UFDC Home  myUFDC Home  Help 



Full Text  
SEARCH FOR HEAVY RESONANCES DECAYING INTO tt PAIRS By VALENTIN NECULA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Valentin Necula I dedicate this work to my parents, MariaDoina and Eugen Necula. ACKNOWLEDGMENTS I take this opportunity to express my deepest thanks to my advisors, Prof Guenakh Mitselmakher and Prof Jacobo Konigsberg, for their guidance, continuous support and patience, which played a crucial role in the successful completion of this work and will continue to be a source of inspiration in the future. I would like to take this opportunity to thank Dr. Roberto Rossin for his important contribution to the success of this analysis, from writing code to running jobs and writing documentation, and nonetheless for all the interesting little chats we had, be it politics, history, finance or sports. I am also grateful for the advice I received and the discussions I had with Prof Andrey Korytov, Prof Konstantin Matchev, Dr. Sergey Klimenko and Prof John Yelton. At last but not at least I would like to thank Prof Richard P. Woodard for making my first years at University of Florida very exciting and rewarding. Sometimes I just miss those exams. My stay at CDF benefitted from the interaction I had with many people, and without making any attempt at an exhaustive list I would mention Dr. Florencia Canelli, Dr. Mircea Coca, Dr. Adam Gibson, Dr. Alexander Sukhanov, Dr. Song Ming Wang, Dr. Daniel Whiteson, Dr. Kohei Yorita, Prof John Conway, Prof Eva Halkiadakis, Dr. Douglas Glenzinski, Prof Takasumi MaL, Ii.1111.. Prof Evelyn Thomson, and Prof YoungKee Kim. Special thanks to Dr Alexandre Pronko, who was my officemate in my early days at CDF, and with whom I had quite interesting discussions and played much fewer chess games that I should have. The more relaxing moments I enjoyed in the company of Gheorghe Lungu and Dr. Gavril A. Giurgiu were very useful as well and I would like to thank them both. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... ...... iv LIST OF TABLES ...................... ......... ix LIST OF FIGURES ..................... .......... xi ABSTRACT ....................... ........... xv CHAPTER 1 INTRODUCTION .................... ....... 1 1.1 Historical Perspective ........... ..... ....... 1 1.2 The Standard Model of Elementary Particles ............ 3 1.2.1 Leptons ................... ........ 3 1.2.2 Quarks ......... .............. .... 5 1.3 Beyond the Standard Model .......... ....... ... 6 2 NEW PHYSICS AND THE TOP QUARK ......... ......... 8 3 EXPERIMENTAL APPARATUS .......... ........ ..... 12 3.1 Tevatron Overview ................... .... 12 3.2 CDF Overview and Design .......... ........ .... 15 3.2.1 Calorimetry ................... ... 17 3.2.2 Tracking System ........... ............ 21 3.2.3 The Muon System .......... ............. 25 3.2.4 The Trigger System .......... ........ .... 26 4 EVENT RECONSTRUCTION ......... ............. 29 4.1 Quark and Gluons ................... ..... 29 4.1.1 Jet Clustering Algorithm ......... .......... 30 4.1.2 Jet Energy Corrections ......... ........ .. 31 4.2 Electrons .................... ........ 33 4.3 Muons ........... ...... ........ ...... 34 4.4 N. ii! i in ...4 ................... ........... 35 5 EVENT SELECTION AND SAMPLE COMPOSITION . ... 37 5.1 Choice of Decay Channel ................ .... .. 38 5.2 Data Samples .................. .......... .. 39 5.3 Event Selection .................. ....... .. .. 40 5.4 Sample Composition .................. ..... .. 41 6 GENERAL OVERVIEW OF THE METHOD AND PRELIMINARY T E ST S . . . . . . . . 44 6.1 Top Mass Measurement Algorithm ...... ......... 45 6.1.1 The Matrix Elements (\I IJ ) ................. 48 6.1.2 Approximations: Change of Integration Variables . 50 6.2 Monte Carlo Generators ............... .. .. 51 6.3 Basic Checks at Parton Level ............. .. .. 52 6.4 Tests on Smeared Partons . . ..... .... 54 6.5 Tests on Simulated Events with Realistic Transfer Functions 55 6.5.1 Samples and Event Selection . . ..... 55 6.5.2 Transfer Functions ...... ......... . .. 55 7 f1., RECONSTRUCTION ............... .... .. 58 7.1 Standard Model tt Reconstruction ..... . . 58 7.2 Signal and other \! Backgrounds ..... . . 63 8 SENSITIVITY STUDIES ............... .... .. 77 8.1 General Presentation of the Limit Setting Methodology . 77 8.2 Application to This Analysis ............. . .. 78 8.2.1 Templates ............... ....... .. 79 8.2.2 Template Weighting ................. . .. 81 8.2.3 Implementation .... . . . .. 82 8.2.4 Cross Section Measurement and Limits Calculation .. 83 8.2.5 Expected Sensitivity and Discovery Potential . ... 85 9 SYSTEMATICS ............... ........... .. 87 9.1 Shape Systematics ............... .... .. 87 9.1.1 Jet Energy Scale ....... ........ .. .. 87 9.1.2 Initial and Final State Radiation . . 88 9.1.3 W Q2 Scale .................. ..... .. 89 9.1.4 Parton Distribution Functions Uncert.ii . ... 91 9.1.5 Overall Shape Systematic Uncertainties . .... 91 9.2 Effect of Shape Systematics ............... . .. 92 9.3 Expected Sensitivity with Shape Systematics . . ... 94 10 RESULTS ... .............. .......... .... 96 10.1 First Results .................. ........... .. 96 10.2 Final Results .................. .......... .. 99 10.3 Conclusions .................. ........... .. 101 APPENDIX CHANGE OF VARIABLES AND JACOBIAN CALCULATION SKETCH ...................... ............... 107 REFERENCES ................... ....... ...... 111 BIOGRAPHICAL SKETCH ................... ........ 113 LIST OF TABLES Table page 11 Properties of leptons. Antiparticles are not listed. . ... 4 12 Properties of quarks. Additionally, each quark can also carry one of three color charges. . . . . . . 5 31 Summary of CDF calorimeters. Xo and Ao refer to the radiation length for the electromagnetic calorimeter and interaction length for the hadronic calorimeter, respectively. Energy resolutions correspond to a single incident particle. .............. .. .. .. 18 51 tt decays ................ .. ......... ..... 38 52 Event Selection. .................. ........... .. 40 53 Crosssections and acceptance ................. .. 42 54 Signal acceptance .................. .......... .. 43 81 Acceptances for background samples. ................ .. 81 82 Acceptances for resonance samples .............. .. .. 82 91 Linear fit parameters describing the uncert.,iiil' due to JES systematic; JES and JES+ labels designate a +0a or a variation in energy scale. The uncertainty on crosssection is parametrized with Joxo = o + a l ax o . . . . . 89 92 Linear fit parameters describing the uncert.iiil, v due to ISR modeling. The uncertainty in cross section is parametrized with &TX0o = o + a l a x o . . . . . 90 93 Linear fit parameters describing the uncert.,iiii v due to FSR modeling. The uncertainty in cross section is parametrized with &TX0o = o + a l a x o . . . . . 90 94 Linear fit parameters describing the uncert.iiili v due to WQ2 scale, The uncertainty in cross section is parametrized with TX0o = o + a l ax o . . . . . 90 101 Expected number of events assuming no signal. WW and QCD numbers are derived based on the total number of events observed in the search region above 400GeV/c2. ............... . .. 97 102 Expected number of events assuming no signal. WW and QCD numbers are derived based on the total number of events observed in the search region above the 400GeV/c2. .................. .... 99 103 Expected and observed upper limits on signal crosssection derived from a dataset with an integrated luminosity of 680 pb1. . 104 LIST OF FIGURES Figure page 21 The CDF Run 1 tt invariant mass spectrum. . . 10 22 The CDF Run 1 upper limits for resonance production crosssection times branching ratio. .................. .... 11 31 Overview of the Fermilab accelerator complex. The pp collisions at the centerofmass energy of 1.96 TeV are produced by a sequence of five individual accelerators: the CockroftWalton, Linac, Booster, Main Injector, and Tevatron. ........ .. 13 32 Drawing of the CDF detector. One quarter view. . .... 16 33 The r z view of the new Run II end plug calorimeter . ... 21 34 Longitudinal view of the CDF II Tracking System . . .... 22 35 Isometric view of the three barrel structure of the CDF Silicon Vertex Detector . .............. ............ .. 23 36 One sixth of the COT in endview; odd superlayers are smallangle stereo layers and even superlayers are axial. ........... ..25 37 CDF II Data flow. .................. ........ 27 61 Main leading order contribution to tt production in pp collisions at s 1.96 TeV ...... ......... ............. 48 62 Gluongluon leading order contribution to tt production in pp collisions at I 1.96 TeV ................. .. ...... 49 63 Reconstructed top mass from 250 pseudoexperiments of 20 events at parton level with mt=175 GeV/c2. The left plot is derived using only the correct combination, while the right plot uses all combinations 52 64 Reconstructed top mass vs. true top mass from pseudoexperiments of 20 events using all 24 combinations, at parton level . ... 53 65 Reconstructed top mass vs. true top mass from pseudoexperiments of 20 events with smearing. The left plot is derived using only the correct combination, while the right plot uses all combinations 54 66 Light quarks transfer functions (x 1 E Er ), binned in three absolute pseudorapidity regions [0, 0.7], [0.7, 1.3] and [1.3, 2.0] 56 67 bquarks transfer functions (x 1 E E ), binned in three absolute pseudorapidity regions [0, 0.7], [0.7, 1.3] and [1.3, 2.0] . ... 57 71 ,, reconstruction for the correct combination and for events with exactly four matched tight jets. .................. 59 72 H.,, reconstruction including all events ............. .. 60 73 Examples of if., reconstruction, event by event. . .... 61 74 H.., template for Standard Model tt events. . . 62 75 Reconstructed invariant mass for a resonance with Mxo 650 GeV. The left plot shows all events passing event selection, while the right plot shows only matched events ................ . 64 76 Reconstructed invariant mass for a resonance with Mxo 650 GeV. The left plot shows mismatched k l1i'l I ii events and the right plot shows nonlI. ,I l I 1 events ..... .......... 65 77 W+4p template (electron sample) .... . ... 67 78 W+4p template (muon sample) ................ . 68 79 QCD template ............... ........... .. 69 710 WW template ....... ....... ............... 70 711 W+2b+2p template (electron sample) . . ..... 71 712 W+2b+2p template (moun sample) ..... . . 72 713 W+4p template with alternative Q2 scale (electron sample) . 73 714 All Standard Model background templates used in the analysis 74 715 W+2b+2p template vs W+4p template. W+2b+2p was ignored since the expected contribution is at the level of 12% and the template is very similar to the W+4p template . . ..... 75 716 Signal templates .................. .......... .. 76 81 Signal and background examples. The signal spectrum on the left (Mxo 600 GeV/c2) has been fit with a triple Gaussian. The background spectrum from Standard Model tt has been fit with the exponentiallike function. Fit range starts at 400GeV/c2. ..... 80 82 Linearity tests on fake (left) and real (right) templates. As test fake signal templates we used Gaussians with 60 GeV/c2 widths and means of 800 and 900 GeV/c2. We used also real templates with masses from 450 to 900 GeV/c2. The top plots show the input versus the reconstructed cross section after 1000 pseudoexperiments at integrated luminosity f L = 1000pb1. Bottom plots show the deviation from linearity in expanded scale, with reddotted lines representing a 2% deviation .................. ..... 83 83 Example posterior probability function for the signal cross section for a pseudoexperiment with input signal of 2 pb and resonance mass of 900 GeV/c2. The most probable value estimates the cross section, and 95% confidence level (CL) upper and lower limits are extracted. The red arrow and the quoted value correspond to the 95% CL upper limit ............... ..... ..84 84 Upper limits at 95% CL. Only acceptance systematics are considered in this plot. ............... .......... .. .. 86 85 Probability of observing a nonzero lower limit versus input signal cross section at f L 1000pb1. Only acceptance systematics are included in this plot ............... . ..86 91 Cross section shift due to JES uncertainty for f L = 1000pb1. The shift represents the uncertainty on the cross section due to JES, as a function of crosssection .................. ...... 88 92 Cross section shift due to ISR (left) and FSR (right) uncertainties for f 1000 pb. ............... . .... 89 93 Cross section shift due to WQ2 scale uncertainty for f = 1000pb1 91 94 Total shape systematic uncertainty versus signal cross section. . 92 95 Posterior probability function for the signal cross section. The smeared (convoluted) probability in green, including shape systematics, shows a longer tail than the original (black) distribution. As a consequence the UL quoted on the plot is shifted to higher values with respect to the one calculated based on the original posterior . ... 93 96 Upper limits at 95% CL. The plots show the results for two luminosity scenarios, including or excluding the contribution from shape systematic uncertainties .. ... .. ... .. .. ... ... .. ..... .. 94 97 Probability of observing a nonzero lower limit (LL) versus input signal cross section for f L 1000 pb1. .................. .. 95 101 Reconstructed 1 ., in 320 pb1 of CDF Run 2 data. The plot on the right shows events with at least one SECVTX tag . ... 96 102 Reconstructed it.. in 320 pb1 of CDF Run 2 data, after the 400 GeV cut .......... .......... ................ 97 103 Resonant production upper limits from 320 pb1 of CDF Run 2 data 98 104 KolmogorovSmirnoff (KS) test assuming only the Standard Model. The KS distance distribution from pseudoexperiments is shown in the right plot; the arrow indicates the KS distance between data and the Standard Model template .................. .. 100 105 KolmogorovSmirnoff (KS) test assuming signal with a mass of 500 GeV/c2 and a crosssection equal to the most likely value from the posterior probability. The KS distribution from pseudoexperiments is shown in the right plot; the arrow indicates the KS distance between data and the Standard Model + signal template. . ... 