Indications of Conical Emission of Charged Hadrons at RHIC
By Jason G. Ulery and Fuqiang Wang
When an object moves in a medium at supersonic speed, its interaction with the medium may generate sonic shock waves. The sound wave front forms a cone, called a Mach cone, with an opening angle of π/2-θ with respect the object’s direction of motion, where θ; is called the Mach cone angle. This phenomenon is familiar, whose most common example is a supersonic airplane producing condensation vapor cloud of a Mach cone shape (see Fig. 1).
Now at Brookhaven’s Relativistic Heavy-Ion Collider (RHIC), experimental evidence of conical emission of subatomic particles is observed in high energy nucleus-nucleus collisions, which can presently only be explained by Mach cone shock waves. The STAR collaboration has submitted a paper on their finding to Physical Review Letters .
Earlier experimental measurements suggest a dense and hot medium being produced in heavy-ion collisions at RHIC that exhibits remarkable properties resembling those of a perfect fluid . Energetic partons that are produced early in these collisions and would emerge as collimated jets of particles interact strongly with the medium resulting in jet-quenching . Because the partons move at the speed of light (c), their interactions with the medium can generate supersonic Mach cone shock waves. The shock waves produce a collective emission of particles perpendicular to the wave front. From the measured Mach cone angle in RHIC experiment, one may obtain the speed of sound of the created medium. The speed of sound (cs) is a fundamental property of a medium and means to measure it in nucleus-nucleus collisions has long been sought for. The evidence of conical emission of subatomic particles at RHIC is a big step forward in the quest for the Quark-Gluon Plasma and measurements of its properties.
Generation of sonic shock waves in medium is a natural consequence of interactions of supersonic moving objects with the medium. Formation of Mach cone shock waves in nucleus-nucleus collisions has long been predicted by Greiner et al. in the 70s . Since, experimental evidences of Mach cone shock waves have been sought for but no conclusive evidence has been found. Experiments at RHIC set out to measure azimuthal distribution of energetic particles in coincidence with high transverse momentum (pT) trigger particles. Such correlation is a sensitive probe to jet quenching because high pT particles are thought to mostly come from jets, which are produced back-to-back in azimuthal direction by collisions of partons (quarks and gluons) inside the nucleons. High pT particle coincidence was found to have nearly disappeared on the away-side of the trigger particle, as generally expected from jet-quenching . The energy is transferred into production of low pT particles, and those particles were indeed observed . These particles were found to be enhanced as expected from energy conservation. However, they were observed to be broadly distributed, much broader than measurements in p+p and d+Au collisions [6,7]. For selective kinematic regions, their azimuthal distribution was observed to be even double-peaked about π from the trigger particle’s direction . These observations prompTed renewed interest of Mach cone shock waves. Many authors [8,9] have conjectured that the observed broad distributions were due to conical emission of correlated particles – particles are emitted on a cone around the direction of the away-side partner jet, averaged at π. Hydrodynamic calculations with jet-quenching as a source term support the conjecture . Interestingly, several authors predicted Mach cone shock wave generation by energy loss of heavy quarks using a completely different framework, the String Theory AdS/CFT .
Besides Mach shock waves, Čerenkov gluon radiation by an energetic parton traversing the created medium can also generate conical emission of particles, analogous to Čerenkov photons . This is because the parton’s speed can be larger than the speed of light in the medium, c/n. The index of refraction of the medium, n, decreases with the energy of the emitted gluons. As a result the conical emission angle of the Čerenkov gluons decreases with increasing gluon energy, or the energy of the measured hadrons in the final state .
