Di-jet hadron pair correlation in a hydrodynamical model with a quenching jet A. K. Chaudhuri1,∗ 1Variable Energy Cyclotron Centre, 1-AF, Bidhan Nagar, Kolkata - 700 064, India (Dated: February 5, 2008) In jet quenching, a hard QCD parton, before fragmenting into a jet of hadrons, deposits a frac- tion of its energy in the medium, leading to suppressed production of high-pT hadrons. Assuming thatthedepositedenergy quicklythermalizes, wesimulate thesubsequenthydrodynamicevolution of the QGP fluid. Hydrodynamic evolution and subsequent particle emission depend on the jet trajectories. Azimuthaldistributionofexcessπ− duetoquenchingjet,averagedoverallthetrajec- tories,reasonablywellreproducethedi-hadroncorrelation asmeasuredbytheSTARandPHENIX collaboration in central and in peripheral Au+Aucollisions. 8 PACSnumbers: PACSnumbers: 25.75.-q,13.85.Hd,13.87.-a 0 0 2 Fromthegeneraltheoreticalconsiderations,itwaspre- Evolutionproduces’distorted’Machshockfrontlikesur- n dicted[1]thatinadensedeconfinedmedium,high-speed faces. As predicted in [20], finite fluid velocity and also a partonswillsufferenergyloss,significantlymodifyingthe inhomogeneity of the fluid, distorts the Mach surfaces. J fragmentation function, which in turn will lead to sup- Azimuthal distribution of pions, due to quenching jet 5 pressed production of hadrons. The phenomena called only, do not show the characteristic of ’conical’ flow. 1 ”jet quenching”, was later verified at Relativistic Heavy Rather, depending on the jet trajectory, it show a single ] IonCollider(RHIC),inAu+Aucollisionsat√sNN=200 peakeitheratφ>πorφ<π. Thedistributionaveraged h GeV [2, 3, 4]. However it is unclear how the lost energy overallthejettrajectories,showtwopeakswithadipat t - is transported in the dense medium. It has been sug- φ = π, mimicking the conical flow. However, in [18], we l c gested that a fraction of lost energy will go to collective didnotaccountforthe limited pT range(0.15 pT 4) u excitation, call the ”conical flow” [5, 6, 7]. The parton in the STAR experiment. Moreover, in [18], ≤jet tra≤jec- n moves with speed of light, much greater than the speed tories were characterized by a single parameter, φ , [ prod of sound of the medium (c >> c ), and the quench- while in a two-dimensionalcalculation,asin [18], unique jet s 2 ing jet can produce a shock wave with Mach cone an- characterization of jet trajectories need at least two pa- v gle, θ = cos−1c /c . Resulting conical flow will have rameters. In the present paper, we have corrected the 8 M s jet characteristic peaks at φ = π θ and φ = π + θ . deficiencies in the model. Also, in addition to the STAR 5 − M M 9 Both in STAR [8, 9] and PHENIX [10] experiments, in- data [8], we analyze the PHENIX data [10] on the di- 3 dicationof such peaks are seenin azimuthaldistribution hadron angular correlation in 0-5%, 5-10% and 60-90% . of secondaries associated with high p trigger in central centrality Au+Au collisions. 6 T 0 Au+Aucollisions. Machlikestructure(splitting ofaway 7 side peak) can also be obtained in various other models, 0 e.g. gluon Cerenkov like radiation models [11, 12], the A schematic representation of the jet moving through : v parton cascade model [13], the Markovian parton scat- the medium is shown in Fig.1. We assume that just be- Xi tering model [14], the color wake model [15]. Recently fore hydrodynamics become applicable, a di-jet of high- in [16] energy density wake produced by a heavy quark p partons is produced. Strong jet quenching and sur- r T a movingthroughastronglycoupledN=4supersymmetric, vival of the trigger jet, forbids production in the interior Yang-Mills plasma is computed using ADS/CFT corre- of the fireball. Jet pairs can be produced only on a thin spondence. Mach cone like structures is also observed shell on the surface of the fireball. For Au+Au colli- for quark velocity greaterthan the speed of soundof the sionsatimpactparameterb,weassumethatthe di-jetis medium. produced on the surface of the ellipsoid with minor and Recently, we have numerically solved hydrodynamical major axis, A = R b/2 and B = R 1 b2/4R2, with − − equations with an (time dependent) source,representing R = 6.4fm. One of the jet moves outward and escapes, p the