Conical Correlations, Bragg Peaks, and Transverse Flow Deflections in Jet Tomography
Barbara Betz, Jorge Noronha, Giorgio Torrieri, Miklos Gyulassy, Dirk H. Rischke
aa r X i v : . [ nu c l - t h ] S e p Conical Correlations, Bragg Peaks, and Transverse FlowDeflections in Jet Tomography
Barbara Betz a , Jorge Noronha b , Giorgio Torrieri a , c , Miklos Gyulassy b , Dirk H. Rischke a , c a Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at, Frankfurt am Main, Germany b Department of Physics, Columbia University, New York, 10027, USA c Frankfurt Institute for Advanced Studies (FIAS), Frankfurt am Main, Germany
Abstract
We use (3 + ff erent jet energy lossscenarios for a jet propagating through an opaque medium. The conical correlations obtainedfor fully stopped jets, revealing a Bragg peak, are discussed as well as results from pQCD andAdS / CFT. Moreover, we investigate transverse flow deflection. It is demonstrated that a double-peaked away-side structure can be formed due to the di ff erent contributions of several possiblejet trajectories through an expanding medium.
1. Introduction
The observation of a double-peaked structure in azimuthal di-hadron correlations [1] arose alot of recent interest, since it was suggested [2] that this structure could be related to the emissionangles of Mach cones that are via Mach’s law (cos φ M = c s / v jet ) directly related to the Equationof State (EoS). In general, energetic back-to-back jets produced in the early stages of a heavy-ion collision propagate through the medium, depositing energy and momentum along their path.Certainly, the properties of this deposition depends on the physics of the jet-medium interactions.Recently, di ff erent mechanisms of jet energy loss were analyzed, ranging from weak [3] to strongcoupling [4, 5]. While the static background o ff ers the possibility to compare to results obtainedfrom AdS / CFT, the expansion of a system formed in a heavy-ion collision will certainly influenceany kind of jet deposition scenario [6]. We demonstrate that taking into account various possibletrajectories of jets propagating through the plasma [7] lead to a double-peaked structure on theaway-side of the final particle correlations. Here, the medium is investigated using Glauberinitial conditions, corresponding to a gold nucleus with r = . T max =
200 MeV. We focus on radial flow only and consider most central collisions, thusneglecting any elliptic flow contribution. For simplicity, the medium is always considered as anideal gas of massless SU(3) gluons. In our system of coordinates, the beam axis is pointing intothe z direction and the associated jet moves along the x direction.
2. A Hydrodynamical Prescription of Jets
Assuming that the energy lost by the jet thermalizes quickly [8], we solve the ideal hydrody-namical equations using a (3 + ∂ µ T µν = S ν . (1) Preprint submitted to Nuclear Physics A October 24, 2018 = 200 MeV, v =0.999 (a)-5 -4 -3 -2 -1 0 1x [fm]-4-3-2-1 0 1 2 3 4 y [f m ]
190 200 210 220 230 240 T [ M e V ] (b)-5 -4 -3 -2 -1 0 1x [fm]-4-3-2-1 0 1 2 3 4 y [f m ]
190 200 210 220 230 240 T [ M e V ] Figure 1: Temperature pattern and flow velocity profile (arrows) after a hydrodynamical evolution of t = . ∆ x = . v jet = . The source term for the decelerating jet is given by S ν = τ f Z τ i d τ dM ν d τ δ (4) h x µ − x µ jet ( τ ) i , (2)where τ f − τ i denotes the proper time interval associated with the jet evolution and dM ν / d τ = ( dE / d τ, d ~ M / d τ ) is the energy and momentum loss along the trajectory of the jet x µ jet ( τ ) = x µ + u µ jet τ . We assume that dE ( t ) / dt = a / v jet ( t ), according to the Bethe–Bloch formalism [10], leadingto a Bragg peak as demonstrated in Ref. [11]. For a jet starting at v jet = . ∆ x = . a ≃ − . / fm [11]. Fig. 1 displays the temperature and flow velocity profiles of ajet with an energy loss as determined above and vanishing momentum deposition (left panel)as well as an energy and momentum deposition (right panel). In the latter case, the creationof a di ff usion wake behind the jet is clearly visible, which leads to an away-side peak in theassociated jet direction [11] after performing a Cooper–Frye (CF) [12] freeze-out. Consideringvanishing momentum deposition, the away-side peak is replaced by a conical (double azimuthalpeak) distribution at the expected Mach cone angle [11].The away-side di ff usion peak in the particle correlation also prevails when considering theNeufeld pQCD source term [3, 13], because it involves a large momentum deposition (see Fig.2). The freeze-out results of the AdS / CFT solution, however, show a double-peaked structure inspite of the di ff usion plume due to a novel nonequilibrium strong coupling e ff ect in the “Neck”region as shown in Ref. [5].For the expanding medium, we choose the following ansatz for the energy and momentumdepostition of the jet, scaling with the temperature of the dynamical background S ν = τ f Z τ i d τ dM ν d τ (cid:12)(cid:12)(cid:12)(cid:12) " T ( t , ~ x ) T max δ (4) h x µ − x µ jet ( τ ) i , (3)where dE / dt = / fm and dM / dt = / vdE / dt . Since deceleration does not alter thefreeze-out results significantly [11], we do not include this e ff ect in the present study for the2 π π /2 ππ /20 C F ( φ ) φ [rad]pQCD v=0.580.750.90Neck -0.4-0.2 0 0.2 0.4 0.6 0.8 1 2 π π /2 ππ /20 C F ( φ ) φ [rad]AdS/CFT Figure 2: Normalized (and background subtracted) azimuthal away-side jet associated correlation after Cooper-Fryefreeze-out CF ( φ ) for pQCD [13] (left) and AdS / CFT from [5] (right). Here CF ( φ ) is evaluated at p T = π T ∼ . y = ff erent jet velocities of v = . , . , .
