We study the effects of jet quenching on the hydrodynamical evolution of the
quark-gluon plasma (QGP) fluid created in a heavy-ion collision. In jet
quenching, a hard QCD parton, before fragmenting into a jet of hadrons,
deposits a fraction of its energy in the medium, leading to suppressed
production of high-pT hadrons. Assuming that the deposited energy quickly
thermalizes, we simulate the subsequent hydrodynamic evolution of the QGP
fluid. For partons moving at supersonic speed, v_p > c_s, and sufficiently
large energy loss, a shock wave forms leading to conical flow [1]. The PHENIX
Collaboration recently suggested that observed structures in the azimuthal
angle distribution [2] might be caused by conical flow. We show here that
conical flow produces different angular structures than predicted in [1] and
that, for phenomenologically acceptable values of parton energy loss, conical
flow effects are too weak to explain the structures seen by PHENIX [2].Comment: 4 pages, 3 figures. Last figure changed, now showing angular
  distribution of pions instead of photons. Added comments on "lost jets" and
  pT-dependence of angular correlation

A. K. Chaudhuri

M. Gyulassy

Ulrich Heinz

English

arXiv

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Effects of Jet Quenching on the Hydrodynamical Evolution of Quark-Gluon Plasma

We study the effects of jet quenching on the hydrodynamical evolution of the quark-gluon plasma (QGP) fluid created in a heavy-ion collision. In jet quenching, a hard QCD parton, before fragmenting into a jet of hadrons, deposits a fraction of its energy in the medium, leading to suppressed production of high-p_T hadrons. Assuming that the deposited energy quickly thermalizes, we simulate the subsequent hydrodynamic evolution of the QGP fluid. For partons moving at supersonic speed, v_p>c_s, and sufficiently large energy loss, a shock wave forms leading to conical flow. The PHENIX Collaboration recently suggested that observed structures in the azimuthal angle distribution might be caused by conical flow. We show here that, for phenomenologically acceptable values of parton energy loss, conical flow effects are too weak to explain these structures

Chaudhuri, A. K.

