Revista Mexicana de Física 41, Suplemento 1 (1995) 30-37

Bose-Einstein correlations and multifragmentationin relativistic heavy-ion collisions*

ALBERTO GAGO

Department of Physics, Pontificia Universidad Católica del PerúApartado postal 1761, Lima, Perú

AND

GERARDO HERRERA

Department of PhysicsCentro de Investigación y de Estudios Avanzados del IPNApdartado postal 14-740, 07000 México D.F., México

ABSTRACT. It has been shown that the scale measured through Bose-Einstein correlations (BEC)reflects an intrinsic scale of the order of the meson Compton wavelength. In heavy-ion collisionshowever, the radii of the pion emission region obtained are close to the effective nuclear radius ofthe lighter nucleus. We show that BEC in a multifragmentation picture of heavy-ion collisions iscompatible with the two facts and that it is therefore possible to use Bose-Einstein interferenceto obtain a geometrical description in this kind of reactions. We oulline the possibility to useBose-Einstein correlations to learn something about multifragmentation.

RESUMEN. Se ha demostrado que la escala medida por medio de correlaciones de Base-Einsteinrefleja siempre una escala intrínseca del orden de la longitud de onda de Compton del mesón. Sinembargo en colisiones de iones pesados los radios de la región de emisión obtenida son del ordendel radio nuclear efectivo del núcleo más ligero, usado como proyectil. Aquí mostramos que lascorrelaciones de Bose-Einstein en un marco de multifragmentación en colisiones de iones pesadoses compatible con los dos hechos y que es por eso posible usar interferencia de dos partículas paraobtener una descripción geométrica en este tipo de reacciones. También planteamos la posibilidadde utilizar correlaciones de Bose-Einstein para estudiar el proceso de multifragmentación.

PACS: 25.70.Pq; 25.75.-q

We choosc to examine a phenomenon which is impossible, absolutely impossi-ble, to explain in any cla..o:;sicalway, and which has in it the heart of quantummechanics. In reality it contains the only mystery. We cannot make the mys-tery go away by explaining how it works. We will just tell you how it works.In telling you how it works we will have told you about the basic peculiaritiesof all quantum mechanics.

(R. Feynman, talking about interference [1)).

l. INTRODUCTION

A considerable experimental elfort has been ami is being presentely made to a betterunderstanding of interferometry phenomena in both elementary particle collisions andnuclear reactions at high energies [2,3] .

• This work was supported by CONACYT lInder contract 2146-E.

BOSE-EINSTEIN CORRELATIONS AND MULTIFRAGMENTATION... 31

The Bose-Einstein correlations are commonly described in terms of a two particle cor-relation function RBE which is defined as the ratio of the joint production probabilityP(PI, P2) to the product P(pl )P(p2) of single production probabilities

(1)

where PI and P2 are the pion four-momenta. The correlation function parametrizes theeffect assuming a spatial and temporallocalized source for the bosons [4]. It can be writtenin terms of a density function p(r) which describes the source space-time extension,

(2)

where ,pBE represents the Bose-Einstein symmetrized wave function of the bosons systemand Iv p( r) ér = 1. Taking plane waves to describe the bosons one obtains the correlationfunction for a chaotic pion source:

(3)

where F(p) represents the Fourier transform of the density function p(r).In general the two particle correlation includes a coherence parameter a which multiplies

the second term in Eq. (3). The coherence parameter a takes values between O and 1. Itparametrizes the strength of the correlation and contains information on the chaoticitylevel at which hadronisation occurS.

A description of the space-time evolution of the multiparticle production process isvital in heavy-ion collisions. The formation of a quark-gluon plasma in this reactions hasbeen predicted and Bose-Einstein correlations could be an important tool to study itsproperties. In fact this technique will play an important role in the search of quark-gluonplasma at RHIC and LHC experiments and it is therefore important to understand indetail the different aspects of this effect (see for example [5]).

