Volume 219, number 2,3 PHYSICS LETTERS B 16 March 1989

PAIR C O R R E L A T I O N S O F N E U T R A L S T R A N G E P A R T I C L E S E M I T T E D IN RELATIVISTIC HEAVY I O N C O L L I S I O N S

Carsten G R E I N E R Nuclear Science Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA

and

Berndt M U L L E R lnstitut fiir Theoretische Physik. J. V~i Goethe-Universitiit, Pos(fach 111932, D-6000 Frankfurt am Main 11, Fed. Rep. Germany

Received 19 September 1988; revised manuscript received 28 December 1988

We point out the possible role of neutral strange particles (A, Ks °) as probes for the size of a baryon-rich fireball formed in relativistic nuclear collisions. A striking difference in the size parameters determined by the hyperons on one hand, and kaons or pions on the other hand, may occur in the case of a phase transition to a quark-gluon plasma.

1. Motivation

The H a n b u r y - B r o w n - T w i s s effect has been uti- lized for more than a decade to de te rmine the size of reaction regions in high-energy part icle and nuclear collisions from an analysis of pair correlat ion func- tions. Its principle, which is based on e lementary quantum mechanics, is well known: Part icles with identical quan tum numbers have the tendency to avoid each other if they are fermions, or are effec- t ively at t racted toward each other if they are bosons. Hence, the probabi l i ty to f ind two identical part icles in a localized region of spacet ime is influenced by their quan tum statistics. The two-part icle correlat ion function R (P,~.I) therefore depends sensit ively on the size ro of the emit t ing region, usually called the fireball.

In most high-energy exper iments of this type one has measured correlat ions between charged pions be- cause of their large abundance [ 1,2 ]. However, pions may to a large part originate from decaying reso- nances like the A-baryon or the p- and (o-meson. Thus, due to the nonvanishing l ifetime of these resonances,

Home address: Institut f'tir Theoretische Physik der Universi- t~it. GltickstraBe 8, D-8520 Erlangen, Fed. Rep. Germany.

being in the order of the l ifetime of the fireball, its size ro may be overes t imated in these experiments. This is the case to a lesser degree for the kaons and strange baryons, since product ion of the heavier strange resonances like the Z* or K* is suppressed be- cause of their large masses, and their emission in par- ticle decays is quite rare. Moreover, correlat ions among neutral (s t range) particles are not dis tor ted by Coulomb effects.

Another potent ial ly impor tan t reason for investi- gating the correlation function of strange particles was recently suggested by one of us [ 3 ]. In part icular, the moment of the creation of part icles with posit ive strangeness, i.e. containing a g-quark (mainly the kaons K +, K °, but also the ant ihyperons A, 2;), may significantly depend on the evolution of the fireball. If a baryon-r ich quark-g luon plasma was initially formed in the collision, the kaons K +, K ° with strangeness + 1 can easily be produced because of their large chemical potential ~tK= ~tq--/Zs, where ~tq refers to light quarks u and d. As a consequence the strange- ness content may separate if the phase t ransi t ion of the quark-g luon plasma to confined, hadronic mat- ter is a slow process, occurring in approximate ther- modynamic equi l ibr ium [3,4]. The g-quarks will p redominan t ly move into the hadronic phase along

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with the kaons. Instead, the s-quarks will remain in the deconfined sector of the coexisting phases. Ac- cordingly, the hadrons with negative strangeness A, K- and I(°) are expected to be produced mainly at the last state of the evolving phase transition when the quark phase volume has become quite small.

The produced hyperons will not freeze out imme- diately, but will interact strongly with the surround- ing particles while the surface of the fireball expands with roughly the speed of light. As the thermal veloc- ity of the hyperons is considerably smaller, they will still be localized near the center of the system (ro ~ 2- 3 fm) when the fireball has expanded to the freeze- out density. Strong localization in spacetime implies that a pair of produced A-particles should be strongly correlated. If deconfinement does not occur in a col- lision, i.e. if a highly compressed, thermalized gas of hadrons is formed, by necessity the hyperons are cre- ated and thermalized over a much larger region in spacetime, namely throughout the whole fireball (ro~7 fm). The two-particle correlation function then has a different shape.