Heinz, Ulrich

KnowledgeBank at OSU

Effects of Jet Quenching on the Hydrodynamical Evolution of Quark-Gluon PlasmaA. K. Chaudhuri1,* and Ulrich Heinz21Variable Energy Cyclotron Centre, 1-AF, Bidhan Nagar, Kolkata-700 064, India2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA(Received 10 November 2005; revised manuscript received 12 April 2006; published 9 August 2006)We study the effects of jet quenching on the hydrodynamical evolution of the quark-gluon plasma(QGP) fluid created in a heavy-ion collision. In jet quenching, a hard QCD parton, before fragmenting intoa jet of hadrons, deposits a fraction of its energy in the medium, leading to suppressed production ofhigh-pT hadrons. Assuming that the deposited energy quickly thermalizes, we simulate the subsequenthydrodynamic evolution of the QGP fluid. For partons moving at supersonic speed, vp > cs, andsufficiently large energy loss, a shock wave forms leading to conical flow. The PHENIX Collaborationrecently suggested that observed structures in the azimuthal angle distribution might be caused by conicalflow. We show here that, for phenomenologically acceptable values of parton energy loss, conical floweffects are too weak to explain these structures.DOI: 10.1103/PhysRevLett.97.062301 PACS numbers: 25.75.q, 13.85.Hd, 13.87.aRecent Au Au collision experiments at theRelativistic Heavy-Ion Collider (RHIC) saw a dramaticsuppression of hadrons with high transverse momenta(‘‘high-pT suppression’’) [1], and the quenching of jets inthe direction opposite to a high-pT trigger particle [2,3],when compared with p p and d Au collisions. This istaken as evidence for the creation of a very dense, coloropaque medium of deconfined quarks and gluons [4].Independent evidence for the creation of dense, thermal-ized quark-gluon matter, yielding comparable estimates forits initial energy density (hei * 10 GeV=fm3 at timetherm & 0:6 fm=c [5]), comes from the observation ofstrong elliptic flow in noncentral Au Au collisions [1],consistent with ideal fluid dynamical behavior of the bulkof the matter produced in these collisions.These two observations raise the question of what hap-pens—in the small fraction of collision events where ahard scattering produces a pair of high-pT partons— to theenergy lost by the parton traveling through the medium.The STAR Collaboration has shown that, while in centralAu Au collisions there are no hard particles left in thedirection opposite to a high-pT trigger particle, one seesenhanced production (compared to p p) of soft (low-pT)particles, broadly distributed over the hemisphere diamet-rically opposite to the trigger particle [6]. This shows thatthe energy of the fast parton originally emitted in thedirection opposite to the trigger particle is not lost, butseverely degraded by interactions with the medium. As theimpact parameter of the collisions decreases, the averagemomentum of the particles emitted opposite to the triggerparticle approaches the mean value associated with all softhadrons, i.e., the hpTi of the thermalized medium [6]. Thissuggests that the energy lost by the fast parton has beenlargely thermalized. Nevertheless, this energy is depositedlocally along the fast parton’s trajectory, leading to localenergy density inhomogeneities which, if thermalized,should in turn evolve hydrodynamically. This would mod-ify the usual hydrodynamic expansion of the collisionfireball as observed in the overwhelming number of softcollision events where no high-pT partons are created.Since the fast parton moves at supersonic speed, it wassuggested in Ref. [7] that a Mach shock (‘‘sonic boom’’)should develop, resulting in conical flow and preferredparticle emission at a specific angle away from the direc-tion of the fast parton which lost its energy. This Machangle is sensitive to the medium’s speed of sound, cs, andthus offers the possibility to measure one of its key prop-erties. A recent analysis by the PHENIX Collaboration [8]of azimuthal dihadron correlations in 200A GeV Au Aucollisions revealed structures in the angular distributionwhich might be suggestive of conical flow.The idea of Mach shock waves traveling through com-pressed nuclear matter was first advocated 30 years ago[9,10], but RHIC collisions for the first time exhibit [5] thekind of ideal fluid behavior which might make an extrac-tion of the speed of sound conceivable. An alternate sce-nario, in which the color wake field generated by the fastcolored parton traveling through a quark-gluon plasmaaccelerates soft colored plasma particles in the directionperpendicular to the wake front [11,12], leads to an emis-sion pattern which is sensitive to the propagation of plasmarather than sound waves [12]. In a strongly coupled plasmawith overdamped plasma oscillations, which seems to bethe preferred interpretation of RHIC data [13,14], the wakefield scenario should reduce to the