EVENT-BY-EVENT PHYSICS IN

RELATIVISTIC HEAVY-ION COLLISIONS

Henning Heiselberg

NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

Motivated by forthcoming experiments at RHIC and LHC, westudy event-by-event fluctuations in ultrarelativistic heavy-ion col-lisions. Fluctuations in particle multiplicities, ratios, transverse mo-menta, rapidity, etc. are calculated in participant nucleon as wellas thermal models. The physical observables, including multiplic-ity, kaon to pion ratios, and transverse momenta agree well withrecent NA49 data at the SPS, and indicate that such studies donot yet reveal the presence of new physics. Predictions for RHICand LHC energies are given. The centrality dependence with andwithout a phase transition to a quark-gluon plasma is discussed -in particular, how the physical quantities are expected to displaya qualitative different behavior in case of a phase transition, andhow a first order phase transition can be signaled by anomalousfluctuations and correlations in a number of observables.

Preprint submitted to Elsevier Preprint 4 May 2001

Contents

1 Introduction 3

2 Phase Transitions and Fluctuations 4

2.1 Order of the QCD phase transition 5

2.2 Density, rapidity, temperature and other fluctuations 6

3 Multiplicity Fluctuations in Relativistic Heavy-Ion Collisions 8

3.1 Fluctuations in the participant model 8

3.2 Fluctuations in the thermal model 12

3.3 Degree of thermalization 13

3.4 Enhanced fluctuations in first order phase transitions 15

4 Fluctuations and Correlations in Particle Ratios 17

4.1 K/π ratio and strangeness enhancement 19

4.2 π+/π− ratio and entropy production 19

4.3 π0/π± ratio and chiral symmetry restoration. 21

4.4 J/Ψ multiplicity correlations and absorption mechanisms 21

4.5 Photon fluctuations to distinguish thermal emission fromπ0 → 2γ 23

5 Transverse Momentum Fluctuations 23

6 Event-by-Event Fluctuations at RHIC 25

7 HBT 27

7.1 The correlation function 28

7.2 HBT for droplets 29

7.3 Centrality dependence of HBT radii 32

8 Summary 32

2

9 Appendix A 34

References 35

3

1 Introduction

The importance of event-by-event physics is evident from the following simpleanalogy: Stick a sheet of paper out of your window on a rainy day. Keepingit there for a long time - corresponding to averaging - the paper will becomeuniformly wet and one would conclude that rain is a uniform mist. If, however,onr keeps the sheet of paper in the rain for a few seconds only, one observesthe striking droplet structure of rain. Incidentally, one has also demonstratedthe liquid-gas phase transition! Analysing many events gives good statisticsand may reveal rare events as snow or hail and thus other phase transitions.The statistics of droplet sizes will also tell something about the fragmentation,surface tension, etc. By varying initial conditions as timing and orienting thepaper, one can further determine the speed and direction of the rain drops.

Central ultrarelativistic collisions at RHIC and LHC are expected to produceat least ∼ 104 particles, and thus present one with the remarkable opportu-nity to analyze, on an event-by-event basis, fluctuations in physical observ-ables such particle multiplicities, transverse momenta, correlations and ratios.Analysis of single events with large statistics can reveal very different physicsthan studying averages over a large statistical sample of events. The use ofHanbury Brown–Twiss correlations to extract the system geometry is a fa-miliar application of event-by-event fluctuations in nuclear collisions [1], andelsewhere, e.g, in sonoluminosence [2]. The power of this tool has been strik-ingly illustrated in study of interference between Bose-Einstein condensatesin trapped atomic systems [3]. Fluctuations in the microwave background ra-diation as recently measured by COBE [4] restrict cosmological parametersfor the single Big Bang event of our Universe. Large neutrons stars velocitieshave been measured recently [5] which indicate that the supernova collapseis very asymmetrical and leads to large event-by-event fluctuations in “kick”velocities during formation of neutron stars.

Studying event-by-event fluctuations in ultrarelativistic heavy ion collisionsto extract new physics was proposed in a series of papers [6–8], in which theanalysis of transverse energy fluctuations in central collisions [9] was used toextract evidence within the binary collision picture for color, or cross-section,fluctuations. More recent theoretical papers have focussed on different aspectsof these fluctuations, such as searching for evidence for thermalization [10–12],and critical fluctuations at the QCD phase transition [13].

Results from the RHIC collider this year are eagerly awaited. The hope isto observe the phase transition to quark-gluon plasma, the chirally restoredhadronic matter and/or deconfinement. This may be by distinct signals of en-hanced rapidity and multiplicity fluctuations [14,13] in conjunction with J/Ψsuppression, strangeness enhancement, η′ enhancement, plateau’s in temper-

4

atures, transverse flow or other collective quantities as function of centrality,transverse energy or multiplicity.

Recently NA49 has presented a prototypical event-by-event analysis of fluctu-ations in central Pb+Pb collisions at 158 GeV per nucleon at the SPS, whichproduce more than a thousand particles per event [10]. The analysis has beencarried out on ∼100.000 such events measuring fluctuations in multiplicities,particle ratios, transverse momentum, etc.

The purpose of this review is to understand these and other possible fluctu-ations. We find that the physical observables, including multiplicity, kaon topion ratios, and transverse momenta agree well with recent NA49 data at theSPS, and indicate that such studies do not yet reveal the presence of newphysics. Predictions for RHIC and LHC energies are given. The centralitydependence with and without a phase transition to a quark-gluon plasma isdiscussed - in particular, how the physical quantities are expected to displaya qualitative different behavior in case of a phase transition, and how a firstorder phase transition can be signaled by very large fluctuations.

2 Phase Transitions and Fluctuations

Lattice QCD calculations find a strong phase transition in strongly interactingmatter [15,16]. The Early Universe went through it ∼ 10−4 seconds after theBig Bang and by colliding heavy nuclei we expected to produce this transitionsat sufficiently high collisions energies.

Fig. 1. Scaled energy density, entropy density and pressure vs. scaled temperatureat the continuum limit in pure gauge SU(3) theory [16]. The horizontal dashed lineis the ideal gas limit and the vertical hatched band illustrates the latent heat.

5

2.1 Order of the QCD phase transition

The nature and order of the transition is not known very well. Lattice calcu-lations can be performed for zero quark and baryon chemical potential only,µB = 0, where they suggest that QCD has a first order transition providedthat the strange quark is sufficiently light [15,16], that is for Nf

>∼3. The transi-tion is due to chiral symmetry restoration and occur at a critical temperatureTC ' 150 MeV. In pure SU(3) gauge theory (that is no quarks, Nf = 0) thetransition is a deconfinement transition which also is of first order and occurat a higher temperature Tc ' 260 MeV.

However, when the strange or the up and down quark masses become heavy,the QCD transition changes to a smooth cross over. The phase diagram isthen like the liquid-gas phase diagram with a critical point above which thetransition goes continuously through the vapor phase. For reasonable valuesfor the strange quark mass, ms ∼ 150 MeV and small up and down quarkmasses, the transition is, however, first order [15,16].

For exactly two massless flavors, mu,d = 0 and ms = ∞, the transition issecond order at small baryon chemical potential. Random matrix theory findsa 2nd order phase transition at high temperatures which, however, change intoa 1st order transition above a certain baryon density - the tricritical point.

B

T

Tc

AGS: Au-Au, (past)Eb = 11 AGeVs = 5 AGeV

Eb = 40 AGeV

SPS: Pb-Pb (now)Eb = 158 AGeVs = 17 AGeV

RHIC: Au-Au (now!)s = 200 AGeV

LHC: Pb-Pb (2005)s = 5.5 ATeV

Early Universe

neutron stars?

Hadron Gas

confinement

Quark-Gluon Plasma

deconfinement

Fig. 2. An illustration of the QCD phase diagram, temperature vs. baryon chemicalpotential. The regions of the phase diagram probed by various high energy nuclearcollisions are sketched by arrows. From [17].

6

2.2 Density, rapidity, temperature and other fluctuations

Fluctuations are very sensitive to the nature of the transition. In case of asecond order phase transition the specific heats diverge and have been ar-gued to reduce the fluctuations drastically [13]. For example, the temperaturefluctuations have a probability distribution [18]

w ∼ exp(−CV (∆T

T)2) ; (1)

a diverging specific heat near a 2nd order phase transition would then removefluctuations if matter is in global thermal equilibrium.

