- Email: [email protected]
Interplay between Mach cone and radial expansion in jet events
Available online at www.sciencedirect.com
Nuclear Physics A 956 (2016) 577–580 www.elsevier.com/locate/nuclphysa
Interplay between Mach cone and radial expansion in jet events Y. Tachibanaa,b,c,, T. Hiranoc, a Theoretical
Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan of Engineering, Nishinippon Institute of Technology, Fukuoka 800-0344, Japan c Department of Physics, Sophia University, Tokyo 102-8554, Japan
b Department
Abstract We study the hydrodynamic response to jet propagation in the expanding QGP and investigate how the particle spectra after the hydrodynamic evolution of the QGP reflect it. We perform simulations of the space-time evolution of the QGP in gamma-jet events by solving (3+1)-dimensional ideal hydrodynamic equations with source terms. Mach cone is induced by the jet energy deposition and pushes back the radial flow of the expanding background. Especially in the case when the jet passage is off-central one, the number of particles emitted in the direction of the push back decreases. This is the signal including the information about the formation of the Mach cone and the jet passage in the QGP fluid. Keywords: QGP, jet quenching, relativistic hydrodynamics, Mach cone
1. Introduction The bulk medium of quark-gluon plasma (QGP) produced in heavy ion collisions has turned out to behave like an almost perfect fluid [1, 2, 3, 4, 5]. Many observables in heavy ion collisions are well reproduced by so-called hydro models in which relativistic hydrodynamics is used to describe the space time evolution of the bulk medium. At the same time, the energetic patrons produced in initial hard scatterings lose their energies while traversing the QGP fluid due to the strong interaction (jet quenching) [6, 7, 8, 9, 10, 11, 12]. Assuming that the deposited energy and momentum are transferred to the QGP fluid, the QGP hydrodynamically responds to the jet quenching. Since the jets can be considered as supersonic moving sources, the conical shockwave, so-called Mach cone, is supposed to be induced [13, 14]. The Mach cone can affect various observables in jet events in heavy ion collisions, e.g., the enhancement of low momentum particles far away from the jet direction [15], fragmentation function and jet transverse profile[16, 17]. In this study, we focus on the Mach cone as a hydrodynamic response to the jet quenching. The simulations in the case where the jet traverse the expanding QGP and deposits its energy and momentum into the fluid is performed by using hydrodynamic model. Then we calculate the consequent spectra of particles after the hydrodynamic evolution to investigate how they reflect the formation of Mach cone.
Email addresses: [email protected] (Y. Tachibana), [email protected] (T. Hirano)
http://dx.doi.org/10.1016/j.nuclphysa.2016.01.007 0375-9474/© 2016 Elsevier B.V. All rights reserved.
578
Y. Tachibana, T. Hirano / Nuclear Physics A 956 (2016) 577–580
Fig. 1. (Colour online) Energy density distribution of the medium fluid in transverse plane ηs = 0 at τ = 8.1 fm/c. The jet is produced at (x, y) = (3.0 fm, −4.5 fm) and travels in the negative x-direction.
2. Model To describe the space-time evolution of the QGP with incoming energy and momentum from the jet, we employ relativistic ideal hydrodynamic equations with source terms; ∂μ T μν (x) = J ν (x) ,
(1)
where T μν is the energy momentum tensor of the QGP fluid, J ν is the source term. Here, we assume that the deposited energy and momentum from the jet are equilibrated instantaneously and locally in the fluid. The source term is a density of the energy and momentum deposited from the jets. In this work, we neglect the structure and the mass of the jet and use a simple form of the source terms: J μ (x) = −(d pμjet /dt) δ(3) (x − xjet (t)). We use the jet energy-loss of the form which is proportional to T 3 [18] and the jet
factor of proportionality is set to reproduce the nuclear modification factor for jets around pT ∼ 100 GeV/c √ in Pb-Pb collisions at sNN = 2.76 TeV. In this study, the medium is considered as the (3+1)-dimensional ideal fluid. The initial profile of the medium is generated by the optical Glauber model combined with the BGK model [19]. We numerically solve Eq. (1), which describes both the hydrodynamic response to the jet and the expansion of the background QGP. The interplay between them are naturally included in the calculation. 