2. TUE FRAGMENTATlON-RADIUS PUZZLE

In order to describe the quantum interference during the fragmentation in high energy re-actions, phenomelogical parametrizations of the effect have been proposed [6]. Accordingwith them the scale measured with Bose-Einstein correlations is of the order of the Comp-ton wavelength of the meson used to analyse the correlation [7]. This view is supportedby high energy physics experiments, which do not see any dependence of the fragmenta-tion radius with the energy involved in the collisions. The fragmentation models for thisreactions, on the othcr hand, do prcdict an incrcasc of thc hadronisation volumc.

In nuclear collisions it has been oberved that the radius extracted from two particlecorrelations are consistent with a A 1/3 scaling, where A is the atomic mass of the projectile.

32 ALBERTOGAGOANOGERAROOHERRERA

FIGURE 1. Multilragmentation in nuclear reactions (a). In relativistic heavy-ion collisions mul-tifragmentation can be seen as the appearance oC several fragmenting Quark.antiquark systemsevolving in space-time (b).

Since nuclear radii scale as roA 1/3, the measurement of the same scaling using Bose-Einstein correlations provides an indication that Bose-Einstein interference measurementshave a geometrical interpretation beyond the scale ofthe meson used in the analysis [8-10].\Ve show that these two views are compatible with each other. The dynamics 01 nuclear

reactions is such that Bose-Einstein correlations indeed have sorne geometrical interpre-tation in spite 01 the fact. that. an int.rinsic scale of t.he order of t.he meson size is alsopresen!..

3. BE INTERFERENCE IN MULTIFRAGMENTATION

In int.ermediate energy nucleus-nucleus col1isions highly excit.ed nuclear matt.er is proaucedand t.he emission 01 fragment.s ano it.s subsequent decay becomes import.an!.. The st.udy offragment-fragment. correlations provides a gooa met.hod to understand in more detail howmultiparticle production happens.In relativistic heavy-ion col1isions, one may think of t.he mult.iparticle product.ion pro-

cess as taking place in a temporary arop of compressed and heat.ed nuclear mat.ter. Thedeterminat.ion of the size of this drop 01' "firebal1" would provide fundamental informa-tion about t.his state. In t.his reactions one can think 01 the mult.ipart.icle production asthe result 01 the Iragmentation 01 quark-antiqllark systems in many different places ofspace-time [111 (see Fig. 1).In these two scenarios the two particle correlation function differs from t.he commonly

used to describe multiparticle production in elementary particle reactions. The correlationfunction in nuclear col1isions 01' relativistic heavy ion col1isions should incorporate the factthat the two mesons could eventual1y originate in different. fragmentation processes. Inthis case the funct.ion p(r) of Eq. (2) can be written as a sum of e.g., gaussian terms whichdescribe the many different fragments or distributions in space-time where fragment.ationis taking place.

DOSE-EINSTEINCOIUlELATIONSANDMULTIFRAGMENTATION... 33

fragment1t(k1) detector 1

/"/" //"

/" /, /", /" /,/"/" , /

/",

/", , //' ,

detector 2

fragment

FIGURE 2. Two parlide interference in multiple fragmentation. If there are two fragmentationregioos there exists the possibility that the pions originate in different processes.

If for example there are two gaussian-shaped fragments centered at TI and TZ anddecaying into pions and other hadrons the correlation function RBE of Eq. (2) which takesinto account the two possibilities shown in Fig. 2 will be

(4)

where we have assumeo that the two fragments have the same width parameter a andTíz = TI - T"2 is the distance between the fragments. In general for n fragments with thesame width the correlation function will be,

(5)

If the gaussian-shaped fragmentation regions are distributed in a finite extension mod-eled by a function g(r;).then

(6)

34 ALBERTOGAGOANDGERARDOHERRERA

7

r(lm)rms

•,

•

2

D

p d "He

2

"c "'Ne

3 •"Kr

, •A'/J ( projeclile )

FIGURE3. Shows the radii of the pion emission region VS. the cube root of the mass number of theprojectile nucleus collected in ReL [SI. Only the all inelastic reactions are plotted for consistency.The curves represent the measured radius given in Eq. (S) with Tfea, = Ofm (solid line), Tfea, =0.5 fm (dashed line), Tfea, = 1.0 fm (dotted line) and Tfea, = 1.5 fm (dashed-dotted line).

and the correlation function:

(7)

This can be generalizcd to the continuous case

(8)

where the numbcr of fragmcnts or quark-antiquark systems is very large. \Ve have sccnthat the corrclation function of Eq. (5) convcrges very fas\. It does not change for n > 4in a visible way for reasonable values of the parameters.Although at intcrmcdiate encrgy nuclcar collisions thc number of fragments is finite the

continuos aproadl gives an idea of what the trend is and how the presence of fragmentscan infiucnce the parameters of thc correlation.

