Pseudorapidity and Correlation Measurements in High Energy Collisions Mark Parra-Shostrand August 9, 2013 Introduction My REU project consisted of studying two-particle correlations as a function of pseudorapidity due to jet production in high-energy proton-proton collisions. I had the pleasure working with my faculty mentor Dr. Sean Gavin and graduate student Chris Zin. This work was an extension of the work done by past REU students Mike Catanzaro and Daniel Taylor (see their reports on the REU website). Specifically, I have studied three observables R, D, and δp t1 δp t 2 (see correlation measurements). These three observables allow us to get a better understanding of pseudorapidity dependence. Also, it helps to explain the effects jets have on transverse momentum over a wide range of rapidities. In the beginning, I studied proton-proton collisions with simple analysis measurements of the charged multiplicity. I started to learn how to use the Monte Carlo event generator PYTHIA 8 (see Software). I then created a program through PYTHIA 8 that could generate our data and study the three observables as a function of pseudorapidity. Additionally, I had used a program called ROOT to analyze this data by making plots and histograms (see Software). When I generated my one million events through PYTHIA 8 it had a strict criteria. In order for the particles to be counted they had to be charged, in a pseudorapidity cut from -6 to 6, and a transverse momentum cut from.15 to 2 GeV. If all the criteria were met, then PYTHIA 8 counted the particle and stored it. Pseudorapidity When two particles collide and separate they create a flux tube between them. The two particles travel near the speed of light in opposite directions, stretching the flux tube until it fragments, creating particles. In the center of the flux tube is the center of mass and as you deviate from the center of mass characteristics of the particles change. For instance, as you travel away from the center of mass, along the beam axis, the fragmented particles begin to increase in velocity towards the speed of light. Also, pseudorapidity increases as you move away from the center of mass and θ decreases. However, near the center of
mass, velocity and pseudorapidity approaches zero and θ becomes much larger. Additionally, these characteristics are the same on both sides of the spectrum (see Figure 1). Figure 1: eta. Image credit Wikipedia Pseudorapidity is a function of a particle s angle ( θ ) from the beam axis. We can think of pseudorapidity as the speed of the particle along the beam axis and this gives us a relation between position and momentum. We can describe pseudorapidity in terms of momentum as η = 1 2 ln P + P z P P z. P z is the momentum directed down the beam axis and P t is the transverse momentum perpendicular to the beam axis. Therefore, you have tanθ = P t. However, in a more simplified form η = ln tan θ P z 2. Jets Jets occur in both particle and nuclear physics experiments. However, it is rare that jets would occur. In the case we do have jets, they occur only from a head-on collision between two quarks or gluons. At the collision point, two sprays of particles split off from the collision and produce a narrow cone of particles with high momentum. Mini-jets are similar and occur at much lower energies than jets. For our purpose, minijets behave exactly like jets. Radial Flow Radial flow is a phenomenon that occurs mostly in heavy ion collisions. During the collision, flux tubes form along the beam axis. As the two nuclei separate, flux tubes start to fragment and form particles with low transverse momentum. This process continues to happen and more particles are created near the center of the beam axis. Since there is a
difference in the amount of particles a pressure builds up and pushes particles outward radially from the center. These particles burst out in a random direction with a high transverse momentum. Correlation Measurements Correlation measurements have seen a great deal of focus in understanding the importance of nuclear collisions. We know that some of the earliest evidence of jet quenching came from measurements of two particle correlations. I now reference some studies on two-body correlation functions. For simplicity, we used pseudorapaidity dependence throughout the experiment over a rapidity range [1]. The statistical quantities used for one-body and two-body densities are: ρ 1 ( η) = dn dη and ρ 2 ( η 1,η 2 ) = d 2 N (1) dη 1 dη 2 From Eq. (1) to write multiplicity in a range of pseudorapidity N = ρ 1 ( η)dη (2) Δη Here... represents the average of the number of charged particles per event and fluctuations of the particle number are found by integrating the two- particle density to get the average number of pairs per event, ( ) = ρ 2 η 1,η 2 N N 1 ( )dη 1 dη 2 (3) Δη You may recognize a statistical measurement for the particle fluctuation as the variance from (2) and (3). In equilibrium these events follow a Poisson distribution so σ 2 eq = N. So we define this robust variance as [1]: R = σ 2 N = N 2 N 2 N 2 N N 2 (4) This tells us how far the events are from being in equilibrium. D is the main observable that we used to analyze our data. Professor Sean Gavin and Mike Catanzaro proposed this observable a few years ago. D allows us to distinguish between the effects of jets and radial flow. We argue that if D is positive than we expect jets to dominate. However, if D is negative we say that the system exhibits radial flow.
