e

a methodmedium.lyzing the

Physics Letters B 574 (2003) 21–26

www.elsevier.com/locate/np

Finding the remnants of lost jets at RHIC

Subrata Pala, Scott Prattb

a National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University,East Lansing, MI 48824, USA

b Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA

Received 29 May 2003; received in revised form 31 July 2003; accepted 11 August 2003

Editor: W. Haxton

Abstract

A fast parton propagating through partonic matter loses energy by medium-induced gluon radiation. We proposeto detect the energy loss of jets and the final-state interaction of the radiated gluons and the jets with the partonicBy embedding jets and radiated energy into a parton cascade, we find that the fate of lost jets can be studied by anamomentum distribution of soft hadrons relative to the jet axis. 2003 Elsevier B.V. All rights reserved.

PACS: 12.38.Mh; 24.85.+p; 25.75.-q

Dudyex-

ol-re-he

d tovo-e-a

arytal

jets

ightof

e atheonretheur

ost-y of

The attenuation of hard jets produced in pQCprocesses provides an excellent opportunity to stthe properties of hot and dense partonic matterpected to be formed in ultrarelativistic heavy ion clisions [1–4]. Due to the energy lost by the jets as asult of multiple elastic and inelastic interactions in tpartonic matter, the final hadronpT distribution fromthe fragmenting jets will be suppressed comparethat from nucleon–nucleon reference spectra conluted with the number of binary collisions. Measurment of highpT hadrons is thus expected to providediagnostic tool for the total energy loss�E of jets inthe dense partonic medium [5,6]. Relative to the bincollision scaledpp reference spectrum, experimendata [7–10] for central Au+ Au collisions at RHIC

E-mail address: pal@nscl.msu.edu (S. Pal).

0370-2693/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2003.08.064

indeed showed a suppression forπ0s andh±, wherethe suppression is smallest atpT ≈ 2 GeV/c and in-creases to an approximately constant value forpT =4–10 GeV/c. Apart from the suppression of highpT

hadrons at RHIC, the suppression of the away-sidecompared to the near-side jets at highpT [11] is an in-dication of jet quenching for the away-side jet.

Even the disappearance of away-side jets mbe considered as only circumstantial evidencejet-energy loss. For instance, one could imaginscenario where monojets are produced due to2 → 1 gluon fusion processes, e.g., in the glusaturation model [12]. If the away-side jet wetruly produced then absorbed by the medium,extra momentum and energy should survive. Oaim is to investigate the degree to which the lenergy and momentum can be extracted from lowpT

observables. We propose a straightforward strateg

.

22 S. Pal, S. Pratt / Physics Letters B 574 (2003) 21–26

henw-ofas

yet,e

linetsts oe aich

ionsdnd

onPC

s,

be

sedrton

singsring

all,cena

se

ofhason,at

onde is

ini-

ar

a-

tsn-ast

oper-

m-–

sticer,ede

loss

yred

ith

gating on the observation of a high-energy parton, tanalyzing the accompanying alteration of the loenergy spectra relative to the azimuthal directionthe jet. The distribution of these extra particles,a function of pT and as function of their rapiditand azimuthal angle relative to the triggering jmight provide critical insight for understanding thmechanism for jet-energy loss.

To investigate these issues, we consider a basedynamical model of the collision, onto which hard jehave been embedded. The base distribution consisthe HIJING model [13,14], which is used to generatset of initial partons, and ZPC, a parton cascade whsimulates the subsequent elastic partonic interact[15]. HIJING provides the initial momentum anspacetime coordinates of partons from minijets astrings which originate from soft nucleon–nuclecollisions. The parton–parton elastic scattering in Zis determined by the cross section,

(1)dσ

dt= Ca4πα2

s

(1+ µ2

s

)1

(t −µ2)2,

where t and s are the usual Mandelstam variableCa = 9/8,1/2,2/9 for gg,gq, qq scatterings, andαs is the strong coupling constant assumed to0.30. The effective screening massµ is assumed toindependent of temperature and density and is uas a parameter to obtain the desired parton–paelastic cross sectionσ ≈ Ca4πα2

s /µ2. We have used

a typical value forµ = 0.50 GeV. After the partonfreeze out, they are traced back to their parent strand are converted to hadrons using the Lund stfragmentation model [16].

