Jet and Minijet Contributions to Transverse Momentum Correlations in High Energy Collisions

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1 Jet and Minijet Contributions to Transverse Momentum Correlations in High Energy Collisions Mike Catanzaro August 14, Intro I have been studying the effects of jet and minijet production on momentum fluctuations in proton-proton and heavy-ion collisions. Specifically, I ve been studying transverse momentum and multiplicity correlations using the Monte Carlo event generator PYTHIA [1]. There are many different types of correlations that occur in nuclear collisions and I ve been focusing on two: radial flow and jets. These two phenomena cause particles to have large transverse momenta, although they re very different effects physically. We have derived a quantity, which we call D, which allows us to distinguish between the effects of radial flow and jets. We argue that D is negative for systems that exhibit radial flow, and positive for collisions which only have jets. I begin by arguing from general principles that the jet contribution to D is positive while the flow contribution is negative in sec. 2. In sec. 5, I use PYTHIA simulations to show that D is indeed positive when jets and minijets are present, and estimate values for proton collisions. to check our calculation for jets. PYTHIA is an event generator for proton-proton collisions at RHIC energies that includes minijet and jet production, but omits flow and other complicating factors that emerge in nucleus collisions. In sec. 4 I use an independent source model to relate the PYTHIA results to nuclear collisions. Throughout the summer, I ran these simulations in PYTHIA and calculated D for various ranges in transverse momenta and pseudorapidity. I am working in collaboration with faculty mentor Sean Gavin and graduate student George Moschelli. In progress are efforts to combine my efforts with hydrodynamic flow calculations as in [2]. 1.1 Radial Flow Radial flow is a phenomena that occurs mostly in heavy ion collisions. During the collision, flux tubes arise down the beam axis due to confinement of quarks. As the two nuclei separate, the flux tubes begin to fragment, creating particles with very low tranvserse momenta. As more and more of these types of particles are created, there s more particle creation near the center of the beam (smaller radius) rather than toward the circumfrence. This difference in the number of particles acts like as a pressure, pushing particles outward radially from the center of the beam. Thus, we see particles emerging from the collision with a very large component of transverse momenta. Radial flow, as described above, is a statistical phenomena. It only begins to influence the physics of the collision when there are high numbers of particles. Thus, we do not expect radial flow to play a large role in proton-proton collisions. However, when there are a large number of particles, like in Au-Au collisions, radial flow should play a much larger role. 1.2 Jets Jets occur in nuclear and particle physics experiments. These are rare processes in which quarks or gluons scatter head-on in a Rutherford-like fashion. High momentum transfers of the order of a few GeV are required to resolve the individual quarks or gluons within the target and projectile nucleons. The high momentum 1

2 partons so produced fragment to form jets of high momentum hadrons. Properties of jets have been computed to high precision using perturbation theory applied to Quantum Chromodynamics. Minijets refer to jet-like processes that occur at lower momentum scales, where multi-parton interactions are important. For our purposes, minijets behave exactly like jets, the difference being that QCD perturbation theory is no longer on firm footing. When one nucleus crashes through the other, the quarks inside each scatter off one another. This causes flux tubes to arise between the quarks. These flux tubes are in a different orientation than those from radial flow, since the quarks which cause them have scattered. As the nuclei separate, the flux tubes formed from the jets begin to stretch and eventually, fragment into particles with very high transverse momenta. These particles can appear quite similar to those affected by radial flow due to their momenta. However, the physics is completely different compared to that of radial flow. 1.3 PYTHIA PYTHIA is a computer program used for simulations of collisions of single particles. It relies on the use of Monte Carlo methods and interactions between charged particles to re-create the physics of elementary particle collisions. Along with PYTHIA, I also used ROOT this summer. ROOT is a data analysis program developed by CERN. I used ROOT to create histograms and graphs of data from PYTHIA, which we used to test our calculations. The PYTHIA model includes minijet and jet processes in a phenomenological description of proton collisions that has successfully described a range of data. These data include multiplicity and transverse momentum correlations [1]. Observe that PYTHIA does not incorporate radial flow as might occur in nucleus collisions. This approximation is ideal for our effort, because we want to disentangle the effects of flow and minijets. I remark that other event generators exist that can simulate both minijet and flow effects. 2 Covariance of p t and Multiplicity Correlation measurements have drawn wide and well-deserved attention. Measurements of two particle correlations provided the first striking evidence of jet quenching. New studies indicate further effects associated with the passage of partons through the high density medium [3, 4]. The success of these studies has recently inspired broader interest in correlations of all particle types, with and without triggers. Such studies focus on two-body correlation functions r(p 1, p 2 ) = ρ 2 (p 1, p 2 ) ρ 1 (p 1 )ρ 1 (p 2 ), (1) where ρ 2 (p 1, p 2 ) = dn/dy 1 d 2 p t1 dy 2 d 2 p t2 is the density of particle pairs and ρ 1 (p) = dn/dyd 2 p t is the single particle density. Fluctuation studies provide an alternative set of experimental techniques for attacking the same correlation physics. Such studies focus on measuring a particular quantity in each collision event and studying its variation in an ensemble of events. Typical experiments report quantities such as the multiplicity N averaged over events; here always denotes the average over events. In fluctuation studies, one can also study the variance N 2 N 2. These quantities are related to (1) by R := N 2 N 2 N N 2 = 1 N 2 r(p 1, p 2 )dp 1 dp 2, (2) see ref. [5]. This result follows because the integral of ρ 2 gives the average number of pairs N(N 1). Historically, we have viewed event-by-event fluctuations of the multiplicity, mean p t, net charge, and baryon number as probes of the QCD phase transition, but signature behavior has yet to be seen. Jet production and hydrodynamic transverse flow both enhance the transverse momenta of particles. The occurrence of a jet adds both particles and p t to an event, increasing both its total multiplicity N and total transverse momentum P t. In p t is uncorrelated with multiplicity, then we can write P t N P t N = p t ( N 2 N 2 ) uncorrelated (3) 2