First order phase transition are contrarily expected to lead to large fluctua-tions due to droplet formation [19] or more generally density or temperaturefluctuations. In case of a first order phase transition relativistic heavy ion col-lisions lead to interesting scenarios in which matter is compressed, heated andundergoes chiral restoration. If the subsequent expansion is sufficiently rapid,matter will pass the phase coexistence curve with little effect and supercool[20]. This suggests the possible formation of “droplets” of supercooled chiralsymmetric matter with relatively high baryon and energy densities in a back-ground of low density broken symmetry matter. These droplets can persistuntil the system reachs the spinodal line and then return rapidly to the now-unique broken symmetry minimum. A large mismatch in density and energydensity seems to be a robust prediction for a first order transition in randommatrix theory [20].

If the transitions is first order, matter may supercool and subsequently createfluctuations in a number of quantities. Density fluctuations in the form ofhot spots or droplets of dense matter with hadronic gas in between is a likelyoutcome. We shall refer to these regions of dense and hot matter in space-timeas well as in momentum space as droplets. If we assume that hadrons emergeas a Boltzmann distribution with temperature T from each droplet and ignoretransverse flow, the resulting particle distribution is

dN

dyd2pt∝∑

i

fi e−mt cosh(y−ηi)/T . (2)

Here, y is the particle rapidity and pt its transverse momentum, fi is thenumber of particles hadronizing from each droplet i, and

ηi =1

2log

ti + ziti − zi

=1

2log

1 + vi1− vi

(3)

is the rapidity of droplet i.

When mt/T � 1, we can approximate cosh(y− ηi) ' 1+ 12(y− ηi)2 in Eq.(2).

The Boltzmann factor determines the width of the droplet rapidity distribu-

7

0 2 4 6 8 10y

dN/dy

Fig. 3. Sketch of droplet formation (top) and corresponding rapidity distribution(bottom) excluding the continuous hadronic background.

tion as ∼√T/mt. The rapidity distribution will display fluctuations in rapid-

ity event by event when the droplets are separated by rapidities larger than

|ηi−ηj |>∼√T/mt. If they are evenly distributed by smaller rapidity differences,

the resulting rapidity distribution (2) will appear flat.

The droplets are separated in rapidity by |ηi − ηj | ∼ ∆z/τ0, where ∆z is thecorrelation length in the dense and hot mixed phase and τ0 is the invarianttime after collision at which the droplets form. Assuming that ∆z ∼ 1fm — atypical hadronic scale — and that the droplets form very early τ0<∼1fm/c, we

find that indeed |ηi−ηj |>∼√T/mt even for the light pions. If strong transverse

flow is present in the source, the droplets may also move in a transverse direc-tion. In that case the distribution in pt may be non-thermal and azimuthallyasymmetric.

Even if the transition is not first order, fluctuations may still occur in thematter that undergoes a transition. The fluctuations may be in density, chiralsymmetry [21], strangeness, or other quantities and show up in the associatedparticle multiplicities. The “anomalous” fluctuations depends not only on thetype and order of the transition, but also on the speed by which the collisionzone goes through the transition, the degree of equilibrium, the subsequent

8

hadronization process, the amount of rescatterings between hadronization andfreezeout, etc. It may be that any sign of the transition is smeared out anderased before freezeout. This is the case for the event-by-event fluctuationsmeasured at CERN [14] within experimental accuracy. Whether they remainat RHIC is yet to be discovered and we shall provide some tools for the analysisin the following sections.

3 Multiplicity Fluctuations in Relativistic Heavy-Ion Collisions

In order to be able to extract new physics associated with fluctuations, itis necessary to understand the role of expected statistical fluctuations. Ouraim here is to study the sources of these fluctuations in collisions. As weshall see, the current NA49 data can be essentially understood on the basisof straightforward statistical arguments. Expected sources of fluctuations in-clude impact parameter fluctuations, fluctuations in the number of primarycollisions, and the results of such collisions, nuclear deformations [9], effectsof rescattering of secondaries, and QCD color fluctuations. Since fluctuationsin collisions are sensitive to the amount of rescattering of secondaries takingplace, we discuss in detail two limiting cases, the participant or “woundednucleon model” (WNM) [22], in which one assumes that particle productionoccurs in the individual participant nucleons and rescattering of secondariesis ignored, and the thermal limit in which scatterings bring the system intolocal thermal equilibrium. Whether rescatterings increase relative fluctuationsthrough greater production of multiplicity, transverse momenta, etc., or de-crease fluctuations by involving a greater number of degrees of freedom, is notimmediately obvious. Indeed VENUS simulations [23] showed that rescatter-ing had negligible effects on transverse energy fluctuations. As we shall see,both models give similar results for multiplicity fluctuations. In the woundednucleon model fluctuations arise mainly from multiplicity fluctuations for eachparticipant and from impact parameter fluctuations. Limited acceptance alsoinfluences the observed fluctuations. We calculate in detail statistical fluctua-tions in multiplicity, K/π ratios, and transverse momentum. Finally, we showin a simple model how first order phase transitions are capable of producingvery significant fluctuations.

3.1 Fluctuations in the participant model

Let us first calculate fluctuations in the participant model, which appears todescribe well physics at SPS energies [10]. In this picture

N =Np∑i

ni, (4)

9

Global Fluctuations

Accepted Particles0 100 200 300 400 500

Eve

nts

1

10

102

103

104

<pT> per Event (MeV/c)250 300 350 400 450 500

Eve

nts

1

10

102

103

104

Fig. 4. Event-by-event fluctuations of multiplicity (top) and pt (bottom) measuredby NA49 in central Pb+Pb collisions at the SPS [10].

10

where Np is the number of participants and ni is the number of particlesproduced in the acceptance by participant i. In the absence of correlationsbetween Np and n, the average multiplicity is 〈N〉 = 〈Np〉〈n〉. For example,NA49 measures charged particles in the rapidity region 4 < y < 5.5 and finds〈N〉 ' 270 for central Pb+Pb collisions. Finite impact parameters (b<∼3.5 fm)as well as surface diffuseness reduce the number of participants from the totalnumber of nucleons 2A to 〈Np〉 ' 350 estimated from Glauber theory; thus〈n〉 ' 0.77. Squaring Eq. (4) assuming 〈ninj〉 = 〈ni〉〈nj〉 for i 6= j, we find themultiplicity fluctuations

ωN = ωn + 〈n〉ωNp, (5)where in general we write

σ(x) = 〈x2〉 − 〈x〉2 ≡ 〈x〉ωx (6)for any stochastic variable x (see Appendix for a derivation).

A major source of multiplicity fluctuations per participant, ωn, is the lim-ited acceptance. While each participant produces ν charged particles, only asmaller fraction f = 〈n〉/〈ν〉 are accepted. Without carrying out a detailedanalysis of the acceptance, one can make a simple statistical estimate assum-ing that the particles are accepted randomly, in which case n is binomiallydistributed with σ(n) = νf(1− f) for fixed ν. Including fluctuations in ν weobtain, similarly to Eq. (5),

ωn = 1− f + fων . (7)In NN collisions at SPS energies, the charged particle multiplicity is ∼ 7.3and ων ' 1.9 [24]; thus 〈ν〉 ' 3.7 and f ' 0.21 for the NA49 acceptance.Consequently, we find from Eq. (7) that ωn ' 1.2. The random acceptance as-sumption might be improved on by correcting for known rapidity correlationsdue to resonances, etc.

Multiplicity generally increase with centrality of the collision. We will use theterm centrality as impact parameter b in the collision. It is not a directly mea-surable quantity but is closely correlated to the transverse energy producedET , the measured energy in the zero degree calorimeter and the total particlemultiplicity N measured in some large rapidity interval. The latter is againapproximately proportional to the number of participating nucleons

Np(b) =∫

overlap

[ρ(r +

b

2) + ρ(r− b

2)

]d3r . (8)

The number of participants for colliding two spherical nuclei of radius R isshown in Fig. (6) as function of impact parameter.