3. Simulations and Results We perform simulations of the case where one jet travels through the expanding QGP formed in a √ perfectly central Pb-Pb collision at sNN = 2.76 TeV. The jet is assumed to be produced in the transverse plane ηs = 0 at τ = 0 and travel in a straight line in the transverse plane. At τ = 0.6 fm/c, the medium starts to evolve as a fluid and the interaction between the jet and the QGP is turned on. The case corresponds to a gamma-jet event where a pair of a photon and a parton is produced at initial hard scatterings. The photon has the same initial energy as the parton and propagates freely in the opposite direction. In the following, we set the direction of the photon propagation to φ = 0 and the jet direction to φ = π without a loss of generality. Figure 1 shows the energy density distribution of the medium in the transverse plane at ηs = 0 at τ = 6.0 fm/c for a event with a jet produced at (x, y) = (3.0 fm, −4.5 fm). We can see V-shaped regions having relatively higher energy density. This is the Mach cone asymmetrically distorted by the radial flow driven by the background expansion. To see how the resulting spectra reflect the Mach cone, we calculate the azimuthal angle distribution of the particles emitted form the bulk medium by using Copper-Frye formula [20]. Here we present the increase of the charged pions with 1 < pT < 2 GeV/c compared to the case without jet propagation; ΔdNπ± /dηdφ = dNπ± /dηdφ − dNπ± /dηdφ|w/o jet . This does not include the contribution of the particles from jet fragmentation. The solid line in Fig. 2 (a) shows the azimuthal angle distribution after taking the event
Y. Tachibana, T. Hirano / Nuclear Physics A 956 (2016) 577–580
579
jet
average over all jet production points. The trigger for the jet pT is set to 80 GeV/c. In the direction of the jet propagation π = φ, a peak can be seen due to the jet-induced flow. On the photon propagation side, the distribution is almost flat. However, the result for a single event looks different. The dashed line is the azimuthal angle for a single event where the jet is produced at (x = 3 fmy = −4.5 fm). In this case, the peak is shifted by the radial flow of the back ground. Furthermore, we can see a dip in a direction almost perpendicular to the peak. The dip is caused by the wave front of Mach cone. In this event, the Mach cone propagates against the radial flow. Especially, the wave front of the Mach cone on the center side of the medium largely develops due to the higher temperature and strongly pushes back the radial flow (Fig. 1). As a result, the particle emission in the direction of the push back is suppressed. We saw that the dip appears in the events of jet with the off-central path in the medium as a consequence of the interplay between the Mach cone an the radial flow. Here we constrain the jet path by setting the trigger to extract small energy loss events [21, 22]. Figure 3 (a) shows the jet production point distributions in the case without the trigger for the photon. Figure 3 (b) shows the jet production point distributions in jet the case where the trigger for pγT is set to 110-120 GeV/c and the trigger for pT is set to 100-110 GeV/c. When the trigger for photon is introduced to extract small energy loss events, the jet path is more likely to be off-central in the medium. Shown in Fig. 2 (b) is the azimuthal angle distribution when small energy loss events is extracted. In this case, two dips can be seen at both ends of the peak. The origin of these dips is same as the dip in the single event shown by the dashed line in Fig. 2 (a). The dips appear because the contribution of the events with off-central jet path becomes more dominant. (b)
0.6
Event-averaged pjet T > 80GeV/c
0.4
Single event xjet T (τ = 0) = (3 fm,- 4.5 fm)
0.2 0 -0.2 -0.4 -0.6
−π/2
0
0.5 0.4
Δ dNπ± /dφdη|η=0
Δ dNπ± /dφdη|η=0
(a)
π/2
π
0.3
pγT 110-120 GeV/c pjet T 100-110 GeV/c
0.2 0.1 0 -0.1 -0.2
3π/2
−π/2
φ
0
π/2
π
3π/2
φ
Fig. 2. (Colour online) Azimuthal angle distributions of charged pions with 1 < pT < 2GeV/c for gamma-jet events. The distribution in the case without jet propagation is subtracted as a background. The solid line in (a) is the result averaged over the events with the jet jet transverse momentum pT > 80 GeV/c in the final state. The dashed line in (a) is the result for the single event with the jet production point at (x, y) = (3.0 fm, −4.5 fm). The solid line in (b) is the result averaged over the events with the photon transverse momentum jet γ 110 < pT < 120 GeV/c and the jet transverse momentum 100 < pT < 110 GeV/c.
y fm
pjet T > 80 GeV/c
8 6 4 2 0 -2 -4 -6 -8
-8 -6 -4 -2 0 2 4 6 8 x fm
pγT 110-120 GeV/c
(b)
No trigger for photon
pjet T 100-110 GeV/c
Event Fraction
0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0
y fm
(a)
8 6 4 2 0 -2 -4 -6 -8
-8 -6 -4 -2 0 2 4 6 8
Event Fraction
0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0
x fm
Fig. 3. (Colour online) Distributions of the jet production point in the transverse plane for gamma-jet events. The trigger is set to (a) jet jet γ pT > 80 GeV/c and (b) 100 < pT < 110 GeV/c and 110 < pT < 120 GeV/c.
