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Physics Letters B 298 (1993) 456-468 PHYSICS LETTERS B North-Holland

A study ofK°K ° Bose-Einstein correlations in hadronic Z ° decays

OPAL Collaboration

P.D. Acton a G. Alexander b, J. Allison c p.p. Allport d K.J. Anderson e, S. Arcelli f, A. Astbury g, D. Axen h, G. Azuelos i'1, G.A. Bahan c, J.T.M. Baines c, A.H. Ball j, J. Banks c, R.J. Barlow c, S. Barnett c, J.R. Barley d, G. Beaudoin i, A. Beck b, J. Becket k, T. Behnke ~' K.W. Bell m, G. Bella b, p. Bentkowski i, p. Berlich k S. Bethke n O. Biebel o, U. Binder k, l.J. Bloodworth P, P. Bock n, B. Boden °, H.M. Bosch n, S. Bougerolle h, H. Breuker q, P. Bright-Thomas a, R.M. Brown m, A. Buijs q, H.J. Burckhart q, C. Burgard ~', P. Capiluppi f, R.K. Carnegie r, A.A. Carter s, J.R. Carter d, C.Y. ChangJ, D.G. Charlton q, P.E.L. Clarke a I. Cohen h, J.C. Clayton P, W.J. Collins d, J.E. Conboy t, M. Cooper u M. Coupland v, M. Cuffiani f, S. Dado u, G.M. Dallavalle f, S. De Jong q, L.A. del Pozo a, H. DengJ, A. Dieckmann n, M. Dit tmar w, M.S. Dixit x, E, do Couto e Silva Y, J.E. Duboscq q E. Duchovni z G, Duckeck n, l.P. Duerdoth c, D.J.P. Dumas r P.A. Elcombe d, P.G. Estabrooks r E. Etzion b, H.G. Evans e, F. Fabbri f, M. Fincke-Keelerg, H.M. Fischer °, D.G. FongJ, M. FoucherJ, A. Gaidot aa O. Ganel z J.W. Gary w j. Gascon i R.F. McGowan c N.l. Geddes m C. Geich-Gimbel °, S.W. Gensler e, F.X. Genti t aa, G. Giacomelli f, V. Gibson d, W.R. Gibson s J.D. Gillies m, J. Goldberg u, M.J. Goodrick d, W. Gorn w, C. Grandi f, F.C. Grant d J. Hagemann ~, G.G. Hanson Y, M. Hansroul q, C.K. Hargrove x, P.F. Harrison s, j. Hart q P.M. Hattersley p, M. Hauschild q, C.M. Hawkes q, E. Heflin w R.J. Hemingway r, R.D. Heuerq, J.C. Hill a, S.J. Hillier p, T. Hilse k, D.A, Hinshaw i, J.D. Hobbs q, P.R. Hobson a, D. Hochman Z. R.J. Homer p, A.K. H o n m a g,l R.E. Hughes-Jones c, R. Humber t k p. lgo-Kemenes n H. lhssen n. D.C. Imrie a, A.C. Janissen r, A. JawaheryJ, P.W. Jeffreys m, H. Jeremie i M. Jimack f, M. Jobes p, R.W.L. Jones s p. Jovanovic P, C. Jui w D. Karlen r, K. Kawagoe ab, T. Kawamoto ab R.K. Keeler g, R.G. Kellogg J, B.W. Kennedy t, S. Kluth d, T. Kobayashi ab, D.S. Koetke q, T.P. Kokott °, S. Komamiya ab, L. K6pke q, J.F. Kral q, R. Kowalewski r j. von Krogh n J. Kroll e, M. Kuwano ab p. Kyberd s, G.D. Lafferty c R. LahmannJ, F. Lamarche i J.G. Layter w P. Leblanc i, A.M. Lee j, M.H. Lehto t, D. Lellouch z, C. Leroy i, J. Letts w, S. Levegrtin ° L. Levinson z, S.L. Lloyd s, F.K. Loebinger c, J.M. LorahJ, B. Lorazo i M.J. Losty X, X.C. Lou Y, J. Ludwig k M. Mannelli q, S. Marcellini f, G. Maringer o, C. Markus o A.J. Martin s j .p. Martin i T. Mashimo ab, P. Mfittig °, U. Maur °, J. McKenna g, T.J. McMahon p, J.R. McNutt a, F. Meijers q, D. Menszner n, F.S. Merritt e, H. Mes x, A. Michelini q, R.P. Middleton m G. Mikenberg Z, J. Mildenberger r, D.J. Miller t, R. Mir y, W. Mohr k, C. Moisan i A. Montanari f. T. Mori ab M. Morii ab T. Mouthuy y,2 B. Nellen o H.H. Nguyen c, M. Nozaki ab S.W. O'Neale p F.G. Oakham x, F. Odorici f, H.O. Ogren Y, C.J. Oram g,i, M.J. Oreglia e S. Orito ab J.P. Pansart aa B. Panzer-Steindel q, P. Paschievici z, G.N. Patrick m N. Paz-Jaoshvili b P. Pfister k, J.E. Pilcher e, J. Pinfold ac D. Pitman g, D.E. Plane q, P. Poffenberger g, B. Poli f A. Pouladdej r E. Prebys q, T.W. Pritchard s H. Przysiezniak i G. Quast*, M.W. Redmond c, D.L. Rees q, G.E. Richards c, D. Robinson q, A. Rollnik °, J.M. Roney g'3, E. Ros q, S. Rossberg k. A.M. Rossi f,4 M. Rosvickg, P. Routenburg'-, K. Runge k O. Runolfsson q, D.R. Rust Y M. Sasaki ab C. Sbarra q, A.D. Schaile k, O. Schaile k W. Schappert r, p. Scharff-Hansen q, P. Schenkg, B. Schmitt °, H. yon der Schmitt n S. Schreiber °. C. Schwick ~, J. Schwiening °,

