Resurrecting the Dead-Cone: EEC Analysis of the Dead-Cone in Simulated proton-proton Collisions
ABSTRACT
The dead-cone effect is a fundamental feature of quantum chromodynamics dictating the angular suppression of gluon radiation for massive quarks. While the dead-cone has been studied at Large Hardon Collider energies, analysis of this phenomenon in low energy jets at the Relativistic Heavy Ion Collider (RHIC) have yet to be conducted. This investigation visualizes the dead-cone through the Energy-Energy Correlator (EEC) for proton-proton (pp) collisions at RHIC. Simulations were conducted using PYTHIA for 200 GeV pp. The dead-cone’s mass dependence was observed through differences in angular distances of particles in jets (∆R). These differences were specifically measured in the transition region (or peaks) of the EEC for these jets. The EEC peaks for heavy flavor jets (charm and bottom) are at larger ∆R indicating higher gluon emission suppression at small angles. Additionally, as jet energy increases, the EEC transition region was measured at smaller ∆R for all jet flavors. This effect is observed to be more enhanced for heavy flavor jets than light flavor jets corroborating with the dead-cone’s angular relationship of m/E. These findings suggest both the mass and energy dependency of the dead-cone while also demonstrating their detectability through the EEC.
INTRODUCTION.
Extremely small segments of matter, called protons, make up most matter within the universe. However, even smaller than protons and neutrons are quarks. First theorized in 1964 [1], quarks make up the substructure of protons, neutrons, and every other particle. There are 6 “flavors” of quarks of varying known masses [2], the behaviors of which are not entirely understood. To study this incredibly small packet of matter, particle colliders are used to break apart protons into their smaller components and study the behaviors of matter at this scale. Colliders accelerate protons to extremely high speeds and then smash them into each other. Detectors placed around the collision point absorb the particles ejected from the collisions and record their energy and momentum in three dimensions.
The Relativistic Heavy Ion Collider (RHIC) collider at Brookhaven National Laboratory in Upton, New York, is used to accelerate protons to relativistic speeds[3] and then collide them within one of the two main detectors, sPHENIX and STAR. Most notably, RHIC has been vital to the discovery and exploration of quark gluon plasma [4].
In this paper, we simulate an effect called the Dead-Cone to improve understanding of the strong force within high energy nuclear physics. This fundamental understanding can then be applied to the sectors of materials science, energy, and defense for their further advancement [5].
Up and down quarks are the most common as they are the lightest of their brethren. Strange and charm quarks are much heavier than up and down quarks and are thus seen less often and the heaviest quarks, top and bottom, are the rarest. Quarks, also referred to as partons, are extremely unstable as singular particles, so they form hadrons like protons. Also referred to as confinement, this phenomenon makes it impossible to directly measure partons. Within these hadrons, quarks are held together by gluons which are massless particles that operate as the “glue” holding together normal matter [6].
Quantum Chromodynamics (QCD).
The strong force is responsible for the behavior of quarks within hadrons. This force, mediated through the exchange of gluons, is holds partons together and does so through a 3-charge system. To describe these charges, each has been assigned a color red, blue, or green. Strong force attractions operate similarly to electromagnetic force interactions with the exception of three charges instead of two. QCD, however, does not only govern the charge laws for hadron construction. Other behaviors, like the dead-cone effect, are also included within QCD due to their direct relationship to the strong nuclear force.
When two hadrons collide, the quarks inside them scatter off one another transferring the fraction of the hadron’s momentum that they possess and then scatter. The scattered quarks go through a process called parton showering, in which there are nearly parallel streams of particles, called jet. Jets preserve the structure of quark showering and are a powerful tool for parton showering analysis [7].
Parton Showering.
When partons first shower within a jet, one quark or gluon often contains most of the momentum within the event system. This parton is described as the leading parton, the flavor and energy of which depend on many factors. Primarily, the energy of the collisions. In collisions of particles with equal mass, the momentum of the entire system will be zero. Therefore, during a collision event, because of the conservation of momentum, often two leading partons will emerge with back-to-back or collinear geometry.
