturegradients anddynamicallygeneratedspace- momentum correlations(e.g. radial expansion afterthe compres- sion phase or resonance decays). Thus, intensity interferometry generallydoesnot yield thepropersource size,butratheran ef- fective“lengthofhomogeneity”[1].Itmeasures sourceregionsin whichparticlepairsarecloseinmomentum,sothatthey arecor- relatedasaconsequenceoftheirquantumstatisticsorduetotheir two-bodyinteraction.Ingeneral, thesignandstrengthofthecor- relationisaffected by (i) thestrong interaction,(ii) the Coulomb interactionifchargedparticlesareinvolved,and(iii) thequantum statisticsinthecaseofidenticalparticles(Fermi–Diracsuppression forfermions,Bose–Einsteinenhancement forbosons).Inthecase of

π π

correlations,themutualstronginteractionwasfoundtobe minor[4] comparedtotheeffects (ii)and (iii).

Pionfreeze-outdynamicsmaybe relevanttoongoingsearches forthe QCD critical point in the T –

μ

B plane, where T and

μ

B are the temperature and the baryon-chemical potential. Systems with

μ

B abovethe criticalpointare expectedtoundergo afirst- orderphasetransitionwhichmightbevisibleinanon-monotonic behavior of various source parameters. However, it is also con- ceivablethattheinitialtemperaturesofthesystem,whichcanbe reachedin heavy-ioncollisions athigh

μ

B,are not high enough tocreateadeconfinedpartonicstate.Inthisscenarioafirstorder phaseboundarycannotbereachedexperimentally.Arecentlypub- lishedexcitationfunctionofHBTsourceradii[5] fromthedomain oftheRelativistic Heavy IonCollider(RHIC) downto lower colli- sionenergiesindicates sucha non-monotonicenergydependence aroundcenter-of-massenergiesof√

sNN<10 GeV.Eventhougha partof thisbehavior can be relatedto the strong impact ofdif- ferentpairtransversemomentum intervalsinvolvedinthesource parametercompilationofRef. [5],toacertainextentthedeviation ofthe datapoints froma monotonic trend remains atlow ener- gies.Here,newprecisiondata,especiallyatlow collisionenergies of√

s_NN<5 GeV,cancontribute tothe clarificationofthisexcit-

ingobservationbeforedefiniteconclusionsonachangeinphysics canbedrawn.

It is worth emphasizing that only preliminary data [6] of identical-pion HBT dataexist fora largesymmetric collision sys- tem (like Au + Au or Pb + Pb) at a beam kinetic energy of about 1 A GeV (fixedtarget,√

s_NN=².3 GeV).⁶Forthesomewhatsmaller system La + La, studied at 1.2 A GeV with the HISS spectrometer at the Lawrence Berkeley Laboratory (LBL) Bevalac, pioncorrela- tion data were reported by Christie etal. [7,8]. An oblate shape of the pionsource and a correlation ofthe source size with the system size were found. Also, pion intensity interferometry for smallsystems (Ar + KCl,Ne + NaF)was studiedat1.8 A GeV atthe LBL Bevalac usingthe Janus spectrometer by Zajcet al.[9]. Both groups madefirst attempts to correctthe influence of the pion- nuclear Coulombinteraction onthe pionmomenta.Theeffecton the sourceradii, however,were foundnegligible fortheir experi- ments.