456 Elsevier Science Publishers B.V.

Volume 298, number 3,4 PHYSICS LETTERS B 14 January 1993

W.G. Scott m, M. Settles Y, T.G. Shears d, B.C. Shen w C.H. Shepherd-Themistocleous x P. Sherwood t, R. Shypit h A. S imon o p. Singh s, G.P. Siroli f, A. Skuja j, A.M. Smith q, T.J. Smith q, G.A. Snow j, R. Sobie g'3, R.W. SpringerJ, M. Sproston m, K. Stephens c, J. Steuerer g, R. Str6hmer n, D. Strom ad, T. Takeshita ab,5, P. Taras i, S. Tarem z, M. Tecchio e, P. Teixeira-Dias n N. Tesch o N.J. Thackray P, M.A. Thomson t E. Torrente-Lujan u G. Transtromer a, N.J. Tresilian c T. Tsukamoto ab, M.F. Turner q, G. Tysarczyk-Niemeyer n, D. Van den plas i, R. Van Kooten q, G.J. VanDalen w, G, Vasseur aa, C.J. Virtue x, A. Wagner e, D.L. Wagner e, C. Wahl k, J.P. Walker p, C.P. Ward d, D.R. Ward d, P.M. Watkins p, A.T. Watson p, N.K. Watson q, M. Weber n, P. Weber r, S. Weisz q, P.S. Wells q, N. Wermes °, M.A. Whalley p, G.W. Wilson w, J.A. Wilson p, V.-H. Winterer k, T. Wlodek z, S. Wotton n, T.R. Wyatt c, R. Yaari z, A. Yeaman s, G. Yekutieli z, M. Yurko i, W. Zeuner q and G.T. ZornJ

a Brunel Universit)', L~rbridge, Middh'sex UB8 3Ptt, UK b Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, lsrael c Department of Physics, Schuster Laboratoo,, The University, Manchester MI3 9PL, UK d Cavendish Laboratory, Cambridge CB30HE, UK e Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA f Dipartimento di Fisica dell'Universita di Bologna and INFN, 1-40126 Bologna, Italy g Department ~fPh.vsics, University of Victoria, P.O. Box 3055, I)ctoria, B.C., Canada V8W 3P6 h Department of Physics, University of British Columbia, 6224 Agriculture Road, Vancouver, BC, Canada V6T IZI i Laboratoire de Physique NuclOaire, Universit~ "~ de Montr['al, Montreal, Quebec, Canada H3C 3J7 J Department of Physics and Astronomy, Universio~ of Mao4and, College Park, MD 20742, USA k Fakultiit fiir Physik. Albert Ludwigs Universitiit, I4"-7800 Freiburg, FRG

Universitiit Hamburg/DESY, I1. Institut fiir Experimental Physik, 14-2000 Hamburg 52, FRG m Ruthoford Appleton Laboratoo,, Chillon, Didcot, O.rfordshire OXI 10QX, UK n Physikalisches Institut, Universit~it lteidelberg, W-6900 Heidelberg, FRG o Physikalisches lnstitut, Universitiit Bonn, W-5300 Bonn 1, FRG P Sehool of Physics and Space Research, University of Birmingham, Birmingham Bl5 2TT, UK q CERN, European Organisation for Particle Physics, CH-1211 Geneva 23, Switzerland r Department of Physics, Carleton University, Colonel By Drive, Ottawa, Ontario, Canada KIS 5B6 s Queen Mao: and Westfield College, University of London, London E1 4NS, UK t UniversiO, College London, London [~'C1E 6BT, UK u Department of Physics, Teehnion-lsrael Institute ~)f Technology, fta~fa 32000, Israel v Birkbeck College, London 14"CIE 7HV, UK w Department of Physics, University qfCal~fornia, Riverside, CA 92521, USA × Centre for Research in Particle Physics, Carleton University, Ottawa, Ontario, Canada KIS 5B6 Y Department of Physics, Indiana University, Swain Hall gest 117, Bloomington, IN 47405, USA z Nuclear Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel aa DPhPE, CEN-Saclay, F-91191 Gff:sur-Yvette, France ab International Centre for Elementao, Particle Physics and Department of Physics,

University of Tokyo, Tokyo 113, Japan and Kobe University, Kobe 657, Japan

a c Department of Physics, University qf AIberta, Edmonton, Canada AB T6G 2Jl ad Department of Physics, University qf Oregon, Eugene, OR 97403, USA

Received 23 November 1992

457


Bose-Einstein correlations in o o KsK s pairs have been studied for the first time in e + e annihilations. These correlations were measured through the quantity Q, the four momentum difference of the pair. Out of about half a million Z ° hadronic decay events a total of about 6000 events with two or more identified K ° mesons ,,,,'ere used for the analysis. We observe a threshold enhancement in -0 0 KsK s pairs originating flom a mixed sample of K°K ° and K°K ° (K ° K °) pairs. Although we can not presently exclude that a part of this effect is due to scalar meson decays, our analysis attributes the enhancement to Bose-Einstein correlations. Wc find the values for the strength of the effect and the radius of the K ° emitting source to be 2 = 1.12 ± 0.33 :k 0.29 and R 0 = {0.72 ± 0.17 ~* 0.19) fro, respectively, where the first error is statistical and the second systematic.

1. I n t r o d u c t i o n

In part icle react ions leading to mul t ihadron ic fi-

nal states the interference between a pair of identi-

cal bosons, the so called Bose-Eins te in Corre la t ion

(BEC), is well known. These corre la t ions lead to an

enhancemen t of the n u m b e r o f identical bosons over

that o f non- ident ical bosons when the two part icles

are close to each other in phase space. Exper imen-

tally this effect, also known as the G G L P effect, was

first observed by Go ldhabe r et al. [ 1 ], m their study

of like-sign charged pions in p~ annih i la t ions at ,,/s

= 2.1 GcV. In addi t ion to the q u a n t u m mechanica l

aspect o f the BEC, these correla t ions can be used to

es t imate the d imens ion o f the emi t t ing source o f the

identical bosons [2]. Some recent reviews which sum-

marize the under lying theoret ical concepts o f the BEC

and the exper imenta l results are given, for example,

in refs. [3,4].