The energy that each leading parton will have will be in some proportion of momentum and mass. Thus, more massive leading partons (charm, bottom, and top flavor quarks) will tend to carry less momentum, and lighter flavor leading partons (gluons and up, down, and strange flavor quarks) will often have more energy. Leading partons radiate energy in the form of gluon emissions (to reach a more stable lower energy state). This process of gluon emission is then repeated, and the gluons that have been emitted decay into more quarks that then radiate more gluons in the same way forming the multibranched geometry of jet substructure. At the end of each branch partons undergo the process of hadronization through which hadrons are formed, and then said hadrons reach detectors. While the actual process of hadronization is still not yet entirely understood, this study implemented one theory of hadronization within simulation, string based “Lund hadronization,” included in PYTHIA [8] the collision simulation software implemented in this investigation.
After the parton showering is complete and hadronization has occurred, hadrons then reach detectors built to measure a variety of attributes of each particle detected. Firstly, detectors measure the trajectory or “track” of the particles. Secondly, detectors called calorimeters measure the energy of the ejected hadrons [9]. With the energy and momentum of the hadron calculated from the trajectory, it can then be recombined to find the heavy flavor hadrons, when they are present, before decay.
The Dead-Cone.
The dead-cone is a feature of QCD directly stemming from the angular emission of gluons during parton showering. Through this emission, scattered quarks reach a lower energy state while conserving the system’s momentum. The amount of energy radiated is proportional to the angle of gluon emission. Consequently, for more massive quarks with more of their total momentum in the form of mass, this angular emission is suppressed at smaller angles. This occurs because gluon emission angle is related to the total energy of the quark (momentum and mass energy combined), and for heavier quarks, more of that total energy is in the form of mass. Therefore, heavy quarks no longer have energy to radiate when they reach momentums at which low emission angles commonly occur causing lower emission angles to be much less likely. This general trend of suppressed small angle radiation forms a cone around the small angle region of heavy flavor jets in which far fewer particles are found thus known as the dead-cone. Because the angle of emission is suppressed both at high masses and at low energies m/E can be used to describe the angle of the dead cone [10].
Energy Correlators.
The two point energy correlator, or Energy-Energy Correlator (EEC), is reemerging observable within the field that provides insight into the substructure of jets. The EEC is a histogram of angular distances between particles (∆R) within a jet, weighted by the energies of these particles. Within EECs there are three distinct regions, the free hadron region, perturbative region, and transition region [11]. The free hadron region appears at small ∆R thus representing later splittings within the collision and in this region the EEC is almost always increasing. This distribution in this region is uniform, but due to geometric effects, is constantly increasing. The perturbative region, found at large ∆R, is where the perturbative effects of quark and gluon interactions take effect as larger splitting angles occur earlier in the collision event. Finally, the transition region is in between the free hadron and perturbative region denoted by the peak in ∆R for the EEC.
In this study, the two point energy correlator was used to examine the impact of the dead-cone on jet substructure, and the ways in which leading parton flavor impacts jet substructure through EEC distribution at RHIC energies. Because the dead-cone directly impacts the substructure of jets, which is observed through the EEC, it is expected that the dead cone will cause mass-based EEC distribution differences, specifically through the suppression of the EEC for small ∆R, and an enhancement at large ∆R for heavy flavor quarks. In future studies, the two point energy correlator can then be used to attempt to observe said impact in data. While Energy Correlators have been used to investigate the dead-cone at LHC energies [12], an investigation of RHIC has yet to be conducted. This analysis will allow for a new understanding of the ways in which the dead-cone functions at lower energies than previously investigated through the EEC.
METHODS.
Simulation Specifications.
To examine the impact that jet flavor has on the EEC, and specifically the mass effect known as the dead-cone, proton-proton collisions were simulated using the Monte Carlo [13] based program PYTHIA 8.315 [14] with the Fastjet 3.5.0 package installed [15]. Data management and figure generation was performed using ROOT, a C based architecture developed by CERN.
Jet Selection.
Jets were reconstructed using the Anti-kt algorithm included in the Fastjet package [15]. This algorithm creates jets by first finding the highest momentum particle (leading particle), and then adding the surrounding particles to the list of jet constituents in order from highest momentum to least within a set radius in the eta-phi plane until the jet is fully constructed. For this study, all jets were selected with a radius of 0.4. Then the leading partons of the collision were geometrically matched to jets within a radius of 0.3, as these jets are extremely likely to have been initiated by these partons [16]. Once a jet was identified, all particles within the jet radius were iterated through and their distance from every other particle within the jet is then stored within a root histogram weighted by the product of the two particles momentum (an EEC) and sorted by the flavor of the initiating parton associated with that jet.