As a consequence of nuclear correlations, which strongly reduce density fluc-tuations in the colliding nuclei, the fluctuations ωNp(b) in Np are very smallfor fixed impact parameter b [8]. Almost all nucleons in the nuclear overlapvolume collide and participate. [By contrast, the fluctuations in the number of

11

−12 −8 −4 0 4 8 12X [fm]

−10

−5

0

5

10

Y [

fm]

Ry

Rx

P t

φ

b

Fig. 5. Reaction plane of semi-central Pb + Pb collision for impact parameterb = RPb ' 7fm. The overlap zone is deformed with Rx ≤ Ry. The reaction plane(x, z) is rotated by the angle φ with respect to the transverse particle momentumpt which defines the outward direction in HBT analyses.

binary collisions is non-negligible.] Cross section fluctuations play a small rolein the WNM [8]. Fluctuations in the number of participants can arise when thetarget nucleus is deformed, since the orientations of the deformation axes varyfrom event to event [26]. The fluctuations, ωNp, in the number of participantsare dominated by the varying impact parameters selected by the experiment.In the NA49 experiment, for example, the zero degree calorimeter selects the5% most central collisions, corresponding to impact parameters smaller thana centrality cut on impact parameter, bc ' 3.5 fm. We have

ωNp〈Np〉 =1

πb2c

bc∫0

d2bNp(b)2 − 〈Np〉2 , (9)

where 〈Np〉 = (1/πb2c)∫ bc0 d2bNp(b). The number of participants for a given

centrality, calculated in [25], can be approximated by Np(b) ' Np(0)(1−b/2R)for 0 ≤ b<∼3.5 fm; thus

ωNp =Np(0)

18

(bc2R

)2

. (10)

For NA49 Pb+Pb collisions with Np(0) ' 400 and (bc/2R)2 ' 5% we findωNp ' 1.1. Impact parameter fluctuations are thus important even for the

12

0.0 0.5 1.0 1.5 2.0b/R

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Npart/2ARx/RRy/Rδv2(b)/v2

s(b=R)

Fig. 6. Number of participants, transverse radii of nuclear overlap R=x 〈x2〉 and

R2y = 〈y2〉, deformation δ = (R2

y − R2x)/(R

2y + R2

x), and elliptic flow parameterv2 versus impact parameter. From [25].

centrality trigger of NA49. Varying the centrality cut or bc to control such im-pact parameter fluctuations (10) should enable one to extract better any moreinteresting intrinsic fluctuations. Recent WA98 analyses confirm that fluctua-tions in photons and pions grow approximately linearly with the centrality cut[27]. The Gaussian multiplicity distribution found in central collisions changesfor minimum bias to a plateau-like distribution [9].

Calculating ωN for the NA49 parameters, we find from Eq. (5), ωN ' 1.2 +(0.77)(1.1) = 2.0, in good agreement with experiment, which measures a mul-tiplicity distribution ∝ exp[−(N − 〈N〉)2/2〈N〉ωexpN ], where ωexpN is of order2.01 [10].

3.2 Fluctuations in the thermal model

Let us now consider, in the opposite limit of considerable rescattering, fluctua-tions in thermal models. In a gas in equilibrium, the mean number of particlesper bosonic mode na is given by

〈na〉 = (exp (Ea/T )− 1)−1 , (11)

13

with fluctuationsωna = 1 + 〈na〉 . (12)

The total fluctuation in the multiplicity, N =∑a na, is

ωBEN = 1 +∑a

〈na〉2/∑a

〈na〉. (13)

If the modes are taken to be momentum states, the resulting fluctuations areωBEN = ζ(2)/ζ(3) = 1.37 for massless particles, while for pions at temperatureT = 150 MeV ωBEN = 1.11 [28].

Resonances add to fluctuations in the thermal limit whereas they are im-plicitly included in the WNM fluctuations. In high energy nuclear collisions,resonance decays such as ρ → 2π, ω → 3π, etc., lead to half or more of thepion multiplicity. Only a small fraction r ' 10% produce two charged parti-cles in a thermal hadron gas [30] or in RQMD [29]. Including such resonancefluctuations in the BE fluctuations gives, similarly to Eq. (5),

ωBE+RN = r

1− r

1 + r+ (1 + r)ωBEN . (14)

With r ' 0.1 we obtain ωBE+RN ' 1.3. If not all of the decay particles fall into

the NA49 acceptance the fluctuations from resonances will be reduced. In [13]the estimated effect of resonances is about twice ours: ωN ' 1.5, not includingimpact parameter fluctuations.

Fluctuations in the effective collision volume add a further term 〈N〉σ(V )/〈V 〉2to ωBE+R

N . Assuming that the volume scales with the number of participants,ωV /〈V 〉 ' ωNp/〈Np〉, we find from Eq. (5) that ωN = ωBE+R

N + 〈n〉ωNp ' 2.1,again consistent with the NA49 data. Because of the similarity between themagnitudes of the thermal and WNM multiplicity fluctuations, the presentmeasurements cannot distinguish between these two limiting pictures.

3.3 Degree of thermalization

It is very unfortunate that the WNM and thermal models predict the samemultiplicity fluctuations in the NA49 acceptance - and that they agree withthe experiment. If the numbers from the two models had been different andthe experimental number in between these two, then one would have hadquantified the degree of thermalization in relativistic heavy ion collisions.

By varying the acceptance one can, however, distinguish between the twomodels (see Fig. (7)). In the analysis of thermal fluctuations the acceptancedid not enter in any way - except indirectly through the small contributionfrom resonances. In the WNW the acceptance was important in reducing themultiplicity fluctuations, ων ' 1.9, from each source to that in the acceptance,ωn ' 1.2, as given by Eq. (7). Unfortunately, the NA49 acceptance happen toreduce the fluctuations in the WNM to almost the same as expected from the

14

0.0 0.2 0.4 0.6 0.8 1.0Acceptance fraction (f)

1.0

1.2

1.4

1.6

1.8

2.0

ωN (

with

out i

pf) ωN

WNM=1−f+fων

ωNBE+R

NA49 and WA98

Fig. 7. Multiplicity fluctuations vs. acceptance fraction f = 〈N〉/NTot. Fluctuationwith impact parameter fluctuations subtracted are shown for in the participantnucleon (WNM, Eq. (5)) and thermal (BE+R, Eq. (14)) models. Experimentaldata are shown for NA49 [10] and WA98 [27].

thermal limit, ωBE+R = 1.3. Future experiments should vary the acceptancewhen measuring multiplicity fluctuations or simply divide the acceptance intosegments.

A measure of the degree of thermalization can then be defined

Degree of thermalization ' ωWNMN − ωexpN

ωWNMN − ωBE+R

N

, (15)

which ranges from unity in the thermal limit to zero in the WNM. Whereasboth ωWNM

N and ωexpN may depend on the acceptance the degree of thermal-ization Eq. (15) should not depend on the acceptance.

The ratio will, however, depend on centrality. For peripheral collisions, whereonly few rescatterings occur, we expect the WNM to be approximately validand the degree of thermalization to be small. For central collisions, wheremany rescatterings occur among produced particles, we expect to approachthe thermal limit and the degree of thermalization should be close to unity.

WA98 have analysed fluctuations as function of centrality also varying the ETbin size. This way the impact parameters, which scale with bin size, can beextracted and is found to agree well with those predicted from Eq. (10). As im-

15

pact parameter fluctuations are a major contribution to the total fluctuations,it is important to extract and remove these “trivial” fluctuations in order tomeasure the “bare” fluctuations more precisely. Furthermore, because impactparameter fluctuations are proportional to the acceptance fraction through〈n〉 = f〈ν〉, they must be subtracted for the thermal fluctuation to be inde-pendent of the acceptance (see Fig. (7)).

3.4 Enhanced fluctuations in first order phase transitions

First order phase transitions can lead to rather large fluctuations in physicalquantities. Thus, detection of enhanced fluctuations, beyond the elementarystatistical ones considered to this point, could signal the presence of such atransition. For example, matter undergoing a transition from chirally sym-metric to broken chiral symmetry could, when expanding, supercool and formdroplets, resulting in large multiplicity versus rapidity fluctuations [38]. Letus imagine that ND droplets fall into the acceptance, each producing n par-ticles, i.e., 〈N〉 = 〈ND〉〈n〉. The corresponding multiplicity fluctuation is (seeAppendix A)

ωN = ωn + 〈n〉ωND . (16)As in Eq. (7), we expect ωn ∼ 1. However, unlike the case of participantfluctuations, the second term in (16) can lead to huge multiplicity fluctuationswhen only a few droplets fall into the acceptance; in such a case, 〈n〉 is largeand ωND of order unity. The fluctuations from droplets depends on the totalnumber of droplets, the spread in rapidity of particles from a droplet, δy ∼√T/mt, as well as the experimental acceptance in rapidity, ∆y. When δy �

∆y and the droplets are binomially distributed in rapidity, ωND ' 1−∆y/ytot,which can be a significant fraction of unity.