These corre la t ions have been observed in like-sign

charged pion-pairs over a wide energy range and for

many different initial state reactions leading to mult i-

pion final states. More recently, two charged pion cor-

relat ions have been s tudied in high energy e+e anni-

hi la t ion at LEP, on and a round the Z ° mass, where the

radius o f the pion emi t t ing source was measured by

O P A L to be (0.928 ± 0.019 ± 0.150) fin [51 whereas

A L E P H reported (0.65 ± 0.04 ± 0.16) fm [6] and

D E L P H I obta ined (0.62 ± 0.04 J_ 0.20) fm [7].

1 Also at TRIUMF, Vancouver, Canada V6T 2A3. 2 Present address: Centre de Physique des Particulcs de

Marseille, FaculI~ des Sciences de Luminy, Marseille, France.

3 And IPP, University of Victoria, Departmentt of Physics, P. O. Box 3055, Victoria, Canada BC V8W 3P6.

4 Present address: Dipartimento di Fisica, Universitfi della Calabria and INFN, 1-87036 Rende, Italy.

5 Also at Shinshu University. Matsumoto 390, Japan.

In contrast to tile ample informat ion published on

BEC in pion pairs, very few studies have been re-

ported on correlat ion effects in charged kaon-pairs

[8,9]. This is also the case for the K°K ° correlat ion

studies [10,11], which had relatively meager statis-

tics in compar i son to the pion samplcs. An apparent

reason for this si tuation stems from the fact that the

product ion rate o f strange particles at center of mass

energies so far explored is o f the order of 10 to 20%

of that o f pions so that the n u m b e r o f events with

two or more ident i f ied K-mesons is rather rare. Fur-

Ihcrmore, the exper imenta l methods for separat ing

charged kaons from pions or protons were often lack-

ing. In the case of neutral kaons, most exper iments

stud}' o n l y l h c K ° -Tr+zr decay. m o d e r e s u h i n g i n a

further reduct ion in statistics. It is expected that the

identical boson systems, K°K ° and K ° K °. will man-

ifest Bose-Eins te in correlat ions, s imilar to neutral el-

like-sign charged pion pairs, i r respect ive o f ttle spe-

cific decay state chosen. O f special interest however,

is the casc o f the non-ident ical K°K ° system. Although

this is a boson -an t iboson system, a BEC-like enhance-

ment is nevertheless expected, as will be shown later,

p rovided one selects the ( ' = + 1 charge conjugat ion e igcnvalue state o f k'0k-0 W0k'0

In the present paper we report on a stud,,' of the K 0 K 0 Bose-Eins te in correlat ions in the s s syslcm pro-

duced in about ha l f a mil l ion Z ° mul t ihadronic decay'

events measured with the O P A L detector at LEP. The

exper imenta l set-up and data selection are described

in section 2. Section 3 is devoted to the descript ion

o f the analysis technique, also utilized previously to

measure the Bose-Eins te in effect in like-sign pion-

pairs. In the same section we show that the BEC effect IgO kzt) should also bc seen in the , - s ' - s system, independent

of whether its origin is a K°K g or a K°K ° (K ° K '))

pair. The physics analysis, the choice o l ' a reference

sample and the BE( ' results are given in section 4. In

scction 5 ~c discuss the possible contr ibut ion fi'om

458


the decays of the f0(975) and a0(980) resonances to the BEC effect and compare our results to former BEC studies carried out with kaon and pion-pairs.

2. The experiment

2.1. Experimental set-up

The OPAL detector is designed to measure outgoing particles coming from e+e annihilations at high energies. Details concerning the OPAL detector and its performance are given elsewhere [ 12 ] so here we will describe briefly only those detector elements pertinent to the present analysis, namely, the central tracking

chambers. These consist of a precision vertex detector, a large jet chamber and additional z-chambers surrounding the jet chamber. The vertex detector, a I m long cylindrical drift chamber of 470 mm diameter, surrounds the beam pipe and consists of an in- ner layer of 36 cells each with 12 sense wires and an outer layer of 36 small angle (40 ) stereo cells each with 6 sense wires. The jet chamber has a length of 4 m and a diameter of 3.7 m. It is divided into 24 sectors, each equipped with 159 sense wires ensuring a large number of measured points even for particles emerg- ing from a secondary vertex. The z-chambers consist of 24 drift chambers, 4 m long, 50 cm wide and 59 mm thick. They are subdivided in 8 cells each with 6 sense wires perpendicular to those of the jet chamber and provide an exact measurement of the : coordinate along the beam direction ~ . They cover polar angles from 44 ° to 136 ° and 94% of the azimuthal angle. All the chambers are contained in a solenoid providing an axial magnetic field of 0.435 T.

2.2. Data selection

The present study was carried out with 499000 hadronic Z ° decay events corresponding to an inte- grated luminosity of about 20.8 pb -~ collected dur- ing 1990 and 1991 at center of mass energies on and

A right-handed coordinate system is adopted by OPAL, where the x axis points to the center of the LEP ring, and positive z is along the electron beam direction. The angles 0 and 4 ~ are the polar and azimuthal angles, respectively.

around the Z ° mass, The criteria used for the selection of the Z ° hadronic decay event sample were described previously [13 ]. In addition, we have accepted only multi-hadron events recorded while the jet and z- chambers were fully operational. At least 5 well reconstructed charged tracks were required for each event.

The method for finding K ° decays into ~+~r- was essentially the same as that used in a former OPAL stud}' of inclusive Z ° decays into K~ mesons [14].

It started by systematically pairing tracks of oppo- site charge. These tracks had to fulfil the following conditions: a min imum transverse momentum of 150 MeV/c with respect to the beam direction and more than 50% of the possible jet chamber hits (at least 20). Each track was also required to have more than 3 z- chamber hits or a reconstructed end point inside the jet chamber, determined by using the last wire with a hit. These requirements ensure a good mass resolution by improving the measurement of the polar an-

gle. Furthermore, the radial distance of the track to the beam axis at the point of closest approach was required to exceed 3 mm to reduce the large combina- torial background.