100 106 i ,, spectrum in data vs. Standard Model + 2 pb signal contribution from a resonance with a mass of 500 GeV/c2 . . .... 101 107 Reconstructed it., in CDF Run 2 data, 680 pb1 . . ... .. 102 108 Resonant production upper limits in CDF Run 2 data, 680 pb 102 109 KolmogorovSmirnoff test results are shown together with the reconstructed 1t., using 680 pb1 and the corresponding Standard Model expectation template ................. ...... ........ 103 1010 Posterior probability distributions for CDF data and masses between 450 and 700 GeV ................. ... ...... 105 1011 Posterior probability distributions for CDF data and masses between 750 and 900 GeV. ................... ....... 106 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SEARCH FOR HEAVY RESONANCES DECAYING INTO tt PAIRS By Valentin Necula August 2006 Chair: Guenakh Mitselmakher Cochair: Jacobo Konigsberg M. i' r Department: Physics We performed a search for narrowwidth vector particles decaying into topantitop pairs using 680 pb1 of data collected by the CDF experiment during 20022005 Run 2 of the Tevatron. The center of mass energy of the pp collisions was 1.96 TeV. Model independent upper limits on the production crosssection times branching ratio are derived, at 95% confidence level. We exclude the existence of a leptophobic Z' boson in a topcolorassisted technicolor model with a mass Mz, < 725GeV/c2, and our results can be used to constrain any other relevant theoretical model. CHAPTER 1 INTRODUCTION 1.1 Historical Perspective The science of Physics investigates the laws governing the behavior of matter, from the smallest subnuclear scales to the largest astronomical spacetime regions and even the nature of the universe as a whole, as in cosmology. In High Energy Physics we are concerned with understanding the socalled fundamental "bricks" of matter or elementary particles and their interactions. It is not easy to ascertain elementariness, in fact it is quite impossible, and history shows us that more often than not what was considered elementary at one point was found later to be a composed system: molecules, which are the smallest units of substance possessing specific 1.li,i .,1 and chemical properties, were found to be made up of smaller units, atoms. A huge variety of organic matter with quite different 1.i i. ... I. 1mi, .i1 properties is composed of just three atoms, hydrogen, carbon and oxygen. For some time atoms were considered to live up to their ancient meaning of indivisible units of matter, until the end of the 19th century when the ini,'i. ius cathode rays puzzled ]'li,i. i1i with their properties. As J.J. Thomson correctly predicted, the cathode rays were actually streams of subatomic particles known today as electrons. It wasn't long until Rutherford proved in his famous scattering experiments that the positive charge inside atoms is confined to a pointlike core, or nucleus, a discovery which led to the classic planetary model of the atom. The elementariness of the atom vanished, and the focus moved to the structure of the nucleus. At first it was thought that the nucleus contained electrons and protons, but eventually the neutron (postulated by Rutherford) was discovered and the picture of matter had been simplified even more: just three particles, the proton, the neutron and the electron, were enough to build all known atoms. They were the new elementary particles, however soon they were joined by a large number of new particles with strange names like pions, kaons, eta and rho particles. The simple and maybe beautiful picture of three elementary particles at the basis of all matter had to be abandoned. Both experimental and theoretical breakthroughs lead to the understanding that protons, neutrons and the vast majority other particles are composed of smaller and stranger units, called quarks. Two different developments took place during this time though. First, one of the most brilliant lli, i. i1 of all times, P.A.M. Dirac, predicted in 1928, solely on theoretical grounds, the existence of a new particle which was later called the positron. It was supposed to be just like the electron, but positively charged, an antielectron. Amazingly, positrons were in fact observed only four years later and then it was found that other particles had antiparticles. It was an universal phenomenon. Secondly, searching for a particle postulated in the Yukawa theory of nuclear forces, experimentalists found something else, as it is often the case: a new negatively charged particle which behaved just like an elecron except it had much higher mass and it was unstable. It was called a muon. This phenomenon was found to have its own kind of universality and lead to the classification of elementary particles in three generations, as it will be detailed later. Particle 1.]li,i also investigates the interactions or forces between the elementary constituents of matter. By mid 20th century 1li, i. i1, counted four distinct forces: the gravitational force, the electromagnetic force, the strong nuclear force responsible for holding quarks together inside a proton or neutron for instance, and the weak nuclear force responsible for / decays and other phenomena. The early picture of classical "force" fields mediating the interactions was abandoned after Dirac quantized the Maxwell's equations successfully, laying the foundation for quantum field theory and introducing the idea that interactions are mediated by exchanges of virtual particles. Later it was discovered that indeed the strong and weak nuclear forces are mediated by virtual particles, the gluon and the massive W+, W and Z bosons respectively. However, even though we have a classical set of equations describing gravitation and powerful formalisms for quantizing fields, all attempts at quantum gravity failed. Delving into that iniiI ry is not the purpose of this dissertation though, and now we will proceed to a more formal presentation of the theoretical framework underlying our current understanding of elementary particles and their interactions. 1.2 The Standard Model of Elementary Particles The Standard Model is a quantum field theory which is based on the gauge symmetry SU(3)c x SU(2)L x U(i)y [1]. This gauge group includes the symmetry group of the strong interaction, SU(3)c and the symmetry group of the unified electroweak interaction, SU(2)L x U(i)y. As pointed out earlier, gravitation didn't fit the scheme and it is not part of the Standard Model. All the variety of phenomena is the result of the interactions of a small number of elementary particles, classified as leptons, quarks and force carriers or mediators. They are also classified in three generations with similar properties. 1.2.1 Leptons All leptons and hadrons have spin 1/2, and all force mediators have spin 1. There are three six charged leptons, the electron (e), the muon (pt), the tauon (r) and their positively charged antiparticles. For each charged lepton there corresponds a neutral lepton, called a neutrino (u). Even though neutrinos do not carry electric charge, they have distinct antiparticles due to the fact that they possess a property called lepton number. There are three lepton numbers, the electronic lepton number, the muonic lepton number and the tauonic lepton number. An electron carries a +1 electronic lepton number and an electronic neutrino (ve) also carries a +1 electronic lepton number. Similarly a muon and a muon neutrino (uP) carry a +1 muonic lepton number, a tauon and a tau neutrino (VP) carry a +1 tauonic lepton number. The antiparticles of these particles carry 1 leptonic numbers and in the Standard Model each lepton number is conserved such that in any reaction the total lepton numbers of the initial state particles should be equal to the total lepton numbers of the final state particles. It should be noted that significant evidence has been gathered during the last decade indicating that neutrinos oscillate, thus violating the leptonic number conservation. Table 11: Properties of leptons. Antiparticles are not listed. Particle Spin Charge Mass 1st generation e 1/2 1 0.510998920.00000004 MeV/c2 Ve 1/2 0 < 3 eV/c2 2nd generation I 1/2 1 10:..... ;..',+0.000009 MeV/c2 uP 1/2 0 < 0.19 MeV/c2 3rd generation 1/2 1 1776.99_+:. MeV/c2 U, 1/2 0 < 18.2 MeV/c2 The interactions of leptons are described by the electroweak theory which unifies electromagnetism and the weak force. In this gauge theory there are three massive force carriers, the W+, W and Z bosons and one massless force carrier, the photon(7). In fact a pure gauge theory of leptons and gauge bosons would lead to massless particles, so in order for the particles to "c,. llii mass the spontaneous symmetry breaking mechanism was proposed. This adds an extra spin 0 boson to the picture, the Higgs boson, by which all gauge bosons except one (7) acquire mass, and leptons can acquire mass simply by coupling to the scalar Higgs field. Even though the massive bosons [2, 3, 4, 5] have been discovered at CERN more than 20 years ago, the Higgs boson has not been discovered. It is also possible that the mass problem is solved by some other mechanism. 1.2.2 Quarks There are six types of quarks and their antiparticles, commonly referred to as the up (u), down (d), strange (s), charm (c), bottom(b) and top(t) quarks. They carry fractional electrical charges and a new property called color, which is responsible for the strong interactions of quarks. Each quark can carry one of three colors, red, blue and green. The antiquarks carry anticolors, antired, antiblue and antigreen. Quarks' properties are summarized in Table 12. Quarks also take part in electroweak processes and that led to some remarkable predictions. It was found that in order to be able to renormalize the electroweak theory an equal number of generations of quarks and leptons was needed, but when these ideas appeared only three quarks were known, the u, d and s. Few years later in 1974 the c quark was discovered, thus completing the second quark generation as expected. Another three years later a third generation charged lepton was discovered, r, and in the same year a third generation quark was discovered, the b. The interesting part is that the massive bosons themselves were not discovered until 1983 The quest for the last missing pieces in the generation picture ended with the top quark discovery in 1994 at Fermilab and the V, discovery in 2000, also at Fermilab. Table 12: Properties of quarks. Additionally, each quark can also carry one of three color charges. Particle Spin Charge Mass 1st generation u 1/2 +2/3 1.54 MeV/c2 d 1/2 1/3 48 MeV/c2 2nd generation c 1/2 +2/3 1.151.35 GeV/c2 s 1/2 1/3 80130 MeV/c2 3rd generation t 1/2 +2/3 178.0+4.3 GeV/c2 b 1/2 1/3 4.14.4 GeV/c2 The strong interactions of quarks are mediated by eight massless gluons (g) which carry double color charge, thus being able to interact among themselves. The theory of strong interactions is known as Quantum Chromodynamics (QCD) and it is a gauge theory based on the SU(3) Lie group. It has two characteristics not found in the electroweak theory, called color confinement and ..1,lnil.1 i' freedom. The interaction between colored particles is found to increase in strength with the distance between them, therefore quarks do not appear as free particles. Instead they form color singlet states either by combining three quarks with different colors (barions) or combining a quark and an antiquark (mesons). This is "color confinement". Conversely, at smaller and smaller distances the interaction strength decreases and the coupling constant as becomes small enough for perturbative methods to work. This feature is known as '. i',i' ii..1 i' freedom." 1.3 Beyond the Standard Model The Standard Model has managed to explain very well a vast amount of experimental data, however there are reasons to believe it is an incomplete theory : As mentioned earlier, gravity is left out altogether Possibly connected to the previous point, the observed masses of particles are completely unexplained. The Higgs mechanism is just a way by which particles would ., Iiiiure" mass, both bosons and fermions, but it does not predict their values. The gauge anomaly of the electroweak theory is canceled only if we have an equal number of quark and lepton generations, and the charges of the particles within one generation obey a certain constraint equation. This implies that there is some deeper connection between quark and leptons which might also explain why we have only three generations. Besides particles' masses, there are still quite many arbitrary parameters in the Standard Model, like the relative strengths of the interactions, the Weinberg angle sin Ow, the elements of the CabibboKobayashiMaskawa matrix which describe the strength of crossgeneration direct coupling of quarks via charged currents. There are significant indications that neutrinos oscillate. The amount of known matter in the Universe is less than what would be necessary to produce a flat geometry as observed, and it is believed that there must exist other types of matter, dark matter, besides a nonzero cosmological constant or dark energy, which would explain the discrepancy. But these conclusions rely on the validity of General Relativity in describing the Universe as a whole, which is not quite obvious. Many theories beyond the Standard Model have been proposed, like Supersymmetry, String theories, Grand Unified Theories (GUTs), extra dimensions theories, Technicolor, quark compositeness theories and others. Some are basically impossible to test at current available energies, but most have a large parameter space and it is difficult to rule them out completely. In this work we decided to adopt a model independent approach to our search for Physics beyond the Standard Model, at least as much as it is possible. CHAPTER 2 NEW PHYSICS AND THE TOP QUARK The top quark is so much heavier than the other quarks, including its 3rd generation sibling the b quark, that it is natural to ask whether this fact is related to its possible coupling to New Physics. This idea was explored in a theory called "topcolorassisted technical' ,'" [6, 7] which introduces new strong dynamics coupling preferentially to the third generation, thus making the tt and bb final states of particular interest. This theory introduces a topcolor heavy Z' and "topgli, ,,", both decaying into tt and bb pairs. There are other