Y. Tachibana, T. Hirano / Nuclear Physics A 956 (2016) 577–580
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4. Summary In this work, the Mach cone as a hydrodynamic response to the jet quenching in the QGP fluid was √ studied. We performed simulations of the gamma-jet events in central Pb-Pb collisions at sNN = 2.76 TeV by using the hydrodynamic model. In the model, the relativistic hydrodynamic equations with source terms is used to describe the space-time evolution of the bulk medium with incoming energy and momentum from the jet. The energy-momentum deposition from jet induces a Mach cone. In the expanding QGP fluid, the Mach cone is asymmetrically distorted by the radial flow in the case where the jet travels through the off-central passage in the medium. Then we showed the azimuthal-angle distribution of the particles emitted from the medium modified by the jet propagation. In the distribution after taking the average over the events triggered by the jet transverse momentum, only a peak can be seen in the direction of the jet. However, in the distribution for a single event with the off-central jet passage, the dip was seen on the jet passage side in the medium. The dip appears as a result of the interplay between the Mach cone and the radial flow. The Mach cone pushes back the radial flow and the particle emission is suppressed in the direction in which the radial flow is pushed back. Furthermore, we showed that when the trigger is introduced to extract the small energy loss events, two dips were seen in the azimuthal angle distribution after taking the event average. The origin of these dips is the same as the one in the single event with the off-central jet passage. They appear because the contribution of the events with off-central jet path becomes dominant. This can be the direct signal of the Mach cone and also includes the information about the jet path in the medium. Acknowledgments The authors would like to thank Y. Hirono, G.-Y. Qin and X.-N. Wang for helpful discussions and comments. This work was supported by JSPS KAKENHI Grant Numbers 25400269. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
U. W. Heinz and P. F. Kolb, Nucl. Phys. A 702, 269 (2002). T. D. Lee, Nucl. Phys. A 750, 1 (2005). M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30 (2005). E. V. Shuryak, Nucl. Phys. A 750, 64 (2005). T. Hirano and M. Gyulassy, Nucl. Phys. A 769, 71 (2006). J. D. Bjorken, Fermilab, Report No. FERMILAB-PUB-82-059-THY 1982 (unpublished). D. A. Appel, Phys. Rev. D 33, 717 (1986). J. P. Blaizot and L. D. McLerran, Phys. Rev. D 34, 2739 (1986). M. Rammerstorfer and U. W. Heinz, Phys. Rev. D 41, 306 (1990). M. Gyulassy and M. Plumer, Phys. Lett. B 243, 432 (1990). M. H. Thoma and M. Gyulassy, Nucl. Phys. B 351, 491 (1991). M. Gyulassy and X. -n. Wang, Nucl. Phys. B 420, 583 (1994). H. Stoecker, Nucl. Phys. A 750, 121 (2005). J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney, J. Phys. Conf. Ser. 27, 22 (2005) [Nucl. Phys. A 774, 577 (2006)]. Y. Tachibana and T. Hirano, Phys. Rev. C 90, no. 2, 021902 (2014). X. N. Wang and Y. Zhu, Phys. Rev. Lett. 111, no. 6, 062301 (2013) . Y. He, T. Luo, X. N. Wang and Y. Zhu, Phys. Rev. C 91, 054908 (2015) . B. Betz, J. Noronha, G. Torrieri, M. Gyulassy and D. H. Rischke, Phys. Rev. Lett. 105, 222301 (2010) . T. Hirano and Y. Nara, PTEP 2012, 01A203 (2012) . F. Cooper and G. Frye, Phys. Rev. D 10, 186 (1974). T. Renk, Phys. Rev. C 74, 024903 (2006) doi:10.1103/PhysRevC.74.024903 [hep-ph/0602045]. G. Y. Qin, Eur. Phys. J. C 74, 2959 (2014) doi:10.1140/epjc/s10052-014-2959-3 [arXiv:1210.6610 [hep-ph]].