In this letter we report on the first investigation of

π

⁻

π

⁻

and

π

⁺

π

⁺ correlationsatlow relativemomenta inAu + Au colli- sionsat1.23 A GeV,continuingourpreviousfemtoscopicstudiesof smallercollisionssystems[10–12].Theexperimentwasperformed with the High Acceptance Di-Electron Spectrometer (HADES) at theSchwerionensynchrotronSIS18atGSI,Darmstadt.HADES[13], although primarily optimizedto measure di-electrons[14],offers alsoexcellent hadronidentificationcapabilities[15–18].HADES is achargedparticledetectorconsistingofasix-coiltoroidalmagnet centeredaroundthebeamaxisandsixidenticaldetectionsections locatedbetweenthecoils andcovering polaranglesbetween18^◦ and85^◦.Eachsector is equippedwithaRing-Imaging Cherenkov (RICH) detector followed by four layers of Mini-Drift Chambers (MDCs), two in front of and two behind the magnetic field, as well asascintillator Time-Of-Flight detector(TOF)(45^◦–85^◦) and Resistive Plate Chambers (RPC) (18^◦–45^◦). Both timing detectors, TOF and RPC, allow for good particle identification, i.e. proton- pion separation. (Due to their low yield, kaons hardly affect the pionselectionatSISenergies.)TOF,RPC,andPre-Showerdetectors (behind RPC, for e^± identification) were combined into a Multi- plicityandElectronTriggerArray (META).Severaltriggersareim- plemented. The minimumbiastrigger isdefined bya signal ina diamond START detector in front of the 15-fold segmented gold target.Inaddition,onlinePhysicsTriggers(PT)areused,whichare basedonhardware thresholdsontheTOFsignals, proportionalto

6 ThroughoutthispublicationA GeV referstothemeankineticbeamenergy.

the eventmultiplicity, corresponding to atleast 20 (PT3) hits in the TOF. About 2.1 billion PT3 triggered Au + Au collisions corre- sponding to the 40% most central events are taken into account forthecorrelation analysis. The centralitydeterminationisbased onthesummednumberofhitsdetectedby theTOFandtheRPC detectors. The measured events are divided in centrality classes correspondingto successive10% regionsofthetotalcrosssection [19].Here,wereportonlyonresultsofthe0–10%class;theentire centralitydependenceofpionsourceparametersanalyzed asfunc- tionofazimuthalanglew.r.t.thereactionplanewillbepartofan extendedforthcoming paper,while yieldsandphase-spacedistri- butionsofchargedpionsaretobepresentedinaseparatereport.

Generally,thetwo-particlecorrelationfunctionisdefinedasthe ratiooftheprobability P2(p₁,p₂)tomeasuresimultaneouslytwo particleswithmomenta p₁ and p₂ andtheproductofthecorre- spondingsingle-particleprobabilities P1(p₁)andP1(p₂)[1],

C

(

p₁

,

p₂

) =

^P2

(

p₁

,

p₂

)

P1

(

p₁

)

P1

(

p₂

) .

(1) Experimentallythiscorrelationisformedasafunctionofthemo- mentumdifference betweenthetwo particlesofagivenpairand quantified by taking the ratio ofthe yields of‘true’ pairs (Ytrue) anduncorrelatedpairs (Ymix). Ytrue is constructedfromall parti- clepairsintheselectedphasespaceintervalfromthesameevent.

Y_mix isgenerated by eventmixing, whereparticle 1andparticle 2aretakenfromdifferentevents.Care wastakentomixparticles fromsimilareventclassesintermsofmultiplicity,vertexposition andreactionplane angle.Theeventsare allowed todifferby not morethan10unitsinthenumberoftheRPC + TOFhit multiplic- ityof≥^{182 (i.e.}correspondingtotheuncertaintyofthecentrality class0–10%[19]),1.2 mminthez-vertexcoordinate(amountingto lessthanone thirdofthespacingbetweentarget segments),and 30degreesinazimuthalangle(tobecomparedtotheeventplane resolutionof^cos=⁰.612),respectively.