theoretical avenues for producing heavy resonances, like Universal Extra Dimension models [8, 9, 10]. The simpler versions [8, 9] assume only one extra dimension of size R, and lead to new particles via the KaluzaKlein(KK) mechanism. In the minimal UED model [9] only one more parameter is needed in the theory, the cutoff scaleA. An interesting feature is the conservation of the KK number at tree level, and in general the conservation of the KK parity defined as (1)" where n is the KK number. As a consequence the lightest KK partner at level 1 has negative KK parity and it is stable, therefore possible candidates for our search are level 2 KK partners. These can couple to Standard Model particles only through loop diagrams, given the need to conserve KK parity. Another UED model [10] assumes that all known particles propagate in two small extra dimensions, also leading to new states viathe KaluzaKlein mechanism. Resonance states below 1 TeV are predicted in this model, and they have significant couplings to tt pairs. From a purely experimental point of view the tt production mechanism is an interesting process in which to search for New Physics since the full compatibility of tt candidate events with the Standard Model is not known with great precision due to quite limited statistics. There is room to explore for possible nonStandard Model sources within such an event sample. In this dissertation we focus on the search for a heavy resonance produced in pp collisions at s = 1.96 TeV which decays into tt pairs. The basic idea is to compute the tt invariant mass spectrum and search for indications of unexpected resonance peaks. We will implement the tools needed to set lower and upper limits for the resonance production crosssection times branching ratio at any given confidence level. A discovery would amount to a nonzero lower limit at a significant confidence level. A similar search was carried out at the Tevatron by the CDF [11] and DO [12] collaborations on the data gathered in "Run 1", the period of operation between 19921995. The tt invariant mass as reconstructed by the CDF analysis in the "lepton plus j. I channel is shown in Figure 21. There are only 63 events for the entire Run 1 dataset, which corresponds to an integrated luminosity of 110 pb1. About half of them were tt events. Based on this distribution the 95% confidence level upper limits on tt resonant production crosssection times branching ratio were computed, as a function of resonance mass (Figure 22). The main challenge of this analysis is the reconstruction of the tt invariant mass spectrum. In this analysis we use an innovative approach which includes matrix element information to help with the reconstruction, as it will be explained in later chapters. S20 I f X tt Simulation 18  Mx=500GeVc2 . ri500S F= 0.012 MN, 16  250 S14  12 o S400 (600 00 Reconstructed M (GeV c2) 10 CDF Data (63 events) i tt and W+jets Simulations (63 events) 6 S i W+jets Simulation (31.1 events) 4 I  2  300 400 500 600 700 800 900 1000 Reconstructed Mt (GeV/c2) Figure 21: The CDF Run 1 tt invariant mass spectrum. 4000 0 00 600 700 800 900 1000 Mx (GeV/c ) Figure 22: The CDF times branching ratio. Run 1 upper limits for resonance production crosssection CHAPTER 3 EXPERIMENTAL APPARATUS The Fermi National Accelerator Laboratory (FNAL, Fermilab) has been a leading facility in experimental particle phl, i, , for the last 30 years. The hadron collider, called the Tevatron, is the world's most powerful accelerator where protonantiproton collisions are investigated. While many measurements and searches have been carried out, probably the most famous results out of the Tevatron program are the discovery of the bottom quark in 1977 and the discovery of the top quark in 1994, during the 19921995 Tevatron operation period known as "Run I". At the moment of this writing we are in the middle of Run 2, the second Tevatron operation period which started in the spring of 2001. Record instantaneous luminosities ( ~ 1.7 1032 cm2S1) have been achieved recently, which makes the search for new particles including the last missing block of the Standard Model, the Higgs boson, a lot more interesting. The Collider Detector at Fermilab (CDF) and DO are two general purpose detectors built at almost opposite collision points along the accelerator. In this analysis we use data collected by the CDF collaboration during the period 20022005. The center of mass energy in Run 2 is s 1.96 TeV, the highest collision energy ever achieved. 3.1 Tevatron Overview The Fermilab accelerator complex is shown on a schematic drawing in Fig. 31. In order to produce such high energy pp collisions a sequence of five individual accelerators is needed. TFFRMIn.ARS A C'F!1FR..TOR CHMIN TEVATAOft *l 0 "r i3 .i 'BECvCLER 1JLLl tIM ~ ~ ~ ~ ~ ' ~ ' ,; ii .*'~II x Lv 1 i, ""F,WW mwk& = Figure 31: Overview of the Fermilab accelerator complex. The pp collisions at the centerofmass energy of 1.96 TeV are produced by a sequence of five individual accelerators: the CockroftWalton, Linac, Booster, Main Injector, and Tevatron. First, the CockroftWalton accelerator boosts negative hydrogen ions to 750 KeV energy. Then, the ions are directed to the second stage of the process provided by the 145 m long linear accelerator (Linac) which further increases the energy of ions up to about 400 MeV. Before the next stage the ions are stripped of their electrons when they pass through a carbon foil, leaving a pure proton beam. These protons move to the next stage, the Booster, which is a synchrotron accelerator of about 150 m in diameter. At the end of this stage the protons reach an energy of 8 GeV. Next, protons are injected into another circular accelerator called the Main Injector. The Main Injector serves two functions. It provides a source of 120 GeV protons needed to produce antiprotons. It also boosts protons and antiprotons from 8 GeV up to 150 GeV before injecting them into the Tevatron.  eBJrr r~ii =,. ,1%E~ In order to produce antiprotons, 120 GeV protons are transported from the Main Injector to a nickel target. From the interaction sprays of secondary particles are produced, including antiprotons. Those antiprotons are selected and stored into the Debuncher ring where they are stochastically cooled to reduce the momentum spread. At the end of this process, the antiprotons are stored in the Accumulator, until they are needed in the Tevatron. The Tevatron is a protonantiproton synchrotron collider situated in a 1 km radius tunnel. It accelerates 150 GeV protons and antiprotons up to 980 GeV, leading to a pp collision centerofmass energy of 1.96 TeV. Inside the Tevatron the beams are split into 36 "biii. li, which are organized in three groups of 12. Within each group the bunches are separated in time by 396 ns. Collisions take place bunch by bunch, when a proton bunch meets an antiproton bunch at the interaction point. Just for clarity we should add that the beams are injected bunch by bunch. The collisions do not take place at the exact same location each time but are spread in space, according to a Gaussian distribution with a sigma of about 28 cm along the beam direction and also extending in the transverse plane with a circular crosssection defined by a radius of about 25 pm The instantaneous luminosity of the Tevatron is given by inst N f (31) where Np and Np are the numbers of protons and antiprotons per bunch, f is the frequency of bunch crossings and A is the effective area of the crossing beams. A compact period of time during which collisions take place in the Tevatron is called a "store" and it can last from few hours to over 24 hours. During a store the instantaneous Illilii ..il is decreasing exponentially due to collisions and transverse spreading of the beams which leads to losses of protons and antiprotons. The instantaneous luminosity can drop one order of magnitude during one store. Run 2 initial instantaneous luminosity ranged from about 5 1030 cm2S1 in 2002 to the record 1.7 1032 cm211 in 2006 and there are hopes for even higher values in the future. 3.2 CDF Overview and Design The Collider Detector at Fermilab (CDF) is a general purpose detector located at one of the two beam collision points along the Tevatron known as "BO". The idea of a general purpose detector is to allow the study of a wide range of processes occurring in pp collisions. For that purpose CDF is designed such that it can identify electrons, muons, photons and jets. It is indirectly sensitive to particles which escape detection, like the neutrinos. A schematic drawing of the CDF detector is shown in Fig. 32. It is cylindrically symmetric about the beam direction with a radius of about 5 m and a length of 27 m from end to end, and weighs over 5000 metric tons. The CDF collaboration uses a righthanded Cartesian coordinate system with its origin in the center of the detector, the positive zaxis along the proton beam direction, the positive xaxis towards the center of the Tevatron ring and the positive yaxis pointing upward. The azimuthal angle o is defined counterclockwise around the beam axis starting from the positive xaxis. The polar angle 0 is defined with respect to the positive zaxis. However, another quantity is widely used instead of the polar angle. It is called pseudorapidity and it is defined by the formula r= ln(tan(0/2)). The reason is that in the massless approximation, which is a very good one at these energies, relativistic boosts along the zaxis are additive in the pseudorapidity variable and this property is important, for instance in the consistent definition of jet cones. The pseudorapidity can also be defined with respect to the actual position of the interaction vertex, in which case it is called event pseudorapidity. CENTRAL DRIFT CHAMBER ELECTROMAGNETIC SCALORI METER CHAMBER HADRON IC CALORIMETER MUON DRIFT CHAMBERS STEEL SHIELDING MUON SCINTILLATOR S COUNTER ISL (3 LAYERS) F_ SVX I (3 BARRELS) INTERACTION POINT (BO)  ~ SOLENOID COIL ,PRESHOWER DETECTOR .   I I SHOWERMAX DETECTOR FT. m0 1 2m 3m 4m 5m S 4 6 8 10 12 14 16 ft1. Figure 32: Drawing of the CDF detector. One quarter view. The detector is composed by a series of subdetectors. Closest to the beam is the silicon vertex detectors which are surrounded by charged particle tracking chambers. The silicon vertex detectors are used to reconstruct the position of the collision vertex and particle moment. Next are the electromagnetic and hadronic calorimeters used for energy measurements and at last the muon chambers. There is also a timeofflight system used for charged hadrons identification and the Cherenkov Luminosity Counters (CLC) which measure Illiiii il v. For this analysis we use all major parts of the detector. The calorimetry is necessary for jet reconstruction, energy measurements for electrons, muon identification and also for the calculation of missing transverse energy. The tracking system plays a major role in electron and muon identification and in momentum measurement, and the muon chambers are important for muon identification. In this section we will provide a general description of the major components of the detector, mainly emphasizing the parts used for this analysis. A more comprehensive description can be found in the published literature [13] 3.2.1 Calorimetry The purpose of the calorimeters is to measure the energy depositions of particles passing through them. However not all particles interact in the same way. N' i, iiii li escape without any interaction at all, and high energy muons also escape the calorimeters without losing much energy. Apart from that, the rest of the particles leave their entire energy in the calorimeter with some exceptions in the case on pions for instance which can travel, rarely, beyond the calorimeter. Even though neutrinos do not interact with the calorimeter, by applying the conservation of momentum in the transverse plane one can calculate the total transverse momentum of the neutrinos. Since the calorimeter measures energy this inferred (qi .illi ', is known as missing transverse energy. In case the event contained high energy muons it needs further corrections before it can be identified as neutrino transverse momentum since, as mentioned before, the muons also do not leave much energy in the calorimeter. The electromagnetic calorimeter is designed such that it can measure well the energy of photons and electrons positronss). Electrons above 100 MeV lose their energy mostly through bremsstrahlung or photon radiation. High energy photons produce electronpositron pairs in the nuclear electromagnetic fields of the material, thus restarting the cycle and leading to the development of an electromagnetic shower of electrons, positrons and photons. At the last stage, low energy photons unable to create electronpositron pairs lose their energy by Compton scattering and photoelectric processes, while low energy electrons lose their energy by ionization. For simplicity we will assume that the initial particle moves perpendicular to the detector. Then as the shower develops in the calorimeter more and more energy is deposited, but at different depths or in different layers of the detector. However at some point the number of new shower particles starts to decrease and then later no new particles will be created. After this point the energy deposited per layer starts to decrease, exponentially. The depth of the maximum energy deposition layer is called the shower maximum and can be used for particle identification. Other charged particles like muons behave differently because the energy loss via radiation starts to dominate energy loss via ionization at much higher energies, higher by a factor of (m/mr)2, approximately. Given the energy scale at the Tevatron, a typical muon leaves roughly 10% of its energy in the electromagnetic calorimeter and thus it is not possible to identify and measure muon moment using the calorimeter. Table 31: Summary of CDF calorimeters. Xo and Ao refer to the radiation length for the electromagnetic calorimeter and interaction length for the hadronic calorimeter, respectively. Energy resolutions correspond to a single incident particle. Calorimeter ill,.1 im rp coverage Depth Energy resolution a(E)/E CEM p < 1.1 18 Xo 13.5%//ET e 2% PEM 1.1 < Irl < 3.6 21 Xo 16%/ET 1% CHA Ir\ < 0.9 4.5 Ao 75%/ e 3% WHA 0.7 < The hadronic calorimeter functions on similar principles, it