3 We expect jets to enhance the covariance of P t and N so that the left side is larger than the right side of (3). In contrast, flow pushes particles from low to high p t without changing N. The left side of (3) is therefore unaffected by flow, while the right side increases because p t increases. We therefore propose that the quantity D := 1 N 2 [ ( NPt N P t ) p t ( N 2 N 2 ) ]. (4) We expect this quantity to increase due to jet or minijet production. On the other hand, transverse flow should decrease D. Following section 2.2 of ref. [6], one can show that (4) vanishes in thermal equilibrium. This quantity also vanishes when N is the same in all events. We will show in the next section that D vanishes when we integrate over all momentum. To relate D to the pair density, we now show that (4) satisfies D = 1 N 2 dp 1 dp 2 (p t1 p t ) r(p 1, p 2 ). (5) We define N = ρ 1 dp and p t = P t / N. We can also write these quantities as the ensemble average of sums over particles produced in each event, e.g., P t = i p ti. Similarly, the integral of the pair density satisfies ρ 2 dp 1 dp 2 = N(N 1), as mentioned earlier. We then expand N 2 D as the sum of three terms, the first of which is p t1 ρ 2 (p 1, p 2 )dp 1 dp 2, = (N 1)P t = NP t P t = NP t p t N. The second term is p t1 ρ 1 ρ 1 dp 1 dp 2 = dpp t ρ 1 (p) dpρ 1 (p), which is P t N by definition. The third term reduces to p t R N 2. Now, we combine these three pieces to find: N 2 D = dp 1 dp 2 (p t1 p t )(ρ 2 ρ 1 ρ 1 ) = NP t N P t p t N p t ( N(N 1) N 2 ) = NP t N P t p t ( N 2 N 2 ) This form of D in (4) is easier for us to compute in almost all cases, and is also the form we implemented in PYTHIA. However, in the following calculations, we will use both. 3 A Useful Theorem We now turn to show that D vanishes when we integrate over the full range of p. We arm ourselves with some useful facts. First, recall that the correlation function ρ 2 := dn/dp 1 dp 2 satisfies dn dp 1 dp 2 = N(N 1). (6) dp 1 dp 2 Next, observe that completeness requires that the integral of the pair distribution over the momentum of the second particle be proportional to the single particle distribution. The condition (6) then implies: dn dp 1 dp 2 dp 2 = With these equations in mind, we now analyze D in (5). N(N 1) ρ 1 (p 1 ). (7) N 3

4 Now that we know this claim must be true, we may continue in our calculation of D. From (7), we see the following also holds: N(N 1) dp 1 dp 2 p t1 ρ 2 = dp 1 p t1 dp 2 ρ 2 = dp 1 p t1 ρ 1 (p 1 ). N Once we know this, it s a simple calculation to show that D is identically zero: N 2 D = dp 1 dp 2 (ρ 2 ρ 1 ρ 1 )(p t1 p t ) = dp 1 dp 2 ρ 2 p t1 dp 1 dp 2 ρ 1 ρ 1 p t1 p t dp 1 dp 2 (ρ 2 ρ 1 ρ 1 ) = p t ( N(N 1) N 2) p t N 2 R 0, where we have used (2). We emphasize that D need not vanish if it is evaluated over a finite interval of rapidity y = (1/2) ln((e + p z )/(E p z ))) or pseudorapidity η = (1/2) ln((p + p z )/(p p z ))). Nevertheless, we will find D to be small in PYTHIA simulations even in the range 1 < η < 1. We will find that D sharply deviates from zero when the lower limit of the p t integrals are increased from zero. 4 Independent Source model The independent source model is crucial in our work, since it gives us a way of taking our results for proton-proton collisions, and scaling it to Au-Au collisions. This model assumes that nuclear collisions are a superposition of independent proton-proton collisions. This, however, neglects the rescattering of hadrons and implies that charged particle pairs can only be correlated if they re produced in the same collision [5]. Suppose that we have M independent sources, which fluctuate from event to event. We then write ρ 1 = ˆρ 1 M. Using this notation, we can re-write N as ( ) dpρ 1 (p) = dp ˆρ 1 (p)m = dp ˆρ 1 (p) M To make the computation simpler, we define µ := dp ˆρ 1 (p). Hence, N = µ M and ρ 2 = ˆρ 2 M + ˆρ 1 ˆρ 1 M(M 1). It also follows that ( ) P t = dp ˆρ 1 (p)p t M Now, we re-write R using the above formulation, starting from the definition of R: R = dp 1 dp 2 (ρ 2 ρ 1 ρ ) ( ρ 1 ) 2 = d 3 p 1 d 3 p 2 [ ( ˆρ2 M + ˆρ 1 ˆρ 1 M(M 1)) ˆρ 1 ˆρ 1 M 2] ( ρ 1 ) 2 = d 3 p 1 d 3 p 2 ˆρ 2 M + d 3 p 1 d 3 p 2 ˆρ 1 ˆρ 1 M(M 1) d 3 p 1 d 3 p 2 ˆρ 1 ˆρ 1 M 2 ( ρ 1 ) 2 = a 0 M + µ 2 M(M 1) µ 2 M 2 ( ˆρ 1 M) 2 4