Dihadron azimuthal correlations, however, cannot prove conical emission because other mechanisms can also generate broad or double-peak structure in dihadron correlations [13,14,15]. For example, jets may be deflected by transverse radial flow , or jet particles can come out more easily when they are directed out-words than inwards, resulting in a larger survival probability of those side pointed particles . Because jets are side-directed in each event, and the side-direction varies from event to event, the net effect after summing over many events would be a broad or double-peaked distribution of correlated particles. On the other hand, Mach cone shock waves or Čerenkov gluon radiation yields conical emission in a single event; correlated particles are emitted along a well defined azimuthal angle at both sides of π. In order to tell which mechanism is responsible for the observed dihadron correlation structure, one has to use an additional correlated particle. If the second particle is always relatively close to the first particle in each event, then it is due to deflected jets. If the second particle, on the other hand, is sometimes on the opposite side of π from the first particle, then it is the evidence of conical emission, and Mach cone shock waves or Čerenkov gluon radiation would be the underlying mechanism. The pT dependence of the emission angle can further distinguish between Mach cone shock wave and Čerenkov gluon radiation .
Several approaches can be taken to measure three-particle correlations using a pair of associated particles correlated with a high pT trigger particle. One approach is the three-particle cumulant , and the other is the background subtraction method . The STAR collaboration at RHIC has used both methods. The trigger and associated particle pT ranges are 3-4 GeV/c and 1-2 GeV/c, respectively. Both trigger and associated particles are restricted to the pseudo-rapidity range of -1 to 1 determined by the detector accepTance. This news article focuses on the background subtraction method and remarks briefly on its comparison to the three-particle cumulant result.
In the background subtraction method, the event with a high pT trigger particle (triggered event) is assumed to be composed of two components: one component is made of particles that are not correlated with the trigger particle excepT the indirect flow correlation with the reaction plane, and the other component is made of those particles that are correlated with the trigger particle beyond the indirect flow correlation . Figure 2(a) shows these two components, where the red points include all associated particles, and the solid histogram is the distribution of the background particles. The background distribution is obtained by mixing a trigger particle with an associated particle from another, inclusive event of the same centrality bin. The modulation in the background is due to the flow correlation. The mixed-event background is scaled by a factor a. The correlated component is show in Fig. 2(b) in red. The separation between the two components (i.e. the determination of the factor a) is done by requiring the final three-particle correlation result to have zero yield averaged over 10% lowest data points of total 24×24 data points. The systematic uncertainty in the separation is indicated by the two dashed histograms (in both panels), one of which is from the requirement that the correlated particle distribution is positive definite, and the other is obtained from the requirement that the single lowest data point in the final three-particle correlation is zero. The systematic uncertainty in the flow subtraction is indicated in the blue histograms.
In the three-particle correlation analysis, pairs of associated particles are studied with their azimuthal angles relative to that of the trigger particle, ΔΦ1 and ΔΦ2. The distribution of these pairs is shown in Fig. 1(c). These pairs are composed of three sets . One set is made of pairs where both particles are correlated with the trigger particle beyond the flow correlation. This is the three-particle correlation signal that is of interest. The other two sets are backgrounds. One is made of pairs where both particles come from the flow modulated background (see Fig. 1a). This set is obtained by mixing the trigger particle with an inclusive event of the same centrality bin as the triggered event. A scaling factor, a2b, is applied where a takes care of the associated multiplicity difference between the inclusive events and the underlying background of triggered events, and b ≈ 1 takes into account the non-Poisson statistics . This background is shown in Fig. 2(d), where the flow correlation due to the trigger particle anisotropy is not included but is added in Fig. 2(e). The other background is made of pairs where one particle is correlated with the trigger particle (beyond the flow correlation) and the other is from the flow modulated background. This is obtained by the product of the two-particle correlation function and the flow-modulated background. This background (together with the flow correlation) is shown in Fig. 2(e). The construction of this background assumes that the two-particle correlation and its background are not correlated. However, data show that both vary with the reaction plane [18.19] and thus are indirectly correlated. This effect is taken into account by using dihadron correlation measurements with respect to the reaction plane [18,19].