Since the jet production cross section is smwe include additional hard jets in order to enhanstatistics. The multiplicity of the additional jets in aevent is obtained from a Poisson distribution withmean〈n〉 = 10. The underlying dynamics of the badistribution (dNparton/dy ≈ 250 at

√s = 200AGeV)

is not significantly affected by the smaller numberthe embedded jets. This technique of oversamplingbeen widely used to study rare particle productisuch asK− at SIS/GSI [17], multistrange particlesSPS [18], andJ/ψ at RHIC [19]. Inp + p collision,the lowest order (LO) pQCD momentum distributiof hard partons that propagate in the parton casca

f

given by

Edσ

ppj

d3p=K

∑a,b

∫dxa dxb

×∫

d2kTa d2kTb g(kTa )g(kTb )

(2)× fa(xa,Q

2)fb(xb,Q2)Edσab

d3p.

In Eq. (2),xa , xb are the momenta fractions andkTa ,kTb are the intrinsic transverse momenta of thetial scattered partons. A constant factorK = 2 is usedto account for the NLO corrections. For the collineparton distribution function (PDF)fa(xa,Q2), we em-ploy the LO Glück–Reya–Vogt (GRV94) parametriztion [20], and for the transverse PDFg(kTa ), we useGaussian intrinsickT distribution of width 〈k2

Ta〉 =

1 GeV2/c2. We neglect nuclear shadowing for the jeas its effect is rather small at mid-rapidity at RHIC eergies [21,22]. Since we explore events with at leone triggered hadron ofptrig

T > 4 GeV/c, a minimummomentum transfer ofpmin

T = 4 GeV/c for the addi-tional jets is assumed. In the hard processes, a prformation time ofτ0 = 1/mT is used for parton production with transverse massmT . The initial trans-verse coordinate of a hard jet is specified by the nuber of binary collision distribution for two WoodsSaxon geometries.

The hard jets can suffer energy loss by elascattering with the partons from HIJING. Howevthe dominant energy loss is via medium-inducgluon radiation along the trajectory of the jet. Wconsider here the dominant first-order total energyexpression in opacity,χ = L/λg , from the reactionoperator approach [4,23]:

�E = 2πα3s CaCA

∞∫τ0

dτ ρ(r(τ ), τ

)

(3)× (τ − τ0) log2E

µ2L.

Here,r(τ ) andE are the position and initial energof the jet,ρ(r(τ ), τ ) is the parton density. The colofactor Ca is given in Eq. (1) and may be expressasCa = CRCT /dA, whereCR andCT are the colorCasimirs of the jet and the target parton, anddAis the dimension in the adjoint representation w

S. Pal, S. Pratt / Physics Letters B 574 (2003) 21–26 23

e

tion

fve

soft.tis-

us--

i-takecalm,alal

n-

ton

ss-di-the

ossngr viar asintoer-glti-

6],con-ee

gyic

r

k

t-

ichan

ive

han

lossltperergy

fter

CasimirCA. The effective path length required by thradiated gluons for traversing the matter,L, is taken tobe 4 fm. Because of jet quenching the fragmentafunction (FF) for a jetc to a hadronh in vacuum,Dh/c(zc,Q

2c), is altered due to the modification o

the kinetic variables of the jet, and the effectifragmentation function follows the relation [4],

zcD′h/c

(zc,Q

2c

) = z′cDh/c

(z′c,Q

2c′)

+NgzgDh/g

(zg,Q

2g

),

zc = ph

pc

, z′c = ph

pc −�E(pc),

(4)zg = ph

�E(pc)/Ng

.