9. The line with triangles represents the Neck contribution(which is a region close to the head of the jet) for a jet with v = . expanding medium. Below, we consider a 5 GeV trigger parton which corresponds to trigger- p T of p trig T = . ∼
70% of its energy.Experiments can only trigger on the jet direction, thus one has to consider di ff erent startingpoints for the jet which is done according to x = r cos φ, y = r sin φ , where r = φ =
90, 120 , , , , ,
270 degrees. To modelthe experimental situation, the CF freeze-out results are convoluted by a Gaussian representingthe near-side jet, leading to a background subtracted, normalized, and jet-averaged CF signal for b = h CF ( φ ) i = R π h N back ( φ ) i d φ " d h N con i ( φ ) p T d p T dyd φ − d h N back i ( φ ) p T d p T dyd φ . (4)This CF signal (see solid lines in the upper panels of Fig. 3) displays a broad away-side peak for p assoc T = p assoc T = ff erent paths (for φ = ...
180 degrees see lower panels of Fig. 3) add upto two peaks in the left and in the right part of the away-side (dashed lines in the upper panel ofFig. 3).It is important to notice that the main contributions to the peaks in the left and right part ofthe away-side come from non-central jets (see lower panel of Fig. 3).Thus, we have shown, using a full (3 + ff erent contributions of severalpossible jet trajectories through an expanding medium [3, 7]. Therefore, it seems natural toconclude that this shape, interpreted as a conical signal, does not result from a “true” Mach cone,but is actually generated by the averaging of distorted wakes. Clearly, the emission angle of sucha structure is not connected to the EoS. However, these results do not imply that Mach cones arenot formed in heavy-ion collision. The e ff ects of longitudinal expansion, finite impact parameter,and di ff erent freeze-out prescritiptions (like coalescence [14]) remain to be considered.3 π /2 0 π /2 π π /2 C F 〈 ba ck 〉 ( φ ) φ [rad]Jet 180Jet 150Jet 120Jet (cid:176)(cid:176)90-0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 〈 C F ( φ ) 〉 p Tassoc =1.0 GeVtotalleft partright part -0.001 0 0.001 0.002 0.003 0.004 0.005- π /2 0 π /2 π π /2 C F 〈 ba ck 〉 ( φ ) φ [rad]Jet 180Jet 150Jet 120Jet (cid:176)(cid:176)90 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 〈 C F ( φ ) 〉 p Tassoc =2.0 GeVtotalleft partright part
Figure 3: The normalized, background-subtracted, and path-averaged azimuthal two-particle correlation after performingan isochronous CF freeze-out (solid lines in the upper panels) for 5 GeV jets depositing energy and momentum for a p assoc T = p assoc T = ff erent jet paths. The lower panel displays the contribution from the di ff erent jettrajectories with φ = ...
180 degrees.
Acknowledgments
The work of B.B. is supported by BMBF and by the Helmholtz Research School H-QM. J.N.and M.G. acknowledge support from DOE under Grant No. DE-FG02-93ER40764. G.T. was(financially) supported by the Helmholtz International Center for FAIR within the framework ofthe LOEWE program (Landeso ff ensive zur Entwicklung Wissenschaftlich- ¨Okonomischer Exzel-lenz) launched by the State of Hesse. References [1] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. , 152301 (2005); S. S. Adler et al. [PHENIX Collabora-tion], Phys. Rev. Lett. , 052301 (2006).[2] H. Stoecker, Nucl. Phys. A , 121 (2005); J. Casalderrey-Solana, E. V. Shuryak and D. Teaney, J. Phys. Conf.Ser. , 22 (2005) [Nucl. Phys. A , 577 (2006)].[3] R. B. Neufeld, B. Muller and J. Ruppert, Phys. Rev. C , 041901 (2008); R. B. Neufeld, Phys. Rev. D , 085015(2008).[4] P. M. Chesler and L. G. Ya ff e, Phys. Rev. Lett. , 152001 (2007); S. S. Gubser, S. S. Pufu and A. Yarom, Phys.Rev. Lett. , 012301 (2008).[5] J. Noronha, M. Gyulassy and G. Torrieri, Phys. Rev. Lett. , 102301 (2009).[6] L. M. Satarov, H. Stoecker and I. N. Mishustin, Phys. Lett. B , 64 (2005).[7] A. K. Chaudhuri, Phys. Rev. C , 057902 (2007); A. K. Chaudhuri, Phys. Rev. C , 027901 (2008).[8] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. , 152301 (2005).[9] D. H. Rischke, Y. Pursun, J. A. Maruhn, H. Stoecker and W. Greiner, Heavy Ion Phys. , 309 (1995).[10] W. H. Bragg and R. Kleemann, Philos. Mag. , 318 (1905).[11] B. Betz, J. Noronha, G. Torrieri, M. Gyulassy, I. Mishustin and D. H. Rischke, Phys. Rev. C , 034902 (2009).[12] F. Cooper and G. Frye, Phys. Rev. D , 186 (1974).[13] B. Betz, M. Gyulassy, J. Noronha and G. Torrieri, Phys. Lett. B , 340 (2009).[14] G. Torrieri, V. Greco, J. Noronha, M. Gyulassy, this proccedings., 340 (2009).[14] G. Torrieri, V. Greco, J. Noronha, M. Gyulassy, this proccedings.