The momentum difference isdecomposed intothree orthogo- nalcomponents assuggestedby Podgoretsky [20], Pratt [21] and Bertsch [22].The three-dimensionalcorrelationfunctionsarepro- jections of equation (1) into the (out,side, long)-coordinate sys- tem, where ‘out’ means along the pair transverse momentum, kt= (^pt,1+^pt,2)/2,‘long’isparalleltothebeamdirection z,and

‘side’isorientedperpendicular tothe other directions.Theparti- cles forminga pair are boostedinto the longitudinallycomoving system(LCMS), wherethe z-components of the momenta cancel eachother, pz1+^pz2=^{0.Notethat inother} publicationsalsothe paircomovingsystem(p₁+p₂=0)isfrequentlyused.TheLCMS choice allows for an adequate comparison with correlation data taken at very different, usually much higher, collision energies, wherethedistributionoftherapidity,y=^tanh−1(βz),ofproduced particlesis foundto be notasnarrow asin thepresentcasebut largelyelongated.(Here,βz=^pz/E, E=

p²+^m2₀ ^andm0 arethe longitudinal velocity, the total energy and the rest mass of the particle, respectively. We use units with ^h¯ =^c2=^{1.) Hence, the} experimentalcorrelationfunctionisgivenby

C

(

qout

,

q_side

,

q_long

) =

N ^Ytrue

(

q_out

,

q_side

,

q_long

)

Ymix

(

qout

,

q_side

,

q_long

) ,

(2) whereqi= (^p1,i−^p2,i)/2 (i=^{‘out’,‘side’,‘long’)aretherelative} momentumcomponents,andN ^isanormalizationfactorwhichis fixedbytherequirementC→^{1 atlargerelativemomenta,where} the correlation function is expected to flatten out at unity. Note that,asinourprevious intensityinterferometryanalyses[10–12], we use theabove low-energy convention of q whichis common

also in studies of proton-proton correlations, in contrast to the high-energy convention of

π π

correlations, Q =^{2q. The statisti-} calerrorsofequation(2) aredominatedbythoseofthetrueyield, sincethemixedyieldisgeneratedwithmuchhigherstatistics.

Two-trackreconstructiondefects(e.g.tracksplittingandmerg- ing effects) that are particularlyimportant to HBTanalyses were correctedbyappropriateselectionconditionsontheMETA-hitand MDC-layerlevels,i.e.bydiscardingpairswhichhitthesameMETA cell, and by excluding forparticle 2 three successive wires sym- metrically around theMDC wirefired by particle 1. Thismethod was tested with simulations carrying neither quantum-statistical nor Coulomb effects, based on UrQMD [23], Geant [24] and a detailed description of the detector response, to firmly exclude anyclose-track effect.Alsobroaderexclusionwindows havebeen tested, but nosignificant improvement was found.These simula- tionsalsoshowedthattherearenosignificantlong-rangecorrela- tions,usuallyattributedeithertoenergy–momentumconservation incorrelationanalysesofsmallsystemsortominijet-likephenom- enaathighenergies.

Thedataaredividedintosevenkt binsfrom50to400 MeV/c.

The three-dimensional experimental correlation function is then fittedwiththefunction

C_fit

(

qout

,

q_side

,

q_long

) =

N



(

1

− λ) + λ

^KC

(ˆ

q

,

R_inv

)

C_qs

(

q_out

,

q_side

,

q_long

) 

,

(3)

where

Cqs

(

qout

,

q_side

,

q_long

) =

¹

+

exp

(−(

^2qoutR_out

)

²

− (

^2qsideR_side

)

²

− (

^2qlongR_long

)

²

)

(4) represents the quantum-statistical part of the correlation func- tion. The parameters N and λ in Eq. (3) are a normalization constant and the fraction of correlated pairs, respectively, and ˆ

q=^qinv(q_out,q_side,q_long,k_t) is the average value of the invari- ant momentum difference, q_inv=¹₂

(p₁−^p2)²− (^E1−^E2)²,for givenintervalsoftherelativemomentumcomponentsandkt.The rangeoftheone- andthree-dimensionalfits extendsinqinv from 6 MeV/c to 80 MeV/c.Log-likelihood minimization[25] wasused in all fits to the correlation functions. The influence of the mu- tual Coulomb interaction in Eq. (3) is separated from the Bose–

EinsteinpartbyincludinginthefitsthecommonlyusedCoulomb correction by Sinyukov et al. [26]. The Coulomb factor K_C re- sults from the integration of the two-pion Coulomb wave func- tion squared over a spherical Gaussian source of fixed radius.