The second term on the right eventually gives thehadrons from the fragmentation ofNg radiated gluonsNote that the modified fragmentation function safies the sum rule

∫dzc zcD

′h/c(zc,Q

2c)= 1. The initial

hard jets after the partonic phase are fragmenteding the LO Binnewies–Kniehl–Kramer (BKK) parametrization [24] of the fragmentation functions.

In absence of any definitive relation for how radated energy should be translated into particles, werecourse to a statistical prescription. Assuming lothermodynamical equilibrium of the parton mediuif �E(r,�τ) is the energy radiated in a time interv�τ (evaluated from Eq. (3)), the number of thermgluons emitted in�τ can be related to the entropy icrease�S of the plasma,

(5)Ng(r,�τ)≈ 1

4�S = 1

4

�E(r,�τ)

T (r, τ ),

where the local temperatureT ≈ ε(r, τ )/3ρ(r, τ ) isextracted from the cascade by viewing the local parnumber and energy densitiesρ(r, τ ) andε(r, τ ). Thefactor of 1/4 comes from considering an ideal maless gas of classical particles in equilibrium. The raated gluons are formed at spacetime points alongparent jet’s trajectory consistent with the energy ldescribed in Eq. (3). They are initially directed alothe axis of the parent jet and subsequently scatteelastic collision with the neighboring partons afteformation timeτ0 as used in HIJING. Effectively, thiprocedure converts 100% of the radiated energythermal energy, while maintaining momentum consvation. A less-efficient prescription for thermalizinthe radiated energy would result in a reduced mu

plicity. Based on local parton–hadron duality [25,2each of these radiated gluons after freeze-out areverted to a pion with equal probability for the thrcharge states.

The inset of Fig. 1 shows the fractional enerloss�E/E for the fast gluons traversing the partonmedium for central Au+ Au collisions at a centeof mass energy

√s = 200 AGeV. In contrast to

the asymptotic BDMS [2] energy loss for a thicstatic plasma,�EBDMS = πα3

s CRCA/2 · v · L2ρ (thefactor v ∝ log(L/λg) ∼ 1–3), where�E/E increasesrapidly with decreasing energy, the GLV [4,23] firsorder radiative energy loss�E (dashed line) exhibitsalmost a linear energy dependence forE = 4− 7 GeVwhile at higher energy�E/E reveals a log(E)/E

decrease. Elastic collision with other partons, whis modeled with the parton cascade, leads toadditional energy loss by 25% (solid line).

Fig. 1. The production rate of gluons at all rapidities from radiatjet energy loss in central (b = 0 fm) Au + Au collisions at RHICc.m. energy of

√s = 200 AGeV. The results are for jets wit

different pT (dashed lines) and for the total jets (solid line) inevent, normalized to the average number of jets〈Nj 〉 with the givenpT . The dash-dotted line gives the rate of radiative energy(d�E/dt)/〈Nj 〉 in GeV/fm. The rise at small times is the resuof performing the calculation in Cartesian time, rather than protime. The inset shows the energy dependence of fractional enloss�E/E of gluon jets due to radiation (dashed line) and aelastic scattering (solid).

24 S. Pal, S. Pratt / Physics Letters B 574 (2003) 21–26

rgys),einanlly

calts,

ltsithwted

itht

eyave

ichatftiond in

er-henl asm-u-

pid-

r-thetedl-rt-

r-

intk-t

en-

es,be

r

heis

y

iclespen

re

The dynamical evolution of the average eneloss per jet in an event (for jets at all rapiditie(d�E/dt)/〈Nj 〉, rises for small times due to thlongitudinal expansion of the system as shownFig. 1. If proper time had been used rather thCartesian time, the rate would have monotonicafallen due to the falling density.