In the extreme case where none or only one droplet falls into the acceptancewith equal probability, we have ωND = 1/2 and 〈n〉 = 2〈N〉. The resultingfluctuation is ωN ' 〈N〉, which is more than two orders of magnitude largerthan the expected value of order unity as currently measured in NA49. Thissimple example clearly demonstrates the importance of event-by-event fluctu-ations accompanying phase transitions, and illustrates how monitoring suchfluctuations versus centrality becomes a promising signal, in the upcomingRHIC experiments, for the onset of a transition. The potential for large fluc-tuations (orders of magnitude) from a transition makes it worth looking forat RHIC considering the relative simplicity and accuracy (percents) of multi-plicity measurements.

Let us subsequently consider a less extreme model in which a transition leadsto enhanced fluctuations of some kind. Specifically, assume that the totalmultiplicity within the acceptance arise from a normal hadronic component

16

E1 E2

Centrality (ET)

0

1

2

3

4

5

ωN/ω

HM

ωΗΜ

ωQM= 2ωHM

ωQM= 5ωHM

ωQM=10ωHM

Fig. 8. Multiplictity fluctuations vs. centrality (total multiplicity or ET ). Anomalousfluctuations appear when a transition to a new state of matter (QM) starts atcentrality E1 (see text).

(NHM) and from a part (NQM) that underwent a transition:N = NHM +NQM ; (17)

Its average is 〈N〉 = 〈NHM〉 + 〈NQM〉. Assuming that the multiplicities fromeach of the components are independent, the multiplicity fluctuation becomes

ωN = ωHM + (ωQM − ωHM)〈NQM〉〈N〉 , (18)

where ωX ≡ (〈N2X〉− 〈NX〉2)/〈NX〉 as earlier with X = HM,QM . Here, ωHM

is the standard fluctuation in hadronic matter ωHM ' 1 − 2, rather modelindependent as discussed above. The fluctuations within the component thatunderwent a transition, ωQM depends on the type and order of the transition,the speed by which the collision zone goes through the transition, the degreeof equilibrium, the subsequent hadronization process, the amount of rescatter-ings between hadronization and freezeout, etc. If any sign of the transition issmeared out and erased then ωQM ' ωHM , and we will not be able so see thetransition in fluctuations. At the other extreme, we find the droplet scenariodiscussed above where ωQM ∼ 〈N〉 ∼ 102 − 103.

17

The amount of QM and thus 〈NQM〉 depends on centrality, energy and nuclearmasses in the collision. For a given centrality the densities vary from zeroat the periphery of the collision zone to a maximum value at the center.Furthermore, the more central the collision the higher energy densities arecreated. The transverse energy, ET , the total multiplicity and/or the energyin the zero-degree calorimeter, EZDC , have been found to be good measuresof the centrality of the collision at SPS energies. Therefore, it would be veryinteresting to study fluctuations vs. centrality which are proportional to energydensity. By varying the binning size for centrality one can also remove impactparameter fluctuations as discussed above.

If the energy density in the center of the collision zone exceeds the criticalenergy density for forming QM at a certain centrality, E1, then a mixed phaseof QM and HM is formed. At a higher energy density, where the criticalenergy density plus the latent heat for the transition is exceeded, which weshall assume occur at a centrality E2, then a pure QM phase is produced inthe center. These quantities will depend on the amount of stopping at a givencentrality, the geometry, Tc, etc. In the mixed phase, E1 ≤ ET < E2, therelative amount of QM, 〈NQM〉/〈N〉, is proportional to both the volume ofthe mixed phase. and the fraction of the volume that is in the QM phase. Thelatter varies in the volume such that it vanishes at HM/QM boundary.

In Fig. (8) a schematic plot of the fluctuations of Eq. (18) is shown as func-tion of centrality for various ωQM . Up to centrality E1 the fluctuations areunchanged. Above the central overlap zone undergoes the transition to theQM/HM mixed phase and fluctuations start to grow when ωQM > ωHM . Atthe higher centrality, E2 the central overlap zone is in the pure QM phase butthe maximum fluctuations ωQM are not reached because the surface regionsof the collision zone is still in the HM phase.

The multiplicity fluctuations can be studied for any kind of particles, total orratios. Total multiplicities describe total multiplicities whereas, e.g. the ratioπ0/(π+ + π−) can reveal fluctuations in chiral symmetry. The onset and mag-nitude of such fluctuations would reveal the symmetry and other propertiesof the new phase.

4 Fluctuations and Correlations in Particle Ratios

By taking ratios of particles, e.g. K/π, π+/π−, π0/π±, ..., one removes volumeand impact parameter fluctuations to first approximation. Simply increas-ing/decreasing the volume or centrality, the average number of particles ofboth species scales up/down by the same amount and thus cancel in the ratio.

18

Fig. 9. Event-by-event fluctuations in the K/π ratio measured by NA49 in centralPb+Pb collisions at the SPS [10].

Fluctuations in various particles can reveal very different physics as will bedemonstrated in the following.

19

4.1 K/π ratio and strangeness enhancement

To second order in the fluctuations of the numbers of K and π, we have

〈K/π〉 =〈K〉〈π〉

(1 +

ωπ〈π〉 −

〈Kπ〉 − 〈K〉〈π〉〈K〉〈π〉

). (19)

The corresponding fluctuations in 〈K/π〉 are given by

D2 ≡ ωK/π〈K/π〉 =

ωK〈K〉 +

ωπ〈π〉 − 2

〈Kπ〉 − 〈K〉〈π〉〈K〉〈π〉 . (20)

The fluctuations in the kaon to pion ratio is dominated by the fluctuations inthe number of kaons alone. The third term in Eq. (20) includes correlationsbetween the number of pions and kaons. It contains a negative part fromvolume fluctuations, which removes the volume fluctuations in ωK and ωπ sincesuch fluctuations cancel in any ratio. In the NA49 data [10] the average ratioof charged kaons to charged pions is 〈K/π〉 = 0.18 and 〈π〉 ' 200. Excludingvolume fluctuations, we take ωK ' ωπ ' 1.2 − 1.3 as discussed above. Thefirst two terms in Eq. (20) then yield D ' 0.20−0.21 in good agreement withpreliminary measurements D = 0.23 [10]. Thus at this stage the data gives noevidence for correlated production of K and π, as described by the final termin Eq. (20), besides volume fluctuations. The similar fluctuations in mixedevent analyses Dmixed = 0.208 [10] confirm this conclusion.

Strangeness enhancement has been observed in relativistic nuclear collisionsat the SPS. For example, the number of kaons and therefore also 〈K/π〉 isincreased by a factor of 2-3 in central Pb+Pb collisions. It would be interestingto study the fluctuations in strangeness as well. By varying the acceptance onemight be able to gauge the degree of thermalization as discussed above. Thefluctuations in the K/π ratio as function of centrality would in that casereveal whether strangeness enhancement is associated with thermalization orother mechanisms lie behind. In a plasma of deconfined quarks strangeness isincreased rapidly by qq → ss processes and lead to enhancement of averagestrangeness as well as fluctuations in case of density fluctuations or dropletformation at very early times around chemical freezeout. Such strangenessfluctuations would survive even if the density fluctuations are erased duringhadronization and thermal freezeout.

4.2 π+/π− ratio and entropy production

The fluctuations in the π+/π− ratio have been studied in detail in [30]. Theyfind that resonances such as ρ, ω, ... decaying into two or three pions tendto correlate the π+ and π− production. Consequently, the fluctuation in theπ+/π− ratio is reduced by ∼ 30% in good agreement with NA49 data. Inturn the reduced fluctuations can be exploited to reveal the precise amount of

20

resonances decaying into charged pions, kaons, etc.