This radius is iteratively approximated by the result of the cor- responding fit to the correlation function. In Eq. (3), the non- diagonal elements comprising the combinations ‘out’–‘side’ and

‘side’–‘long’ vanish for symmetry reasons [27] whenazimuthally andrapidity integratedcorrelations functionsarestudied [28,29], as it is done in the present investigation. The ‘out’–‘long’ com- ponent, however, can have a finite value depending on the de- gree of symmetry of the detector-accepted rapidity distribution w.r.t. midrapidity ( ycm=⁰.74). We studied thiseffectby includ- ing in Eq. (4) an additional term −^2qout(2R_{out long})²q_long, where theprefactoraccountsforbothnon-diagonalterms,‘out’–‘long’and

‘long’–‘out’. We foundonly marginal differencesinthefits which delivered,foralltransverse-momentumclasses,rathersmallvalues of R²_{out long}<1 fm². Forall results presentedhere, we restricted the pair rapidity to an interval |^y−^ycm|<0.35, within which dN/dy doesnot varybymore than10%,andlimitedourselvesto thefitfunctionwiththeBose–Einsteinpart(Eq. (4))consistingof diagonalelementsonlyandaddedthesmalldeviationstothesys- tematic errors. The effectof finite momentum resolutions of the

Fig. 1. ProjectionsoftheCoulomb-correctedthree-dimensionalπ⁻π⁻ correlation function(dots)andoftherespectivefits(dashedcurves)for the ktintervals of 100–150 MeV/c (top),200–250 MeV/c (middle),and300–350 MeV/c (bottom).The left,center,andrightpanelsgivethe‘out’,‘side’,and‘long’directions,respectively.

Theunplottedq componentsareintegratedover±^{12 MeV/c.}

HADEStrackingsystemisstudiedwithdedicatedsimulations.Typ- icalGaussian resolutionvaluesof

σ

q(qinv=^{20 MeV/c})^{2 MeV/c} areestimated.Incorporatingacorrespondingcorrectionintothefit functionbyconvolutionofEq. (3) withaGaussianresolutionfunc- tionleadstoradiusshiftsofaboutδR/R +^2%.

Fig. 1 shows one-dimensional projections of the Coulomb- corrected

π

⁻

π

⁻correlationfunctiontogetherwithcorresponding fitswithEq. (3) forvariousk_t intervals.(Due tothepermutability ofparticles 1and 2, oneoftheq projectionscan berestricted to positivevalues.) Thepeak duetothe Bose–Einsteinenhancement becomesevidentatlow|^q|^{.Itswidthincreaseswithincreasingk}t. Thecorrelation functions for

π

⁺

π

⁺ pairs looksimilar. The main systematicuncertaintiesoftheresultspresentedbelowarisefrom theslightfluctuationsofthefitresultswhenvaryingthefitranges (∼⁰.1–0.3 fm),fromthe forward–backwarddifferencesof thefit resultsw.r.t.midrapiditywithin similartransversemomentumin- tervals(∼⁰.03–0.1(0.2)fmforRinv,Rside, Rlong (Rout)),andfrom thedifferenceswhen switchingon/off the‘out’–‘long’component in the fit function (∼⁰.05–0.2 fm). Finally, all systematic error contributionsareaddedquadratically.InFig.2they areshownas hatchedbands.