Fig. 1 also shows the rapidity integrated dynamirate of gluon radiation per average number of je(dNg/dt)/〈Nj 〉, as evaluated from Eq. (5). The resuare for total jets in an event, and also for jets wdefinitepT . The emission pattern is found to follothat for the energy loss. Though the number of radiagluons per〈Nj 〉 is seen to increase withpT , therapidly decreasing jet production cross section wincreasing jet energyE actually leads to smallesnumber of gluon contribution from jets withpT �12 GeV/c. Although higher-energy jets are rare, thradiate more particles due to fact that they hmore available energy and due to the logarithmdependence of the radiation in Eq. (3). We find tpartons with initialpT ∼ 9 GeV/c produce the bulk othe radiated gluons. Using the statistical prescripdescribed above, about 5.5 soft gluons are emittean event per jet.

Since jets are typically formed in pairs, the obsvation of a jet can be used as a trigger. One can tsearch for the remains of the balancing jet, as welany particle whose existence is derived from mediuinduced radiation from the triggered jet. For our calclations, the trigger is defined as a hadron at midraity |y| < 1 with its transverse momentumptrig

T above4 GeV/c. The x direction is then defined to be pependicular to the beam axis and pointing alongdirection of the triggering hadron. Particles radiafrom the trigger jet would typically have positive vaues ofpx while particles associated with the jet’s paner would have negative values ofpx . For a perfectdetector, momentum conservation implies

(6)∫

pxρ(px) dpx = −PT ,

wherePT ≡ ptrigT is the momentum of the trigger pa

ticle andρ(px) = dN/dpx refers to the momentumdensity of all other particles. Of course, this constrais significantly relaxed by finite acceptance. By maing a similar distributionρ0 from other events, withou

the requirement of a trigger, one can define

(7)�ρ(px) = ρ(px)− ρ0(px),

which describes the variation of the momentum dsity due to the jet. Two quantities related to�ρ are ofparticular interest:

�N ≡∫

�ρ(px) dpx,

(8)�Px ≡∫

px�ρ(px) dpx.

�N counts the net number of additional particlwhile�Px describes the momentum, which shoulda fraction of−PT , determined by the acceptance.

Fig. 2 showsρ(px) for events with a triggemomentum, 4< PT ≡ p

trigT < 6 GeV/c. For large

negative values ofpx , ρ(px) is suppressed in Au+Aucollisions relative topp results as expected due to tsuppression of away-side jets [11]. The situationreversed for|px | < 600 MeV/c, where the spectrum

Fig. 2. The distributionρ(px) for particles produced at mid-rapidit

|y| < 1 in events with a triggered hadron 4< ptrigT

< 6 GeV/c inthe same rapidity range. The results are for total number of part(solid circles) and for particles from radiative jet energy loss (ocircles) in central (b = 0 fm) Au + Au collisions at RHIC energyof

√s = 200 AGeV. Thep + p results at the same energy a

shown by open squares. The ratioRAA(px) for the multiplicitiesin Au + Au collisions to that fromp+p collisions for the triggeredjet event is shown in the inset.

S. Pal, S. Pratt / Physics Letters B 574 (2003) 21–26 25

cles.anfortra

thetedby

iallytio

iler

u-

ex-

e

ar-id-

in-

de-iesim-is-e of.g.,iones-ee

tumted

themmnetog

ly-

adssi-

teder-

tlye-to

hatsure

filarForto

befsoias

uiteuchthe

re-the

osstedalfhernced beldthe

ithnse-g.,ionu-

ofatldate

is dominated by the radiated and rescattered partiThe momentum constraint, Eq. (6), is satisfied byincrease of low momentum particles to compensatethe loss of the away-side jet. The net number of exparticles,�N , is higher in the Au+ Au case as wouldbe expected from the induced radiation and fromstatistical nature of Eq. (5) which converts the radiaenergy into particles with momenta characterizedthe temperature. The alteration of the jets is especapparent in the inset of Fig. 2 which shows the raof the Au+ Au andpp results. At large negativepx

the suppression is approximately a factor of 3, whthe enhancement at lowpx is approximately a factoof two.