Physics Specifications.
The √s=200 GeV proton-proton collisions simulated in this investigation were run with parameters similar to RHIC. Simulations were run to recreate jets at different pT ranges of 10-20 GeV, 20-30 GeV, 30-40 GeV, and 40-60 GeV, which we can expect at RHIC, to investigate Dead-Cone energy dependence. A collision modifier, Hard QCD, was turned on, and a pTHat for interactions was set to 5 GeV less than the specified pTjet range. Parton initiated jets were selected based upon the aforementioned cuts, and all constituents of these jets for EEC analysis were required to fall within the detector η of the sPHENIX detector (<1.1) [17], one of the principal detectors at RHIC ensuring that any selected jets could be reasonably detected experimentally. Jets were then filtered to ensure they fell into the correct pT ranges. The EEC was then calculated for all jets selected, and the energy weighted pairs were plotted based on leading parton flavor. The EEC was then normalized by the free hadron region highlighting possible differences in the transition and perturbative regions. This analysis includes 1 billion events that passed the cuts stated above.
Because experimental data includes inherent variability, χ2 minimization functions, included within Root, were applied to each curve. This allowed for the easy extraction of information like the peak of the transition region via a gaussian fit and the slope of the perturbative region via a power law fit the boundaries of which were based on the range in which bottom quarks were successfully fit.
RESULTS.
The EEC calculated for the √s = 200 GeV proton-proton collisions simulated in PYTHIA is shown in Figure 1 and a distinct difference is visible between the EEC for the different quark masses across all jet pT ranges. In the transition region, or the peak, of each EEC, there is a strong differentiation across the jet flavors where more massive jet flavors peak at larger ∆R. At all four pT ranges, the peak for light flavor jets is at smaller ∆R than for heavy flavor jets.
When comparing the fitted gaussian peak positions of the EEC as a function of jet pT, it can be seen that as the pT range of the jet increases, the peak position of the EEC decreases. Another trend visible is that the peak ∆R positions become progressively more similar as jet pT increases. This same trend indicates that heavier jet flavors experience the impacts of higher jet energies more than lighter flavors. Additionally, the mass stratification of peak ∆R can be clearly seen indicating both a mass and energy dependence visible in the EEC peak.
The perturbative region of the proton-proton EEC was also examined through a power law fit plotted in figure 3. Figure 3 follows a similar trend to Figure 2 where as jet energy increases, the slopes became more similar. Additionally, at higher jet pTs, the slope of the perturbative region becomes more negative indicating a greater drop-off in larger ∆R particle-pairs as energy within the jet increases and thus more pairs at a small ∆R.
DISCUSSION.
It can be concluded based on the results found in this study that the Energy-Energy Correlator is an effective tool in the investigation of the dead cone effect in proton-proton collisions at RHIC and EIC energies. We ran 1 billion simulated events for quark flavor, and after plotting and normalizing the EEC for each of them, multiple trends emerged. The mass and energy dependence of the dead cone predicted by previous literature was clearly apparent and followed their predicted trends. Overall, our research effectively tested and found success in using the EEC to identify the Dead-Cone in simulated RHIC energy events.
Figure 2 demonstrates how in each jet momentum range, as the mass of the quark increases, the position of the EEC peak for that flavor moves towards larger ∆R. This result aligns with the relationship of dead cone angle being proportional to m/E, because as the mass increased, so did the dead cone. The increase in the size of the dead cone is what is likely causing the EEC to move towards larger ∆R. Because the dead-cone diminishes the number of particles radiated by the leading particle of jet after that leading particle has reached a lower energy, there would be less small angle pairs. Therefore, peaks that are toward larger ∆R also represent jets with less pairs at small ∆R directly indicating the dead-cone. The proportional difference in EEC distribution across initiating parton masses directly relates to the prediction of m/E because as the mass is increasing, the impact that the dead cone has on the jet substructure as measured via the EEC can be clearly observed.