To separate a potential source radius bias introduced by the Coulomb force the charged pions experience in the field of the chargedfireball,wefollowtheansatzusedinRef. [30],

E

(

p_f

) =

^E

(

p_i

) ±

^Veff

(

r_i

),

(5) whereE isthetotalenergy,p_i(p_f)istheinitial(final)momentum andr_iistheinitial position ofthepionintheCoulomb potential Veffwithpositive(negative)signfor

π

⁺ (

π

⁻).With

R_π±π^± R_π˜0π˜⁰

≈

^qi q_f

= |

^pi

|

|

p_f

| =

 

 

₁

∓

2V_eff

|

p_f

|

1

+

^m2π p²_f

+

^Veff2

p²_f

,

(6)

where qi (q_f) is the initial (final) relative momentum, and with Veff/kt^{1,itturnsoutthatthe}constructedsquaredsourceradius forpairs of neutral pions (denoted by

π

˜⁰

π

˜⁰ in the following in contrasttothecasewhere

π

⁻

π

⁻and

π

⁺

π

⁺dataarecombined) issimply thearithmetic mean ofthe corresponding quantitiesof thechargedpions,

R²_π˜0π˜⁰

=

¹ 2

R²_π+π⁺

+

R²_π−π⁻

 ,

(7)

Fig. 2. Sourceparametersasfunctionofpairtransversemomentum,kt,forcentral (0–10%)Au + Aucollisionsat1.23 A GeV.Theupperleft,upperright,centerleft,and centerrightpanelsdisplaytheinvariant,out,side,andlongradii,respectively.The lowerleftandlowerrightpanelsshowthecorrespondingλparametersresulting fromthefitstotheone- andthree-dimensionalcorrelationfunctions,respectively.

Blacksquares(redcircles) arefor pairsofnegative(positive)pions.Bluedashed linesrepresentconstructedradii ofneutralpionpairs(seetext).Error barsand hatchedbandsrepresentthestatisticalandsystematicerrors,respectively.

whichisvalidforallradiuscomponents(eventhoughinthe‘out’

direction, Eq. (6) looks slightlydifferent).Finally, theconstructed

π

⁰

π

⁰ correlation radii are derived from cubic spline interpola- tionsofthektdependenceofboththecorrespondingexperimental

π

⁻

π

⁻ and

π

⁺

π

⁺data.Thisinterpolationisnecessarybecause– asresultofdifferentdetectoracceptances–thechargedpionpairs exhibitslightlydifferentaveragetransversemomenta,eventhough theyaremeasuredinidenticalkt intervals.

Fig. 2 shows the dependence on average k_t (determined for q_inv<50 MeV/c) of the one-dimensional (invariant) and three- dimensional source radii for

π

⁻

π

⁻ (black squares) and

π

⁺

π

⁺

(red circles) pairs. While for low transverse momentum the Coulomb interaction withthe fireball leads to an increase (a de- crease) of the source size derived for negative (positive) pion pairs, atlarge transversemomentum apparently theCoulomb ef- fectfadesaway.TheeffectissmallestforR_out.Notethatthecharge splitting of the source radii was early predicted by Barz [31,32]

who investigated the combined effects of nuclear Coulomb field, radial flow, andopaqueness on two-pion correlationsfor a large collision system such as Au + Au in the 1 A GeV energy regime.

EarlierexperimentalworksattheBevalacemployingathree-body Coulomb correction found the effect negligible for their studies of smaller systems [7–9]. The parameter λ derived fromthe fits with Eq. (3) appears rather independent of charge sign and de- creases only slightly with increasing transverse momentum, cf.

lowerrightpanelofFig.2.Itfitswellintoapreliminaryevolution with√

sNN establishedpreviously[5],exceptthelowestE895data point.Incontrast,λresultingfromthefitstotheone-dimensional