Also shown in Fig. 2 (open circles) is the distribtion of the particles in Au+ Au collisions that stemsfrom the induced radiation described in Eq. (3). Aspected, these particles are at much lowerpT than thosefrom pp collisions due to their thermalizing with thmedium.

For this range of triggers the number of extra pticles, both charged and neutral, produced at mrapidity was approximately�N = 19, versus 14 forpp collisions. We emphasize that this is a relativecrease and that determiningρ0 is difficult, as discussedlater. The numbers increase by∼ 20% if all rapidi-ties are evaluated. Of course, experiments wouldtect a reduced number after accounting for efficiencand acceptances. By itself, this number providesportant insight into the driving force behind the dappearance of away-side jets. If the disappearancaway-side jets was largely an initial-state effect, e2 → 1 gluon fusion processes in the gluon saturatmodel [12], few extra particles would be created,pecially at mid-rapidity. One would then expect to sa reduction in�N for Au + Au vs.pp rather than theincrease found here.

If we assume a perfect detector, the momenintegration described in Eqs. (6) and (8), resulin �Px = −4.2 GeV/c, after triggering on leadinghadrons in theptrig

T = 4–6 GeV/c range. This verifiesmomentum conservation in the procedure. As withcalculation of�N , most of this balancing momentuwas found in the mid-rapidity cut. If the mechanisfor jet quenching were an initial-state effect, owould expect less of the balancing momentumbe found with a rapidity near that of the triggerinhadron.

It would be straightforward to expand this anasis to include a binning in rapidity as well aspx . Onecould then analyze the degree to which a jet spreits energy and momentum in rapidity. Another posbility would be to bin withpT or ET which wouldallow the determination of the extra energy relato the trigger. The calculations shown here were pformed with approximately 4× 104 trigger particles.Two-dimensional binnings would require significanhigher statistics. We would certainly expect largacceptance experiments such as STAR or PHENIXbe able to perform a comparable analysis as to wwas demonstrated here if they were able to meaon the order of 105 events with trigger particles.

Given sufficient statistics, the principal difficulty othese sort of analyses lies in defining a class of simevents for the purpose of background subtraction.instance, if one uses multiplicity at mid-rapiditydefine the events, the quantity�N would always bezero. The means for determining centrality mustindependent of the particles used in the analysis oρ

andρ0. Furthermore, it must be sufficiently accuratethat the observation of the triggering jet does not bthe event towards being more central. It might be qpossible to quantitatively estimate and correct for sa bias. Fortunately, these effects do not determineextraction of�Px since the base distribution,ρ0 doesnot contribute toPx .

We also investigated the degree to which our pdictions are sensitive to the parameters used inmodel. By reducing the medium-induced energy lby a factor of two, the net number of extra radiaparticles was only reduced from 5.5 to about hthat number. The energy loss is found to be ratinsensitive to the parton elastic cross section. Othe medium becomes opaque, the behavior shoulfairly insensitive to details as thermalization shoumask any details of the microscopic physics, andlinear dependence of the induced multiplicity wthe energy loss parameters should dampen. Coquently, with a much higher initial gluon density, e.dNg/dy ≈ 1000 is suggested in some gluon saturatmodels [25], it is not clear to what degree the distribtion of produced particles would be increased.

In conclusion, we have shown that an analysislow pT observables can unveil the fate of lost jetsRHIC. A similar analysis of experimental data couprovide irrefutable evidence for establishing final-st

26 S. Pal, S. Pratt / Physics Letters B 574 (2003) 21–26

ion.ldnd

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1;7;1)

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8

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jet energy loss as the explanation for jet suppressA high-statistics multi-dimensional analysis wouprovide quantitative insight into the dynamics adissipation of jets in high-density matter.

Acknowledgements

The authors wish to thank Ivan Vitev for explaiing the finer points of medium-induced radiation. Thwork was supported by the National Science Fountion under Grant No. PHY-0070818.

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