Study of dihadron fragmentation function correlations using charged jets in proton-proton collisions at s = 7 TeV

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1 Study of dihadron fragmentation function correlations using charged jets in proton-proton collisions at s = 7 TeV Derek Everett, Claude Pruneau, Sidharth Prasad August, Abstract We present the measurements of dihadron fragmentation function correlations between particle pairs in the fragmentation of particles inside a common jet using the ALICE detector at the LHC in pp collisions at 7 TeV. Jets are reconstructed using the FastJet anti-k T algorithm with resolution parameter R =. in the jet p T range from - GeV/c. The uncorrected dihadron fragmentation function and correlations are compared to the results obtained using Monte Carlo simulatations generated with PYTHIA 6. (tune Perugia-. We have found good agreement between data collected at ALICE and the simulated results. Also, the dihadron fragmentation function correlation shows little dependence on the energy scale. I INTRODUCTION A jet is spray of particles characterized by a large transverse momenta and high degree of collimation resultant from the hard scattering of two partons carrying a large fraction of the nucleon momentum. What results is a shower of hadrons typically with high transverse momenta p T. Many studies have been conducted investigating the momentum fragmentation function for a single particle within a jet, and the fragmentation of jets which are back-to-back. Missing correlations among backto-back jets have revealed properties of the hot and dense QCD medium. Less is known, however, concerning the dihadron fragmentation function and QCD evolution among pairs of particles within a common jet. By examining the fragmentation of particle pairs in simulated pp collisions, and collisions detected with the ALICE detector, we will have a model to study the behavior and structure of jets, and also their DGLAP energy-evolution. II EXPERIMENTAL METHOD Data were collected from the ALICE detector; specifically the Time-Projection Chamber (TPC [] and Internal Tracking System (ITS. The TPC is the main device in the ALICE detector for the detection and identification of charged particles. As charged particles traverse the gas volume, the gas is ionized, leaving a trail. Due to the magnetic field present, the curvature of the trail will depend on the charge and momentum of the particle. The ionized gas will drift at a constant speed until it makes contact with one of two end plates, which house electronics to detect the trail (radius and ionization density. Detection efficiency is nearly ninety percent. The geometry and detector resolution imposes the following cuts on particle detection: η track <.9 p track T >. GeV/c vertex cut of +/- cm III JET RECONSTRUCTION Jets were reconstructed using the FastJet [] Anti-k T algorithm [], with Resolution parameter R =.. The algorithm is briefly described: For each pair of particles, (i,j, we find the distance given by d i,j = min( kti, k tj R ij R ( where R ij = (y i y j + (Φ i Φ j. (

2 with k T : trasnverse momentum, y : pseudorapidity, Φ : azimuthal angle, and R : Resolution parameter. For each particle, we also find the beam distance, d ib, given by d ib = kti ( We find the minimum, d min, of all d i,j and d ib. If d min is a d i,j we merge the two particles i and j by adding their fourmomenta vectors. If d min is a d ib, we declare particle i to be a final jet, and remove it from the list. This process is repeated until there are no particles left. IV MONTE CARLO SIMULATION DETAILS Data were simulated with the PYTHIA 6. Monte Carlo event generator, using the tune Perugia (. This tune is based on data collected from SPS, LEP, and Tevatron []. Proton-proton collisions with center of mass energy of 7 TeV were generated in ten different hard bins. In order to generate significant statistics for jets with high transverse mementum, we fix the momentum transfer of the interaction to be on certain hard scales, which we call hard bins. Very hard interactions (interactions with high momentum transfer between partons are rare, and to generate a significant statistical sample using Minimum Bias events would require much heavier computing resources and time. We choose instead to generate events which have a fixed momentum transfer, and then normalize the statistical sample appropriately according to the hard bin of generation. The p T hard bins were as follows: -, -, - 6, 6-7, 7-8, 8-7, 7-6, 6 -, - 9, and > 9 (GeV/c Given a jet, all particle kinematics of interest were stored according to the jet p T bin. The jet p T bins were: -, -, - 6, 6-8, 8 - (GeV/c The cuts due to detector geometry and resolution have been included when generating data simulated with PYTHIA, and effects due to the detector p T resolution as well as efficiency have also been included by transporting simulated data with the GEANT detector simulation package. The AliRoot FastJet finder was employed with the following parameters: R Parameter (. Particle Pseudorapidity range (-.9,.9 Jet Psuedorapidity range(-.,. Anti-k T algrorithm reconstruction The following cuts were made: only final state, charged particles with transverse momenta greater than MeV/c were included. V Definitions Kinematic variables of interest for this analysis are defined as follows: The transverse momenta p T = p x + p y z = pparticle T p jet T ξ = ln ( z The fragmentation functions and correlation functions are defined as follows: Fragmentation function (FF D = dn dz Dihadron fragmentation function (DFF D, z = Correlation function C, z = D D D d N dz dz We now present the fragmentation functions, dihadron fragmentation functions, framentation function convolutions, and correlation functions for data simulated by PYTHIA, with detector effects simulated by GEANT, as well as data from the ALICE detector.