Fig. 3. Excitationfunction ofthe sourceradii Rout (upperpanel), Rside (central panel),andRlong(lowerpanel)forpairsofidenticalpionswithtransversemassof mt=260 MeV incentralcollisionsofAu + AuorPb + Pb.Squaresrepresentdataby ALICEatLHC(π⁻π⁻+π⁺π⁺)[37],fulltrianglesSTARatRHIC(π⁻π⁻+π⁺π⁺)[5], diamondsareforCERESatSPS(π⁻π⁻+π⁺π⁺)[38],opentrianglesareforNA49 atSPS(π⁻π⁻)[39],opencirclesareπ⁻π⁻databyE895atAGS[1,28],andopen (full)crosses involveπ⁻π⁻ (π⁺π⁺)dataofE866atAGS[40],respectively.The presentdataofHADESatSIS18forpairsofπ⁻π⁻(π⁺π⁺)aregivenasopen(full) stars.Statisticalerrorsaredisplayedaserrorbars;ifnotvisible,theyaresmaller thanthecorrespondingsymbols.

(q_inv-dependent) correlation function, exhibits a significant de- creasewithkt (cf.lower leftpanel),probablypointingtothefact that the one-dimensional fit function is not adequate. Note that deviationsfromGaussiansourceshapeswillbestudiedinaforth- comingpaperby applyingthe methodofsourceimaging [33,34], orbyusingLévysourceparameterizations[35].

TheexcitationfunctionsofR_out, R_side,and R_long forpionpairs produced in central collisions are displayed in Fig. 3. All shown radius parameters havebeenobtained byinterpolating theexist- ing measured data points to the same transverse mass of mt=



k²_t +^m2_π =^{260 MeV at which data points by STAR at RHIC} [5] are available. The statistical errors are properly propagated andquadraticallyaddedwithsystematicdifferencesoflinear and cubic-spline interpolations. Extrapolations were not necessary at thismt value. Corresponding excitation functions at other trans- verse masses show similar dependencies. Surprisingly, Rout and R_side vary hardly morethan 40% over three ordersof magnitude in center-of-mass energy. Only R_long exhibits a systematical in- crease by about a factor of two to three when going in energy from SIS18 via AGS, SPS, RHIC to LHC. Note that in the excita- tionfunctionsofRef. [5] notall,particularlyAGS,datapointswere properlycorrectedfortheirkt dependence.WhiletheHADES Rout and Rside datafornegative pionscompletely agreewiththelow- est E895 data at 2 A GeV, R_long deviates fromthe corresponding

Fig. 4. ExcitationfunctionofR²out−R²_side,ascalculatedfromthedatapointsshown inFig.3.

E895 datapoint. Both data are, however,in accordance withthe overall smooth trendwithin 2

σ

.(Thelow-energy CERES dataof R_out andthe E866 datapoint of R_long for

π

⁻

π

⁻ pairs appearto beoutliers.)

The combinationof R²_out andR²_side canbe relatedto theemis- sion time duration[36],(

τ

)²≈ (^R2out−^R2_side)/βt²^,where βt is thetransversepairvelocity.TheexcitationfunctionofR²_out−^R2_side is shown in Fig. 4. Up to now almost all measurements below 10 GeV are characterized by large errors and scatter sizeably.

(Here, the outlying low-energy CERES data are solely caused by thedeviationinRout,cf.toppanelofFig.3.)ThenewHADESdata show that the difference of source parameters in the transverse plane almost vanishes at low collision energies. With increasing energy, it reaches a maximum at √

sNN∼^{20–30 GeV and after-} wards decreases towards zero at LHC energies. One would con- cludethatinthe1 A GeV energyregionpionsareemittedintofree spaceduringashorttimespanoflessthanonetotwofm/c.How- ever,alsotheopaqueness ofthe sourceaffectsR²_out−^R2_side ^which couldcauseittobecomenegative,thuscompensatingthepositive contributionfromtheemissiontime[32].

The excitation function of the freeze-out volume, V_fo = (2

π

)^3/2R²_sideRlong, is given in Fig. 5. Note that this definition of a three-dimensional Gaussian volume does not incorporate R_out since generallythislength is potentially extendeddueto a finite value of the aforementioned emission duration. From the above HADES data,we estimate a volume ofabout 1300 fm³ for pairs ofconstructedneutralpions.Thevolumeofhomogeneitysteadily increases with energy, butis merely a factorfour larger at LHC.