5 where we have defined a 0 := d 3 p 1 d 3 p 2 ˆρ 2. Now, we see a different but equivalent definition of R: [ a0 µ 2 ] 1 R = µ 2 M + M 2 M 2 M 2 We also wish to write D in an equivalent form, including our M sources. Recalling the definition of D, we then proceed in a similar fashion: d 3 p 1 d 3 p 2 (ρ 2 ρ 1 ρ 1 )(p t1 p t ) D = ( ) 2 ρ1 Let us define b 0 := ˆρ 2 p t1. Now, we may re-write D with a 0 and b 0 as follows: D = b 0 µ 2 p t0 µ 2 M + p t0 [ M 2 M 2 M 2 Canceling out the second and fourth terms, we see a simpler form: D = b 0 µ 2 p t0 µ 2 M = b 0 a 0 p t0 µ 2. M ] [ a0 µ 2 p t0 µ 2 M + M 2 M 2 M 2 p t0(a 0 µ 2 ) µ 2 M To apply these formulas in nuclear collisions, we identify the number of sources M with the number of participant nucleons N part. By definition, the number of participants in a proton proton collision is two. We can then write D AA = 2D pp and R AA = 2R pp + N part 2 N part 2 N part N part N part 2. (8) The second contribution to R, M 2 M 2 M 2 N 2 part N part 2 N part 2, represents volume fluctuations, which must be computed from a nuclear collision model such as HIJING. We stress that D does not depend on volume fluctuations, while R does. 5 Results of simulations. As stated above, I ran proton-proton simulations at 200 GeV this summer in PYTHIA in order to test our various calculations of D. I implemented various p t and η cuts in the simulations, to see how D varied. I also calculated D as a function of minimum p t. The data from these simulations is shown in the tables below: 1 < η < 1, all charge states p t,min R D ] 5

6 1 < η < 1, negatively charged p t,min R D < η < 1, positively charged p t,min R D These tables show various values for D and R, as calculated in PYTHIA at 200 GeV. These values were calculated only using minimum bias, due to the short interval in pseudorapidity. The values in the above tables are plotted in Figure 1. This plot shows the above noted trend as p t,min varies. 6 Outlook I have studied the effect of jet and minijet production on the correlations in hadron collisions. I introduce a quantity D tailored to distinguish the effect of jets from that of flow. General arguments and PYTHIA calculations reported here demonstrate that the contribution of jets and minijets to D is positive. I argued that the flow contribution to D is negative. I mention that flow calculations following ref. [2] confirm this expectation. Gavin, Moschelli and I plan to combine or jet and flow calculations in future work. Joining these two models together would help to increase our current understanding of the roles that jets and radial flow play in momentum correlations. 7 Acknowledgements I would like to thank R. Bruner for discussions. This work was funded under NSF Grant PHY The author was also funded under a research grant from the Wayne State University Honors Department. References [1] T. Sjostrand and M. van Zijl, Phys. Rev. D 36, 2019 (1987). [2] S. Gavin, L. McLerran and G. Moschelli, Phys. Rev. C 79, (2009) [arxiv: [nucl-th]]. [3] N. Armesto, C.A. Salgado, U.A. Wiedemann, Phys. Rev. Lett. 93, (2004); P. Romatschke, Phys. Rev. C 75, (2007); A. Majumder, B. Muller, S. A. Bass, Phys. Rev. Lett. 99, (2007); C. B. Chiu, R. C. Hwa, Phys. Rev. C 72, (2005); C. Y. Wong, arxiv: [hep-ph]; R. C. Hwa, C. B. Yang, arxiv: [nucl-th]; T. A. Trainor, arxiv: [hep-ph]; A. Dumitru, Y. Nara, B. Schenke, M. Strickland, arxiv: [hep-ph]. [4] C. A. Pruneau, S. Gavin and S. A. Voloshin, Nucl. Phys. A 802, 107 (2008) [arxiv: [nucl-ex]]. [5] C. Pruneau, S. Gavin and S. Voloshin, Phys. Rev. C 66, (2002) [arxiv:nucl-ex/ ]. [6] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D 60, (1999) [arxiv:hepph/ ]. 6

7 Figure 1: D and R versus p t,min. 7