The final three-particle correlation results are shown in Fig. 3 for d+Au (left panel) and 12% central Au+Au (right panel) collisions, respectively. The d+Au data exhibit a di-jet structure; no other prominent features are observed. In central Au+Au collisions, however, two distinct peaks are observed along the off-diagonal direction. These peaks provide experimental evidence for conical emission of correlated charged hadrons on the away-side of the trigger particle. For more descripTion of the three-particle correlation results the reader is referred to Ref. .
Figure 4 show in red the off-diagonal projections of the three-particle correlation signals for d+Au and central Au+Au collisions. Two distinct side-peaks are seen for the central Au+Au data. A fit to the combination of a Gaussian at zero and two symmetric side-Gaussians, a side-Gaussian peak position of 1.38 ± 0.02 (stat.) ± 0.06 (syst.) is obtained. This is the conical emission angle. Also shown in the black histogram in the right panel of Fig. 4 is the off-diagonal projection of the central Au+Au data but with a = b = 1. Two side peaks are also observable. The conical emission angle is further found to be independent of the emitted particle pT. This is inconsistent with Čerenkov gluon radiation . The observed conical emission of correlated hadrons in the data is, therefore, likely due to Mach cone shock waves [8,9].
The three-particle correlation analysis is an extremely difficult analysis; the signal to background ratio is of the order 1/1000 in central Au+Au collisions. The result is critically important in the quest to understand the strongly interacting medium created at RHIC. Extraordinary claim requires extraordinary scrutiny, as the STAR collaboration has done. Four major questions come to mind:
- The first is whether other analysis approaches reach the same conclusion. STAR has undertaken a different approach with three-particle cumulant analysis. A cumulant is usually analyzed using event-average quantities. This is done in the STAR cumulant analysis  where the single particle multiplicity is averaged over the azimuthal angle in the laboratory frame which is uniform excepT detector effects. This cumulant presently does not show a conical emission signal. However, this cumulant is complicated by flow correlations and cross terms involving with anisotropic flow and jet-correlation . On the other hand, the single particle multiplicity distribution in the azimuthal angle relative to the reaction plane has a modulation due to anisotropic flow. If one could rotate all the events to align up their reaction planes and then analyze the three-particle cumulant, then one would get another cumulant result. This cumulant should be equivalent to the result shown in the right panel of Fig. 4 with a = b = 1 (black histogram). The two cumulants would be identical if the events were Poisson and there were no correlations between anisotropic flow and jet-correlation. Unfortunately, none of those is fulfilled as aforementioned, resulting in two different cumulants. However, the events, before and after the reaction plane rotation, are clearly identical. In the future, we expect that a detailed understanding of the differences between the two cumulants will teach us even more about the dense matter that we are studying at RHIC.
- The second is whether the conical emission signal could be generated by an incorrect subtraction of the background shown in Fig. 2(d), because this background shows structures along the diagonal caused by correlations (including flow and others) between the two associated particles. This is impossible because the three-particle correlation strength shown in Fig. 3 (right panel) would require a 10% inaccuracy in the scaling factor a2b for this background, which would give a shift in the three-particle correlation pedestal as large as 50.
- The third is whether the conical emission signal is caused by an incorrect subtraction of flow correlation, which affects the background shown in Fig. 2(e) in a non-trivial way. This is unlikely because the conical emission signal persists by using the anisotropic flow measurements from the reaction plane method  and the four-particle cumulant method  as well as the 2D decomposition method . These flow measurements are believed to be the extreme values to be used in jet-correlation background subtraction. These extreme values are included in the systematic uncertainties shown in Fig. 4.
- The fourth is whether the conical emission signal is an artifact of the two-component approach. This is unlikely because any triggered event can be decomposed into two components, one is correlated (beyond flow correlation) with the trigger particle and the other is not. The real question is how well to separate the two components, and this is done by three-particle zero yield at minimum with conservative systematic uncertainties as shown in Fig. 2(a,b).