Our article is organized in the following way: In the next section we present a brief derivation of the two- particle correlation function (HBT-effect), where we also consider the effect of a possible A-A resonance. We then present model calculations for various cor- relation functions and discuss the possible influence of the dibaryonic H-particle on the A-A correlation.

2. The correlation function R(Prel)

The two-particle correlation function R (P~e~), de- fined by

l d60"12 1 d30-1 1 d302 0"12 d3p~d3p2 - [ 1 +R(Prel) ] 0.1 d3pl 02 d3p2 ' (1)

where

Prcl =P2 --PJ ,

measures the difference between the two-particle cross section and the product of the inclusive single-parti- cle cross sections. In thermodynamical equilibrium R should only depend on the size ro of the region con- taining the particles at freeze-out. For relative mo- menta Pre~ smaller than the inverse size of the emit- ting region one expects that R deviates from zero, the

sign depending on the quantum statistics of the ob- served identical particles.

Because we are mainly interested in A-A correla- tions, we use the nonrelativistic expressions for cal- culating the function R(Prel) described earlier by Koonin [ 5 ] for p-p correlations. Let D(rt, p) be the relative probability that a particle with momentum p, which is regarded as an outgoing wave packet, freezes out at a place r at time t, i.e. at the last inter- action point. Here D is normalized on the differential cross section,

f d3rdt D(rt, p)= ! d3~

d3p •

Following the arguments in ref. [ 5 ], the two-particle differential cross section may be approximated as

1 d60.j2

0.12 d3pl d 3p2

= i d t ld t2 f d3rld3r2D(rltl,pl)D(r2t2,P2) - o o

X (2z~h)6[ '''r'vl'v2t'' wJ,2 t ' l , r2) I 2 ,

Pl -]-P2 Pcm - - - - . rq=r~+v(t2-tl), v= 2m 2m (2)

Here '' 'f ...... is the (unknown) exact two-particle W l . 2

Schr6dinger wavefunction in the final channel, where the two particles carry the asymptotic momentapl and Pz, respectively. Thus the physics of the correlation function R is contained in the final wavefunction and therefore R depends, to some extent, on the final-state interaction between both particles.

Because we are not interested in such effects at the moment, but in the influence of the underlying quan- tum statistics, we first take the wavefunction simply as a product of plane waves. For spin 1/2 fermions q/~.: has to be antisymmetric under particle ex- change, where the two fermions can be either in the singlet or the triplet state. We parametrize the distri- bution function D(rt, p) by the following simple gaussian ansatz:

D(rt, p)

-~ydppld3a( 1 lz~J,_r6 2 2 ) ~ - - ~ e x p ( - - r /ro) 6(t-- t0) , (3)

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to denotes the momen t of freeze-out, and ro defines the spatial size. Inserting (3) in (2) and compar ing with ( 1 ), the correlat ion function for noninteract ing part icles is predic ted to be o f the form

RF(Prel ) --½exp 2 2 2 = (-p~elro/2h ) ,

R~ (pr~) = exp ( --p ~, r2/2h 2) . (4)

A possible consequence of final-state interact ion between two emi t ted fermions is the appearance of a quas ibound state. We restrict our considera t ion to an

=Pres/m, which s-wave resonance at the energy Ere s 2

we add to the singlet part of the final state wavefunc- tion [ 6 ]. The resonance width FE =FE ( p ~ ) cannot be assumed to be constant for all relat ive momen ta p ~ ; we take it to be of the form

FE(pr~, ) =rE,w~ (pr~,/Pr~)

X exp [ - ½ (pr21 2 2 --P,-es)/rr~] , (5)

where rr~ is the width of the gaussian wavefunct ion o f the quas ibound state. With these assumpt ions we obta in the following expression for the correlat ion function for two fermions with a resonance in the scattering ampl i tude:

R(pr~l) = - ½exp 2 t 2 ]

1 3/2 1 _ ( 1 1 ~--3/2 +Gj

P~s 1 1 - - ~ Xexp ~ _ + rres

1 1 ( ~ r o 2 1 ~-3/2 + ~ (Trh)3 r~ [a(Prel) ]2 + ~ J '

(6)

where

(2rt) -3/4 /-1/2

d(pre,)-- mx/~lrr3/s2 [m-2 (P~ 2,-pre~)2 2+FE/412 1/2"

We are now in the posi t ion to discuss the influence of the size pa ramete r ro and o f a possible resonance channel on the two-part icle correlat ion function for identical, neutral particles, such as A or K °. (Charged particles, like ~ - , K + or the proton, cannot be de- scribed as above by simple plane waves, because they are subject to the repulsive Coulomb interact ion at

short distances. In that case, the correlat ion function will always decrease for small relat ive momenta Pre~. )

3. Some results and conclusions

Here we are mainly interested in the short-l ived, neutral kaons K ° (because the decay length of the K ° is too long to allow detect ion in present experi- ments ) . However, the part icles in heavy ion colli- sions are first created as eigenstates of the strong in- teraction, i.e. K ° (gd) and I~ ° ( sd ). Consequently, the correlat ion function of the measured K ° is a mixture of those of the K ° and the I~ °. Knowing that K ° and I( ° turns into K ° and K ° half of the t ime, and assum- ing equal m o m e n t u m d i s t r i b u t i o n f ( p ) in the differ- ential cross section (d3tT/dp=o-total f ( p ) ) for the K ° and the I( °, the correlat ion function for the Ks ° takes the form

a~o RKs ° (Prel) = ( 0"Ko + 0"Ko) 2 RK° (Prel)

O'p.o + (ffKO+rYK,~)2RRo(Prel), (7)

where aKO, aKO denotes the total yield of the pro- duced kaons. In the case of a baryon-r ich scenario, which we especially want to examine, the K ° are pro- duced much more abundant ly than the I( ° because o f their f lavour content. Thus Rks o is expected to be

RK ° (Prol) ~ RKo (Prel) •

SO one impor tan t conclusion is the invisibi l i ty of the I~°-component in the correlat ion of the K °. This im- plies that the relevant fireball size is that associated with emission of g-quarks, i.e. s imilar to that valid for pions.

In fig. 1 we show the dependence of RK~ (or RKo ) for different sizes ro of the emitt ing source. I f we adopt some recent results from the 7t -data [1] , the ex- pected radius should be a round 7 fm. The pair cor- relat ion function of noninteract ing A-part icles is shown in fig. 2. (The possible contr ibut ion of the Z °, which decay somewhat rapidly into A, to the corre- lat ion function of all detected A is considered below; for the moment their influence is neglected. )

In both figures one recognizes a sensitive depen- dence o f R on the size ro. I f ini t ial ly a quark-g luon

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1.0

0.8

% 0.6 Q.

or" 0.4

0.2

0.O

, i , i

i

20 40

Kaon-cor re ta t ion

i i i i i

2frn

3

60 80 100 120 140

r e t a t i v e m o m e n t u m ( M e V / c )

Fig. 1. The pair correlation function R (Pr,q) of emitted neutral kaons. The expected size of the fireball lies around 7 fro. This both should be the case, whether originally a quark-gluon plasma (QGP) or an excited hadronic resonance gas (HRG) is created in an uhrarelativistically heavy ion collision.