If a quark-gluon plasma is produced it may have interesting effects on the fluc-tuations in the π+/π− ratio. The particle or entropy density of quarks, anti-quarks and gluons is a factor∼ 5 larger than for a hadron gas at Tc ' 150 MeV.An increased entropy density in the collisions volume will therefore lead toenhanced multiplicity as compared to a standard hadronic scenario if totalentropy is conserved. The associated particle production must conserve netcharge on large rapidity scales (∆y>∼1) due to causality because fields cannotcommunicate over large distances and therefore must conserve charge withinthe “event horizon”. Therefore the net charge, N+ − N−, is approximatelyconserved whereas the total charge, N+ +N−, increase by an amount propor-tional to the additional entropy produced. The fluctuations ωπ+/π− and thedispersion D2

± of Eq. (20) in the π+/π− ratio are both proportional to thecharge difference squared divided by the total charge to the first and secondpower respectively. The dispersion of Eq. (20) is therefore

D2± =

ωπ+/π−

〈π+/π−〉 = D2±,HM

(〈NHM〉〈N〉

)2

, (21)

where D2±,HM is the dispersion in standard hadronic matter [30]. The dis-

persion decrease with the square of the increased total multiplicity 〈N〉 withrespect to the multiplicity in a hadronic scenario 〈NHM〉, which can be esti-mated in the WNM.

As an example, assume that the entropy density, and therefore also the multi-plicity, increase from a hot hadronic gas to a quark-gluon plasma by a factor∼ 5 and approximately half of the volume is involved in this transition. Thedispersion is then according to Eq. (21) suppressed by a factor ∼ 52/2, i.e.about an order of magnitude.

At SPS energies the total particle multiplicity is only slightly larger thanthat predicted from participant nucleon models (WNM) based on multiplici-ties from individual NN collisions. Therefore there is little additional entropyproduction, i.e., 〈N〉 ' 〈NHM〉 and consequently it is no surprise that thefluctuation in π+/π− can be explained by standard hadronic fluctuations cor-rected for resonances [30].

The restriction of vanishing net charge production over large rapidity scalesmay only be partly fulfilled and certainly in small rapidity intervals, ∆y<∼1,it breaks down and fluctuations should increase towards that in standardhadronic matter. Consequently, the fluctuations in π+/π− should depend sen-sitively on the rapidity interval ∆y.

Again, it is important to monitor the fluctuations and dispersion D2± as func-

tion of centrality. An anomalous suppression starting at some centrality wouldbe a signal of additional entropy and particle production from, e.g., a phase

21

transition.

4.3 π0/π± ratio and chiral symmetry restoration.

Fluctuations in neutral relative to charged pions would be a characteristic sig-nal of chiral symmetry restoration in heavy ion collisions. If, during expansionand cooling, domains of chiral condensates gets “disoriented” (DCC), anoma-lous fluctuations in π0/π± ratios could results if DCC domains are large. Forsingle DCC domain the probability distribution of ratios d = π0/(π0+π++π−)is P (d) = 1/

√2d with mean 〈d〉 = 1/3 and fluctuation ωd = 4/15, i.e. much

larger than ordinary fluctuations in such ratios (see Eq. (20)) which decreaseinversely with the number of pions.

Neutral pions are much harder to measure than charged pions but with respectto fluctuations, it does not matter! The anomalous fluctuations in π0 due to aDCC are anti-correlated to π±, i.e. they are of same magnitude but oppositesign.

4.4 J/Ψ multiplicity correlations and absorption mechanisms

J/Ψ suppression has been found in relativistic nuclear collisions [31] and itis yet unclear how much is due to absorption on participant nucleons andproduced particles (comovers). Whether “anomalous suppression” is presentin the data is one of the most discussed signals from a hot and dense phase atearly times [31]. It was originally suggested that the formation of a quark-gluonplasma would destroy the cc bound states [32].

In relativistic heavy ion collisions very few J/Ψ’s are produced in each colli-sion. Of these only 6.9% branch into dimuons that can be detected and so thechance to detect two dimuon pair in the same event is very small. Therefore,it will be correspondingly difficult to measure fluctuations and other highermoments of the number of J/Ψ.

Another more promising observable is the correlations between the multiplic-ities in, e.g., a rapidity interval ∆y of a charmonium state ψ = J/Ψ, ψ′, ..(Nψ) and all particles (N). The correlator: 〈NNψ〉 − 〈N〉〈Nψ〉 also enters thein the ratio ψ/N (see Eq. (19)). The correlator has as good statistics as thetotal number of ψ and it may contain some very interesting anti-correlations,namely that ψ absorption grow with multiplicity N . The physics behind canbe comover absorption, which grows with comover density, or formation ofquark-gluon plasma, which may lead to both anomalous suppression of ψ andlarge multiplicity in ∆y. Contrarily, direct Glauber absorption should not de-

22

pend on the multiplicity of produced particles N since it is caused by collisionswith participating nucleons.

To quantify this anti-correlation we model the absorption/destruction of ψ’sby simple Glauber theory

Nψ

N0ψ

= e−〈σcψρcl〉 ≡ e−γN/〈N〉 , (22)

where N0ψ is the number of J/Ψ’s before comover or anomalous absorption sets

in but after direct Glauber absorption on participant nucleons. In Glauber the-ory the exponent is the absorption cross section times the absorber density andpath length traversed in matter. The density and therefore also the exponentis proportional to the multiplicity N with coefficient

γ = −d logNψ

d logN |N=〈N〉. (23)

In a simple comover absorption model for a system with longitudinal Bjorkenscaling, it can be calculated approximately [33]

γ '∑c

dNc

dy

〈vcψσcψ〉4πR2

logR/τ0 , (24)

where dNc/dy, σcψ, vcψ and τ0 are the comover rapidity density, absorptioncross section, relative velocity and formation time respectively.

On average comover or anomalous absorption is responsible for a suppressionfactor e−γ. It is difficult to determine because only the total ψ suppressionincluding direct Glauber absorption on participants is measured.

The anti-correlation is straight forward to calculate when the fluctuations in

the exponent are small (i.e. γ√ωN/〈N〉 � 1). It is

〈NNψ〉 − 〈N〉〈Nψ〉 = −γωN 〈Nψ〉 . (25)It is negative and proportional to the amount of comover and anomalousabsorption and obviously vanishes when the absorption is independent onmultiplicity (γ = 0). The anti-correlation can be accurately determined as thecurrent accuracy in determining 〈Nψ〉 is a few percent (NA50 minimum bias[31]) in each ET bin.

The anti-correlations in Eq. (25) may seem independent of the rapidity inter-val. However, it should not be less than the typical relative rapidities betweencomovers and the ψ or rapidity range of the droplet.

The anticorrelations of Eq. (25) quantifies the amount of comover or anoma-lous absorption and can therefore be exploited to distinguish between theseand direct Glauber absorption mechanisms. In that respect it is similar to theelliptic flow parameter for ψ [33] for the comover absorption part but differsfor the anomalous absorption.

23

4.5 Photon fluctuations to distinguish thermal emission from π0 → 2γ

WA98 have measured photon and charged particle multiplicities and their fluc-tuations versus centrality and ET binning size. As mentioned above impactparameter fluctuations are proportional to the ET binning size and the WA98analysis nicely confirms this, and can subsequently remove impact parameterfluctuations. The resulting charged particle multiplicity fluctuations with im-pact parameter fluctuations subtracted, ωN − 〈n〉ωNp ' 1.1− 1.2 were shownin Fig. (7).

The fluctuations in photon multiplicities were found to be almost twice as largeas for charged particles ωγ − 〈n〉ωNp ' 2.0. This has the simple explanationthat photons mainly are produced in π0 → 2γ decays. The fluctuations arethen the double of the fluctuations in π0 to first approximation as seen fromEq. (14).

If the photons were produced from a thermal fireball one would expect thatthey would exhibit Bose-Einstein fluctuations, ωγ = ωBEN = 1.37 for mass-less particles. Photon fluctuations can therefore be exploited to quantify theamount of thermal emission vs. π0 → 2γ decay from a hadronic gas.

