Jet Propagation and Mach Cones in (3+1)d Ideal Hydrodynamics
Barbara Betz, Miklos Gyulassy, Dirk H. Rischke, Horst Stöcker, Giorgio Torrieri
aa r X i v : . [ h e p - ph ] A p r Jet Propagation and Mach Cones in (3+1)d IdealHydrodynamics
Barbara Betz , , Miklos Gyulassy , Dirk H. Rischke , , HorstSt¨ocker , , Giorgio Torrieri Institut f¨ur Theoretische Physik, J.W. Goethe-Universit¨at, Frankfurt, Germany Helmholtz Research School, Universit¨at Frankfurt, GSI and FIAS, Germany Department of Physics, Columbia University, New York, USA Frankfurt Institute for Advanced Studies (FIAS), Frankfurt, Germany Gesellschaft f¨ur Schwerionenforschung, GSI, Darmstadt, GermanyE-mail: [email protected]
Abstract.
The observation of jet quenching and associated away–side Mach cone–like correlations at RHIC provide powerful “external” probes of the sQGP producedin A+A reactions [1], but it simultaneously raises the question where the jet energywas deposited. The nearly perfect bulk fluidity observed via elliptic flow suggeststhat Mach cone–like correlations may also be due to rapid local equilibration in thewake of penetrating jets. Multi-particle correlations lend further support to thispossibility [2]. However, a combined study of energy deposition and fluid responseis needed. We solve numerically 3–dimensional ideal hydrodynamical equations tocompute the flow correlation patterns resulting from a variety of possible energy-momentum deposition models. Mach–cone correlations are shown to depend criticallyon the energy and momentum deposition mechanisms. They only survive for aspecial limited class of energy–momentum loss models, which assume significantly lesslongitudinal momentum loss than energy loss per unit length. We conclude that thecorrect interpretation of away–side jet correlations will require improved understandingand independent experimental constraints on the jet energy–momentum loss to fluidcouplings.
1. Introduction
One of the major findings at the Relativistic Heavy Ion Collider (RHIC) is thesuppression of highly energetic particles [1]. Two– and three–particle correlations ofjet–associated intermediate– p ⊥ particles provide an important test of the response ofthe medium to the details of the jet–quenching dynamics [2].The observation of strong flow [3] suggests the possibility that the energy lostis quickly thermalized and incorporated in the local hydrodynamical flow. For aquantitative comparison to data a detailed model of both energy and momentumdeposition coupled with a relativistic fluid model is needed [4, 5, 6].We solve numerically (3 + 1)d ideal hydrodynamics [7], including a Bag ModelEquation of State (EoS) with a critical temperature of 169 MeV to study the interactions et Propagation and Mach Cones in (3+1)d Ideal Hydrodynamics x [fm] y [f m ] T [ M e V ]
208 207 206 205 204 203 202 201 200 199 198 197 196 195 194-5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1 0 1 2 3 4 5 ππ /20- π /2- π d N / d p / ( d N / d p | ( φ = - π )) φ [rad] p T = 8 GeV/cp T = 9 GeV/cp T = 10 GeV/cp T = 11 GeV/c Figure 1. (Left panel) Temperature pattern after a hydrodynamical evolution of t = 7 . dE/dx = 1 . v jet = 0 .
99c along the x –axis. (Right panel) Jet–signal strength determined afterisochronous freeze–out for different p ⊥ values. of the jet with a medium for different energy– and momentum–deposition scenarios andto compute the flow–correlation patterns for the different energy–momentum–depositionmodels. Our focus is to study how hydrodynamical flow profiles (such as Mach cones),defined in configuration space, translate into momentum–space correlation functions viafreeze–out. For this, we investigate the simplest situation of a uniform medium.
2. Jets in Ideal Hydrodynamics
In ideal hydrodynamics, the energy–momentum tensor is locally conserved. Adding ajet to the system, an extended set of equations including a source term S ν has to besolved numerically, ∂ µ T µν = S ν . (1)In this work, we will apply a source term S ν = τ f Z τ i dτ dP ν dτ δ (4) ( x µ − x µ ( τ )) , (2)and assume a constant energy and momentum loss rate dP ν /dτ = ( dE/dτ, d ~M /dτ )along the trajectory of a jet x µ ( τ ) = x µ + u µ jet τ , which moves with nearly the speed of light( v jet = 0 . . t = 7 . /N dN/dφdy | ( y = 0) for the different deposition mechanisms. et Propagation and Mach Cones in (3+1)d Ideal Hydrodynamics ππ /20- π /2- π d N / d p / ( d N / d p | ( φ = - π )) φ [rad] p T = 1 GeV/cp T = 2 GeV/cp T = 3 GeV/cp T = 4 GeV/c -4-2 0 2 4 -4 -2 0 2 4 y [f m ] x [fm] Figure 2.
Jet–signal strength for different p ⊥ values (left panel) and momentumdistribution (right panel) after a hydrodynamical evolution of t = 7 . dM/dx = 1 . v jet = 0 . x –axis.