quenchingjet[17,18, 19]. Itisassumedthatthe lost forming the trigger jet. The other enters into the fire- energyisquicklythermalised. Nevertheless,theenergyis ball. As shown in Fig.1, the trajectory of the quenching deposited locally along the trajectory of the fast parton, jet can be uniquely characterized by two angles, φ , prod leading to local energy density inhomogeneities, which ( π φ π) and φ , ( π/2 φ π/2). The prod jet jet − ≤ ≤ − ≤ ≤ if thermalised should in turn evolve hydrodynamically. fireball is expanding and cooling. The ingoing parton Explicit simulation of hydrodynamic evolution with a travels at the speed of light and loses energy in the fire- quenching jet, indicate that the evolution of the fluid as ball which thermalizes and acts as a source of energy well as subsequent particle emission are strongly influ- and momentum for the fireball medium. We solve the enced by the jet path length in the medium [17, 18, 19]. energy-momentum conservation equation, 2 We solve Eqs.1 in (τ = √t2 z2,x,y,η=1ln t+z ) − 2 t−z ∂ Tµν =Jν, (1) coordinates,assumingboost-invariance. Thesourcheterim µ Eq.3 violate the assumption of boost-invariance. We where the source is modeled as, therefore modify it by replacing the δ-function in (3) by Jν(x)=J(x) 1, cos(φjet), sin(φjet),0 , (2) δ3(r r (t)) 1 δ(x x (τ))δ(y y (τ)) − − jet jet jet − −→ τ − − dE dx J(x)= (x)(cid:0) jet δ3(r rjet(t)). (cid:1) (3) 1 e−(r⊥−r⊥,jet(τ))2/(2σ2) dx dt − (5) (cid:12)(cid:12) (cid:12)(cid:12) −→ τ 2πσ2 (cid:12) (cid:12) (cid:12) (cid:12) Y with σ=0.70fm, r⊥ =(x,y). Intuitively, this replaces the “needle” (jet) pushing ) PF( tprirgogdFe,r jjeett ttthhiorenom.ugRehdaituthhmeeramtlhoeandngiu’icmtosneaincttaiolr’ne,etlehpneogritenhptlaablcoyenmaget“nhktenwbifieela”lmpcruodtditruienccge- quenching jet a’wedge’flow,overestimatingtheeffectofjetquenching. F jet F prod The hydrodynamical equations are solved with the -X X standard initialization described in [22], corresponding to a peak initial energy density of ε =30 GeV/fm3 at 0 τ =0.6 fm/c. We use the equation of state EOS-Q de- 0 scribedin[22]incorporatingafirstorderphasetransition and hadronic chemical freeze-out at a critical tempera- ture T =164MeV. The hadronicsectorofEOS-Qis soft c with a squared speed of sound c2 0.15. -Y s ≈ Some results of our simulations, for Au+Au collisions at impact parameter b=2.3 fm, are shown in Fig.2. In FIG.1: Schematicrepresentationofajetmovingthroughthe medium. ThehighpT pairisassumedtoproduceatP onthe left panels (a),(b) and(c), for a few jet trajectories,con- surface of the fireball characterized by the angle φ . One tour plots of ’excess’ energy density (energy density in prod of the jet escapes forming the trigger jet, the other move in evolution with a quenching jet minus the energy density thefireball at an angle φjet. in evolution without a quenching jet), after 8 fm of evo- lution, are shown. The contour plots of ’excess’ energy Massless partons have light-like 4-momentum, so the density clearly show that the hydrodynamic evolutionof current Jν describing the 4-momentum lost and de- the QGP fluid, in presence of a quenching jet, depend posited in the medium by the fast parton is taken to be on the jet path length. For example, excess energy den- light-like, too. r (t) is the trajectory of the jet moving jet sity distribution in panel (a) and panel (c) are identical with speed dx /dt =c. dE(x) is the energy loss rate | jet | dx except for a rotation about π/2. For those two trajecto- of the parton as it moves through the liquid. It depends ries, the jets traverse a similar path length in the fluid. on the fluid’s local rest frame particle density. Taking Wealsonotethe excessenergydensitydistributioninall guidance from the phenomenological analysis of parton the jet trajectories show Mach cone like surfaces. One energy loss observed in Au+Au collisions at RHIC [21] also notices the distortion of Mach cone like surfaces. In we take panel (b) the excess energy density distribution is sym- dE s(x) dE metric with respect to the jet axis, but not in panel (a) = (4) or (c). Finite fluid velocity and inhomogeneity of the dx s dx 0 (cid:12)0 medium distorts the Mach cone like surfaces [18, 20]. (cid:12) where s(x) is the