Extrapolating Vfo tokt=^{0 yieldsavalueofabout3900 fm3.} ThelargescatterofdatapointsinFig.5below√

s_NN=^{10 GeV} isintriguingandmightindicateanon-trivialenergydependenceof theradius parametersinthisregion.However,the simplestinter- pretationwouldbetoassumeinsteadthattheenergydependence issmooth.(NotethatthedifferenceoftheHADES

π

⁻

π

⁻dataand the lowestE895 datapointat2 A GeV is primarilycausedby the deviation in Rlong.) If, however, the variation ofthe data atlow energies,mostprominentlyseen inthenon-monotonicityof R_side (cf. Fig. 3), isto be taken seriously,new experimental andtheo- reticaleffortsareneededtoclarify thesituation,ascouldbedone withthefutureexperimentsCBMatSIS100/FAIRinDarmstadt[41]

andMPDatNICAinDubna[42] orwiththeSTARfixed-targetpro-

Fig. 5. Excitationfunctionofthefreeze-outvolume,Vfo= (2π)^3/2R²_sideRlong,ascal- culatedfromthedatapointsshowninFig.3.

gram [43].Finally, we wantto recallthat in Figs. 3,4,and5 we displaystatisticaluncertaintiesonly;thesystematiconeswerenot availableforallexperiments.

In summary, we presented high-statistics

π

⁻

π

⁻ and

π

⁺

π

⁺

HBT data for central Au + Au collisions at 1.23 A GeV. The three- dimensionalGaussianemissionsourceisstudiedindependenceon transversemomentumandfoundtofollowthetrendsobservedat higher collision energies, extending the corresponding excitation functionsdowntotheverylowpartoftheenergyscale.Substan- tialdifferencesofthesourceradiiforpairsofnegativeandpositive pionsare found, especially at low transverse momenta,an effect which is not observed at higher collision energies. A clear hier- archyofthe three Gaussianradii is seenin ourdata,i.e. R_long<

R_side≈^Rout,independent oftransverse momentum.Furthermore, asurprisinglysmallvariationofthespace–timeextentofthepion emissionsourceoverthreeordersofmagnitudeincenter-of-mass energy,√

sNN,isobserved.Ourdataindicatethattheverysmooth trendsobservedatultra-relativisticenergiescontinuetowardsvery lowenergies.WhilebothR_outandR_longsteadilydecreasewithde- creasing√

sNN,aweaknon-monotonicenergydependenceofR_side cannotbeexcluded.

Acknowledgements

The HADES Collaboration gratefullyacknowledges the support by the grants SIP JUC Cracow, Cracow (Poland), National Sci- ence Center, 2016/23/P/ST2/040 POLONEZ, 2017/25/N/ST2/00580, 2017/26/M/ST2/00600; TU Darmstadt, Darmstadt (Germany) and Goethe-University, Frankfurt (Germany), ExtreMe Matter Institute EMMIat GSI Darmstadt; TU München, Garching(Germany), MLL München,DFGEClust153,GSI TMLRG1316F,BMBF 05P15WOFCA,

SFB1258, DFGFAB898/2-2; NRNUMEPhIMoscow,Moscow(Rus- sia), inframeworkofRussianAcademic ExcellenceProject 02.a03.

21.0005, Ministry of Science and Education of the Russian Fed- eration 3.3380.2017/4.6; JLU Giessen, Giessen (Germany), BMBF:

05P12RGGHM; IPN Orsay, Orsay Cedex (France), CNRS/IN2P3;

NPI CAS, Rez, Rez (Czech Republic), MSMT LM2015049, OP VVV CZ.02.1.01/0.0/0.0/16013/0001677,LTT17003.

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