5 Transverse Momentum Fluctuations

The total transverse momentum per event

Pt =N∑i=1

pt,i , (26)

is very similar to the transverse energy, for which fluctuations have been stud-ied extensively [9,7]. The mean transverse momentum and inverse slopes ofdistributions generally increase with centrality or multiplicity. Assuming thatα ≡ d log(〈pt〉N)/d logN is small, as is the case for pions [34], the averagetransverse momentum per particle for given multiplicity N is to leading order

〈pt〉N = 〈pt〉(1 + α(N − 〈N〉)/〈N〉) . (27)where 〈pt〉 is the average over all events of the single particle transverse mo-mentum. With this parametrization, the average total transverse momentumper particle in an event obeys 〈Pt/N〉 = 〈pt〉. When the transverse momen-tum is approximately exponentially distributed with inverse slope T in a givenevent, 〈pt,i〉 = 2T , and σ(pt,i) = 2T 2 = 〈pt〉2/2. This latter fluctuation is inprinciple dependent on multiplicity, but as a higher order effect, we ignore itin the following.

24

The total transverse momentum per particle in an event has fluctuations

〈N〉σ(Pt/N) = σ(pt) + α2〈pt〉2ωN + 〈 1

N

∑i6=j

(pt,ipt,j − 〈pt〉2)〉 . (28)

The three terms on the right are respectively:

i) The individual fluctuations σ(pt,i) = 〈p2t,i〉 − 〈pt〉2, the main term. In the

NA49 data, 〈pt〉 = 377 MeV and 〈N〉 = 270. From Eq. (28) we thus obtain

(σ(Pt/N)1/2/〈pt〉 ' 1/√

2〈N〉 = 4.3%, which accounts for most of the experi-

mentally measured fluctuation 4.65% [10]. The data contains no indication ofintrinsic temperature fluctuations in the collisions.

ii) Effects of correlations between pt and N , which are suppressed with respectto the first term by a factor ∼ α2. In NA49 the multiplicity of charged particlesis mainly that of pions for which T ' 〈pt〉/2 increases little compared with ppcollisions, and α ' 0.05−0.1. Thus, these correlations are small for the NA49data. However, for kaons and protons, α can be an order of magnitude largeras their distributions are strongly affected by the flow observed in centralcollisions [34].

iii) Correlations between transverse momenta of different particles in the sameevent. In the WNM the momenta of particles originating from the same par-ticipant are correlated. In Lund string fragmentation, for example, a quark-antiquark pair is produced with the same pt but in opposite direction. Theaverage number of pairs of hadrons from the same participant is 〈n(n− 1)〉and therefore the latter term in Eq. (28) becomes (〈n(n− 1)〉〈n〉)(〈pt,ipt,j 6=i〉−〈pt〉2). To a good approximation, n is Poisson distributed, i.e., 〈n(n− 1)〉/〈n〉 =〈n〉, equal to 0.77 for the NA49 acceptance, so that this latter term becomes' (〈pt,ipt,j 6=i〉− 〈pt〉2). The momentum correlation between two particles fromthe same participant is expected to be a small fraction of σ(pt,i).

To quantify the effect of rescatterings, Ref. [11] suggested studying the differ-ences in 〈N〉σ(Pt/N) and σ(pt) via the quantity

Φ(pt) '√〈N〉σ(Pt/N)−

√σ(pt,i) . (29)

As we see from Eq. (28), in the applicable limit that the second and thirdterms are small,

Φ(pt) ' 1√σ(pt,i)

(α2〈pt〉2ωN + (〈pt,ipt,j 6=i〉 − 〈pt〉2)

).

(30)In the Fritiof model, based on the WNM with no rescatterings between sec-ondaries, one finds Φ(pt) ' 4.5 MeV. In the thermal limit the correlations inEq. (29) should vanish for classical particles but the interference of identicalparticles (HBT correlations) contribute to these correlations by ∼ 6.5 MeV [12]and slightly reduced by resonances. The NA49 experimental value, Φ(pt) =5 MeV (corrected for two-track resolution) seems to favor the thermal limit

25

[10]. Note however that with α ' 0.05−0.1, the second term on the right sideof Eq. (30) alone leads to Φ ' 1− 4 MeV, i.e., the same order of magnitude.If (〈pt,ipt,j 6=i〉 − 〈pt〉2) is not positive, then one cannot a priori rule out thatthe smallness of Φ(pt) does not arise from a cancellation of this term withα2〈pt〉2ωN , rather than from thermalization.

The covariance matrix between multiplicity and transverse momentum hasbeen analysed by NA49 [35]. Strong but trivial correlations is found due tothe fact that higher multiplicity gives larger total transverse momentum event-by-event. This correlation is removed in the quantity Pt/N and its covariancematrix with multiplicity appear diagonal.

6 Event-by-Event Fluctuations at RHIC

The theoretical analysis above leads to a qualitative understanding of event-by-event fluctuations and speculations on how phase transitions may showup. It gives a quantitative description of AGS and SPS data without the needto invoke new physics. We shall here look ahead at RHIC experiments andattempt to describe how fluctuations may be searched for. Specifically, weshall taylor our analysis to the BRAHMS (Broad RAnge Hadron MagneticSpectrometers) experiments as seen in Fig. (10).

Let us specifically study the energy, Ei, deposited in detector i in a givenevent. The total energy deposited in the event is

ETot =D∑i

Ei , (31)

and can be used as a measure of the centrality of the collision. The energydeposited in each element (or group of elements) is the sum over the numberof particle tracks (Ni) hitting detector i of the individual ionization energy ofeach particle (εi)

Ei =Ni∑n

εn . (32)

The average is: 〈Ei〉 = 〈Ni〉〈ε〉. The energy will approximately be gaussiandistributed, dσ/dEi ∝ exp(−(Ei − 〈Ei〉)2/2ωEi〈Ei〉), with fluctuations (seeAppendix A)

ωEi ≡〈E2

i 〉 − 〈Ei〉2〈Ei〉 = ωε + 〈ε〉ωNi . (33)

Here, the fluctuation in ionization energy per particleωε〈ε〉 =

〈ε2〉〈ε〉2 − 1 , (34)

depends on the typical particle energies in the detector and the correspond-ing ionization energies for the detector type and thickness. For the BRAHMSdetectors we estimate ωε/〈ε〉 ' 0.25 [36]. As these are “trivial” detector pa-

26

Fig. 10. Layout of the central multiplicity detector array in the BRAHMS experi-ment consisting of 36 scintillator tiles and 24 silicon wafers. The high resolution butslow wafers are each segmented into 7 strips giving 42 pseudo-rapidity bins alongthe beamline in the midrapidity interval −2.5 ≤ η ≤ 2.5. The fast scintillators areused as triggers for central events. The detectors surround the beamline hexagonallyon 5 sides with the sixth kept open for the central spectrometer (TPC).

rameter, we shall exclude the fluctuations ωε in the following analysis andconcentrate on the second term in Eq. (33) which is the fluctuations in thenumber of particles as was examined in detail above.

More generally we define the multiplicity correlations between two bins alsoreferred to as the covariance

ωij =〈NiNj〉 − 〈Ni〉〈Nj〉√

〈Ni〉〈Nj〉(35)

When i, j refer to two rapidity bins the covariance is also proportional to therapidity (auto-)correlation function C(yi − yj).

It is instructive to consider first the complete random (uncorrelated or statis-tical) particle emission. For a fixed total multiplicity NTot, the probability fora particle to end up in bin i is pi = 〈Ni〉/NTot ' 〈Ei〉/ETot. The distributionis a simple multinomial distribution for which

ωij =

1− pi , i = j

−√pipj , i 6= j

. (36)

27

The i = j result is the well known one for a binomial distribution. The i 6= jresult is negative because particles in different bins are anti-correlated: more(less) particles in one bin leads to less (more) in other bins on average due toa fixed total number of particles.