Nuclear Physics A 956 (2016) 577–580 www.elsevier.com/locate/nuclphysa
Interplay between Mach cone and radial expansion in jet events Y. Tachibanaa,b,c,, T. Hiranoc, a Theoretical
Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan of Engineering, Nishinippon Institute of Technology, Fukuoka 800-0344, Japan c Department of Physics, Sophia University, Tokyo 102-8554, Japan
b Department
Abstract We study the hydrodynamic response to jet propagation in the expanding QGP and investigate how the particle spectra after the hydrodynamic evolution of the QGP reflect it. We perform simulations of the space-time evolution of the QGP in gamma-jet events by solving (3+1)-dimensional ideal hydrodynamic equations with source terms. Mach cone is induced by the jet energy deposition and pushes back the radial flow of the expanding background. Especially in the case when the jet passage is off-central one, the number of particles emitted in the direction of the push back decreases. This is the signal including the information about the formation of the Mach cone and the jet passage in the QGP fluid. Keywords: QGP, jet quenching, relativistic hydrodynamics, Mach cone
1. Introduction The bulk medium of quark-gluon plasma (QGP) produced in heavy ion collisions has turned out to behave like an almost perfect fluid [1, 2, 3, 4, 5]. Many observables in heavy ion collisions are well reproduced by so-called hydro models in which relativistic hydrodynamics is used to describe the space time evolution of the bulk medium. At the same time, the energetic patrons produced in initial hard scatterings lose their energies while traversing the QGP fluid due to the strong interaction (jet quenching) [6, 7, 8, 9, 10, 11, 12]. Assuming that the deposited energy and momentum are transferred to the QGP fluid, the QGP hydrodynamically responds to the jet quenching. Since the jets can be considered as supersonic moving sources, the conical shockwave, so-called Mach cone, is supposed to be induced [13, 14]. The Mach cone can affect various observables in jet events in heavy ion collisions, e.g., the enhancement of low momentum particles far away from the jet direction [15], fragmentation function and jet transverse profile[16, 17]. In this study, we focus on the Mach cone as a hydrodynamic response to the jet quenching. The simulations in the case where the jet traverse the expanding QGP and deposits its energy and momentum into the fluid is performed by using hydrodynamic model. Then we calculate the consequent spectra of particles after the hydrodynamic evolution to investigate how they reflect the formation of Mach cone.
Email addresses: [email protected] (Y. Tachibana), [email protected] (T. Hirano)
http://dx.doi.org/10.1016/j.nuclphysa.2016.01.007 0375-9474/© 2016 Elsevier B.V. All rights reserved.
578
Y. Tachibana, T. Hirano / Nuclear Physics A 956 (2016) 577–580
Fig. 1. (Colour online) Energy density distribution of the medium fluid in transverse plane ηs = 0 at τ = 8.1 fm/c. The jet is produced at (x, y) = (3.0 fm, −4.5 fm) and travels in the negative x-direction.
2. Model To describe the space-time evolution of the QGP with incoming energy and momentum from the jet, we employ relativistic ideal hydrodynamic equations with source terms; ∂μ T μν (x) = J ν (x) ,
(1)
where T μν is the energy momentum tensor of the QGP fluid, J ν is the source term. Here, we assume that the deposited energy and momentum from the jet are equilibrated instantaneously and locally in the fluid. The source term is a density of the energy and momentum deposited from the jets. In this work, we neglect the structure and the mass of the jet and use a simple form of the source terms: J μ (x) = −(d pμjet /dt) δ(3) (x − xjet (t)). We use the jet energy-loss of the form which is proportional to T 3 [18] and the jet
factor of proportionality is set to reproduce the nuclear modification factor for jets around pT ∼ 100 GeV/c √ in Pb-Pb collisions at sNN = 2.76 TeV. In this study, the medium is considered as the (3+1)-dimensional ideal fluid. The initial profile of the medium is generated by the optical Glauber model combined with the BGK model [19]. We numerically solve Eq. (1), which describes both the hydrodynamic response to the jet and the expansion of the background QGP. The interplay between them are naturally included in the calculation. 