Lambda-corretat ion w i thout resonance

o.o , , , , , , , , , ,

-0.2 7frn 5 Q 3 HR

- 0 . 4 2fm

Q- ~:£ -0.6

-0.8

-1.0

0 20 /,0 60 80 100 120 140

r e t a t i v e m o m e n t u m ( M e V / c )

Fig. 2. The pair correlation function of the A depends sensitively on the source radius ro of the spacial volume, where the particles are located at freeze-out. The two different expected scenarios are outlined (i.e. whether the fireball has undergone a phase transition from quark-gluon plasma to hadronic matter (labeled QGP) near chemical equilibrium or whether only a state of strongly interacting hadronic matter (labeled HRG) has been built up in the collision)• One clearly can determine the size ro from analyzing the shape•

plasma was created - as commented above (see ref. [3 ] ) - , the A are expected to be emitted from a strongly localized region with ro lying around 3 fm (labeled " Q G P " in the figure). This possible behav- iour then must be clearly seen in the deduced corre- lation of the produced neutral hyperons. Accord- ingly, if a s-g-separation does occur during a phase

transition from a quark-gluon plasma to a hadron gas, it has to show up in the measured pair correlation of the A. (r0)A then has to be significantly smaller than

(ro)~, (ro)N or (rO)K and thus can be understood as a direct indication for plasma creation.

However, a more detailed analysis must take into account the part of the correlation of the measured A coming from Z°-decays. All Z°-hyperons, decaying after some 10 -2° s, are recognized as A-hyperon, al- though they were originally produced as different particles. Comparing (7) to the total correlation of all A, one similarly may approximate

RAamal (P re l )

O'Tk,direc~ (aA,di~c, + aA,ZO) 2 RA, di~,(P~t)

aTx,ro "~ (0.A,direc t _{_0.A,,;O)2 RA,EO(Pr,,I) • (8)

The Z ° and A are not correlated, because they are dif- ferent particles. Hence, if there are nearly as many E ° as A produced in the fireball, the total value of the correlation function drops to (assuming RA.J~rcc,~

RA,ZO)

RA lolal ~ 1 • ~ RA.dircct .

Although the magnitude of the A-correlation may be smaller, the slope, depending only on the size of the fireball, remains unchanged. In addition, in (8) the original correlation function R~o will be widened (-+RA,zO) due to the recoil of the photon on the A in the order of 60 MeV/c. This leads to a noticeable modification, but does not seriously affect the shape, Consequently, the source size parameter can still be deduced reliably from the total A-A-correlations.

Despite these interesting features and promise for future investigations, some cautioning remarks ap- pear necessary. The neutral particles like the A-hy- peron or the K ° are difficult to see in experiments, one can only identify them by their weak decays. Moreover, the multiplicity of A and K ° is likely to be much smaller than the total multiplicity. Therefore a detailed study of the pair correlation is much more difficult than for pions. Nevertheless we hope that progress in experimental techniques, especially con- cerning the efficiency of detecting neutral strange

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particles, will allow for such measurements t~1. Also note that we have described the correlat ion effects in the center-of-mass frame treat ing part icle k inemat ics nonrelativist ically. Because the fireball i tself moves rapidly in the laboratory frame, the detected mo- menta of the measured part icles have to be Lorentz- boosted to the center-of-mass frame.

We next discuss the influence of a possible low-en- ergy A-A-resonance on the A-correlat ion. Concern- ing the A-A- in te rac t ion there exists some informa- tion from double hypernuclei [7,8]. It appears that an at tract ive A-A-po ten t i a l comparable in strength with the N-N- fo rce is needed to fit the da ta on A6He. Adopt ing these potentials, we find that quas ibound s-resonances, located a few MeV above the A - A - threshold are possible. We have also es t imated the decay of such an s-wave resonance, which can range anywhere from 1-10 MeV, taking Bodmer ' s [7] in- teract ion parameters (range rAA~3 fm) . Here, we take the radius of the resonance as 2.4 fm. Two val- ues of the width and the resonance energy are chosen within the range allowed by the analysis of ref. [ 7 ]. As one sees from fig. 3, the resonance is visible par- t icularly for small fireball sizes ro. This is clear, be- cause the part icles must overlap in space to feel the at t ract ive interact ion for such a state. Note, that the presence of a resonance would simplify the experi- mental deduct ion of the fireball size. The corre- sponding width FE and resonance energy Erc~ could be deduced from the measured correlat ion, and thus also the size ro can be determined. Thus more infor- mat ion about the A-A- in te rac t ion may be obta ined in addi t ion to the double hypernuclei data. Relat iv- istic heavy ion collisions may provide a unique source of informat ion about it.