3 D D...6 Figure : DFF for ALICE data < jet p T <.8 z.8 z Figure : DFF for PYTHIA simulated detector level data < jet p T <.8 z z VI Results VI.I Dihadron Fragmentation Functions and Convolutions Several observations may be made of the DFF, D, z, in Figures -: There is great similarity between the data from ALICE and the Detector Level simulated Data. Also, there is similarity in the shape and form of the function between different jet p T bins. The function is steeply falling as z and z increase; the probability of finding a particle pair, z sharply decreases as z and z increase. This function is necessarily symmetric across the line z = z ; physically, a particle pair, z is equivalent to a pair, z. Also, the region z + z > is physically unrealizable, because by definition, a jet must have two or more constituent particles. There is a kinematic constraint that makes z and z necessarily dependent: By definition: n z i = ( for every n-particle jet. This implies and if we have a jet with only two particles, then VI.II Dihadron Correlation Functions i= z z ( z = z (6 Our statistical sample from ALICE data becomes too sparse for jet p T > 6 GeV/c as evidenced by the increased fluctuations in the DFF and FF Convolutions for jet p T > 6 GeV/c (Figures 9,, 9, and. Due to this statistical inadequacy, the Correlation Functions for these jet p T bins, 6-8 and 8- GeV/c, are unreliable and have not been included. Several observations may be made regarding the Correlation Functions, shown in Figures -6: The Functions C show good agreement between the ALICE Data and PYTHIA simulated data, as expected given the similarity of the DFF. The likelihood of finding a particle with high momentum fraction z decreases as z increases, and vice versa. We are most likely to find another particle with a low momentum fraction z when we have a particle with a low momentum fraction z. We also begin to see statistical fluctuations in the correlation function C from ALICE data at jet p T to 6 GeV/c. Finally, the magnitude and shape of the correlation function C is very similar from the different jet p T bins The similarity in the correlation function between the three jet p T bins is striking, given that as jet p T increases, the available energy for particle creation also increases, which increases the average particle multiplicity. Given an increase in the number of particles in a jet (track multiplicity, one might expect the statistical correlation between any two particles to weaken (approach ; however, this is not observed.

4 D...6 Figure : DFF for ALICE data < jet p T <.8 z.8 z.6.. D...6 Figure : DFF for PYTHIA simulated detector level data < jet p T <.8 z z D...6 Figure : DFF for ALICE data < jet p T < 6.8 z z D...6 Figure 6: DFF for PYTHIA simulated detector level data < jet p T < 6.8 z z.8.6..

5 D...6 Figure 7: DFF for ALICE data 6 < jet p T < 8.8 z.8 z.6.. D...6 Figure 8: DFF for PYTHIA simulated detector level data 6 < jet p T < 8.8 z z D...6 Figure 9: DFF for ALICE data 8 < jet p T <.8 z z D...6 Figure : DFF for PYTHIA simulated detector level data 8 < jet p T <.8 z z.8.6..