In ideal case, the measured conical emission angle is related to the speed of sound via cos(θ;) = cs/c. In relativistic heavy-ion collisions, however, the relationship is complicated by collision dynamics. Renk et al. found that the Mach cone angle is distorted by the hydrodynamic flow in the medium, the size of which depends on the relative orientation between the Mach cone angle and the flow direction . The measured Mach cone angle is a net effect over the collision evolution. In order to extract the speed of sound and the equation of state of the medium from the measured conical emission signal, further studies are needed. However, the measured conical emission signal is a major step forward, providing an in-principle means to measure the speed of sound of the medium.
 B.I. Abelev et al. (STAR Collaboration), arXiv:0805.0622.
 I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys. A757, 1 (2005); B.B. Back et al. (PHOBOS Collaboration), Nucl. Phys. A757, 28 (2005); J. Adams et al. (STAR Collaboration), Nucl. Phys. A757, 102 (2005); K. Adcox et al. (PHENIX Collaboration), Nucl. Phys. A757, 184 (2005).
 M. Gyulassy and M. Plümer, Phys. Lett. B 243, 432 (1990); X.-N. Wang and M. Gyulassy, Phys. Rev. Lett. 68, 1480 (1992); R. Baier, D. Schiff, and B.G. Zakharov, Ann. Rev. Nucl. Part. Sci. 50, 37 (2000).
Scheid, Muller, Greiner, Phys. Rev. Lett. 32, 741 (1974);
Hofmann, Stoecker, Heinz, Scheid, Greiner, Phys. Rev. Lett. 36, 88 (1976).
 C. Adler et al. (STAR Collaboration), Phys. Rev. Lett. 90, 082302 (2003).
J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 95, 152301 (2005).
 M. Horner (STAR Collaboration), J. Phys. G 34, S995 (2007).
H. Stoecker, Nucl. Phys. A750, 121 (2005);
J. Casalderrey-Solana, E. Shuryak, D. Teaney, J. Phys. Conf. Ser. 27, 22 (2005); J. Ruppert, B. Muller, Phys. Lett. B 618, 123 (2005); A.K. Chaudhuri, U. Heinz, Phys. Rev. Lett. 97, 062301 (2006); T. Renk, J. Ruppert, Phys. Rev. C 73, 011901(R) (2006).
 T. Renk, J. Ruppert, Phys. Rev. C 76, 014908 (2007).
 R.B. Neufeld, B. Muller, J. Ruppert, arXiv:0802.2254.
 A. Yarom, Phys. Rev. D 75, 125010 (2007); P.M. Chesler, L.G. Yaffe, Phys, Rev. Lett. 99, 152001 (2007); S.S. Gubser, S.S. Pufu, Phys. Rev. Lett. 100, 012301 (2008).
I.M. Dremin, Nucl. Phys. A767, 233 (2006);
V. Koch, A. Majumder, X.-N. Wang, Phys. Rev. Lett. 96, 172302 (2006).
I. Vitev, Phys. Lett. B 630, 78 (2005);
A.D. Polosa, C.A. Salgado, Phys. Rev. C 75, 041901(R) (2007).
 N. Armesto, C.A. Salgado, U.A. Wiedemann, Phys. Rev. Lett. 93, 242301 (2004).
 C.B. Chiu, R.C. Hwa, Phys. Rev. C 74, 064909 (2006).
 C.A. Pruneau, Phys. Rev. C 74, 064910 (2006).
 J.G. Ulery, F. Wang, arXiv: nucl-ex/0609016.
 J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 93, 252301 (2004).
 A. Feng (STAR Collaboration), Quark Matter 2008 proceedings.
 C. Pruneau (STAR Collaboration), arXiv: nucl-ex/0703010.
 J.G. Ulery, F. Wang, arXiv: nucl-ex/0609017.
 J. Adams et al. (STAR Collaboration), Phys. Rev. C 72, 014904 (2005).
 C. Adler et al. (STAR Collaboration), Phys. Rev. C 66, 034904 (2002).
M. Daugherity (STAR Collaboration), Quark Matter 2008