is designed to interact strongly with hadrons, thus making it possible to measure their energy by measuring the deposited energy. In this case the incoming particle interacts with the nuclei of the material in the detector leading to a similar shower development. The CDF calorimeter system covers the full azimuthal range and extends up to 5.2 in Iq Its components are the Central Electromagnetic Calorimeter (CEM) and the Central Hadronic Calorimeter (CHA) which cover the central region as the name II.., . the Plug Electromagnetic Calorimeter (PEM) and the Plug Hadronic Calorimeter (PHA), which extend the l coverage more; the Endwall Hadronic Calorimeter (WHA), which is located in between the central and plug regions; and finally the Miniplug (\! NP), which is a forward electromagnetic calorimeter which is not used in this analysis. Some technical details are listed in Table 31. Each calorimeter l, I,. II is divided in smaller units called towers and has a projective geometry, which means that all towers point to the center of the detector. Central Calorimeter. Each tower of the central calorimeters covers 150 in AO and 0.11 in Ap and it is composed of alternating layers of absorber and active material. When a particle passes through the dense absorber material it produces a shower of secondary particles which interact with the active material and produce light. The light is collected and converted in a measurement of energy deposition. The CEM is made of 0.5 cm thick pol I vl i.i' scintillator active layers which are separated by 0.32 cm thick lead absorber layers. The CEM extends from the radius of 173 cm up to 208 cm from the beam line and the total thickness of the CEM material is about 18 radiation lengths. It is divided into two identical pieces at q = 0 and both have an one inch thick iron plate at q = 0. This kind of uninstrumented region is commonly referred to as a An important parameter is the energy resolution. The CEM resolution for electrons or photons between 10 and 100 GeV is given by o(E) 13.5% .5% 2% (CEM), (32) E \E where ET (in GeV) is the transverse energy of the electron or photon and the symbol indicates that two independent terms are added in quadrature. Inside the GEM, at a depth of about six radiation lengths or 184 cm away from the beam line, there is the Central Electromagnetic Shower Maximum detector (CES). Its position corresponds to the location of the maximum development of the electromagnetic shower which was described earlier. The CES determines the shower position and its transverse development using a set of orthogonal strips and wires. Cathode strips are aligned in the azimuthal direction providing zview information and anode wires are arranged along the z direction providing the r view information. The position measurement using this detector has a resolution of 0.2 cm for 50 GeV electrons. The CHA is located right after the CEM and its pseudorapidity coverage is Ir\ < 0.9 while WHA calorimeter extends this coverage up to Ir\ < 1.3. It has a depth of about 4.5 interaction lengths and consists of 1 cm thick acrylic scintillator layers interleaved with steel layers 2.5 cm thick. The end wall calorimeter uses 5 cm thick absorber layers. The electromagnetic and hadronic calorimeters were calibrated using electron and respectively pion test beams of 50 GeV. Their performance is described by the energy resolution. For charged pions between 10 and 150 GeV it is given by o(E) 75% E) 7% 3% (CHA, WHA), (33) Plug Calorimeter. The PEM and PHA calorimeters cover an pq range between 1.1 and 3.6 and employ the same principles.The PEM is a lead/scintillator calorimeter with 0.4 cm thick active layers and 0.45 cm thick lead layers. It also includes a shower maximum detector at a depth of about 6 radiation lengths, the PES, but it is not used in this analysis. The PHA contains 0.6 cm thick scintillator layers and 5 cm think iron layers. An r z cross section view of the CENTRAL TRACKING I, , 'HADRON CALORIMETER ....  2 Figure 33: The r z view of the new Run II end plug calorimeter CDF plug calorimeters is shown in 33. In this analysis the calorimeters were used to determine the momentum and direction of electrons and jets. 3.2.2 Tracking System The purpose of the tracking system is to reconstruct trajectories and moment of charged particles and find the location of the primary and secondary vertices. A primary vertex is the location where a pp interaction occurred. A secondary vertex is the location where a decay took place. For instance charm and bottom hadrons have a longer lifetime than light quarks hadrons, long enough that they can travel and decay at a location experimentally discernible from the primary vertex location. Such distances are of the order of hundreds of microns and this feature is exploited in heavy flavor t..'ii; algorithms. The components of the tracking system are the following: superconducting solenoid, silicon detectors and a large opencell drift chamber known as Central Outer Tracker (COT). A diagram is shown in Figure 34. As it can be seen, the CDF Tracking Volume m END WALL 2.0 HADRON _1.0 CAL. 1. 0 .5 1.0 1.5 2.0 2.5 3.0 m 1.5  0 .5 1.0 1.5 2.0 2.5 3.0 m SVX II INTERMEDIATE 5 LAYERS SILICON LAYERS Figure 34: Longitudinal view of the CDF II Tracking System. COT isn't very useful for II > 1 so CDF can rely only on the silicon detectors for that region. But for the II < 1 range both silicon and COT information is used and a full 3D track reconstruction is possible. The Solenoid. This is a superconducting magnet which produces a 1.4 T uniform magnetic field oriented along the zaxis. It is 5 m long and 3 m in diameter and it allows for the determination of the momentum and sign of charged particles. Silicon Detectors. It is composed of three separate parts: Layer 00 (LOO), the Silicon Vertex Detector (SVX) and the Intermediate Silicon Layers (ISL). Layer 00. This is the innermost part of the silicon detectors and is made up by a single layer of radiation hard silicon attached to the beam pipe [14]. Its purpose is to improve the impact parameter resolution for low momentum particles which suffer multiple scattering in the materials and readout electronics found prior to other tracking system components. Also it can help extend the lifetime of the tracking system in general, given that the inner layers of the SVX will degrade due to radiation damage. Silicon Vertex Detector. The SVX is segmented into three barrels along the zaxis and has a total length of 96 cm. Each barrel is divided into 12 wedges in 9, which contain five layers of silicon microstrip detectors. All layers are doublesided (Figure 35). .> . 1 Ii Figure 35: Isometric view of the three barrel structure of the CDF Silicon Vertex Detector. It is located outside the LOO from 2.4 cm to 10.7 cm in radial coordinate. Both r z and r 9 coordinates are determined. This idll1 I. im is used to tri:. r on displaced vertices which are an indication of heavy flavor content and helps with the track reconstruction. It is a complex system involving a total of 405,504 channels and unfortunately it is impossible to present it in any detail without going into too many technicalities. Intermediate Silicon Layers. The ISL is composed of three layers of doublesided silicon with axial and smallangle stereo sides and it is placed just outside the SVX. The geometry is less intuitive but it can be seen in Figure 34: there is one layer in the central region (\rl\ < 1), at a radius of 22 cm. In the plug region (1 < \rl\ < 2) two layers of silicon are placed at radii of 20 and 28 cm, respectively. The SVX and ISL are a single functional system which provides standalone silicon tracking and heavy flavor l..'.ii; over the full region rlK < 2.0. Central Outer Tracker. It is a large opencell drift chamber which provides tracking at relatively large radii, between 44 cm and 132 cm and it covers the region qrl < 1.0. It consists of four axial and four small angle (30) stereo superlayers. The superlayers are divided in small cells 0 and each cell contains 12 sense wires. The endview of the COT detector is shown in Figure 36. The cells are filled with a gas mixture of ArEtCF4 in proportions 50:35:15. The charged particles passing through the chamber ionize the gas and the produced electrons are attracted to the sense wires. When they arrive in the vicinity of the wire a process of avalanche ionization occurs and more electrons are produced and then collected by the wire. The location of the initial electron can be calculated based on the the sense wire which was hit and the drift velocity. This only describes how one 'point' of the trajectory is determined, but the process repeats in other cells and based on the location of many such hits a track trajectory is reconstructed. The important parameter to be reconstructed is the track curvature from which particle momentum is obtained. The COT has a resolution of about 0.7 104cm1, which leads to a momentum resolution of 6pr/pT ~ 0.;;' (GeV/c). The typical drift velocity is about 100k/m/ns. Figure 36: One sixth of the COT in endview; odd superlayers are smallangle stereo layers and even superlayers are axial. The COT allows for the reconstruction of tracks of charged particles in the r and r z planes. 3.2.3 The Muon System The Muon System is positioned farthest from the beam line and it is composed of four systems of scintillators and proportional chambers. They cover the region up to I < 2. In this analysis we only muons detected by the three central muon detectors known as the Central Muon Detector (CMU), Central Muon Upgrade (CMP) and Central Muon Extension (CMX). Since these systems are placed behind the calorimeter and behind the return yoke of the magnet most other particles are absorbed by them. However, an extra layer of 60 cm of steel is added in front of the CMP for the same purpose of absorbing other particles. These three systems cover the region I < 1.0. The 1.0 < I < 2.0 range is covered by the Intermediate Muon System (IMU), but we don't use it in this analysis. 3.2.4 The Trigger System As mentioned earlier in Run II bunches of protons and antiprotons collide every 396 ns. The average number of pp collisions per bunch crossing depends on the instantaneous llijii.il ,I but for typical luminosities in Run II we expect one pp collision or more per bunch crossing therefore if we were to record all events we would need to save 1.7 million events per second. The typical event size is about 250 kB so at such a rate we would need to save 435 GB of data per second. However most pp collisions are diffractive inelastic collisions in which the proton or antiproton is broken into hadrons before the two are close enough such that a l.,d core" interaction between partons can occur. These type of collisions are not of much interest and therefore there is no need to record them. The purpose of the tri.. r system is to filter out these less interesting events, categorize and save the remaining ones. This is achieved through a 3tier architecture shown in Fig. 37. Leveli (LI) and Level2 (L2) tri. r systems use only part of the entire event to make a decision regarding the event. They use dedicated hardware to perform a partial event reconstruction. At Level1 all events are considered. They are stored in a pipeline since the LI logic needs 4 /s to reach a decision, much longer than the 396 ns between two consecutive events. So while the decision making algorithm is executed by the LI hardware the event is pushed down the pipeline, which serves the purpose of temporary memory. When the event reaches the end of Dataflow of CDF "Deadtimeless" ] [ Trigger and DAQ I L1 Storage Pipeline: 42 Clock Cycles Deep L2 Buffers: 4 Events DAQ Buffers Levell: 7.6 MHz Synchronous pipeline 5544ns latency <50 kHz Accept rate Level 2: Asynchronous 2 stage pipeline ~20ps latency 300 Hz Accept Rate L1+L2 rejection: 20,000:1 PJW 10/28/96 Figure 37: CDF II Data flow. the pipeline the decision is made and the event is either ignored or allowed to move on to Level2. It is important to bear in mind that the LI tri2 .r is a synchronous pipeline, with decision making pipelined such that many events are present in the L1 tri2 .r logic simultaneously yet at different stages. Even though it takes 4 fis to reach a decision and even though events come every 396 ns the trij. r analyzes them all, just not one at a time. The LI tri2 .r reduces the initial rate of about 1.7 MHz to below 20 kHz. The Level2 tri. r is an .i..nchronous system with an average decision time of 20 pts. The events passing LI are stored in one of the four L2 buffers waiting for a L2 decision. If an event arrives from LI and all the L2 buffers are full the system incurs dead time and it is recorded during the run. The L2 tri. r has a an acceptance rate of about 300 Hz, another significant reduction. An event that passed L2 is transferred to the data acquisition (DAQ) buffers and then via a network switch to a Level3 CPU node. L3 uses full event reconstruction to make a decision whether to write the event on tape or not. It consists of a "farm" of commercial CPUs, each processing one event at a time. If the event passes this level as well it is sent for writing on tape. The maximum output rate at L3 is 75 Hz, the main limitation being the datalogging rate with a typical value of 18 MB/s. Events are classified according to their characteristics and separated into different tri.2 r paths. Some of these classes of events are produced copiously and in order to leave enough bandwidth for less abundant event types a prescale mechanism is put in place. For example a prescale of 1:20 keeps only one event out of 20 that passed the tri.. r requirements. CHAPTER 4 EVENT RECONSTRUCTION The raw data out of the many subdetectors contains a wealth of information which is not always relevant from a pli, i analysis point of view. For instance, in this analysis we need to know the moment of electrons, among other things. But what we do have in terms of raw data is a series of hits in the tracking system and energy depositions in the electromagnetic and hadronic calorimeters, and these readings could be caused by other particles, or may not be compatible with the trajectory of an electron in the magnetic field of the detector. Therefore detailed studies are necessary in order to find an efficient way of identifying raw data patterns compatible with those produced by an electron passing through the detector and at the same time reject as much fakes as possible. In short the task of the event reconstruction is to identify the particles which were present in the event and measure their 4momenta as well as possible. We will investigate this process in more detail for each kind of particle involved. 