Intersection points of track pairs in the radial plane were considered to be secondary vertex candidates. Additional cuts were then imposed on these pairs. The radial distance from the intersection point to the pri- mary vertex had to be larger than 1 cm and the reconstructed momentum vector of the Ks ° candidate in the plane perpendicular to the beam axis had to point to the beam axis within 20 . In the case where both in-

tersections of the track pair passed these cuts, the one closer to the beam axis was taken.

All the track-pairs which passed these cuts were re- fitted with the constraint that they originate from a common 3-dimensional vertex. Pairs with an invariant mass of less than 100 MeV/c 2, when assuming both tracks to be electrons, were taken to be photon conversions and were rejected.

The mass distribution of the reconstructed K ° decays is shown in fig. [ assuming both tracks to be pions. As can be seen, a prominent peak is observed at the K ° mass value. The distribution was fitted with a gaussian shape plus a polynomial expression for the background. The measured K ° mass value thus obtained was 496.9 ± 0.1 MeV/c 2, consistent =2 with

~2 For footnote see next page.

459


5oo O P A L

K 4000

~ 3500

3000

2500

2(X)0

1500

1 0 0 0

I1 ' 0.3 0.4 0.5 0.6 0.7

M ~ [GeV/c 2]

Fig. 1. The 7r+Tr invariant mass distribution of the K ° candidales m ()PAL data.

the world average o f 497.671 M e V / c 2 [ 15]. The ob-

tained width o f a = 7.0 y 0.1 M e V / c 2, corresponds to

our exper imenta l mass resolution. In order to reduce

the background we have imroduced a cut on lhe mea-

sured K ~* mass of 2 :25 M c V / c 2 from the mean fitted

value. This cut reduced the background to ( 11 m l )%

coming mainly from accidental ly reconst ructed sec-

ondary, vert ices and from some (about 2%) mis idenl i -

fled A-hyperons. Cor respondingly the es t imated back-

ground in this sample in terms of pairs of K ° can-

didates, which in the fol lowing will be referred to as

K~K~ pairs, was then (21 _- 1)% that is, a signal of

(79 t 1)%. In this way we were left with a lotal o f

5501 events having two ident i f ied K ° ~ 7r+zr de-

cays, 465 with three K ° decay's and 23 exents with a

higher K ° cand ida le mul l ip l ic i t ) .

3. Analysis tools

Bose-Eins te in correla t ions in part icle ph,~ sics are in

general described in terms of a normal ized correlat ion

=2 The quoted errors are statistical only. The difference bep, vecn the measured K ° mass and the world average can be explained by the unccrlainly in the mean value of the OPAL magnetic t]c/d.

funct ion ( ' def ined as

( ' ( l q , I;2 ) - p ( l q , P2 ) ( I ) PO(Pl,P2) '

where th and l)2 are the four m o m e n t a o f the two

bosons, l>(lh,p2) = ( l / ( 7 ) ( d S a / d 4 p l d 4 p 2 ) is the

measured density for two identical bosons and

P0(lh,l~2) is the two part icle density in the absence

o f BE(? which we will refer to as the reference sample

dis t r ibut ion.

In conjunc t ion with the studies o f BEC in pion pairs

several parametr iza t ions of the correlat ion funct ion

( ' can be found [3,4, 16]. For the correlat ion funct ion

(" we have used the Lorentz invar iant variable O

Q2 = (pl - 1 ' 2 ) 2 , (2)

which was also used in previous studies o f the BEC in

c+e - mul t ihadron ic annihi la t ions [4]. For two iden-

tical particles eq. (2) can be rewrit ten as

Q2 = 312 4t//2 , (3)

where 312 is the invar iant mass squared of the two

particles and m is the rest mass o f each of them.

The corre la t ion function ( ' ( (2 ) at (2 = 0 at tains the

value 1 in the absence of BEC and the value 2 in the

presence o f BEC of two identical bosons. However ,

in the case o f like-sign charged pions, i.e. 2z+~ ÷ or

~ pairs, the exper imenta l values [4] are found to

be between 1 and 2. This devia t ion from the expected

theoret ical value o f 2 may indicate that not all pairs o f

particles are emi t ted from purely incoherent sources.

Fur thermore , )3oi all particle pairs used are formed

from identical bosons which further reduces the value

of ( ' ( (2 = 0). If the particle source is assumed to

have a gaussian shape, C ( Q ) can be parametr izcd as

follows [ 3 ]:

( ' ( (2 ) = I + 2 c x p ( - Q 2 R 2 ) , (4)

where the paramete r R is related to the boson emi t te r

size R0 in fm through the relation Ro = heR. The

paramete r 2, often called the strength or chaot ici ty

parameter , measures the strength of the effect and can

vary between 0 and 1. In practice the data arc fitted

to an expression o f the type

( ' ( Q ) = N [ I +. / ( (2 )2exp ( -Q2R2)]©(Q) , (5)

where N is a normal iza t ion factor and 4)(Q) is a term of the t )pc (I + (~Q + ~ Q2). It accounts for the rise

460


of the correlation function at large Q values due to long range two-particle correlations, imposed, for example, by charge and energy conservation and phase space constraints. In order to keep 2 as the strength parameter only for the boson under study, one in- troduces a function f ( Q ) , to be determined experi- mentally. This function should account tbr the background effects and their possible Q dependence. Here it should also be noted that, unlike the case of charged pion pairs, no Coulomb corrections are needed since

K°K ° pair is a system of neutral particles. the s s For the BEC study of charged pion or charged kaon

final states, the identical boson-pair system is sim- ply guaranteed by the requirement that they have the same charge sign. In the case of the BEC study' of

0 0 KsK s pairs the origin may come either from a pair of identical K°K ° (K ° K °) mesons or from a non- identical K°K 6 boson-ant iboson system. From a sam-

ple of Monte Carlo generated events, to be discussed later, we have estimated that about 3 (3 for Q < 1 GeV ) of our data sample of two K ° events originate from a K°K 6 pair and the rest from pairs of K ° or K ~ mesons. This being the case, when trying to estimate the expected BEC effect one needs to discuss sepa-

0 0 rarely these two sources of the KsK s pairs. In the following discussion we neglect effects due to CP viola- tion.