As shown above there are nonstatistical fluctuations due to sources, Bose-Einstein fluctuations, resonances, etc., and — in particular — density fluctua-tions. As in Eq. (17) we assume that the multiplicity consist of particles froma HM and a QM phase. The covariances in Eq. (18) are derived analogouslyto Eq. (18)

ωij = ωij,HM + (ωij,QM − ωij,HM)〈Ni,QM〉〈Ni〉 , (37)

when 〈Ni〉 = 〈Nj〉; when different the general formula is a little more compli-cated. Now, the hadronic fluctuations ωij,HM is of order unity for i = j, smallerfor adjacent bins and vanishes or even becomes slightly negative according to(36) for bins very different in pseudorapidity or azimuthal angle φ. The QMfluctuations are much larger: ωi,QM ∼ 〈Ni,QM〉 (see the discussion after Eq.(16)). In order to discriminate the QM fluctuations from the hadronic ones,Eq. (37) requires

〈Ni,QM〉>∼√

(ωij − ωij,HM)〈Ni〉 . (38)If we assume NTot ∼ 8000 and group the BRAHMS detector segments into 8along the beamline, 〈Ni〉 ∼ 1000. To see a clear increase in fluctuations, say∆ω ≡ ωij −ωij,HM ∼ 1, a density fluctuation of only 〈Ni,QM〉>∼

√Ni ∼ 30 par-

ticles are required corresponding to a few percent of the average. By analysingmany events (of the same total multiplicity) the accuracy by which fluctu-ations are measured experimentally is greatly improved. Generally, ∆ω ∼1/√Nevents, and so fluctuations can in principle be determined with immense

accuracy.

It may be advantageous to correlate bins with the same pseudorapidity butdifferent azimuthal angles since the hadronic correlations between these aresmall whereas QM fluctuations remain.

7 HBT

Particle interferometry was invented by Hanbury-Brown & Twiss (HBT) forstellar size determination [1] and is now employed in nuclear collisions [1,10,37].It is a very powerful method to determine the 3-dimensional source sizes, life-times, duration of emission, flow, etc. of pions, kaons, etc. at freeze-out. Sincethe number of pairs grow with the multiplicity per event squared the HBTmethod will become even better at RHIC and LHC colliders where the multi-plicity will be even higher.

28

The use of HBT to extract the system geometry is a familiar application ofevent-by-event fluctuations in nuclear collisions [1]. We shall here describe howHBT in combination with density fluctuations can be exploited to reveal thesize of the sources for fluctuations as well as being a complementary signal[38].

7.1 The correlation function

The standard HBT method for calculating the Bose-Einstein correlation func-tion from the interference of two identical particles is now briefly discussed. Fora source of size R we consider two particles emitted a distance ∼ R apart withrelative momentum q = (k1 − k2) and average momentum, K = (k1 + k2)/2.Typical heavy ion sources in nuclear collisions are of size R ∼ 5 fm, so thatinterference occurs predominantly when q<∼h/R ∼ 40 MeV/c. Since typicalparticle momenta are ki >∼K ∼ 300 MeV/c, the interfering particles travelalmost parallel, i.e., k1 ' k2 ' K � q. The correlation function due to Bose-Einstein interference of identical spin zero bosons as π±π±, K±K±, etc. froman incoherent source is [1]

C2(q,K) = 1 ±∣∣∣∣∣∫d4x S(x,K) eiqx∫d4x S(x,K)

∣∣∣∣∣2

, (39)

where S(x,K) is the source distribution function describing the phase spacedensity of the emitting source.

Experimentally the correlation functions are often parametrized by the gaus-sian form

C2(qs, qo, ql) = 1 + λ exp[−q2sR

2s − q2

oR2o − q2

lR2l − 2qoqsR

2os − 2qoqlR

2ol] .

(40)Here, q = k1−k2 = (qs, qo, ql) is the relative momentum between the two parti-cles and Ri, i = s, o, l, os, ol the corresponding sideward, outward, longitudinal,out-sideward and out-long HBT radii respectively. We have suppressed the Kdependence. We will employ the standard geometry, where the longitudinaldirection is along the beam axis, the outward direction is along Kt, and thesideward axis is perpendicular to these. Usually, each pair of particles is lorentzboosted longitudinal to the system where their rapidity vanishes, y = 0. Theiraverage momentum K is then perpendicular to the beam axis and is chosen asthe outward direction. In this system the pair velocity βK=K/EK points in

the outward direction with βo = pt/m⊥, where m⊥ =√m2 + p2

⊥ is the trans-verse mass, and the out-longitudinal coupling Rol vanish at midrapidity [41].Also Ros vanishes for a cylindrically symmetric source or if the azimuthal angleof the reaction plane is not determined and therefore averaged over — as hasbeen the case experimentally so far. The reduction factor λ in Eq. (40) may

29

be due to long lived resonances [40,29], coherence effects, incorrect Coulombcorrections or other effects. It is λ ∼ 0.5 for pions and λ ∼ 0.9 for kaons.

7.2 HBT for droplets

Droplets lead to spatial fluctuations in density which can be probed by cor-relations between identical particles [1]. For simplicity we parametrize thesedroplets by spatial and temporal gaussians of size Rd and duration of emissiontd. The distribution of particles in space and time — usually referred to as thesource — for droplets situated at xi = (ri, ti) is

S(x,K) ∼∑i

S(xi, K) exp

[−(r− ri)

2

2R2d

− (t− ti)2

2t2d

], (41)

where S(xi, K) is the distribution of sources. Normalizations of S cancel whencalculating correlation functions, Eq.(39). Other effects as transverse flow,opacities [42], resonances [29], etc. can be included but we shall ignore themhere for simplicity. Likewise, the extension to droplets of different size andduration of emission is straight forward but unnecessary for our purpose. Weexpect the scale of S is the nuclear overlap size RA of order several fermi’swhereas the droplet size Rd is smaller — of order a few fermi’s.

From (39) and (41) we obtain

C2(q) ' 1 + exp[−q2R2d − (q · β)2t2d] |S(q,K)|2 , (42)

where

S(q,K) =∑i

S(xi, K)eiq·xi/∑i

S(xi, K) (43)

is the Fourier transform of the distribution of sources and β = K/K0 is theaverage velocity of the pair.

If there is only a single droplet S(q,K) = 1 and the correlation function issimply given by the droplet gaussian. If there are many droplets, the sum canbe replaced by an integral.

For non-expanding sources we can assume that the particle momentum distri-bution is the same for all droplets, i.e., independent of K.

For many droplets we assume that the droplets are distributed by a gaussian

S(xi, K) ∼ exp(−r2i /2R

2A − t2i /2R

2A) , (44)

30

where tA is the spread in droplet formation times and RA is the size of thenuclear overlap zone. It increases with decreasing impact parameter - roughly

as RA(b) '√RA(0)2 − b2, where RA(0) ' 1.2A1/3fm is the nuclear size.

The resulting correlation function becomes

C2(q) ' 1 + exp[−q2(R2A +R2

d)− (q · β)2(t2A + t2d)] . (45)

Since RA and ctA are typically of nuclear scales ∼ 5−10 fm whereas we expectRd ∼ 1 fm, the droplets are simply “drowned” in the background of the otherdroplets.

In the case of only two droplets (of same size) the resulting correlation functionis

C2(q) ' 1 + exp[−q2R2d − (q · β)2t2d] cos(

1

2q · (x1 − x2)) . (46)

However, as the xi’s differ from event to event the oscillation in (46) will,when summing over events, result in a Fourier transform over the distributionof xi’s in different event and the end result will be similar to (45). The numberof pairs in a single event at RHIC energies is probably not large enough todo event-by-event HBT. The oscillation will also be smeared by resonances,Coulomb effects, a hadronic background, etc.

We conclude that it will be very difficult to see the individual droplets throughHBT when the sources do not expand.

At RHIC energies the collision regions are rapidly expanding particularly inthe longitudinal direction. Consequently the droplets may have different ex-pansion velocities and rapidities and the distribution S(xi, K) will depend onK and this difference will now be exploited.

When the droplets are separated in rapidity such that |ηi−ηj |>∼√T/m⊥, only

particles within the same droplet contribute to the correlation function. Ifwe transform to the droplet center-of-mass system, i.e. ηi = 0, the correlationfunction is then given by (45) with S = 1. In terms of qs, qo, ql it can be written

C2(q) = 1 + exp[−q2

sR2d − q2

o(R2d + β2

o t2d)− q2

l (R2d + β2

l t2d)− 2qoqlβoβlt

2d

],

(47)where βo = p⊥/m⊥ cosh(Y ) and βl = tanh(Y ), are the transverse (outward)and longitudinal velocities respectively of the pair.