4.1 Quark and Gluons Quarks and gluons produce a spray of particles via parton showering, hadronization and decay. Therefore they do not interact with the detector directly but appear as a more or less compact set of tracks and calorimeter towers in which energy has been deposited. By "(,'Ip.... I we mean compact in the r]  plane. Such a detector pattern is called a jet and in this case the purpose of the reconstruction is to identify jets consistent with quark or gluon origins and estimate their overall energy and momentum. 4.1.1 Jet Clustering Algorithm There are a couple of algorithms to identify these jets and estimate their energy. In this analysis we used an iterative "fixed c, algorithm (JETCLU) for jet identification [15]. The idea is to find something like the center of the jet and then assign all towers within a given radius R in the 9q plane around this center to that jet. The algorithm begins by creating a list of all seed towers, or the towers with transverse energy above some fixed threshold (1 GeV). Then, for each of the seed towers starting with the highest ET tower, a precluster is formed by all seed towers within radius R of the seed tower. In this iterative process the seed towers already assigned to a precluster are removed from the list of available seed towers. For each precluster a new center is found by doing an ET weighted average of the 9q positions of the towers pertaining to the precluster. This is called centroidd". Now using the centroids as origin we can recluster the the towers, this time allowing for the inclusion of towers with energy above a lower threshold (100 MeV). Again we compute the centroid and the process is repeated until it converges, when the latest centroid is very close to the previous centroid. In the iterative procedure it is possible to have one tower belonging to two jets. But this would lead to inconsistencies because the total energy of the jets would not be equal to the total energy of the towers. Therefore after the iterative procedure is finished we have to resolve this double counting issue. One way is to merge the clusters that share towers. This happens if the overlapping towers' energy is more than 75% of the energy of the smaller cluster. But if this requirement is not satisfied each shared tower is assigned to the closest cluster. In order to find the 4momenta of the particles we assign a massless 4momenta for each electromagnetic and hadronic tower based on the measured energy in the tower. The direction is given by the unit vector pointing from the event vertex to the center of the calorimeter tower at the depth that corresponds to the shower maximum. The total jet 4momenta is defined by summing over all towers in the cluster in the following way: N S= (Ef+ Ead) (41) i= 1 N m + Ehad Sn h d COS ... 2) p = y,(E7 sin 61 cos em + E sin ad cos .C) (42) i=1 N py = E sin m sin m + Eha sin 0d had) (43) i= 1 N pz = (Eem cos 0 + E s ad) (44) i= 1 where E7, E^a, [ 0,T, 0,hd are the electromagnetic and hadronic tower energies, azimuthal and polar angles for the ith tower in the cluster. The jet 4momentum depends on the choice of R. For small values towers pertaining to the original parton are not included in the cluster, while for large values we risk merging jets pertaining to separate partons. A compromise used in many CDF analysis is R = 0.4, and this is the value used here as well. 4.1.2 Jet Energy Corrections The algorithm just presented returns an energy value that needs further corrections in order to reflect, on average, the parton energy. The reasons for the discrepancy are many, some instrumental and some due to underlying ]li,i, .,l processes. A few important instrumental effects are listed below: Jets in regions less instrumented, like in between calorimeter wedges or in the r = 0 region will naturally measure less energy. It is known that for low energy charged pions (ET < 10GeV) the calorimeter response is nonlinear, while in the energy measurement procedure it is assumed linear. Charged particles with transverse moment below 0.5 GeV/c are bent by the magnetic field and never get to the calorimeter. Fluctuations intrinsic to the calorimeter response. Important pl,' i. .,l effects are the following: The jet can contain muons which leave little energy in the calorimeter, and neutrinos which escape undetected. Therefore the cluster energy underestimates the parton energy. Choosing a radius R = 0.4 in the clustering algorithm we lose all towers rightfully pertaining to the jet but laying outside that radius. Extra particles can hit the same towers, coming either from other interactions present in the event or from the underlying event (the interaction of the proton and antiproton remnants, i.e. the quarks that did not take part in the hard process). CDF developed a standard procedure [16] to correct for such effects. The user can choose to correct only for certain effects using the standard corrections and correct other effects with more analysisspecific corrections. This is also the case for this analysis, so we are using the standard corrections only for the instrumental effects. From there we use Monte Carlo simulations to map the correlation between the parton energy and the (partially) corrected measured jet energy. 4.2 Electrons In this analysis we are using only electrons detected in the central calorimeter. Most if not all of an electron's energy is deposited in the electromagnetic calorimeter, therefore the reconstruction algorithm starts by identifying the list of seed towers, which are towers with electromagnetic energy greater than 2 GeV. Then, towers .,li.1:ent to the seed towers are added to the cluster if they have nonzero electromagnetic or hadronic energy and are located in the same ) wedge and nearest in r] direction. At the end only clusters with electromagnetic ET greater than 2 GeV and electromagnetic to hadronic energy ratio smaller than 0.125 are kept. However this last requirement regarding the ratio is ignored for very energetic electrons with energy greater than 100 GeV. What has been described above is just an "electromagnetic object" candidate. It serves as basis for identifying both electrons and photons. Further selection criteria [17] are necessary to identify electrons and separate them from photons or isolated charged hadrons, 7r mesons and jets faking leptons. These other criteria are listed below: A quality COT track with a direction matching the location of the calorimeter cluster must be present. The ratio of hadronic energy to calorimeter energy (HADEM) satisfies HADEM < 0.055 + 0.00045 E, where E is the energy. Compatibility between the lateral shower profile of the candidate with that of test beam electrons. Compatibility between the CES shower profile and that of test beam electrons. The associated track's z position should be in the luminous region of the beam, which is within 60 cm of the nominal interaction point. The ratio of additional calorimeter transverse energy found in a cone of radius R=0.4 to the transverse energy of the candidate electron is less than 0.1 (isolation requirement). 4.3 Muons Muons leave little energy in the calorimeter but they can be identified by extrapolating the COT tracks to the muon chambers and looking for matching stubs there [18]. A stub is a collection of hits in the muon chambers that form a track segment. The muon candidates are preselected by requiring rather loose matching criteria between the COT track and the stubs. As for electrons, we apply a set of identification cuts [17] to separate muons from cosmic rays and hadrons penetrating the calorimeter: Energy deposition in the calorimeter consistent with a minimum ionizing particle, usually hadronic energy less than 6 GeV and electromagnetic energy less than 2 GeV. Small energydependent terms are added for very energetic muons with track momentum greater than 100 GeV. The distance between the extrapolated track and the stub is small, compatible with a muon trajectory. The actual value depends on the particular muon detector involved (CMP, CMU, CMX) but it is around 5 cm. The distance of closest approach between the reconstructed track to the beam line (do) is less than 0.2 cm for tracks containing no silicon hits and less than 0.02 cm for tracks containing silicon hits (which provide better resolution). As for electrons, the associated track's z position should be in the luminous region of the beam, within 60 cm of the nominal interaction point. The ratio of additional transverse ET in a cone of radius R = 0.4 around the track direction is less than 0.1 4.4 N ii i i , NX IlIi ii' 'i escape detection entirely but since the transverse momentum of the event is zero, and that includes neutrinos, we can indirectly measure their total PT by summing all the transverse energy (momentum) measured in the detector and assigning any imbalance to neutrinos or other (undiscovered) long lived neutral particles escaping detection. This (Ill.ili il' is called iiiiig transverse (i i I[ and it is defined N = (Ecsin Oe" + Etdsin O@ad)cosQi (45) i=1 N Y = (E sin O" + Eadsin 08Id);:,,., (46) i=1 where E d, E" is the hadronic and respectively electromagnetic energy of the ith caloritemeter tower, 0' is the the polar angle of the line connecting the event vertex to the center of the ith tower and qi is a weighted average defined by: E Osin Orcosmn + hadEsn had( , i n Eemsn Om + Ehadsin hand (7) with '"",. ." weighted averages themselves but intratower. In the calculation of fi using the formulae above only towers with energy above 0.1 GeV are used. This requirement is applied individually to hadronic and electromagnetic components. The magnitude FT is given by _VT '+ 1+ (48) Since muons do not leave much energy in the calorimeter and raw jet energy measurements are systematically low it follows that the above quantity is only a first order approximation for the neutrinos' Pr and needs further corrections. The first correction is directly related to jet corrections. If we scale the energy of jets by some factor because that is a better match to parton energy then in computing the total measured ET we should replace the raw jet energy measured by the calorimeter with the corrected energy as given by the jet energy corrections. These corrections are applied only to jets with ET above 8 GeV, and therefore all calorimeter towers not included within such jets do not receive any correction. The second correction is related to muons being minimum ionizing particles, leaving little energy in the calorimeter. Therefore a better estimate of the total ET of the event is obtained by removing calorimeter towers associated with muons from the above calculations and replacing their contribution with the measured PT of the muons. In this analysis we use the missing ET value only for event selection. It plays no role in the reconstruction of the invariant mass and therefore more detailed studies on missing ET resolution are not included here. CHAPTER 5 EVENT SELECTION AND SAMPLE COMPOSITION The top quark decays so quickly that it does not have time to form any top hadrons and therefore a tt final state appears under different signatures based on the decay chain of the top quark: t W+b (51) W+ + 1 W+ qq (52) where I stands for one of the charged lepton types e, pt or T, q stands for u or c and q' for one of the "down" quarks d, s or b. The top quark can also decay to either a d or a s quark instead of b but the combined branching ratios for these two processes are below 1% and generally ignored. Based on these decay modes we can see that a tt pair decay can appear under three different experimental signatures: Six jets or sometimes more due to radiation, when both W bosons decay hadronically. This is the Ii., Ironic" channel. Four jets or more, a charged lepton and missing ET when only one W boson decays hadronically. This is the "ki .1 Il I i I channel. Two jets or more, two charged leptons of opposite sign and missing ET when both W bosons decay leptonically. This is the ",lil] 1.p1 i" channel. The scheme is complicated a bit because the T lepton also decays before detection and it can either "t i.,,f I.. iI" into a jet, if it decays hadronically, or produce an electron or a muon and more neutrinos, if it decays leptonically. However, regardless of the T decay mode, these events are difficult to identify and we decided to develop an algorithm which should work well with nonr events only. The branching ratios are defined essentially by the W branching ratios and lead to the following numbers: Table 51: tt decays Category Branching Ratio Dilepton (excluding 7) 5% Dilepton (at least one r) 6% Lepton+Jets (excluding r) 30% r+Jets 15% Hadronic 44% 5.1 Choice of Decay Channel The choice for the decay channel has to take into account two more factors, the intrinsic f., reconstruction resolution and the signal to background ratio (S/B). The reconstruction resolution is worse when more information is missing. Let us take a look at each channel individually: In the dilepton channel we measure well the lepton moment, we have some uncert.,iiii on the two b quark moment due to various effects described in the previous chapter, and we don't measure at all the moment of the two neutrinos (6 variables). In the lI i1 ii I i 1 channel we measure well the lepton momentum, we have some uncert.iiiil v on the four quark moment and we don't measure at all the neutrino moment (3 variables). In the hadronic channel we have some uncert.,iil v on the six quark moment. In each case we can reduce the number of unknown variables by applying transverse momentum conservation, which yields two constraints, but since this is the same across the channels we can just compare them based on the facts stated above. If nontt backgrounds were absent we would certainly pick the hadronic channel since it has the highest branching ratio and least loss of information because no neutrinos escape detection. However the S/B ratio for Standard Model tt in the hadronic channel, without any t..'ii,; requirement, is about 1:20 while