Finally we want to comment briefly on the influ- ence of an exotic quas ibound resonance on the A-cor- relation, namely the so-called, possible d ibaryonic state H with quark content ( s s iuudd) . This strange- ness-2 particle may be bound [7,9] or quas ibound [ 10]. It is believed, that if the H-part ic le really exists as a quas ibound state, i.e. i f its mass lying above the A-A- threshold , the width of the resonance should be large ( ~ 10 MeV) [ 10]. In the case o f an H-part icle, one has to use a resonance radius r r ~ 1.3 fm instead

"~ Two experiments at CERN (NA36 and WA85 ) are dedicated to measurements of strange particles, in particular A and K °.

Lambda-corre[ation with resonance

' t , t , t ' I ' I

3 a E 2 f r o P ~ e ~ = 5 0 M e V / c

2 Rres = 2 / ' f m

=

131

0

-1 I I i I i i i i i i

20 40 60 80 100

relative momentum (MeV/c)

Lambda-corretation with resonance ' I , t ' t ' t ' I 1

b 2 f ~ / ~ ~ P~,~ = 50 M e V / c

/ \ Rre s = 2 4 f m

Wid th = 2 7 MeV

0 if:

-1 I i I i I i I i I

20 40 60 80 100

relative momentum (MeV/c)

Fig. 3. The pair correlation function of the emitted A might be drastically changed, if a possible resonance channel contributes

at low relative momenta. If this is the case, the fireball size in a heavy ion collision has a much stronger influence on the shape of the correlation function and thus the size G may be obtained more precisely. Two examples ( (a ) and (b ) ) with the same resonance position, but with different widths are shown.

of the 2.4 fm used above. In a naive MIT-bag model description all six quarks occupy the 1 s-state, thus the size of the quark bag is expected to be small. We found, that these quasibound states with a width lying around 10 MeV do not have any recognizable effect on the correlation. This would be different only if we assume without any further physical justif ication that the width FE might be much smaller (FE~ 1 MeV) , i.e. if the H-part icle is a relative stable, well-estab- lished resonance. We conclude, that the existence of a nonstable H-part icle is unlikely to be established from correlat ion measurements .

Summarizing, we have tried to mot ivate the use of

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Volume 219, number 2,3 PHYSICS LETTERS B 16 March 1989

strange part icles in future corre la t ion exper iments , bu t also have po in ted out some potent ia l difficulties. In addi t ion to the p ion and nucleon one has two other possibi l i t ies (A, K °) for deduc ing the spacet ime structure (size) of the created fireball. Clearly, this new in fo rma t ion may shed fur ther light on the source parameters . I f the s -g-separat ion dur ing a phase t ran- s i t ion does work, as proposed earl ier [ 3,4 ], a s t r iking difference be tween the extrapola ted radius (ro)A an d for example (ro)~, (ro)K~ may be found. This would be a s t rong h in t for the ini t ia l c rea t ion of a q u a r k - gluon plasma. At last, the observa t ion of a resonance in the A-A-co r r e l a t i on would give some exper imen- tal insight on their in teract ion.

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(1988) 2797. [4] C. Greiner, P. Koch and H. St6cker, Phys. Rev. Len. 58

(1987) 1825. [5] S.E. Koonin, Phys. Lett. B 70 (1977) 43. [ 6 ] U. Fano, Phys. Rev. 124 ( 1961 ) 1866. [7] A.R. Bodmer and Q.N. Usmani, Nucl. Phys. A 463 (1987)

221. [ 8 ] K. Miyahra, K. lkeda and H. Bando, Prog. Theor. Phys. 69

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