hydrodynamic Machcone picture. We here study the dynamical consequencesof the latter, going beyond the discussion of linearizedhydrodynamic equations in a static background offeredin [7].We assume that just before hydrodynamics becomeapplicable, a pair of high-pT partons is produced near thesurface of the fireball. One of them moves outward andescapes, forming the trigger jet, while the other enters intothe fireball along, say, the x direction. The fireball isexpanding and cooling. The ingoing parton travels at thespeed of light and loses energy in the fireball, whichPRL 97, 062301 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending11 AUGUST 20060031-9007=06=97(6)=062301(4) 062301-1 © 2006 The American Physical Societythermalizes and acts as a source of energy and momentumfor the fireball medium. We model this medium as an idealfluid with vanishing net baryon density. Its dynamics iscontrolled by the energy-momentum conservation equa-tions @T  J; (1)where the energy-momentum tensor has the ideal fluidform T  " puu  pg, with energy density "and pressure p being related by the equation of state (EOS)p  p", u  1; vx; vy; 0 is the fluid 4-velocity, andthe source current J is given by Jx  Jx1;1; 0; 0; (2) Jx  dEdxxdxjetdt3r rjett: (3)Massless partons have lightlike 4-momentum, so the cur-rent J describing the 4-momentum lost and deposited inthe medium by the fast parton is taken to be lightlike, too.rjett is the trajectory of the jet moving with speedjdxjet=dtj  c. dEdx x is the energy loss rate of the partonas it moves through the liquid. It depends on the fluid’slocal rest frame particle density. Taking guidance from thephenomenological analysis of parton energy loss observedin Au Au collisions at RHIC [15] we take dEdx sxs0dEdx0; (4)where sx is the local entropy density without the jet. Themeasured suppression of high-pT particle production inAu Au collisions at RHIC was shown to be consistentwith a parton energy loss of dEdx j0  14 GeV=fm at a ref-erence entropy density of s0  140 fm3 [15]. For com-parison, we also perform simulations with 10 times largerenergy loss, dEdx j0  140 GeV=fm.For the hydrodynamic evolution we use AZHYDRO[5,16], the only publicly available relativistic hydrody-namic code for anisotropic transverse expansion. Thisalgorithm is formulated in (, x, y, ) coordinates, where t2  z2pis the longitudinal proper time,   12 lntztzis space-time rapidity, and r?  x; y defines the planetransverse to the beam direction z. AZHYDRO employslongitudinal boost invariance along z but this is violatedby the source term (3). We therefore modify the latter byreplacing the -function in (3) by 3r rjett ! 1 x xjety yjet! 1er?r?;jet2=2222(5)with   0:35 fm. Intuitively, this replaces the ‘‘needle’’( jet) pushing through the medium at one point by a‘‘knife’’ cutting the medium along its entire length alongthe beam direction. The resulting ‘‘wedge flow’’ is ex-pected to leave a stronger signal in the azimuthal particledistribution dN=d	 than ‘‘conical flow’’ induced by asingle parton, since in the latter case one performs animplicit 	-average when summing over all directions ofthe cone normal vector. While a complete study of thiswould require a full 3 1-dimensional hydrodynamiccalculation, the present boost-invariant simulation shouldgive a robust upper limit for the expected angular signa-tures. We show that the angular structures predicted fromwedge flow are too weak to explain the experimentallyobserved 	-modulation [8].The modified hydrodynamic equations in (, x, y, )coordinates read [16] @ ~T  @x~vx ~T  @y~vy ~T  p ~J; (6) @ ~Tx  @xvx ~Tx  @yvy ~Tx  @x ~p ~J; (7)0.080.250.420.590.770.941.11.31.51.6-10 -5 0 5 10-50510Y (fm)0.080.130.180.230.290.340.390.440.500.55-10 -5 0 5 10-10-505100.080.090.100.110.130.140.150.160.180.19-10 -5 0 5 10-10-50510 (c)τ=12.6fm,Xjet=-5.6fm,∆E=8 GeV0.080.420.761.1 1.41.8 2.1-10 -5 0 5 10-10-50510(b)τ=8.6fm,Xjet=-1.6fm,∆E=7.1GeV(d)τ=4.6fm,Xjet=2.4fm,∆E=47.6GeV(a)τ=4.6fm,Xjet=2.4fm,∆E=4.8GeVdE/dx|0=14GeV/fm0.080.230.380.530.690.840.991.11.31.3-10 -5 0 5 10-10-50510 (e)τ=8.6fm,Xjet=-1.6fm,∆E=71.3GeV0.080.170.260.350.450.540.63-10 -5 0 5 10-10-50510dE/dx|0=140GeV/fm(f)τ=12.6fm,Xjet=-5.6fm,∆E=79.4GeVX (fm)FIG. 1. Contours of constant local energy density in the x-yplane at three different times,   4:6, 8.6, and 12:6 fm=c. Ineach case the position of the fast parton, along with the inte-grated energy loss E  R Jxdxdyd up to this point, isindicated at the top of the figure. Diagrams (a)–(c) in the leftcolumn were calculated with a reference energy loss dE=dxj0 14 GeV=fm, those in the right column [panels (d)–(f)] with a10 times larger value.PRL 97, 062301 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending11 AUGUST 2006062301-2 @ ~Ty  @xvx ~Ty  @yvy ~Ty  @y ~p; (8)where ~T  T, ~vi  Ti=T, ~p  p, and ~J  J.To simulate central Au Au collisions at RHIC, we usethe