local entropy den(cid:12)sity without the jet. In the right panels (d), (e) and (f) azimuthal distri- (cid:12) Themeasuredsuppressionofhigh-p particleproduction bution of π− due to the quenching jet are shown. Us- T in Au+Au collisions at RHIC was shown to be consis- ing the standard Cooper-Frey prescription, for each jet tent with a parton energy loss of dE =14GeV/fm at trajectory,we havecalculatedthe azimuthaldependence a reference entropy density of s0=1dx40(cid:12)0fm−3 [21]. We do of pT integrated (1 < pT < 2.5GeV) π− yield (ddNφ)jet not considered the possibility that so(cid:12)me jets might be at freeze-out temperature, TF= 100 MeV. Azimuthal stopped in the medium. It is implicitly assumed that angle φ is measured with respect to the quenching jet the ingoing parton has enough energy to pass through axis. We alsocalculatethe π− yield (ddNφ)nojet in anevo- the medium. The energy loss is weighted by the entropy lution with identical conditions but for the quenching density,asthefluidevolve,entropydensitydecreasesand jet. In(d)-(e) azimuthaldistributionofexcesspionyield atthelatestageoftheevolutionenergylossisminimum. (ddNφ)jet −(ddNφ)nojet, are shown. Excess π− distribution 3 do not show a two peak structure, rather it show a sin- shock wave or the deflected jet. Three-particle angular gle peak either at φ > π or at φ < π. Only when the correlationcan possibly discriminate between them. jet is alongthe diameter,one find a broadpeak centered aroundφ=π. Asimilarpictureisseenatothertrajecto- 10 ries also. Indeed, we find that, depending on the trajec- 0.00-205.001.00025 0.015 tory,aquenchingjetproducespeakeitheratφ− 2. 1 -0.0-203.010 0.015 radoratφ 4 1rad. Asitwillbe shownlate∼r,wh±en 5-0.05-0.0-70.04 0.015 + ∼ ± many events are summed up, a two peak structure, akin 0.0025 to conical flow, appear. The conical flow or the splitting of the away side jet is an average effect, not to be seen m) 0 in a single event. Y (f perturbation in T02(x=2.6,y,t ) 150 (a)F prod=0o,0 0.F0.010j0.e0.1t=0.0-204.15o 0.01 0000....01125050 (d)F prod=0o, F jet=-45o -5 F prod=60o, F jet=0o 0 0.00 -10 -0.05 2 4 6 8 10 12 14 16 -5 -0.10 t (fm) -0.15 -10 -0.20 150-10 (b)-F5prod=00 o, F jet=50o 10000...1120500 (e1)F pr2od=03o , F 4jet=05o 6 F dnojet FTI0G2.(T3:02cionnetvooulrutpiloontwofitpheartjuertbmatiinounsiTn02mionmeevnotluumtiondewnistihty- Y (fm) 0 0.002.10.000.10.1.0000..1.011 -000...000055 - dN/jet oionfuttTha0e2jjedetto)iinsnosthτor−wemynapbiynlantcheoe,nafibtnlaeacdkfiaxlilenodnegxw=tith2he.6ajerftmrot.wraT.jhPeceetrtotrruayrj.becattoiorny -5 -0.10 FN/d -0.15 d -10 -0.20 10-10(c)F p-5rod=0o,0 F jet=455o 1000..12500(f)F1prod2=0o3, F je4t=455o 6 relIantiFonig.a4s,mine4aspuarnedelsbywethheavSeTAshRow[8n]tahneddti-hheaPdrHoEnNcoIXr- 5 0.10 [10] collaboration. STAR collaboration measured the 0.05 0 0.00 correlationfunctionin0-5%centralityAu+Aucollisions. 00..00210.10.10.0 -0.05 They measured charged hadrons ( 0.15 p 4GeV) -5 0.0 0.01 -0.10 associated with high p (4 ptrigger ≤6GTeV≤) trigger -0.15 T ≤ T ≤ -10 -0.20 particle. ThecorrelationfunctionshowninFig.4a,barely -10 -5 0 5 10 0 1 2 3 4 5 6 X (fm) F ( rad) showthesplittingoftheawaysidepeak. PHENIXcollab- oration measured the correlation function as a function FIG. 2: In left panels contour plot of excess energy den- of centrality of collisions. They constructed the corre- sity due to a quenching jet in b=2.3 fm Au+Au collisions are shown. The di-jet is produced at φprod = 0◦, but with lation function between the trigger (2.5 ≤ pT ≤ 4GeV) different orientations; (a) φjet = −45◦, (b)φjet = 0◦ and and charged hadrons (1≤pT ≤2.5GeV). PHENIX cor- (c)φjet = 45◦ respectively. In the right panels, azimuthal relation function in 0-5%, 5-10% and 60-90% centrality distribution of π−, due the quenching jet only, from the cor- collisions are shown in Fig.4b,c and d. In 0-5% and 5- responding evolution are shown. 