(1) K° K ° and K ° K ° decays. In this case we deal with identical bosons which should show the BEC effect in all their decay combinations. In particular the BEC effect should also be observed in the wowo pairs

studied in this work. (2) K°Kddecay, The system K ° K 6 is an eigenstate of

the charge conjugation operator C having two possible eigenvalues ( ' = +1 and C = -1. The probability amplitude ]K° ;K d) can be written as

I K ° : K 6)

' 5 = ~2[K°;K6)( '=+I + ]K°;Kd)('=_I . (6)

It consists of a pair of non-identical bosons which should a priori not show any BEC effect. Following Lipkin [17] we can write the probability' amplitude for a given charge conjugation eigenvalue C as follows:

2 IK°;K6)c=±I

= i ! K O ( p ) ; ~ ( p ) ) ± ½[~d(p):KO(_p)), (7)

where p is now the three momentum vector defined in the K°K iT center of mass system. In the limit of Q = 0, where the BEC should be maximal, p = 0 and eq. (6) reads

1 ~/~ PK°: K-@(: ±1

' iK°C0):K°(0)) + l lK°C0) 'K°(0) ) (8) = ~

This means that, at Q = 0. the probability amplitude for the C = - 1 state is zero whereas that of the state C = +1 is maximal. This will be also the case for boson-ant iboson pairs such as ~z+~z or K+K .

It is well known that the K ° and the ~d mesons are

described in terms of the two CP eigenstates, K ° with C P = + 1 and K ° with C P = - I . This being the case, then when the K°K ° pair is detected through the K ° and K ° decays, the eigenvalue C of the K°K 6 system is

determined. Thus, as Q approaches zero, an enhancement would be observed in the probability to detect

0 0 0 0 Ks K~ pairs (C + 1 ) whereas a de- KsK s pairs and = plction would occur in the K°K ° s e pairs (C = 1),

0 0 Since the present work is restricted to KsK s pairs, which picks out only the C = +1 state, a BEC enhancement is expected. Note, however, that if all the decay modes of the K ° ~ - pairs were detected, then according to eq. (7), no BEC effect will be observed at Q = 0. This however should not be a surprise because the K ° K d system is not composed of identical bosons.

In conclusion, when studying the 0 0 KsK s pairs, as in this work, a BEC-like enhancement should be observed similar to that of like-sign charged pions and charged kaons, even if the 0 0 KsK s pairs originate from a K ° ~ boson-antiboson system.

4. Physics analysis and results

The invariant mass distribution MKK of all K°-pairs is shown in fig. 2a. This distribution, which reaches its maximum near threshold, is seen to decrease smoothly as :IIKK increases apart from some concen- tration of events around 1.5 GeV/c 2. This mass enhancement may be associated with the isoscalar jPC = 2 ++ Q(1525) resonance. In addition some indi- cation is also present in the data for the production of the t"o(1710) resonance with the same quantum numbers. Furthermore, two lower mass resonances,

461


.~ 140

120

~. loo

8 0

6 0

4 0

20

a)

¢ +

O P A L

. , , , , I , , , I , , , , I , , , I , a . . I , , [ , 1 .2 1 .4 1 .6 1.8 2 2 .2 2 .4

MKK [GeV/e 2]

2 4 0

~" 2 0 0 .....

160

120

8 0

4 0

b) . OPAL

0 , i , , , i . . . . , . . . . i . . . . J , , ! . . . . i . . . . i . . . . i , , ,

0 . 2 0 . 4 0 . 6 0 .8 1 1.2 1.4 1.6 1.8

Q [GeV]

Fig. 2. (a) The invariant mass distribution MKK and (b) the dN/dQ distribution for OPAL data o 0 KsK s pairs (full circles) compared to the same distribution obtained from the Monte Carlo simulated events (line histogram). Due to the relatively small statistics at Q values near 0 GeV, we have taken a wider bin for the first one in order to keep its statistical error comparable to those at higher Q values. The area under the Monte Carlo distribution is normalized to the data sample.

the f0(975) and a0(980), are known to exist below the KK threshold and their relation to our analysis will be discussed later on. Note that the ~/~(1020)

0 0 does not decay to KsK s. To construct the correlation function C(Q) it is

necessary to have a reference sample distribution po(Q) which should simulate the data distribution p(Q), shown in fig. 2b (full circles), in all its features except the BEC. In the various studies of the ~±~z ± system the immediately available and obvious reference sample was that obtained from the 7r+Tr

pairs in the same data events. Such a data based reference sample does not exist for the BEC investiga- lions of Ks°Ks°-pairs. In particular the system KsK ~ , ° 0 which is identical in its kinematic properties to the K°Ks°-pairs, is excluded not only for experimental difficulties (the long life-time of K °) but also due to the presence of the destructive interference discussed in section 3.

In principle one can consider the data of K±K ° or K+K pairs as reference samples. However in our experiment, as in many others, the identification of charged kaons is limited to a restricted momentum range. Furthermore, the K + K- system is plagued by the presence of the ~h(1020) vector meson lying near Q = 0. For these reasons we have considered in this work only two possible reference samples, namely, a generated Monte Carlo sample and the event mixing method applied to data events.