the S/B ratio for the l1i'. i I ii I channel is roughly 1:2 with a branching ratio (2/3) comparable to the hadronic channel. Even though the resolution analysis would also favor the hadronic channel, with such a large background it has, most probably, less potential than the l( i'1 i I ii I channel. The dilepton channel has most unknown variables leading to poorest reconstruction resolution and significantly lower branching ratio, even though it enjoys the best S/B around 3:1. This qualitative analysis led us to pick the 1 i .i l i I i, channel as best candidate for this analysis at the beginning of Run 2 when we expected less than 1 fb1 of integrated luminosity available for this dissertation. The final dataset on which this analysis is performed corresponds to 680 pb1 of data. 5.2 Data Samples The data used in this analysis was collected between February 2002 and September 2005. A preselection of the data is carried out by the collaboration and bad runs in which various components of the detector malfunctioned are removed. The remaining good data corresponds to a total integrated liluiii ,ii' v of 680 pb1. Two distinct datasets were used, the high PT central electron dataset and the high PT muon dataset. The electron dataset is selected by a tri .r path that requires a Level3 electron candidate with CEM E} > 18GeV, Ehad/Ern < 0.125 and a COT track with pr > 9GeV/c. The muon dataset is selected by a tri. r path that requires a Level3 muon candidate with pr > 18GeV/c. We use only CMX muons or muons with stubs in both CMU and CMP subdetectors. Dilepton e p events can appear in both datasets and one has to be careful to not double count them. 5.3 Event Selection In order to select tt events in the 1: i.i il I I channel we have to require that each event contains at least four jets, an electron or a muon and 'VT consistent with the presence of a neutrino, that is, a fT value well above the fluctuations around the null measurement. Certainly this leaves a lot of space of maneuver with respect to the pr range and the minimum ET threshold required for each object. An exhaustive study for optimizing the cuts has not been done independently, however we adopted the widely used cuts for Standard Model tt selection in the l l 'i I ii 1 channel which can be found in most CDF top analyses. These cuts are the result of a great amount of work throughout Run 1 and Run 2 and are doing a fine job at separating signal (Standard Model tt in this case) from backgrounds. There could be better cuts that improve the resonant tt S/B but further studies would be necessary to understand the overall effect on sensitivity, and what would be an optimum for a 400 GeV/c2 mass resonance may not be so for a 800 GeV/c2 resonance. The task of studying in detail the impact of selection criteria on sensitivity will have to be addressed in a later version of the analysis. However we did compare the sensitivity among three versions of jet selections and chose the best, as it will be explained later. Table 52: Event Selection Object Requirements Electron GEM, fiducial, not from a conversion ET > 20 GeV + ID cuts Muon CMX or (CMU and CMP) detectors, not cosmics PT > 20 GeV + ID cuts T Corrected T > 20 GeV Tight Jets Corrected ET > 15 GeV, rjl < 2.0 at least four tight jets Loose Jets Corrected ET > 8 GeV, Ir p < 2.4 not used for selection per se, but counted as jets In table 52 we present in a succinct form the requirements [19] for the selection of electrons, muons, jets and the f cut used. Positrons and antimuons follow the same selections, of course. By "fiduciality" of electrons it is meant that they are located in well instrumented areas of the towers, not near tower edges for instance. Conversion removal algorithms are used to remove electrons or positrons that come from photons hitting the various materials found before the calorimeter and producing ee+ pairs. We are not interested in such electrons. The removal per se is done by a standard CDF algorithm [20]. There is also an algorithm for eliminating cosmic ray muons [21] and it is used to veto on such muons in our selection. We also require one and only one lepton and that the distance between the lepton's track ZO coordinate and the jets' vertex position is less than 5 cm, since consistency with tt production requires that all our objects must come from the same interaction point. The identification criteria complete the event selection rules and were discussed in the previous chapter, together with the corrections for FT and jets. A simple study was performed in which we compared the sensitivities of three jet selection criteria: exactly tight four jets four tight jets + extra jets (or none) three tight jets + extra jets (> 0). The first option provided the best sensitivity and we adopted it for our selection. 5.4 Sample Composition The leading Standard Model processes that can produce events passing these selection criteria are the following: W production associated with jets ( W+jets). The W decays leptonically producing a lepton and 1T . tt events. Multijet events where one jet fakes an electron. Will will refer to these generically as QCD. Diboson events such as WW, WZ and ZZ. The relative contribution of these processes can be derived if we know the theoretical crosssection and the acceptance for each of them. Table 53: Crosssections and acceptance Process crosssection Acceptance SI\! tt 6.7 pb 4.5% WW 12.4 pb 0.14% WZ 3.7 pb (I 11' . ZZ 1.4 pb 0.02% W+jets ? 0.7% QCD ? 0.7% However the W+jets and QCD crosssections are not known theoretically with good precision, but in other CDF top analyses the number of events from these processes is extracted from the data. For this analysis we decided to use only the ratio of the expected number of events as derived by these analyses and fit for the absolute normalization since in those analyses no room was left for any nonStandard Model process, and that could bias our search. The constraint used is given below: NQCD 0.1 (53) Nw where N represents the expected number of events. Resonant tt acceptance are listed for comparison in Table 54. The search algorithm finds the most likely values for Nw and signal crosssection as a function of resonance mass, and it is also able to compute the statistical Table 54: Signal acceptance Mxo (GeV/c2) 450 500 550 600 650 700 750 800 850 900 Acceptance 0.047 0.051 0.055 0.057 0.059 0.062 0.062 0.063 0.063 0.061 relevance of the most likely signal crosssection value. We will explore it in detail in the next chapters. CHAPTER 6 GENERAL OVERVIEW OF THE METHOD AND PRELIMINARY TESTS This analysis contains two major pieces, one is the tt invariant mass (3 [,, ) reconstruction and the second is the search for a nonStandard Model component in that spectrum, in particular a resonance contribution. The reconstruction is complicated because our parton level final state, after the top decay chain, is composed of two bquarks, two light quarks, a neutrino and a charged lepton. Experimentally, we measure accurately only the lepton, which makes the task of reconstructing the tt invariant mass spectrum with good precision nontrivial. There are a total of seven poorly measured or unmeasured variables: four quark energies and three components of neutrino moment. In fact the jet direction is also smeared compared to the parton direction, but this is considered a second order effect compared to the above mentioned effects. Throughout the remaining of this dissertation we will always assume that the jet direction is a good approximation for the parton direction. In the CDF Run 1 analysis [11] a somewhat straightforward approach was used to reconstruct the invariant mass spectrum. A X2 fit was constructed based on jet resolutions and the knowledge of W and t masses and it was used to weight the unknown parton values. Minimizing the X2 with respect to the free parameters (the unknowns listed above) provided an estimate for their most probable values. Then those values were used to compute the invariant mass of the system, 1i ... In this dissertation we use an innovative approach using matrix element information to reconstruct the tt invariant mass spectrum. The maximum information about any given process is contained in its differential crosssection and it is therefore natural to think that by making use of more information in the analysis one can improve resolution and therefore sensitivity. Since we decided to pursue a model independent search we will not be able to use any resonance matrix elements. We will use Standard Model tt matrix element to help with weighting the various possible parton level configurations and extract an average value for the invariant mass, event by event. The invariant mass distribution obtained in such a way follows closely the Standard Model tt spectrum at parton level and it is also a good estimator for the resonant tt events as it will be shown later. In order to validate the matrix element machinery we performed a series of tests by implementing a conceptually simpler matrix element analysis, which is the top mass measurement using matrix elements. Our tests include only Monte Carlo simulation studies but they played a crucial role in pushing this analysis forward since our results were very similar to those of groups actually working on the top mass measurement using matrix element information. The remainder of this chapter will present these studies which will also familiarize to reader with the technical details common to both analyses. In the next chapter we will show how to extend the algorithm in order to reconstruct the I,, spectrum. 6.1 Top Mass Measurement Algorithm The purpose of this algorithm is to build a top mass dependent likelihood for each event using the differential crosssection for the I\ tt process. We will use the leading order (LO) term in the Standard Model tt crosssection formula. The final state is made up of the 6 decay products of the tt system. Let pi be their 3momenta. We have the following equation representing the conservation of the transverse momentum of the system: 6 =0 (61) i=1 This is a constraint on the seven unknown variables mentioned in the previous chapter and it will be used in all the top mass tests we will show in this chapter. In reality we have initial and final state radiation (ISR and FSR) which leads to a nonzero PT value. Still, the average PT is null so constraining it to 0 should not bias the result for top mass but maybe only increase the statistical error. For the resonance search analysis though we will use the PT distribution from Monte Carlo simulation and integrate over it since it helps narrow the reconstructed resonance peak. The probability of a given parton level final state configuration pi relative to other configurations is given by: 1 F  dP(pA'ntop) dZ= t dZbfk(Za)fl(Zb)d(Tkl (A (p top, ZaP, ZbP) (62) (Tm(fMtop) or in short dP (1imt) op= part(im t JJ dtop 3} (63) Indices k, I cover the partons types in the proton and antiproton respectively. Summation over both indices is implied. The parton distribution functions (PDFs) are given by fk(z) and P, P designate the proton and antiproton momentum. Plugging in the differential crosssection formula d(Tu / I' ) Ak ( 2 (2)44 11(d3E (64) 4EkE, n, ,  (27)32Ei one can obtain an explicit form for 7part(Almtop). The top mass (rtop) enters as a parameter. We combine the probability densities (7) of all events in the sample into a joint likelihood which is a function of mtop: (65) L(mtop) = X1iX2...7x We expect that maximizing this likelihood with respect to the parameter (mtop) yields its correct (input) value, as it should. The algorithm presented above is only a first step, since it assumes we know the parton level moment which is not true experimentally. But the treatment of more realistic situations in which we don't measure the final state completely or accurately enough follows the same line of thought, basically we compute the probability density of observing a 1: ili Ii I event: l7obs(Jlj2,J3,j4,I' 'II 4 = P~prt(ip(1), p(2), Pp(3), pp(4)', 1, P top)d37 Ti(jp) p(i))d (66) p i=1 In this formula we assume that the first two arguments of the parton density (7prt) function represent the bquark moment, the jet 3momenta are denoted by j and the parton 3momenta by p1. T4(jp) is the probability density that a parton with 3momenta ; is measured as a jet with 3 moment j. These functions are called partontojet transfer functions. We use different transfer functions for b quarks and lighter quarks, so we added an index to differentiate the two. With our conventions T1 = T2 Tb and T3 = T4 = Tight. In practice we approximate the parton direction with the jet direction, as mentioned earlier, which simplifies the calculations a bit. Even with bt..._.in._; information available, there is no unique assignment of jets to partons. This indistinguishability is addressed by summing over all allowed permutations using the p E S4 permutation variable. A permutation is allowed if it doesn't contradict available bt... i._; information. The procedure to extract the top mass is the same as in the idealized case of a perfect measurement of the final state discussed before, that is, combine all events in a joint likelihood and maximize it with respect to the parameter mtop. Figure 61: Main leading order contribution to tt production in pp collisions at s = 1.96 TeV 6.1.1 The Matrix Elements (\II) The leading order matrix element for the process qq tt W+bWb qqblvb (Fig. 61) is not easily calculable analytically without making any approximation. We found it useful to compute the ME directly using explicit spinors and Dirac matrices because this allows us to compute new, nonStandard Model matrix elements very easily in case we wanted to incorporate them in the algorithm later on. Dedicated searches for specific models (spin 0 resonance, spin 1 resonance, color octet resonance) would be interesting as well, but we will not address them in this dissertation. Ignoring numerical factors the quark annihilation diagram amplitude is given by MA qq V(Pq)pMU(Pq) u(p)y'(1 nj)v(pd) u(I )'(1 5)(p,) pt2 2 m+ mtp Pt p m,+ nimtFt Q 0 PW p /m2 p PW P /m2 67) (pq + Pq)2 p2+ 2w + imwFw PW M 2 imwFw If we consider the masses of the light quarks and leptons negligible we can simplify the expression of the W propagators so the ME reads V (pq ," "(P) u(pun)7(l 5)v(pd) u( )y7(l y5)V(p,) qq (pPq + Pq)2 P2'+ m + imww PW m2 + ww W W 1:, rnt r f. +rnt (p2 m 2 + imtFt 2pT Mi + MtFt We tested our numerical calculation using explicit Dirac matrices and spinors with the analytical calculation for the squared amplitude by Barger [22] and we found the two calculations in good agreement. That calculation uses the narrow width approximation (NWA) in treating the top quark propagators and therefore the two methods are not equivalent when one or both of the top masses are offshell. We also tested our implementation on simpler QED matrix element calculations and it produced results identical with their exact analytical expressions. 