10% centrality collisions, the away side peak is splitted into two peaks, but not in 60-90% centrality collisions. In Fig.3, we have shown the perturbation produced in In 60-90% centrality collisions show the usual structure the momentum density ∆T02 = T02(x,y) T02 (x,y) in jet event, two peaks, one at φ = 0 and the other at jet − nojet duetoaquenchingjet. Thetrajectoryofthejetisshown φ = π. We also note that , splitting of the away side by the straight line. ∆T02 do not remain confined along jet is more prominent in PHENIX than in STAR experi- the jet trajectory,it deviates sideward. Eventhough the ment,presumablyduetomorehardhadronsinPHENIX jet is restricted to move parallel to the x-axis and ex- measurements. pected to produce a peak at φ = π, due to finite fluid In Fig.4, solid lines are the p integrated azimuthal T velocityandinhomogeneity,themomentumperturbation distribution of excess pions in Au+Au collisions at im- moves sideward and produce a peak at φ < π. We may pact parameter b=2.3 fm (panel (a) and (b)), b=4.1 fm mention that ∆T02 rather look like that due to a de- (panel(c)) and b=12.1 fm (panel (d)) Au+Au collisions. flectedjet[14]. Inthedeflectedjetpicturealso,inasin- They roughly corresponds to 0-5%, 5-10% and 60-80% − glejetevent,theawayjetisdeflectedsideward,resulting centralityAu+Aucollisions. Theπ productionwasav- inaside wardpeak. Whenmanyeventsaresummedup, eragedoverall possible jet trajectories. For each jet tra- a double peak structure appears in the azimuthal distri- jectory (φ , φ ), we calculate the p integrated ex- prod jet T bution. Two-particleangularcorrelationcannotdiscrim- cess pions dN (φ ,φ ) and average over all the pos- d∆φ prod jet inate betweenthe two scenarios,e.g. the distortedMach sible jet trajectories, 4 dN 1 π 1 π/2 dN < >= dφ dφ (φ ,φ ) (6) prod jet prod jet d∆φ 2π Z−π "π Z−π/2 d∆φ # − Azimuthal distribution of excess π , normalized by a jectories produces the splitting. Mechanism of splitting factorN 2.5 3.5,reasonablywellreproducesthe ofthe awayside peak incentralandmid-centralAu+Au norm ≈ − shapeofthedi-hadroncorrelationinSTARandPHENIX collisions can now be understood in terms of jet quench- measurements in central Au+Au collisions. It must be ing. An individual jet event does not produce ’conical’ emphasized, the present model do not contain any pa- flow. Depending upon the trajectory, an individual jet rameter,otherthanN . ConsideringthatSTAR and produces associated particles either at φ δφ > π or norm + ± PHENIXmeasuredchargedhadron,ofwhichonly 70% φ− δφ<π. Whenaveragedoverallthejettrajectories, ∼ ± are charged pions, N 2.5-3.5, seems reasonable. theassociatedparticlesshowatwopeaksstructure,much norm ≈ The subtle difference between the STAR and PHENIX akin to the ’conical’ flow. However, we also notice that 0-5% centrality data (splitting of the away side peak is though the model correctly predict the splitting of the prominentinPHENIXbutnotinSTARmeasurement)is away side jet in to two peaks, the peaks in experiments also well reproduced in the model. The difference is due aresharperthaninthemodelcalculations. Thereasonis to p range of associated particles. STAR data contain possiblytheassumptionofboost-invariance. The’needle’ T more soft hadronsthan in PHENIX data. Hydrodynam- likejet is replacedby a ’knife’, flattening the peaks. The ics is not very successful in peripheral collisions. Still model calculations also do not reproduce the exact the the model reproduces PHENIX data in 60-90% central- experimentalpeakpositions. Simulatedpeaksareshifted ity collisions. from the experiment by about 0.4 rad. As shown in ∼ [18] the peak in the azimuthal distribution of associated 0.4 particles, depend, non-trivially, on the equation of state (a) STAR 0-5%(x0.1) (b) PHENIX 0-5% 0.3 Nnorm=3.5 Nnorm=2.5 (speed of sound of the) medium. Deviation reflect our 0.2 far from satisfactory knowledge of equation of state of DFN/d0.1 DFN/d the quarks matter for of the QGP/hadronic matter. d d 0.0 -0.1 0.4 (c) PHENIX 5-10% (d) PHENIX 60-90% 0.3 Nnorm=2.5 Nnorm=0.5 ∗ Electronic address: [email protected] 0.2 [1] M.Gyulassy,I.Vitev,X.-N.Wang,andB.-W.Zhang,in Quark-Gluon Plasma 3, edited by R. C. Hwa and X.-N. 0.1 Wang (World Scientific, Singapore, 2004), p. 123. 0.0 [2] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. -0.1 Lett. 88, 022301 (2002) [arXiv:nucl-ex/0109003]. [3] C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 0 1 2 3 4 5 60 1 2 3 4 5 6 89, 202301 (2002) [arXiv:nucl-ex/0206011]. D F (rad) [4] J. 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