For the generation of a Monte Carlo reference sample we have used the JETSET 7.3 program [18] with its BEC option switched offand its free parameters set to values which best describe the OPAL multi-hadron event shape distributions [ 19]. The generated events

were passed through a detailed detector simulation [20] and were analyzed with the same programs as our data events. A JETSET generated sample has previously been successfully applied to the OPAL inclusive K~ analysis [14]. In this version of the Monte Carlo program however, no provision has been made

k " 0 I ~ 0 to account for the decay of resonances into a "-s,-s final slate. For lhe present analysis we utilized a sample of about 570000 events. To verify that the Monte Carlo events are adequate to be used as a reference sample, we have checked that its features pertaining to events with more than one K ° candidate in the final state, which are relatively insensitive to the BEC effect, were in good agreement with the data. This agreement is seen for example in fig. 3a where a comparison is made between the Monte Carlo distribution of the angle between the 0 0 KsK s pair and that obtained from the dala. A similar agreement is seen in fig. 3b where the sum of the momentum values of the two K ° candidates is plotted. In tile insert of the same figure this sum of momenta is plotted for the low pair masses with MKK ~< 1.1 GeV/c2 where the BEC is cxpecled to be concentrated. Finally in fig. 2b we show the Monte Carlo Q distribution in the range 0 ~ Q ~< 2.0 GeV where it is compared with the cor-

462


0.32

~ 0 . 2 8 $

0.24

Z 0.2

0.16

0.12

0.08

0.04

0 -1

a) I

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 O O cos(K~,K~)

2.2

~_ 2

2 .~ 1 . 8

¢a L6

1.4

1.2

0.8

a) OPAL

r e s o n a n c e r e g i o n

i ~ :.~.~!~.~.: .~ :.~.o~ ~,.

0,2 0,4 0.6 0.8 1 1.2 1.4 1,6 1.8

Q [GeV]

Z 0.12

0.1

0.08

0.06

0.04

0.02

0

, ) ....

I I , ' , ' ' 2 4 6 8 110 1L2 ' 1'4 ' 1'6 ' 118 20

sum of K~ momenta p [GeV/c]

1.4

1.2

1

0.8

0.6

0.4

0.2

0

OPAL

0.25 0.5 0.75 1 1.25 1.5 1.75 2

Fig. 3. Comparison of the data to JETSET Monte Carlo generated events: (a) The distribution of the cosine of the angle between the two K ° candidates, cos(K °, Ks ° ). (b) The

sum of the momentum values of the two K ° candidates. The insert illustrates the same distribution for the low mass range MKK ~< 1.1 GeV/c 2 only. The areas are normalized to unity. The data points (full circles) are compared with the corresponding distribution (lined histogram) obtained from the Monte Carlo sample void of the BEC effect. The statistical errors for the Monte Carlo sample are of about the same size as that of our data.

Fig. 4. (a) The measured correlation function C(Q) in the range 0.0 ~< Q ~< 2.0 GeV. The error bars represent the combined statistical uncertainty of the data and Monte Carlo samples. The solid line represents the best fit to the data using eq. (5) with a 2-parameter X 2 fit. The region of the f ; (1525) and the f0(1710) has been omitted from the fit

(see text). (b) 2 versus R0 obtained in a 2-parameter X 2 fit of C(Q) to the data. The cross represents our best values and the contours around it represent the allowed regions for a 50% and 90% confidence level using the statistical errors only.

r e s p o n d i n g d i s t r i b u t i o n for the data .

W e have also c o n s i d e r e d the use o f the H E R W I G

M o n t e Car lo gene ra to r [21] wi th the de ta i l ed O P A L

de tec to r s i m u l a t i o n wh ich p rev ious ly was able to de-

scr ibe the inc lus ive K°s p r o d u c t i o n seen in the O P A L

da t a [ 14 ]. H o w e v e r we f o u n d t ha t th i s g e n e r a t e d sam-

ple fails to desc r ibe m u l t i - K ° sys tems. In pa r t i cu l a r 0 0 the d i s t r i b u t i o n o f the angle be t w een the K s K s pa i r

shows m a r k e d d i f fe rences be t w een H E R W I G a n d the

da ta ove r the whole angula r region. A s imi la r fa i lure o f

H E R W I G to descr ibe the two par t ic le sys tem in O P A L

da ta has a l ready been obse rved in the e n e r g y - e n e r g y

co r re l a t ion func t i on s tudy [22] . W e have the re fo re

not used the H E R W I G M o n t e Car lo as a re ference

sample . 0 0 In the even t mix ing m e t h o d one c o m b i n e s a K s K s

pa i r o r ig ina t ing f rom two d i f fe ren t e + e - a n n i h i l a t i o n

even t s which the re fo re shou ld be vo id of BEC. T h i s

463


method, in its simplest application, is unsuitable at

high energies where a distinct hadron jet structure is

present. In order to try and correct the method for

this deficiency we have adopted the following pro-

cedure. First we have utilized only pairs of multi-

hadronic events which had two identified K ° decays

each. Next we rotated the 0 0 KsKs pair of one event

so that its higher momentum K ° coincided in its di-

rection with the higher momentum K ° of the second

event. We then calculated the Q values from the two

possible mixed 0 0 KsKs pairs. In order to veril3, whether

the mixed event method is able to serve as a reference

sample, we have first applied it to the generated sam-

pie. A comparison between the Monte Carlo mixed

event (2 distribution to that obtained from the Monte

Carlo sample using 0 0 KsK s pairs of the same event re-

vealed a large and significant difference in particu-

la ra t the low (2 values (< 0.2 GeV) which are mosl

sensitive to the BEC. This discrepancy arises because

event mixing not only destroys the Bose-Einstein cor-

relations, but also all other conceivable correlations

such as those coming from gluon emission and the

fragmentation process. Difficulties of the event mix-

ing method due to the hadron jet topology at high en-

ergies have already been noted previously in a BEC

study of like-sign charged pion pairs •6]. In kaon pairs

these difficuhies may well be aggravated due to the

lower number of pair combinations per event. The

reference sample was thus chosen to be derived from

about 570 000 JETSET Monte Carlo events which did

not include the BEC, using the 0 0 KsK s pairs of the same

event, as displayed in fig. 2b.

The correlation function C ( Q ) , defined by eq. (1),

was obtained by dividing the data distribution, bin

by bin, by the reference sample distribution shown

in fig. 2b. The reference sample is normalized to the

total number of the K°K~ pairs in the data within the

range of 0.65 ~ Q 4 1.1 G e V a n d 1.55 ~< Q ~< 2.0

GeV. This range was chosen in such a way as to ax old,

on one hand, the resonance region from 1.1 to 1.55

GeV and, on the other hand, the BEC effect range.