When the droplets overlap in rapidity, we need to consider the distribution ofdroplets in more detail. At ultrarelativistic energies the rapidity distribution isexpected to display approximate Bjorken scaling, i.e., the rapidity distributionis approximately given by (2) where the droplet rapidities ηi are more or less

31

0 2 4 6RA(b) [fm]

0

2

4

6

RH

BT

[f

m]

Ro

RA

Rl

Rs

Fig. 11. The HBT radii as function of nuclear overlap RA which is proportionalto centrality, dN/dy or E⊥. Droplet formation is assumed to set from semicentralcollisions. A long mixed phase will then lead to large outward HBT radius Ro

whereas triggering on large rapidity fluctuations corresponding to small dropletsleads to smaller longitudinal Rl and sideward Rs HBT radii.

evenly distributed between target and projectile rapidities. Parametrizing thetransverse and temporal distribution as gaussians, we arrive at the dropletdistribution

S(xi, K) ∼ exp

[−m⊥T

cosh(Y − ηi)−r2⊥,i

2R2A

− (τi − τf )2

2t2A

], (48)

where τi =√t2i − z2

i is the invariant time and τf the average freeze-out time.

The resulting correlation function becomes (in the system Y = 0, where βl =tanhY = 0 and βo = p⊥/m⊥)

C2(q) = 1 + exp[−q2s (R

2A +R2

d)− q2o(R

2A +R2

d + β2o(t

2A + t2d))

−q2l (τ

2f

T

m⊥+R2

d)] . (49)

We observe that the larger nuclear size RA dominates the smaller droplet sizeRd when droplets overlap.

7.3 Centrality dependence of HBT radii

We can now study the consequences of forming droplets at RHIC energies. Theonset of large rapidity fluctuations is one signal but it should be accompanied

32

by the following behavior of the HBT radii. In peripheral collisions, wherethe nuclear overlap and stopping is small, we do not expect sufficient energydensities to build op and form droplets. Thus the HBT radii should simplygrow with the size of the geometrical nuclear overlap, centrality, dN/dy andtransverse energy ET . At SPS energies the HBT radii are indeed found to scaleapproximately with the geometrical sizes of the colliding systems as given by(49).

If energy densities achieved in RHIC collisions are sufficient to form dropletsin central collisions, the HBT radii will deviate from the geometrical overlapif triggered on large rapidity fluctuations (see Fig. 11) as given by Eq.(47).Comparing the theoretical predictions of Eqs. (47) and (49) with the experi-mentally measured HBT radii of Eq.(40) we see that the sideward and longi-tudinal HBT radii decrease from the nuclear size RA, tA to the droplet sizesRd, td. Thus at a certain semi-centrality, where energy densities achieved innuclear collisions start becoming large enough to create droplets, the sidewardand longitudinal HBT radii should bend over and start decreasing with cen-trality. The outward HBT radius may behave differently depending on theduration of emission. The droplets may emit hadrons for a long time, as isthe case for a long lived mixed phase (referred to as the “burning log”). Inthe hydrodynamic calculation of Ref. [43]), the duration of emission and con-sequently the outward HBT radius increase drastically up to five times largerthan the transverse size of the system, i.e. RA. This scenario is indicated inFig. 11.

Besides droplets we may expect some hadronic background. It is straight for-ward to include such one in the correlation function. As its spatial extend isexpected to be on the scale ∼ RA, it will reduce the correlation function atsmall q ∼ h/RA. However, the droplets will still lead to correlations at largerelative momenta of order q ∼ h/Rd. The large q correlations are suppressed bythe square of the fraction of pions emerging from droplets at a given rapidity.

8 Summary

If first order transitions occur in high energy nuclear collisions, density fluc-tuations are expected which may show up in rapidity and multiplicity fluc-tuations event-by-event. The fluctuation can be enhanced by several ordersof magnitude in case of droplet formation as compared to that from an or-dinary hadronic scenario. Likewise a number of other observables as chargedand neutral pions, kaons, photons, J/Ψ, etc., and their ratios can show anoma-lous correlations and enhancement or suppression of fluctuations as discussedabove. This clearly demonstrates the importance of event-by-event fluctua-tions accompanying phase transitions, and illustrates how monitoring such

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fluctuations versus centrality becomes a promising signal, in the upcomingRHIC experiments, for the onset of a transition. The potential for enhancedor suppressed fluctuations (orders of magnitude) from a transition makes itworth looking for at RHIC considering the relative simplicity and accuracy ofmultiplicity fluctuation measurements.

An analysis of fluctuations in central Pb+Pb collisions as currently measuredin NA49 do, however, not show any sign of anomalous fluctuations. Fluctu-ations in multiplicity, transverse momentum, K/π and other ratios can beexplained by standard statistical fluctuation and additional impact parame-ter fluctuations, acceptance cuts, resonances, thermal fluctuations, etc. Thisunderstanding by “standard” physics should be taken as a baseline for fu-ture studies at RHIC and LHC and searches for anomalous fluctuations andcorrelations from phase transitions that may show up a number of observables.

By varying the acceptance one should be able to determine quantitativelythe amount of thermalization in relativistic heavy ion collisions as function ofcentrality. In the thermal limit the fluctuations are independent of the accep-tance and given by Eq. (13). In the opposite limit of few rescatterings amongproduced particles, the participant nucleon model gives acceptance dependentfluctuations, Eq. (7), and the ratio of Eq. (15) then gives the degree of ther-malization. For peripheral collisions, where only few rescatterings occur, weexpect the WNM to be approximately valid and the degree of thermalizationto be small. For central collisions, where many rescatterings occur among pro-duced particles, we expect to approach the thermal limit and the degree ofthermalization should be close to 100%.

HBT interferometry adds an important space-time picture to the purely mo-mentum space information one gets from single particle spectra. Triggeringon rapidity fluctuations, the HBT radii may display a curious behavior. Theoutward HBT radius Ro may increase drastically with centrality due to along lived mixed phase. The longitudinal Rl and sideward Rs HBT radii will,however, saturate and decrease for the very central collisions because a largerapidity fluctuation signals a hot and dense droplet of small size. The pre-dicted behavior for the sideward and longitudinal HBT radii is opposite tothat predicted in cascade and hydrodynamic calculations. It would be a cleansignal of a first order phase transition in nuclear collisions.

Acknowledgement

I am much grateful to my collaborators G. Baym and A.D. Jackson on someof the work described in this report. Discussion with NA44 and Brahms (J.J.Gardhøje, C. Holm, et al.), WA98 (T. Nayak), NA49 (G. Roland), J. Bondorf,

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S. Jeon, V. Koch, are gratefully acknowledged.

9 Appendix A

As fluctuations for a source model appears again and again (see Eqs. 5,7,14,18)we shall derive this simple equation in detail.

We define the fluctuations for any stochastic variable x as

ωx =〈x2〉 − 〈x〉2

〈x〉 . (50)

It is usually of order unity and therefore more convenient than variances. Fora Poisson distribution, PN = e−ααN/N !, the fluctuation is ωN = 1. For abinomial distribution with tossing probability p the fluctuation is ωN = 1− p,independent of the number of tosses. In heavy ion collisions several processesadd to fluctuations so that typically ωexpN ∼ 1− 2.

Generally, when the multiplicity (N) arise from independent sources (Np) suchas participants, resonances, droplets or whatever,

N =i=Np∑i=1

ni, (51)

where ni is the number of particles produced in source i. In the absence ofcorrelations between Np and n, the average multiplicity is 〈N〉 = 〈Np〉〈n〉.Here, 〈..〉 refer to averaging over each individual (independent) source as wellas the number of sources. The number of sources vary from event to eventand average is performed over typically Nevents ∼ 100.000 events as in NA49or Nevents ∼ 106 in WA98.

Squaring Eq. (51) assuming that the source emit particles independently, i.e.〈ninj〉 = 〈ni〉〈nj〉 for i 6= j, the square consists of the diagonal and off-diagonalelements:

〈N2〉 = 〈Np〉〈n2i 〉+ 〈Np(Np − 1)〉〈ni〉 . (52)

With (50) we obtain the multiplicity fluctuations

ωN =〈N2〉 − 〈N〉2

〈N〉 = ωn + 〈n〉ωNp,as in Eq. (5).

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