6 z z z z Figure : The fragmentation function convolution for ALICE data < jet p T < Figure : The fragmentation function convolution for PYTHIA simulated detector level data < jet p T < z z z z Figure : The fragmentation function convolution for ALICE data < jet p T < Figure : The fragmentation function convolution for PYTHIA simulated detector level data < jet p T < z z z z Figure : The fragmentation function convolution for ALICE data < jet p T < 6 Figure 6: The fragmentation function convolution for PYTHIA simulated detector level data < jet p T < 6 6

7 z z z z Figure 7: The fragmentation function convolution for ALICE data 6 < jet p T < 8 Figure 8: The fragmentation function convolution for PYTHIA simulated detector level data 6 < jet p T < z z z z Figure 9: The fragmentation function convolution for ALICE data 8 < jet p T < Figure : The fragmentation function convolution for PYTHIA simulated detector level data 8 < jet p T < C. C z z z z Figure : The correlation function for ALICE data < jet p T < Figure : The correlation function for PYTHIA simulated detector level data < jet p T < 7

8 C. C z z z z Figure : The correlation function for ALICE data < jet p T < Figure : The correlation function for PYTHIA simulated detector level data < jet p T < C z z C z z Figure : The correlation function for ALICE data < jet p T < 6 Figure 6: The correlation function for PYTHIA simulated detector level data < jet p T < 6 8

9 VI.III Efficiency and Robustness Particle detectors, including the TPC in ALICE, have a limited detection efficiency, which we can define as the probability (density that a particle will be detected. At ALICE, the average efficiency is about 9 percent. Often, this probability depends on the momentum of the particle. So we may write efficiency, ɛ = ɛ(p track T. Then for each particle in a jet, we may write the detection efficiency as a function of the momentum fraction z, ɛ = ɛ p jet T, where pjet T may be considered a parameter which is a constant for each jet. Briefly, we write ɛ = ɛ. ɛ N det N prod where we have detected N det particles out of the N prod that nature has produced. Then, we may write (7 N det = ɛn prod (8 If the probability that a particle will be detected is independent of the detection of any other particle, then we may write for the detection of particle pairs: ɛ, z = ɛ ɛ (9 We now make a small change in notation: and Recall how we have defined our correlation function C, C, z = Now if particle detection is indeed independent, we may write, C, z = N prod N ( N det Ñ ( < N, z > < N >< N > ( < ɛ Ñ, z > < ɛ Ñ >< ɛ Ñ > ( = ɛ ɛ < Ñ, z > ɛ ɛ < N >< N > ( = < Ñ, z > < N >< N > ( That is, if particle detection is independent, then the correlation function C does not depend on the efficiency of the detector, making C a robust quantity of study. In Figures and we show two histograms, which are the ratio of the Correlation Function C from the detector transported simulated data and C from the simulated data without detector effects included. If C is indeed a robust measurement, we should expect this ratio to be very close to unity. We define R = Cdet C part (6 where C det is the correlation function from the detector transported simulated data, and C part is the correlation function from the particle level simulated data (no detector effects. We see this ratio is very close to unity in most bins, for both < jet p T < and < jet p T < 6 (GeV/c. There are some spikes in the ratio, particularly in the very first bin (, in Figure 7, and as, z increases towards the boundary z + z = in Figure 8. However, the relative error for these bins is much higher than that of bins with content near unity. In Figure 7, The relative error of bin (, is 7 percent, with bin (, having only a percent relative error. In Figure 8, the relative error of the two largest spikes are percent and percent, with bin (, having only percent relative error. Therefore, we presume these spikes are due to statistical inefficiency, and that with a larger statisical sample, these ratios should approach unity and become increasingly smooth. The robustness of the correlation function will be studied in further detail, with an increased statistical sample in future efforts. 9