This conclusion can be further confirmed when the peaks for the EEC gaussian fits are plotted as a function of jet pT as seen in Figure 4. At every jet pT range, the same trend occurs where light flavor jets have the smallest peak ∆R, increasing to that of charm flavor jets, and finally to the highest ∆R for bottom flavor jets. Interestingly though, as jet pT increases, the peak ∆R for each flavor jet becomes lesser and more similar. That finding again aligns with the predicted dead-cone relationship of m/E because the pT of the jet is directly linked to the energy of the jet, and as jet pT increased the peak ∆R decreased. An inverse relationship was exactly what was expected between the energy of, and the angular suppression within, the jet, because more energy for leading quarks is radiated before they have reached a stable state.
When a power law fit was applied to the perturbative region, as seen in Figure 3, the mass and energy dependence of the dead-cone could be investigated further. The slope of the perturbative region provides additional insight as it quantifies the impact that the dead cone has on the EEC for large ∆R. After fitting, we found a similar trend to that of the peaks for the different flavor jets. This finding aligns with the current literature predicting a direct relationship between the mass and the size of the dead cone. A larger dead-cone would lead to more particle pairs at large ∆R thus creating less of a fall in the perturbative region at large ∆R. And conversely, light flavor jets show a much more negative slope at large ∆R. Because the EECs plotted are normalized, far fewer particle pairs at large ∆R is directly indicative of more pairs at small ∆R demonstrating the impact of the dead-cone on jet substructure.
The examination of the EEC for electron-proton collisions simulated to match specifications of the theorized Electron Ion Collider yielded extremely similar results to those found at RHIC energies. Similarities were expected though because, while an entirely different collision occurs, the way in which leading quarks distribute their energy is not. Thus, the same trend confirming the predicted mass dependency of the dead was found in the peak position, and in the slope of the power law fit for the perturbative region. The creation of jets is very different in proton-proton collisions when compared with electron-proton collisions, the biggest difference being the that two jets are present after the former, while singular jets are much more likely in the latter.
While this novel application of the EEC does provide valuable insight into the dead cone and its measurement, there are some limitations to our approach. Primarily, all collisions in this investigation were simulated. Therefore, while the conclusions made via this investigation are still valid, they require further analysis using collision data. Additionally, because the collisions were simulated, certain tools and techniques implemented within the simulations would not be possible in experimentation. The geometric matching strategy used to find the jet associated with the leading quark of a collision for example, is not feasible as it requires the ability to identify the leading quark. Copious jet reconstruction is required to do so, rendering that method infeasible. Many decays for the hadrons formed via hadronization were also turned off in this investigation to understand solely the impact that the dead-cone has on jet substructure. However, this cannot be easily achieved in experimentation without intensive jet reconstruction and analytical methodology. Other limitations include the lack of variety in the energy ranges presented. Because these collisions were simulated with the same parameters capable used in RHIC collisions, the maximum collision energy reached was 200 GeV. Due to the conservation of energy, this limits the energies that jets measured can reach, therefore limiting the upper bound to which mass dependency and energy dependency trends can be measured.
Future research in this area could investigate the ways in which different effects such as jet quenching, or multi-parton interactions and initial state radiation impact the EEC measurement of the dead-cone. Ion collisions are also possible and frequently run at RHIC thus, investigations of the ways in which jet quenching in QGP impacts the dead cone, and the EEC are necessary. Also, EECs should be used to analyze experimental data at RHIC to confirm the predictions formed within this study. This should be coupled with further simulation research in which experimentally measurable hadrons, that are traditionally used to tag jets as containing heavier flavor quarks, are allowed to form, to assess the experimental application of the heavy flavor results found in our investigation.
Overall, this investigation provided valuable information as to the applications of the Energy-Energy Correlator specifically at RHIC energies. Simultaneously, the dead-cone and its properties were also explored allowing for a greater understanding of quark behavior in jets. These findings, when combined with previous research, provide a holistic understanding of the ways in which matter functions at high energies and can be applied to future research and exploration.
ACKNOWLEDGMENTS.
Thank you to my mentor Dr. Benjamin Kimelman, Dr. Rithya Kunnawalkam, and Dr. Pamela Popp for their support throughout the project.
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Posted by buchanle on Friday, May 15, 2026 in May 2026.
Tags: Dead-Cone, Energy-Energy Correlator, Particle Physics, Proton, RHIC, simulation