9 t 9 S t/ ^AAVVV AW/ t Figure 62: Gluongluon leading order contribution to tt production in pp collisions at = 1.96 TeV The gluongluon production mechanism is described by three diagrams in Fig. 62, in which the top decays have not been depicted explicitly. The matrix element needed in the crosssection formula for the gluongluon production mechanism has the structure: IM,,2 A1 A2 A32 (69) color where Ai are the amplitudes corresponding to the three diagrams. The color sum covers all possible color configurations for the gluons and quarks. This expression is not optimal with regard to CPU time if we were to do these sums as they stand. We can rewrite it as IM,,g2 = Y( 1 2 + I12 + IA32 2* {A I} + 2 2. 1R{AAX 2 R{A2A*N10) color This form is very convenient, the color sums can be evaluated for each individual term regardless of the kinematics because the amplitudes are factorized as A Akin Color We can write again IMg2 l ki22 f in 12 . JA1 A I' f A I'+ Re{f1 A + fkin A + 23 A A, } (611) All the color summing is encoded in the six constants fi, fij. We found these to be 3/16, 1/12, 1/12, 3i/16, 3i/16 and 1/48 respectively. We crosschecked against the analytical formula available for the 2  2 process described in the diagrams above (ignoring the top decays) and found them in perfect agreement. The procedure just presented works as well for the 2 6 process and this is how we compute it. 6.1.2 Approximations: Change of Integration Variables The method as presented involves seven integrals (three over neutrino 3momentum and four over quark moment) and summing over combinatorics. If for instance we choose to set the tt transverse momentum to zero that would amount to two constraints reducing the number of integrals by two. Or we could choose to set the W or top on shell, depending on the level of precision and speed desired. Even from a purely numerical point of view, it would be easier to integrate only around the top and W mass poles rather than over the large range of the original variables mentioned before. For all these reasons a change of variable was performed. The new variables are the tt transverse momentum and the intermediate particle masses mwi, mw2, mT1, mT2. This is a set of only six new variables, which means we need to keep one of the initial variables unchanged (one of the light quarks' energy). The change of variable and the associated Jacobian calculations are detailed in the Appendix. Since the calculations are a bit lengthy we wanted to make sure no mistake was made so we used simulated events where all variables are available and any change of variables can be readily checked. We found that the change of variable implementation works very well. In the implementation of the algorithm we always use these variables, both for these preliminary top mass tests and for the [,, reconstruction. 6.2 Monte Carlo Generators For some of the top mass tests we used CompHep 4.4 [23], which is a matrix element based event generator. One can select explicitly which diagrams to use for event generation. CompHep preserves all spin correlations and offshell contributions since it doesn't attempt to simplify the diagrams in any way. CompHep generates events separately for each diagram uu  tt dd  tt and gg  tt. We also used Pythia [24] and Herwig [25] official CDF samples ("Gen5") but the first tests for top mass were done with parton level CompHep events and then with Gaussian smeared partons. The Gaussian smearing of parton energies is meant to simulate the relationship between the jet and parton energies. 6.3 Basic Checks at Parton Level Most Likely Top Mass mpv Most Likely Top Mass mpv Entries 250 Entries 250 j Mean 175 j Mean 175 70 RMS 0.1709 5 70 RMS 0.1721 > X2 1 ndf 11.37/9 > :2 /ndf 9.428 9 60 Prob 0.2512 Prob 0.3987 Constant 58.42 4.925 60 Constant 56.3 4.884 0 Mean 175 0.01045 Mean 175 0.01081 0 Sigma 0.159 0.008629 50 Sigma 0.1664 0.009748 40 I 40 30 30 20 20 10 10 3 173.6 174 174.6 176 176.6 176 176.5 177 W3 173.5 174 174.5 175 176.6 176 176.5 177 GeV GeV Figure 63: Reconstructed top mass from 250 pseudoexperiments of 20 events at parton level with mt=175 GeV/c2. The left plot is derived using only the correct combination, while the right plot uses all combinations Finding the top mass when the final state is known or measured perfectly is straightforward so we expect our method to produce the correct answer without any bias. Using uu  t CompHep events, we performed 250 pseudoexperiments of 20 events each. Which means that we extracted the top mass from a joint likelihood of 20 events each time. We repeated this exercise for various generator level top masses to make sure there is no mass dependent bias. First, we used only the correct combination in the likelihood, that is, we not only assumed to have measured the parton 3momenta ideally, but also identified the quark flavors. For mt = 175 GeV the reconstructed mass is shown in the right plot of Figure 63. As it can be seen, we get back the exact input mass. Similarly good results were obtained for other masses. Next we let all 24 combinations contribute to the event likelihood by summing over all permutations and repeated the same exercise. The reconstructed top mass is barely modified by the inclusion of all combinations, as shown in the second plot of Figure 63. Again, tests on other samples with different top masses didn't produce any surprise. These results are summarized in Figure 64 showing the output (reconstructed) mass vs input mass when using all combinations. The slope is consistent with 1.0 and the intercept is consistent with 0, which proves that there are no mass dependent effects, at least not in the mass range of interest. Perhaps it would be useful to remind the reader that the purpose of these studies is to establish the validity of the matrix element calculations and overall correctness of implementation of a nontrivial algorithm. Otherwise they are quite simple. We also looked at the rms of the pull distributions for each mass and it was found to be 1.0 within errors, which is a more compelling indication that we are modeling these events very well with our likelihood. Top Mass : Reconstructed vs True S2 / ndf 1.251/3 0 1 8 5 ................... Prob 0.7408........ ....... p0 0.1337 0.1102 p1 1.001+ 0.0006387 180 175 170 1 6 5 ............................................................I.............................. ...............................I........... 165 165 170 175 180 185 GeV Figure 64: Reconstructed top mass vs. true top mass from pseudoexperiments of 20 events using all 24 combinations, at parton level 6.4 Tests on Smeared Partons A more realistic test involves a rudimentary simulation of the calorimeter response obtained by smearing the parton energies (the four final state quarks' energies). Also, the neutrino 3momentum information is ignored in reconstruction. We used 20% Gaussian i' .iiir._. which is quite realistic when compared to partontojet transfer functions' rms. The tt transverse momentum was taken to be zero and also the top quark was forced on shell, thus the number of integrals was reduced to just three. We used the same uu  t CompHep events for these tests but later we did check with Herwig events and the results were similar. The same pseudoexperiments of 20 events were performed and in Figure 65 we show the reconstructed mass vs the true mass for the right combination and for all 24 combinations. Top Mass : Reconstructed vs True I X2 / ndf 1.421 3 / 0 185 Prob 0.7007 p0 0.214 1.996 p1 1.001 0.01149 180  175 170 165  165 170 175 180 185 GeV Top Mass : Reconstructed vs True I 0 185 180 175 170 165 X2 / ndf 4.742 / 3 Prob 0.1917 pO 1.33 2.384 p1 1.009 0.01374  165 170 175 180 185 GeV Figure 65: Reconstructed top mass vs. true top mass from pseudoexperiments of 20 events with smearing. The left plot is derived using only the correct combination, while the right plot uses all combinations We fit the pulls from pseudoexperiments with a Gaussian and the returned width was 1.09 0.07 for the 175 GeV sample, again consistent with 1. We observed similar pulls for other masses as well. The purpose of this set of tests was to validate the new additions to the algorithm implementation: transfer functions, transformation of variables and integration over unmeasured quantities. The success of this tests gives us confidence that the more realistic version of the algorithm is well designed and well implemented. 6.5 Tests on Simulated Events with Realistic Transfer Functions 6.5.1 Samples and Event Selection We used CDF official tt samples generated with Pythia and Herwig event generators. We apply the event reconstruction and event selection described in the previous chapters requiring for each event to contain one and only one reconstructed charged lepton, at least four tight jets and missing ET > 20 GeV. 6.5.2 Transfer Functions Transfer functions are necessary when we run over simulated events or data in order to describe the relationship between final state quark moment and jet moment. In this case we are interested in the probability distribution of the jet energy given the parton energy. This distribution varies with the energy and pseudorapidity of the parton, so we bin it with respect to these variables. Since the detector is forwardbackward symmetric we only need to bin in absolute pseudorapidity. We have only three bins in absolute pseudorapidity, with the boundaries at 0 0.7, 1.3 and 2. The parton energy bins are determined based on the statistics available, requiring minimum 3000 partonjet pairs per energy bin. This allows for a rather smooth function which can be fit well. For example the central region bquark energy bin boundaries are chosen to be 10 GeV, 37 GeV, 47 GeV, 57 GeV, 67 GeV, 77 GeV, 87 GeV, 97 GeV, 107 GeV, 117 GeV, 128 GeV, 145 GeV, and 182 GeV. Anything above 182 GeV is considered part of one more bin. We should perhaps emphasize that these are parton energy bins. In order to derive the transfer functions we need to match jets to partons first. For matching purposes we require that all four final state quarks are matched uniquely to jets in a cone of 0.4, that is, the AR distance between the parton direction and jet direction is less than 0.4. If this requirement is not met, we do not use the event for deriving transfer functions. The direction smearing is considered a second order effect and ignored, which amounts to identifying the quark direction with the jet direction. This approximation can be corrected to some degree by using "effective wi li h for W and top instead of theoretical values. In other words the smearing in direction leads to a smearing of the mass peak even when there is no energy smearing. The effect can be quantified based on simulation and a corresponding larger width can be en'pl'v.d in the analysis. In fact we do use such a larger width (4 GeV) for the hadronic W mass in our resonance search analysis. Our studies showed that it narrows the resonance peak a bit, but no such tests were performed for top mass. 4500 2200 1200 2000 4000 1800 1000  3500 1600 3000 1400 800 2500 1200 2000 1000 1500 400 1000  1000 200 500 200 I I 10 0 1I I I . I 0 I 1 08 06 04 02 0 02 04 06 08 1 1 08 06 04 02 0 02 04 06 08 1 1 08 06 04 02 0 02 04 06 08 1 Figure 66: Light quarks transfer functions (x 1 ), binned in three Eparton absolute pseudorapidity regions [0, 0.7], [0.7, 1.3] and [1.3, 2.0] In Figures 66 and 67 we show examples of transfer functions for both light quarks and bquarks, respectively. We fit the shape with a sum of three Gaussians, which works fine. The variable plotted is 1 Ejet/Eparton, since it varies less with 57 4500 2200 1200 4000 2000 0 3500 1800 3000 1600 800  1400 2500 1200 600 2000 1000 1500 800 400 600 100 400 200 500 200 1 ,I0 ,, ,I ,,I I  I l 0 ,.I,..L. 0 0I 1 8 06 4 02 0 02 04 06 08 1 1 08 406 4 4 2 0 02 04 06 08 1 1 08 06 04 4 2 0 2 4 06 08 Figure 67: bquarks transfer functions (x 1 Ejet ), binned in three absolute Everton pseudorapidity regions [0, 0.7], [0.7, 1.3] and [1.3, 2.0] the parton energy. It is related to the distribution we introduced as '1.I,I. r f',ll, l i i via a simple change of variable. Our transfer functions are between parton energy and corrected jet energy, as explained in chapter 4. With these tools in place we ran similar pseudoexperiments on the Herwig sample. The returned mtopvalue was 178.1 0.4 GeV/c2 and the pulls' width was 1.05 0.09. The correct (generated) mass for this sample is 178 GeV/c2. We did not run any other tests because the only change we made in the algorithm at this stage was to plug in realistic transfer functions and run it over fully simulated events. As such, the only new thing that needed testing was the derivation of the realistic transfer functions based on Monte Carlo simulation. This is by far a simpler business than the implementation of matrix elements calculations and change of variables together with the rest of the machinery. Based on the results presented above we concluded that our transfer functions' implementation is fine and the algorithm as a whole works very well, is properly constructed and implemented. Also, our top mass results on Monte Carlo were very similar to those of analyses doing the top mass measurement using matrix elements. In the next chapter we will show how the top mass matrix element algorithm can be extended to compute the tt invariant mass, i,, CHAPTER 7 ,1 ., RECONSTRUCTION 7.1 Standard Model tt Reconstruction All the tools developed for the top mass can be turned around to reconstruct any kinematical variable of interest, in particular f.,, Let's assume for simplicity of presentation that we know which is the right combination, that is, we know how to match jets to partons. In that case P({p}, {j}) 7 prt({p}) T({j} {p}) (71) defines the probability