10 R R z z z z Figure 7: The ratio of detector level simulated data and particle level data < jet p T < Figure 8: The ratio of detector level simulated data and particle level data < jet p T < 6 VII SUMMARY AND CONCLUSION In conclusion, the similarity between the correlation functions from the three jet p T bins show that the behavior and structure of particle production, with respect to the momentum fraction z, has little dependence on the energy scale. This conclusion should be checked against higher ranges of jet p T which could not be included in this analysis (up to GeV/c. Also, the intra-jet DFF is a robust measurement for further study of jet behavior. There remains much to study regarding the form and energy-evolution of the DFF and correlation function. A larger statistical sample would allow the analysis to include higher jet p T ranges, which had to be exluded in this analysis. Further, larger statistics should confirm our conclusions about the robustness of the correlation function. Also, these results should be compared with current theoretical predictions regarding the DFF []. VIII Acknowledgments I would like to thank Dr. Claude Pruneau for many hours spent teaching physics and programming, as well as Dr. Sidharth Prasad for the generation of PYTHIA data, transport through GEANT, collection of ALICE data, and countless hours of help with programming and error handling. I would also like to thank Dr. Abhijit Majumder for time spent discussing the theory behind this study, and for the theoretical publication related to this study, as well as the ALICE collaboration for the generation of simulated data and collection of real data. References [] Joachim Baechler, The ALICE TPC [] M. Cacciari and G. P. Salam, FastJet.. user manual [] M. Cacciari, G. P. Salam, and Gregory Soyez, The anti-k T jet clustering algorithm, JHEP (8 6 [] P. Z. Skands, Tuning Monte Carlo Generators : The Perugia Tunes, Phys.Rev.D8:78, [] A. Majumder and Xin-Niang Wang, The dihadron fragmentation function and its evolution, Phys.Rev. D7 ( 7 A NORMALIZATION SCHEME Simulated data was normalized in the following way. All kinematics for particles within jets were histogrammed according to the jet p T, resulting in a single particle histogram and particle-pairs histogram for each kinematic of interest, in every hard bin. The contributions of each hard bin then had to be summed according to the p T bin with proper weight factors. First, the proper jet p T differential cross-section was calculated by combining contributions from hard bins with proper weights. The weight factor, w i is given by w i = σi n trials, where σ i is the average cross-section of hard bin i. Then, the final jet p T spectrum is given by the sum i= w ij i, where j i is the jet p T spectrum of hard bin i. The final jet spectrum is shown against the contributions from each hard bin in Figure 9.

11 Leading Jet p_t Spectrum diff. cross section - Sum > p_t (GeV/c Figure 9: The jet p T spectra from each hard bin and their weighted sum

12 Fragmentation Function Fragmentation Function /N dn/dz - /N dn/dz Track p_t / Jet p_t Track p_t / Jet p_t Figure : The fragmentation function for PYTHIA simulated particle level data from all jet p T bins Figure : The fragmentation function for PYTHIA simulated particle level data, with jet p T bins scaled for distinguishability This histogram was rebinned according to the p T bins of interest (listed above. The normalization for track kinematics, the fragmentation in particular, were then calculated using the content of the rebinned jet p T spectrum. The final fragmentation function for a jet p T bin of interest is given by 6 c i i= c sum z i where c i is the content of the p T bin of interest from the weighted jet spectrum from hard bin i, c total is the content of the p T bin of interest from the final jet spectrum, and z i is the spectrum of the track kinematic (for instance, the fragmentation from hard bin i. Mathematically, the normalization scheme is given: where S p = n i= p J = n i= σ i n trials j i J p σ i n trials j i (7 sp i p j i = n i= σ i n trials J sp i (8 p i is the hard bin index p is the jet p T bin j i is the jet p T spectrum of hard bin i J is the jet differential cross-section dσ dp T s p i is the particle kinematic spectrum from hard bin i in jet p T bin p S p is the final particle kinematic spectrum for jet p T bin p The fragmentation function is shown in Figures and. In Figure, the fragmentation functions have been scaled by increasing multiples of for distinguishability. The two dimensional track kinematic-pairs histograms were normalized in a similar fashion to the one-dimensional track kinematic histograms, summing over contributions from all hard bins with the weight factors given above.

13 B Correlation Function We have chosen to define the correlation function, C, in the following manner: cov(x, Y =< (X µ x (Y µ y > (9 =< XY > µ x µ y ( C (X, Y = cov(x, Y µ x µ y = < XY > µ xµ y µ x µ y ( = < XY > µ x µ y ( The pairs histograms were filled; then, the pairs histogram was divided by the convolution of the kinematic singles histogram with itself and subtracted by, to yield the correlation function.