standard initialization described in [5] and provided inthe downloaded AZHYDRO input file [16], corresponding toa peak initial energy density of "0  30 GeV=fm3 at 0 0:6 fm=c. We use the equation of state EOS-Q described in[5,16] incorporating a first order phase transition and had-ronic chemical freeze-out at a critical temperature Tc 164 MeV. The hadronic sector of EOS-Q is soft with asquared speed of sound c2s  0:15.In our study the quenching jet starts from xjet  6:4 fmat 0  0:6 fm, moving left towards the center with con-stant speed vjet  c. For an upper limit on conical floweffects, the fast parton is assumed to have sufficient initialenergy to emerge on the other side of the fireball. Tosimulate cases where the fast parton has insufficient energyto fully traverse the medium we have also done simulationswhere the parton loses all its energy within an (arbitrarilychosen) distance of 6.4 fm. We further compared with a runwhere the parton moved at (constant) subsonic speed(vjet  0:2c < cs).The resulting evolution of the energy density of the QGPfluid is shown in Figs. 1 and 2. The left column of Fig. 1shows results for a phenomenologically acceptable valuedEdx j0  14 GeV=fm [15] for the reference parton energyloss whereas in all other columns we use a 10 times largerenergy loss. The width of the Gaussian source [see Eq. (5)]is   0:7 fm. In the left column of Fig. 1 the effects of theenergy deposition from the fast parton are hardly visible.Only for a much larger energy loss (right column) werecognize a clear conical flow pattern. The accumulatingwave fronts from the expanding energy density wavesbuild up a sonic boom shock front which creates a Machcone. The right columns in Figs. 1 and 2 show that the conenormal vector forms an angle M with the direction of thequenching jet that is qualitatively consistent with expecta-tions from the theoretical relation cosM  cs=vjet.However, this angle is not sharply defined since the conesurface curves due to inhomogeneity and radial expansionof the underlying medium. This differs from the statichomogeneous case [7].When the parton travels at a subsonic speed vjet 0:2c < cs (left column in Fig. 2), it does not get very farbefore the fireball freezes out due to longitudinal expan-sion. In this case one only observes an accumulation ofenergy around the parton but no evidence of Mach coneformation. When the parton travels with vjet  c but loosesall its energy after 6.4 fm before fully traversing the fireball(right column in Fig. 2), the fireball evolution beyond  7 fm is not affected by the fast parton directly but onlyindirectly through the propagation of earlier depositedenergy. Still, Figs. 1(f) and 2(f) show that the late timeevolution of the fireball is quite similar in both cases,demonstrating that energy deposition by the fast partonduring the late fireball stages is small, due to dilution of thematter, and can almost be neglected.Tests with different values for the width  of theGaussian profile in Eq. (5) for the deposited energy showthat the cone angle gets better defined for smaller sourcesize . Note that the quenching jet destroys the azimuthalsymmetry of the initial energy density distribution butleaves the azimuthally symmetric energy contours to theleft of the jet unaffected.We close with a discussion of observable consequencesof conical flow. One expects [7] azimuthally anisotropicparticle emission, peaking at angles 	  	 M relativeto the trigger jet where M is the Mach angle. Using thestandard Cooper-Frye prescription, we have computed theangular distribution of directly emitted pions at a freeze-out temperature Tfo  100 MeV [5].Figure 3 shows the azimuthal distribution of  from avariety of different simulations. [The 	-independent con-stant dN=dyd	no jet  27 from central collisions with-out jets has been subtracted.] For dE=dxj0  14 GeV=fmthe azimuthal modulation is very small. In none of the0.080.250.420.600.770.941.11.11.31.31.51.6-10 -5 0 5 10-10-505100.080.170.260.360.450.450.540.63-10 -5 0 5 10-10-505100.080.100.120.140.160.180.200.22-10 -5 0 5 10-10-505100.080.420.761.11.41.8 2.12.5-10 -5 0 5 10-10-505100.080.250.420.600.770.941.11.11.11.31.3-10 -5 0 5 10-10-505100.080.190.300.420.530.64-10 -5 0 5 10-10-50510dE/dx|0=140 GeV/fmdE/dx|0=140 GeV/fm(d)τ=4.6fm, Xjet=2.4fm∆E=47.6GeV(b)τ=8.6fm, Xjet=5.3fm∆E=8GeV(e)τ=8.6fm,∆E=64.7GeV(c)τ=12.6fm, Xjet=4.5fm∆E=9.5GeV(f)τ=12.6fm,∆E=64.7GeV(a)τ=4.6fm, Xjet=6.1fm∆E=5.3GeVX (fm)FIG. 2. As in Fig. 1 but for a parton moving at subsonic speedvjet  0:2c (left column) and for a fast parton (vjet  c) whichloses all its energy within the first 6.4 fm (right column).PRL 97, 062301 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending11 AUGUST 2006062301-3cases studied we find peaks at the predicted Mach anglewith an associated dip in the direction of the quenched jetat 	  . One rather sees a peak at 	  , broadened