3. Jet–Deposition Mechanisms
In a first scenario, we study a source term which describes pure energy deposition, i.e., dP ν /dτ = ( dE/dτ, ~
0) in Eq. (2), with dE/dx = 1 . t = 7 . p ⊥ values are selected. This is due to thefact that thermal smearing washes out the signal for a high background temperature.Therefore, using p ⊥ cuts similar to the experiment (3 ≤ p ⊥ ≤ dE/dx < any jet-quenching mechanisms consistent with energy–momentum conservation, and henceits experimental observation is not enough to show that the jet energy has been locallythermalized. To do this, one might check if the height of the peak rises exponentiallywith the associated p T .As a second scenario, we investigate a source term with pure momentum deposition,i.e., dP ν /dτ = (0 , d ~M /dτ ) in Eq. (2), with dM/dx = 1 . p ⊥ values as comparedto the first deposition scenario (cf. right panel of Fig. 1). The reason is that a diffusionwake is excited, which is indicated by the strong flow in jet direction (see right panel ofFig. 2).The third scenario which we consider is characterized by a source term thatdescribes a combined deposition of energy and momentum for a jet–energy loss of dE/dx = 1 . M jet to the totally distributed energy E jet . The cone–like shape only emerges for a smalljet–momentum loss dM/dx and – due to the small value of the jet–energy loss dE/dx – for a high p ⊥ value (see Fig. 3). For a larger jet–momentum loss, this structure isdissolved (caused by the creation of a diffusion wake) and a peak occurs in jet direction. et Propagation and Mach Cones in (3+1)d Ideal Hydrodynamics ππ /20- π /2- π / N d N / d φ d y | ( y = , p T = G e V / c ) φ [rad] dM/dx = 0.18 GeV/fmdM/dx = 0.54 GeV/fmdM/dx = 0.89 GeV/fmdM/dx = 1.25 GeV/fm Figure 3.
Azimuthal two–particle correlation, assuming a jet depositing both energyand momentum, for a jet–energy loss of dE/dx = 1 . M jet to the totally distributed energy E jet for avalue of p ⊥ = 10 GeV/c.
4. Summary
We found that the fluid response to a jet critically depends on the energy–momentum–deposition mechanism. A Mach cone–like pattern occurs in the azimuthal two–particlecorrelation only if the longitudinal jet–momentum loss is significantly less than the jet–energy loss ( dM/dx ≪ dE/dx ), since otherwise the diffusion wake kills the Mach cone–like signal. This result is consistent with Refs. [4]. Moreover, applying p ⊥ cuts similarto the experimentally used values (3 ≤ p ⊥ ≤ dE/dx < References [1] M. Gyulassy, P. Levai and I. Vitev, Nucl. Phys. B , 197 (2000), H. St¨ocker, Nucl. Phys. A ,121 (2005), F. Antinori and E. V. Shuryak, J. Phys. G , L19 (2005), J. Adams et al. [STARCollaboration], Phys. Rev. Lett. , 072304 (2003), S. S. Adler et al. [PHENIX Collaboration],Phys. Rev. C , 054903 (2006).[2] J. G. Ulery [STAR Collaboration], arXiv:0704.0224 [nucl-ex], N. N. Ajitanand [PHENIXCollaboration], Nucl. Phys. A , 519 (2007), C. A. Pruneau [STAR Collaboration], J. Phys.G (2007) S667.[3] P. F. Kolb and U. W. Heinz, arXiv:nucl-th/0305084.[4] J. Casalderrey-Solana, E. V. Shuryak and D. Teaney, Nucl. Phys. A , 577 (2006);arXiv:hep-ph/0511263; arXiv:hep-ph/060218; A. K. Chaudhuri and U. Heinz, Phys. Rev. Lett. , 062301 (2006).[5] M. Gyulassy and X. N. Wang, Nucl. Phys. B , 583 (1994). R. Baier, Y. L. Dokshitzer,A. H. M¨uller, S. Peigne and D. Schiff, Nucl. Phys. B , 291 (1997), U. A. Wiedemann,Nucl. Phys. B , 303 (2000), X. N. Wang and X. F. Guo, Nucl. Phys. A , 788 (2001),H. Liu, K. Rajagopal and U. A. Wiedemann, Phys. Rev. Lett. , 182301 (2006), A. Majumder,B. M¨uller and X. N. Wang, Phys. Rev. Lett. , 192301 (2007).[6] T. Renk and J. Ruppert, Phys. Rev. C , 014908 (2007), T. Renk, J. Ruppert, C. Nonakaand S. A. Bass, Phys. Rev. C , 031902 (2007), J. Noronha, G. Torrieri and M. Gyulassy,arXiv:0712.1053 [hep-ph].[7] D. H. Rischke, Y. P¨urs¨un, J. A. Maruhn, H. St¨ocker and W. Greiner, Heavy Ion Phys.1