D = NP t N P t p t ( N 2 N 2 ) (5) N 2 Finally, the last correlation measurement we used is δp t1 δp t 2. We can think of δp t1 δp t 2 as the covariance per particle pair. My goal is to see how the particles characteristics change together. In this case, it measures the transverse momentum fluctuations in the collision δp t1 δp t 2 = i j N N 1 δp ti δp tj ( ) (6) where δp ti = p ti p t and the average transverse momentum is p t = P t N for P t = p i ti the total momentum in the event[2]. This quantity will vanish if particles i and j are uncorrelated. If δp t1 δp t 2 approaches zero, then we say all the particles transverse momentum is about average. If it s not zero, then there may be explosive jets. RESULTS If you look at figure 2, you can see the plots are not flat. This implies that our data does not follow Poisson distribution. Since jets are present in our events we do not expect the observables to be uniform in pseudorapidity, and our findings support this. If you examine figure 3, I illustrate how jets have a more Gaussian distribution over a range Δη. Jets are peaked in Δη and flux tubes are not. My plots show a curve in the data and I believe this comes from the production of jets. When you interpret the plots at η = 0 you can see that R ~.854, D ~.068 GeV, and δp t1 δp ~.0067 t 2 GeV2. You can also see that on either side of η = 0 they are pretty symmetric, which is expected, due to the symmetric nature of the collision. In addition, the statistical uncertainties are so small that the plotted line covers them up. Software PYTHIA 8, a Monte Carlo event generator, is the program I used to simulate collisions of single particles. It replicates interactions between charged particles to illustrate the elementary particles during collisions [3]. PYTHIA 8 only uses proton-proton collisions, therefore only a few dozen particles result from the collision. Since PYTHIA 8 produces a small amount of particles it does not incorporate radial flow. In order for radial flow to
be significant thousands of particles would need to be present. For a future project, one could use other programs such as HIJING to simulate heavy-ion collisions. Additionally, I used a program developed by CERN called ROOT. ROOT is a data analysis program that creates histograms, plots and many more data analysis methods. I ran ROOT through PYTHIA 8 and analyzed this vast amount of data to create histograms that could illustrate many significant details. ACKNOWLEDGEMENTS I would like to thank George Moschelli and Rajendra Pokharel for discussions and their additional guidance. This work was funded under NSF Grant PHY-0851678. Figure 2: plot observables on one
Jets vs. Flux Tube/String Fragmentation 2.5 2 # of pairs 1.5 1 0.5 0-8 - 6-4 - 2 0 2 4 6 8 Flux tube/string Fragmentation jets Δη Figure 3: Jets vs. Flux Tube/String Fragmentation Figure 4: Close up on D
Figure 5: Close up on <dptdpt> REFERENCES [1] C. Pruneau, S. Gavin and S. Voloshin, Phys. Rev. C 66, 044904 (2002) [arxiv:nuclex/0204011]. [2] S. Gavin and G. Moschelli, (2011) [arxiv:1107.3317v2 [nucl-th]]. [3] T. Sjostrand and M. van Zijil, Phys. Rev. D 36, 2019 (1987).