3. Simulations and Results We perform simulations of the case where one jet travels through the expanding QGP formed in a √ perfectly central Pb-Pb collision at sNN = 2.76 TeV. The jet is assumed to be produced in the transverse plane ηs = 0 at τ = 0 and travel in a straight line in the transverse plane. At τ = 0.6 fm/c, the medium starts to evolve as a fluid and the interaction between the jet and the QGP is turned on. The case corresponds to a gamma-jet event where a pair of a photon and a parton is produced at initial hard scatterings. The photon has the same initial energy as the parton and propagates freely in the opposite direction. In the following, we set the direction of the photon propagation to φ = 0 and the jet direction to φ = π without a loss of generality. Figure 1 shows the energy density distribution of the medium in the transverse plane at ηs = 0 at τ = 6.0 fm/c for a event with a jet produced at (x, y) = (3.0 fm, −4.5 fm). We can see V-shaped regions having relatively higher energy density. This is the Mach cone asymmetrically distorted by the radial flow driven by the background expansion. To see how the resulting spectra reflect the Mach cone, we calculate the azimuthal angle distribution of the particles emitted form the bulk medium by using Copper-Frye formula [20]. Here we present the increase of the charged pions with 1 < pT < 2 GeV/c compared to the case without jet propagation; ΔdNπ± /dηdφ = dNπ± /dηdφ − dNπ± /dηdφ|w/o jet . This does not include the contribution of the particles from jet fragmentation. The solid line in Fig. 2 (a) shows the azimuthal angle distribution after taking the event
Y. Tachibana, T. Hirano / Nuclear Physics A 956 (2016) 577–580
579
jet
average over all jet production points. The trigger for the jet pT is set to 80 GeV/c. In the direction of the jet propagation π = φ, a peak can be seen due to the jet-induced flow. On the photon propagation side, the distribution is almost flat. However, the result for a single event looks different. The dashed line is the azimuthal angle for a single event where the jet is produced at (x = 3 fmy = −4.5 fm). In this case, the peak is shifted by the radial flow of the back ground. Furthermore, we can see a dip in a direction almost perpendicular to the peak. The dip is caused by the wave front of Mach cone. In this event, the Mach cone propagates against the radial flow. Especially, the wave front of the Mach cone on the center side of the medium largely develops due to the higher temperature and strongly pushes back the radial flow (Fig. 1). As a result, the particle emission in the direction of the push back is suppressed. We saw that the dip appears in the events of jet with the off-central path in the medium as a consequence of the interplay between the Mach cone an the radial flow. Here we constrain the jet path by setting the trigger to extract small energy loss events [21, 22]. Figure 3 (a) shows the jet production point distributions in the case without the trigger for the photon. Figure 3 (b) shows the jet production point distributions in jet the case where the trigger for pγT is set to 110-120 GeV/c and the trigger for pT is set to 100-110 GeV/c. When the trigger for photon is introduced to extract small energy loss events, the jet path is more likely to be off-central in the medium. Shown in Fig. 2 (b) is the azimuthal angle distribution when small energy loss events is extracted. In this case, two dips can be seen at both ends of the peak. The origin of these dips is same as the dip in the single event shown by the dashed line in Fig. 2 (a). The dips appear because the contribution of the events with off-central jet path becomes more dominant. (b)
0.6
Event-averaged pjet T > 80GeV/c
0.4
Single event xjet T (τ = 0) = (3 fm,- 4.5 fm)
0.2 0 -0.2 -0.4 -0.6
−π/2
0
0.5 0.4
Δ dNπ± /dφdη|η=0
Δ dNπ± /dφdη|η=0
(a)
π/2
π
0.3
pγT 110-120 GeV/c pjet T 100-110 GeV/c
0.2 0.1 0 -0.1 -0.2
3π/2
−π/2
φ
0
π/2
π
3π/2
φ
Fig. 2. (Colour online) Azimuthal angle distributions of charged pions with 1 < pT < 2GeV/c for gamma-jet events. The distribution in the case without jet propagation is subtracted as a background. The solid line in (a) is the result averaged over the events with the jet jet transverse momentum pT > 80 GeV/c in the final state. The dashed line in (a) is the result for the single event with the jet production point at (x, y) = (3.0 fm, −4.5 fm). The solid line in (b) is the result averaged over the events with the photon transverse momentum jet γ 110 < pT < 120 GeV/c and the jet transverse momentum 100 < pT < 110 GeV/c.
y fm
pjet T > 80 GeV/c
8 6 4 2 0 -2 -4 -6 -8
-8 -6 -4 -2 0 2 4 6 8 x fm
pγT 110-120 GeV/c
(b)
No trigger for photon
pjet T 100-110 GeV/c
Event Fraction
0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0
y fm
(a)
8 6 4 2 0 -2 -4 -6 -8
-8 -6 -4 -2 0 2 4 6 8
Event Fraction
0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0
x fm
Fig. 3. (Colour online) Distributions of the jet production point in the transverse plane for gamma-jet events. The trigger is set to (a) jet jet γ pT > 80 GeV/c and (b) 100 < pT < 110 GeV/c and 110 < pT < 120 GeV/c.