BOSE-EINSTEINCORRELATIONSANDMULTlFRAGMENTATION... 35

7

r(lm)rms

•

•

•

od "He

,I)c l'ONe ooAr MFe

3

UNb '.'Au

•A'/J ( projectile )

FIGURE4. Shows a plot similar to the one in Fig. 3 with the results collected in [12J.As in Fig. 3,only the inelastic reactions are plotted. The curves are the same than those of Fig. 1.

This picture on the other hand, is realistic for the case of higher energies like relativisticheavy-ion collisions where the number of fragmenting quark-antiquark systems is expectedto be very large. Multiparticle production at RHIC and LHC energies would resemble thispicture.As a crude approach we can model the density g( r) of fragments or fragmenting quark-

antiquark systems with a gaussian of the form

( ) = (_1_) 3/2 -r' /2b'9 r 2nb2 e ,

with (r2) = 3b2• The correlation function will be

R (.' /2+b' /2)Q'BE"""" 1 + e 1

(9)

(10)

which implies (r2) = 3(a2 + b2).If the density of fraglllents given by g(r) keeps melllory of the original nuclear density,

the root-lIlean-squared radius of g(r) ",ould be given by r = 1.2 Al/3. Tbe lIleasured rmeas

36 ALBERTOGAGOANDGERARDOHERRERA

7

r(lm)rms

•

•

•

od 'He

2

13 • •

'''Au

•A'/O ( projectile )

FIGURE 5. Same plot than in Fig. 4 but here reactions with the same projeetile and eventuallydifferent target have been averaged. The curves are the same than those in Fig. 1.

(rms) would then be

rmeas = Jrfrag + 1.22 A2/3, (11)

where rfrag (rms) is the size of the fragments or fragmentation regions. It reAeets a sealeof the order of the hadron size [6,7]. The seeond term aeeounts for the distribution offragments i.e. it eontains the geometry of the reaetion. In Fig. 3 we show the curve givenby Eq. (11) with rfrag(rms) = O, 0.5, 1.0 and 1.5 fm. Figure 4 shows a mOfe reeenteompilation of experimental results [12J and Fig. 5 shows the average values when thereare more than one measurements with the same reaetion.This deseription of the phenomena offers an alternative view of the interference which

explains why nuclear physicists observe a scaling in agreement with their expeetationswhile high energy physieists do noto

BOSE-EINSTEIN CORRELATIONS AND MULTIFRAGMENTATION... 37

4. SUMMARY

In elementary particle collisions the measurement of the radius for the fragmentationvolume using Bose-Einstein correlations seems to give the Compton wavelength of theboson used to construct the correlation function. This explains why the fragmentationradius obtained does not depend upon the center-of-mass energy of the reaction. Howeverin nuclear reactions the measurement of Bose-Einstein correlations is useful to extract thegeometry of the fragmentation region. The puzzle is sol ved by considering the fact thatfragmentation in nuclear collisions occurs in a multifragmentation process.

Bose-Einstein interference does offer a technique to study the fragmentation geometryin heavy-ion collisions. This technique wil! be very useful once experiments at RHIC andLHC start looking into reactions at very high energies.

The fragmentation of exeited nuclei is usually stndied by looking at single fragment ob-servables (e.g. mass distribution of fragments) or two-fragment correlations (for instancein terms of the relative veloeities of fragment pairs or angular correlations between frag-ments). BE correlations are only used to obtain information of the global geometry of theprocess. \Ve believe that by learning how the BE interference behaves in a multifragmen-tation frame we could eventually exploit the technique to learn more abont the particleproduction process.

ACKNOWLEDGEMENTS

\Ve would like to thank Jeff Appel for his warm hospitality during our stay at Fermilab.AG wishes to thank Fermilab and Pontifieia Universidad Católica de Perú for financialsupport.

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