byshoulders on both sides which turn into small peaks at(relative to the quenching jet) backward angles if thermalsmearing is reduced by considering only high-pT pions.The peak at 	   is absent when the parton loses all itsenergy halfway through the fireball or is too slow to get tothe other side before freeze-out, suggesting that it reflectsthe directed momentum imparted on the medium by thefast parton. It is slightly more accentuated for smaller and higher pT . We also found that the width of the should-ers is almost independent of the speed of sound of themedium and can not be used to diagnose the stiffness of itsequation of state. The shoulders exist even for subsonicparton propagation (vjet  0:2c, upright triangles), show-ing that other mechanisms (such as back splash from thehard parton hitting the fluid and a general bias of the energydeposition towards the right side of the fireball due to thehigher density of the medium at early times) have a stronginfluence on the angular distribution of the emitted parti-cles which interferes with the position of the Mach peaks.The shoulder resulting from this combination of effects ismuch broader than the angular structures seen in the data[8]. The absence of a clear dip at 	   in our simulationsis all the more troubling since it should have been strongerfor the wedge flow studied here than for real ‘‘conical’’flow.Our calculation does not average over the initial produc-tion points of the trigger particle, i.e., it ignores that in mostcases its quenching partner does not travel right throughthe middle of the fireball cylinder, but traverses it semi-tangentially. This should further decrease the prominenceof the shoulders in dN=d	. We conclude that conical flowmay be able to explain the broadening of the away sidepeak in the hadron angular correlation function around	   pointed out by the STAR Collaboration [6], but isunlikely to be responsible for the relatively sharp structuresnear 	  	 1 seen by PHENIX [8]. This conclusionextends to other conical flow phenomena, such as thosegenerated by color wake fields [11,12].This work was supported by the U. S. Department ofEnergy under Contract No. DE-FG02-01ER41190.*Corresponding author.Electronic address: akc@veccal.ernet.in[1] I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys.A757, 1 (2005); B. B. Back et al. (PHOBOS Collabora-tion), ibid. A757, 28 (2005); J. Adams et al. (STARCollaboration), ibid. A757, 102 (2005); K. Adcox et al.(PHENIX Collaboration), ibid. A757, 184 (2005).[2] C. Adler et al. (STAR Collaboration), Phys. Rev. Lett. 90,082302 (2003); J. Adams et al. (STAR Collaboration),Phys. Rev. Lett. 91, 072304 (2003); Phys. Rev. Lett. 93,252301 (2004).[3] J. Rak et al. (PHENIX Collaboration), J. Phys. G 30,S1309 (2004).[4] M. Gyulassy, I. Vitev, X.-N. Wang, and B.-W. Zhang, inQuark-Gluon Plasma 3, edited by R. C. Hwa and X.-N.Wang (World Scientific, Singapore, 2004), p. 123.[5] P. F. Kolb and U. Heinz, in Ref. [4], p. 634.[6] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 95,152301 (2005).[7] J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney,J. Phys.: Conf. Ser. 27, 22 (2005).[8] S. S. Adler et al. (PHENIX Collaboration), nucl-ex/0507004.[9] G. F. Chapline, M. H. Johnson, E. Teller, and M. S. Weiss,Phys. Rev. D 8, 4302 (1973).[10] W. Scheid, H. Mu¨ller, and W. Greiner, Phys. Rev. Lett. 32,741 (1974).[11] H. Sto¨cker, Nucl. Phys. A750, 121 (2005).[12] J. Ruppert and B. Mu¨ller, Phys. Lett. B 618, 123 (2005).[13] U. Heinz and P. F. Kolb, Nucl. Phys. A702, 269 (2002).[14] M. Gyulassy and L. D. McLerran, Nucl. Phys. A750, 30(2005); E. Shuryak, Nucl. Phys. A750, 64 (2005).[15] X. N. Wang, Phys. Rev. C 70, 031901 (2004) and (privatecommunication).[16] P. F. Kolb, J. Sollfrank, and U. Heinz, Phys. Rev. C 62,054909 (2000); P. F. Kolb and R. Rapp, Phys. Rev. C 67,044903 (2003). The computer code can be downloadedfrom URL http://nt3.phys.columbia.edu/people/molnard/OSCAR/.dN/dΦ−dN/d Φnojet-0.04-0.020.000.020.04 ΦdN/dΦ-dN/dΦnoje t-10-505101520140 0.35 c0.7 .2c0.7 c (dE/dx=0 for τ>7 fm)0.7 cdE/dx(GeV/fm) σ(fm) vjet14 0.35 c(X10)14 0.7 c(X10)π/20 π 3π/2 2ππ-, pT >0π-, pT >1 GeV/c140140140FIG. 3. Azimuthal distribution dN=dyd	 of negative pions perunit rapidity. In the upper panel we integrate over all pT whilethe lower panel shows only pions with pT > 1 GeV=c. Differentsymbols refer to different parameters as indicated. For bettervisibility the 	-independent rate in the absence of the quenchingjet has been subtracted. Filled symbols show the realistic casedE=dxj014GeV=fm, enhanced by a factor of 10 for visibility.PRL 97, 062301 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending11 AUGUST 2006062301-4

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Effects of jet quenching on the hydrodynamical evolution of quark-gluon plasma

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