Y. Tachibana, T. Hirano / Nuclear Physics A 956 (2016) 577–580
580
4. Summary In this work, the Mach cone as a hydrodynamic response to the jet quenching in the QGP fluid was √ studied. We performed simulations of the gamma-jet events in central Pb-Pb collisions at sNN = 2.76 TeV by using the hydrodynamic model. In the model, the relativistic hydrodynamic equations with source terms is used to describe the space-time evolution of the bulk medium with incoming energy and momentum from the jet. The energy-momentum deposition from jet induces a Mach cone. In the expanding QGP fluid, the Mach cone is asymmetrically distorted by the radial flow in the case where the jet travels through the off-central passage in the medium. Then we showed the azimuthal-angle distribution of the particles emitted from the medium modified by the jet propagation. In the distribution after taking the average over the events triggered by the jet transverse momentum, only a peak can be seen in the direction of the jet. However, in the distribution for a single event with the off-central jet passage, the dip was seen on the jet passage side in the medium. The dip appears as a result of the interplay between the Mach cone and the radial flow. The Mach cone pushes back the radial flow and the particle emission is suppressed in the direction in which the radial flow is pushed back. Furthermore, we showed that when the trigger is introduced to extract the small energy loss events, two dips were seen in the azimuthal angle distribution after taking the event average. The origin of these dips is the same as the one in the single event with the off-central jet passage. They appear because the contribution of the events with off-central jet path becomes dominant. This can be the direct signal of the Mach cone and also includes the information about the jet path in the medium. Acknowledgments The authors would like to thank Y. Hirono, G.-Y. Qin and X.-N. Wang for helpful discussions and comments. This work was supported by JSPS KAKENHI Grant Numbers 25400269. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
U. W. Heinz and P. F. Kolb, Nucl. Phys. A 702, 269 (2002). T. D. Lee, Nucl. Phys. A 750, 1 (2005). M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30 (2005). E. V. Shuryak, Nucl. Phys. A 750, 64 (2005). T. Hirano and M. Gyulassy, Nucl. Phys. A 769, 71 (2006). J. D. Bjorken, Fermilab, Report No. FERMILAB-PUB-82-059-THY 1982 (unpublished). D. A. Appel, Phys. Rev. D 33, 717 (1986). J. P. Blaizot and L. D. McLerran, Phys. Rev. D 34, 2739 (1986). M. Rammerstorfer and U. W. Heinz, Phys. Rev. D 41, 306 (1990). M. Gyulassy and M. Plumer, Phys. Lett. B 243, 432 (1990). M. H. Thoma and M. Gyulassy, Nucl. Phys. B 351, 491 (1991). M. Gyulassy and X. -n. Wang, Nucl. Phys. B 420, 583 (1994). H. Stoecker, Nucl. Phys. A 750, 121 (2005). J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney, J. Phys. Conf. Ser. 27, 22 (2005) [Nucl. Phys. A 774, 577 (2006)]. Y. Tachibana and T. Hirano, Phys. Rev. C 90, no. 2, 021902 (2014). X. N. Wang and Y. Zhu, Phys. Rev. Lett. 111, no. 6, 062301 (2013) . Y. He, T. Luo, X. N. Wang and Y. Zhu, Phys. Rev. C 91, 054908 (2015) . B. Betz, J. Noronha, G. Torrieri, M. Gyulassy and D. H. Rischke, Phys. Rev. Lett. 105, 222301 (2010) . T. Hirano and Y. Nara, PTEP 2012, 01A203 (2012) . F. Cooper and G. Frye, Phys. Rev. D 10, 186 (1974). T. Renk, Phys. Rev. C 74, 024903 (2006) doi:10.1103/PhysRevC.74.024903 [hep-ph/0602045]. G. Y. Qin, Eur. Phys. J. C 74, 2959 (2014) doi:10.1140/epjc/s10052-014-2959-3 [arXiv:1210.6610 [hep-ph]].