The resulting C ( Q ) distribution is shown in fig. 4a

in the range 0 ~< Q 4 2.0 GeV and in table 1. The Q resolution, as found in the Monte Carlo gen-

erated sample, was better than 20 McV over the whole Q range under study. In the data sample this reso-

lution is expected to be slightly wider but still much

narrower than the chosen Q bin size. It can be seen

Table 1 The measured correlation function ( '(Q) in l 3 bins for 0. ~< Q 4 2.0 GeV. The errors represent the combined statistical uncertainty of the data and Monte Carlo samples.

Q [GeV] Cnrrelation function ( ' (Q)

0. -0.2 1.71 = 0.36 0.2 -0,35 1.29 ± 0,14 0.35-0.5 1.09 :L 0.09 0.5 -0.65 1.01 i 0.08 0.65-0.8 0.92 ~* 0.06 0.8 -0.95 1.02 ± 0.07 0.95-1.1 0.98 ± 0.07 1.1 -1.25 1.10 ± 0.08 1.25-1.4 1.00 ± 0.08 1.4 -1.55 1.31 ± 0.11 1.55-1.7 1.11 ~+ 0.10 1.7 -1.85 1.00 ± 0.10 1.85-2.0 1.07 ± 0.11

from fig. 4a that, for (2 >~ 0.5 GeV, the distribution

is consistent with unit}' if the (2 range of the f~ ( 1525 )

and f0 ( 1710) mesons is disregarded. At smaller (2 val-

ues an enhancement is seen as the correlation func-

tion C((2) increases and reaches a value of about 1.7

at (2 ~ 0.1 GeV. This behavior is very similar to the

BEC seen in the 7t~z~ ± systems -~3 . We then fitted the

expression given in eq. (5) to tile data. Since it was

found that </)((2) was cssentially independent of(2 in

the range of 0 4 (2 4 2.0 GeV where the fits were

carried out, wc have set ~ = 0, so that eq. (5) reduces

to

c(0)

= N [ l + j ( ( 2 ) 2 e x p ( - ( 2 2 R e ) ] ( l + <$Q)- (9)

For the C (Q) fits to the data we have used the least Z

method and set f((2) to a constant, namely f ( ( 2 ) = 0.79, since it was found from the Monte Carlo stud-

ies that the non K°K ° background was essentially in-

dependent of (2. In the first fit we have left the four quantities 2, R0, and N free. Next we reduced the free parameters to

three, namely, to 2, R0 and (5 where the factor N was

determined by the requirement that the area under the

~3 A similar behavior is also seen at the JETSET 7.3 gen- erator level when the parameters addressed to the BEC

KO K0 in "-s'-s pairs are set to those determined from the pion

studies [5].

464


fitted c ( o ) curve be equal to that given by the data points. Due to the fact that the value obtained for the parameter 6 was found to be consistent with zero both for the 4 and 3-parameter fits, we have further set 6 = 0 and repeated the analysis with a 2-parameter fit to A and R0. The results of these fits are presented in table 2 where DOF stands for the number of degrees of freedom. All fits have acceptable values for x2 /DOF

and, within errors, result in consistent values for the parameters 2 and R0. In the following we will refer to our final 2-parameter result as the reference fit.

To estimate the dependence of our final results on the Ks ° selection criteria we have repeated the analysis modifying these criteria within plausible ranges.

The results of the corresponding 2-parameter fits are summarized in table 3. First, the track selection was changed by limiting the acceptance range to tracks with a polar angle between 44 ° and 136 ° (row [b] in table 3). We then changed the cut on the reconstructed mass of the K ° from ± 25 MeV/c 2 to ± 20 MeV/c 2 (row [c]) and ± 30 MeV/c 2 (row [d]). It is

interesting to note that when we widen this cut to := 75 MeV/c 2 or more, we observe a significant decrease of the enhancement, indicating thereby that the effect does not come from the background.

To examine the dependence of our fit on the O range chosen for normalization and fit, we modified it as follows. First, we select the O range of 0 4 O ~ 1.1 GeV to avoid the range of influence of the f~(1525) and fo(1710) mesons altogether (row [e] ). Next we used for the normalization and the fit the full O-range 0 to 2.0 GeV thereby including the resonance region (row [ f ] ). Finally, in row [g], we used for , [ (Q) , instead of a constant, a linear function in Q. We determined the slope from Monte Carlo studies of the background under the K ° peak and chose the maximum slope allowed within one standard deviation from the mean value. As can be seen from table 3, all the fit results are in good agreement. To estimate the systematic error from the variation of the selection and the fit conditions we assumed the deviations from the reference fit values to be independent. Adding them in quadrature we obtained an uncertainty of ±0.13 and ±0.14 fm for the 2 and R0 values respectively.

As observed in previous BEC studies of charged pion pairs, an obvious important source of systematic errors comes from the particular choice of the reference sample. These systematic errors were estimated

in the di-pion studies from a comparison of several acceptable reference samples. Here however, we are limited to only one reference sample as discussed above.

This being the case we adopted the following method to estimate the systematic uncertainties. Event samples were generated with the JETSET 7.3 program by varying, one at a time, its free parameters by one standard deviation from the optimized values given in ref. [19]. The following parameters were

varied: the QCD cut off parameter AQCD, the Q0 parameter which specifies the min imum parton virtu-

ality to which partons may evolve, aq which controls the transverse momentum spectrum of hadrons and the parameter a which determines the longitudinal momentum spectrum. Finally we varied the ratio of strange vector mesons to vector plus pseudoscalar mesons. The higher value is derived from the mean value and the errors given in ref. [23], while the lower value was presented by this experiment in ref. [24]. The resulting Q distributions were then divided by the standard generated distribution before detector simulation. Through this division we obtained values which have been used as Q dependent weights to modify the standard reference sample with full detector simulation. Finally with these modified distributions we repeated the BEC analysis. The values of the changed parameters and the fit results are given in table 4.

We estimated the systematic uncertainties coming from our choice of the reference sample from the variation of the parameters given in table 4. These were found to bc strongly correlated. Therefore we eval- uated the systematic errors by taking the maximum

deviations from lhe reference fit values. These were found to be ± 0.26 for the ). value and ± 0.13 fm for the R0 value. Due to the unacceptable fit obtained for aq = 0.32 we omitted the values given in row [f] from our systematic errors study. Finally we added in quadrature the contributions from the variations of the selection and the fit conditions to the reference sample uncertainties, such that our final values for the parameters ), and R0 are

), = 1.12 ± 0.33 = 0.29,

Ro = ( 0 . 7 2 ± 0 . 1 7 ± 0 . 1 9 ) fm. (10)

The first quoted error is statistical and the second systematic. The results of the 2-parameter Z 2 reference

465

Volume 298, number 3,4 PHYSICS LETTERS B 14 Janua~ 1993

Table 2 Results of the Z 2 fits of C(Q), as defined in eq. (9), to the data in the range of Q = 0 to Q=2.0 GeV. The region of the f~(1525) and f0(1710) resonances. 1.1 ~ Q ~< 1.55 GeV, has been omitted from the ill. The errors are statistical only.

T y p e o f f i t z 2 / D O F 2 R o (fro) ~5 (GeV - l ) N

4-parameter (2, R o, & N) 1.9/6 1.13 ± 0.47 0.61 ± 0.16 0.10 ± 0.11 0.89 ± 0.11 3-parameter (2, R o, 6) 2.9/7 1.01 ~ 0.49 0.71 ~ 0.18 0.01 3:0.04 - 2-parameter (2, R o) 3.4/8 1.12 i 0.33 0.72 ± 0.17

Table 3 Results from a study of the X 2 fit dependence on the selection criteria and the fit conditions (see text). The errors are statistical only.

Fit conditions z2 /DOF 2 R 0 (fm)

[a] reference fit 3.4/8 1.12 ± 0.33 0.72 ± 0.17

[b] modified track selection 4.5/8 1.10 ± 0.41 0.71 ± 0.22 [c] K °L.~ mass cut: ± 20 MeV/c 2 3.4/8 1.06 ± 0.38 0.78 ± 0.23 [d] K~ mass cut: z~ 30 MeV/c 2 3.9/8 1.11 ± 0.29 0 . 6 4 i 0.13 [e] Q-range: 0 to 1.1 GeV 1.4/5 1.07±0.28 0 .66±0.16 [f] Q-range: 0 to 2.0 GeV 12.0/11 1.18 ± 0.37 0.80 ± 0.18 [g] f ( Q ) = 0 . 7 3 + 0.06.Q 3.4/8 1.20 t_ 0.36 0.73 ± 0.17

Table 4 Results from a study of the Z 2 fit dependence on several parameters of the JETSET 7.3 reference sample which control the momentum distribution of hadrons. The errors are statistical only.

Fit conditions z2 /DOF i R 0 (fm)

[a] reference fit 3.4/8 1.12 2_ 0.33 0.72 ± 0.17

[b] AQCD = 0.28 GeV 3.0/8 1.04 ± 0.32 0.69 ~ 0.16 [c] AQC D = 0.31 GeV 3.4/8 1.32 ± 0.37 0.77 i 0.16 [d] Qo = 0.70 GeV 3.6/8 1.25 ± 0.34 0.75 ± 0.15 [e] Qo = 1.80GeV 4.7/8 1.26~0.34 0.74±0.15 [f] Oq = 0.32 GeV 14.6/8 1.12 ± 0.38 0.91 ± 0.22 [g] aq = 0.40 GeV 2.5/8 1.07 ± 0.30 0.65 • 0.14 [h] a = 0.13 3.3/8 1.31=k0.36 0 .75±0.15 [i] a = 0.30 4.7/8 1.26~0.34 0 .74~0.15 [Jl [ I ' / (V + P ) ] s = 0.43 4.9/8 1.05=t-0.37 0 .65~0.18 [k] [V/ (V + P ) ] s = 0.68 4.6/8 1.38 ± 0.41 0.85 ± 0.19

fit are also i l lustrated in fig. 4b in a 2 -d imens iona l

plot o f 2 versus R0. The con tours represen t the 2 and

R0 values hav ing conf idence levels o f 50% and 90% as

calculated f rom the statist ical errors. As can be seen,

even though the values of 2 and R0 are s o m e w h a t

corre la ted, a value o f 2 -- 0 is exc luded with more

than 90% conf idence level.

5. Summary and conclusions

An e n h a n c e m e n t at small values of Q is seen in

our sample o f o 0 K s K s pairs p roduced in the react ion e+e Z 0 o o ~ K s K s + X. We a t t r ibute this enhance-

ment to Bose -E ins t e in correlat ions. In principle, the

scalar mesons t"o(975) and a0(980) , es tab l i shed via

their decays to ~ r and qn, respectively, can cont r ibu te to the 0 0 K s K s mass d i s t r ibu t ion in the ne ighbourhood

o f the KK threshold . To es t imate the con t r ibu t ion o f

466

Volume 298, number 3,4 PHYSICS LETTERS B 14 Janua~' 1993

also due to T, Sj6strand for helping us to operate

the Monte Carlo program in the presence of two-K °

Bose-Einstein correlations.

It is a pleasure to thank the SL Division for the

efficient operation of the LEP accelerator, the pre-

cise information on the absolute energy, and their

continuing close cooperation with our experimental

group. In addition to the support staff at our own

institutions we are pleased to acknowledge the

Department of Energy, USA,

National Science Foundation, USA,

Science and Engineering Research Council, UK,

Natural Sciences and Engineering Research Council,

Canada,

Israeli Ministry of Science,

Minerva Gesellschaft,

Japanese Ministry of Education, Science and Culture

(the Monbusho) and a grant under the Monbusho

International Science Research Program,

American Israeli Bi-national Science Foundation,

Direction des Sciences de la Mati6re du Commis-

sariat/ l l 'Energie Atomique, France,

Bundesministerium fiir Forschung und Yechnologie,

FRG,

National Research Council of Canada, Canada,

A.P. Sloan Foundation and Junta Nacional de

Inves t iga~o Cient~fica e Tecnol6gica, Portugal.

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468

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