that an event has the parton moment {p} and is observed with the jet moment {j}. In our notation {p} and {j} refer to the set of all parton and jet 3momenta. Integrating on the parton variables, given the observed jets, we obtain the probability used for the top mass measurement. However, the expression provides a weight for any parton configuration once the jets are measured. Any quantity that is a function of parton moment can be assigned a probability distribution based on the ,.,I, r" distribution above, Mt included, and this is our approach. Technically this amounts to the following integration: p(xz{j}) = 7rprt({p}) T({j}{p}) (x if., ({p})){dp} (72) with p(xz{j}) being the I,, probability distribution given the observed jet moment. It should be noted that if we remove the delta function we retrieve the event probability formula used for the top mass measurement method presented before, and therefore all the validation tests presented before are as relevant for S.,, reconstruction. In terms of the modifications in the algorithm these are also minimal, there is nothing much to be added except histogramming .1 ,, during integration. In other words we obtain an invariant mass distribution per event. We will use the mean of this 1,, distribution as our event 1 ,, value. Before running on all events in our various samples and producing templates we want to make sure the .,, reconstruction algorithm works well. We selected events in which we could match uniquely partons to jets and which contained only four tight jets. These are the circumstances that allow full consistency between the reconstruction algorithm and the events reconstructed and that is a selfconsistent test of the method, which is what we intend to show here. hdif mttReco mttHepg Entries 5000 Mean 0.2261 800 RMS 24.93 Smtt HEPG 1200 700 1000 600 mtt RECO 500 800 400 600 300 400 200 200 100  0 100 200 300 400 500 600 700 800 9001000 00 150 100 50 0 50 100 150 200 M, [GeV] GeV Figure 71: ., reconstruction for the correct combination and for events with exactly four matched tight jets. We ran the algorithm on these selected events and we were able to reconstruct S.,, back to the parton level as it can be seen in the left plot of Figure 71. Both plots are produced after running on events selected from the CDF official Pythia sample. Since we use the Standard Model tt matrix element we do expect to reconstruct these events very well and that seems to be the case indeed, as it is shown also in the right plot of Figure 71. There the difference between the reconstructed value and the true value is histogrammed in order to see the intrinsic resolution and check for any bias. The results are very good and we consider the testing and validation part of the analysis ended. 800 mtt RECO 700 600 500 400 300 200 100 0 1 1 11" 1 11 1 1 1' 11 ""1 '" ' 0 100 200 300 400 500 600 700 800 9001000 Mt [GeV] Figure 72: .1f, reconstruction including all events Since in reality we don't know which is the correct combination we adopt the top mass method approach and sum over all allowed combinations in the formula 72. We expect the right combination to contribute more than the others as it happens for the top mass analysis. The ,1 as reconstructed for all events, without any of the requirements mentioned above, is shown in Figure 72. This is what we expect to be the Standard Model contribution to the I,, spectrum in the data. Some examples of event by event reconstruction are shown in Figure 73. The 4th event is a dilepton event and the 8th is a 1.,i, IP iI event. Interestingly Entries 2000 0.14 Mean 396.7 0.14 RMS 20.21 o.12 0.12  0.1 0.1  0.08 0.08  0.06[ 0.06  0.04 0 .04 0.02 0.02 0 .5 .... 7.... 8.... I. 1.. .0 300 400 500 600 700 800 900 1000 300 400 Figure 73: Examples of reconstruction, event by event. they have larger widths than the others which are all 1: i1.1ii I 1 events. Adding combinations together can lead to double or multiple peaks. The top mass used on data is mtop = 175 GeV. Therefore this is the value used in our algorithm when producing I,, templates corresponding to various processes. Figure 74 shows the actual template used for fitting the data, derived by fitting 5000 reconstructed events. Certain approximations were made, since we cannot perform all integrals which appear in the formal presentation because the CPU time involved would be SMttbar Template Mean_SIVM Enties 5000 Mean 456.6  RMS 92.26 4450 S/ ndf 48.09/56 400 Prob 0.7649 Constant 14.65 0.1266 350 Slope 0.2152 0.005337 Expo 0.6148 0.003323 300 250 200 150 100 50 F l l I l + 0 200 400 600 800 1000 1200 Mt [GeV] SMttbar Template o a) D A  a I I I I I I Mean_ Enties Mean RMS 2 / ndf Prob Constant 1' Slope 0.215 Expo 0.614 I l I II^ 0 200 400 600 800 Figure 74: .V, template for Standard Model tt events. astronomical, even using the computing farms commonly available to CDF users. This is so because we need to model the ,, spectrum for 10 signal samples and a couple of backgrounds, and then perform the systematics studies which require recomputing the templates each time. As it was mentioned in the previous chapter, the implementation uses a different set of variables for integration, namely the masses of the two W bosons, the masses of the two top quarks, the total transverse momentum of the tt system and one "W" quark energy. Studies showed that the best approach, given the CPU time limitations, is to set the two top quarks' masses on shell and also set on shell the mass of the W which decays leptonically, leaving us with four integrals to SIVtt 5000 456.6 92.26 48.09 / 56 0.7649 4.65 0.1266 2 0.005337 18 0.003323 1000 1200 M [GeV] ,,,, perform. Even so, for systematics studies we needed about 100,000 CPU hours and we used extensively the CDF computing farms. 7.2 Signal and other Y \ Backgrounds The Monte Carlo samples for signal and all other Standard Model backgrounds (besides tt are run through the same algorithm, thus producing new distributions corresponding to signal and backgrounds respectively. Even though the signal is not 100% correctly modeled by the Standard Model tt matrix element, we expect the reconstruction to work quite well since a significant part of the matrix element is concerned with the top and W decays and that won't depend on the specific tt production mechanism. Especially in the case of a spin 1 resonance the differences between the correct resonance matrix element and the Standard Model matrix element are minimal, since the gluon is a spin 1 particle after all. Even tough the methods presented in this dissertation can be applied to more general cases, the actual limits we are deriving at the end are valid for vector resonances because the Monte Carlo signal samples were generated with a vector resonance model. We want to remind the reader that it was our initial decision to do a model independent search anyway. The results are not completely model independent only because of the Monte Carlo generators used to produce signal samples. Applying the reconstruction to nontt events doesn't produce any particularly meaningful distributions, but they are backgrounds needed to model the data. In what follows we briefly describe the results obtained when running this reconstruction method on the various backgrounds needed in our analysis and presented in a previous chapter. Signal samples We generated signal samples with resonance masses from 450 GeV/c2 up to 900 GeV/c2, every 50 GeV/c2, using Pythia [24]. The reconstructed ,1 for all is shown in Figure 716. The peaks match very well the true value of > >  450o S450 500oo 400 > '>350 400  S300 300 250 S200 200 150o 100 100 50 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Mtt [GeV] Mt [GeV] Figure 75: Reconstructed invariant mass for a resonance with Mxo = 650 GeV. The left plot shows all events passing event selection, while the right plot shows only matched events the resonance mass. In order to better understand the low mass shoulder we split these events in three orthogonal subsamples: events with all four jets matched to partons, mismatched events and fake k1id .1i I I. events (dilepton or hadronic events passing the klid' .1 I i , event selection). The method is expected to work well on matched events and indeed this is what we see in Figures 75 and 76. The shoulder is given by the superposition of mismatched events and fake 1: i..i, I iI . events on top of the nice peak from matched events. The generated width for the resonance was 1.2% of the resonance mass. As it can be seen the reconstructed resonance mass is much wider, due to the relatively large uncertainties in jet measurements and non measuring the neutrino z component at all. However the peak remains prominent enough to be easily distinguished from the exponentially dropping Standard Model processes. SW+jet samples We use the CDF official W + 4 partons ALPGEN [26] samples which are then run through Herwig for parton showering. We looked at W + 2b + 2 S90 80 30 680 S20 50 40 15 30 10 20  5 10  0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Mt [GeV] Mt [GeV] Figure 76: Reconstructed invariant mass for a resonance with Mxo = 650 GeV. The left plot shows mismatched 1: i.1.i, I i I events and the right plot shows nonk i .1 ii I r 1 events partons also but decided not to include it explicitly since the shape is very, very similar and the expected contribution at the level of 12% compared to 60% or more for the W + 4 partons. These can be seen in Figures 77, 711, 712 and a direct comparison of fit templates is shown in 715. So all W+jets events are modeled by the W + 4 partons sample. QCD For QCD we used the data to extract the shape. MIill ijet data is scanned for jets with high electromagnetic fraction which are reinterpreted as electrons based on the assumption that the jets that do fake an electron are very similar to the ones just mentioned. With that said, the usual event selection is applied and the events are reconstructed just like the others. This process produces the template shown in Figure 79. The shape is not much different from W + 4 partons, in fact they are quite close as assumed in the CDF Run 1 analysis when the QCD template was ignored altogether. Dibosons WW, WZ and ZZ The crosssections for the WW, WZ and ZZ processes are 12.4 pb, 3.7 pb and 1.4 pb. The acceptance follow the same trend with 0.1.!' , 0 11,' and 0.02% respectively. Moreover, the WZ and ZZ official samples have fewer events left after event selection and the fits have larger errors. Given that WW dominates anyway we decided to use only that template but increase the acceptance such that the expected number of events will cover the small WZ and ZZ contributions. Since overall the whole diboson part is almost negligible this procedure isn't expected to have any impact other than simplifying the analysis. It can be added that the WW template which is shown in Figure 710 is also very similar to the Standard Model tt W + jets and QCD templates. We put all of them on top of each other for easy comparison in Figure 714. All these templates are used to fit the data and extract limits. The procedure is explained in the next chapter. Wenu+4p Template I I 0 200 Mean_We4p Entries 1856 Mean 445.2 81.35 46.18/41 E 400 600 80( rob 0.2668 constant 12.74 0.205 lope 0.09996 0.004666 xpo 0.724 0.006357 S 1000 1200 M. [GeV] 0 200 400 600 800 1000 1200 Mt [GeV] Figure 77: W+4p template (electron sample) n . m 0 200 400 600 800 1000 1200 Mt [GeV] 68 munu+4p Template MeanWmu4p Entries 2066 Mean 442.6 RMS 80.14 2 / ndf 52.65 / 45 Prob 0.202 Constant 17.03 0.2461 Slope 0.5316 0.01643 Expo 0.5172 0.004151 0 200 400 600 800 1000 1200 Mt [GeV] tt Figure 78: W+4p template (muon sample) QCD Template MeanQCD Entries 2975 Mean 450.7 RMS 87 2 / ndf 31.51/48 Prob 0.9684 Constant 16.06 0.1829 Slope 0.3666 0.01002 Exnn 0.5583 0.003672 * I I II II ItI I II I I11.I11 0 200 400 600 800 1000 1200 Mt, [GeV] Figure 79: QCD template 0 200 400 600 800 1000 1200 Mt [GeV] SM WW Template Mean WW Entries 584 Mean 437.2 RMS 74.6 2 / ndf 20.66 / 29 Prob 0.8711 Constant 12.44 0.4107 Slope 0.13 0.01037 Expo 0.6985 0.01085 0 200 400 600 800 1000 1200 Mt, [GeV] Figure 710: WW template 0 200 400 600 800 1000 1200 Mt [GeV] l 1 II1 0 200 400 600 800 1000 1200 Mt [GeV] 0 200 400 600 800 1000 1200 Mt. [GeV] Figure 711: W+2b+2p template (electron sample) SM Wmunu+2b+2p Template , I I1 .dili ean_Wmu2b2p Entries 1266 Mean 445 RMS 91.64 2 / ndf 34.51/42 Prob 0.7873 Constant 53.72 0.461 Slope 21.62 0.2214 Expo 0.1362 0.001413 V1Th,, ,,, 0 200 400 600 800 1000 1200 Mt [GeV] Figure 712: W+2b+2p template (moun sample) 0 0 200 400 600 800 1000 1200 Mt [GeV] LI 1,,,irt'rl 0 200 400 600 800 1000 1200 Mt [GeV] 0 200 400 600 800 1000 1200 Mt. [GeV] Figure 713: W+4p template with alternative Q2 scale (electron sample) Background templates i00 300 400 500 600 700 800 900 100011001200 Mt [GeV] Figure 714: All Standard Model background templates used in the analysis 0.12 0.1 0.08 0.06 0.04 0.02 SBackground templates I 0.12 0.1 0.08 0.06 0.04 0.02 o00 300 400 500 600 700 800 900 100011001200 Mt [GeV] Figure 715: W+2b+2p template vs W+4p template. W+2b+2p was ignored since the expected contribution is at the level of 12% and the template is very similar to the W+4p template > Mean 44 4 44 Mean . 00 RM 5189  5 400 RM 6363 8250 4 .300 200 250 150 200 1:0 t I 100 0 20o0 4400 600 Oooo G 0 l G 200 400 600 ) 80 lu 0 G 0 Mean >O Mean X 0 S350 RM 95505 RM 9118 300 Ioo . 250 250  200 I 200 20 150 150 2100 4 100 00 >  Mean 597  > r Mean 6302  50 50  I r 4 200 100 Zulu 4uu \u ca luu \ 0 0 ulu 4u bluu 0o 200 400 600 401000 12 5o 20 40051 > 40Me 1 Men 220 200 3 180 R 160 19 100 e80 80 * 60 60 4o 40 20o 0.20 20L 7 200 400 600 800 1000 I G 0 o 200 400 600 800 1000 G,' GeV GeV > M. 160693 220400 120 120 o100 100 80 80 60 60 4O 40 .. 20 20 0 200 400 600 8 00 1000 0 200 400 600 800 1000 20 G.G.eV Figure 716: Signal templates CHAPTER 8 SENSITIVITY STUDIES In this chapter we will present the algorithm used for establishing lower and upper limits for signal crosssection times branching ratio at any desired confidence level (CL). We used a B.,' i.,, approach which was shared with other CDF analyses. The main idea and II,.'. I i. i, for the implementation can be found in [27, 28]. 8.1 General Presentation of the Limit Setting Methodology For generality we will assume that the observed data quantities are contained in a vector n = (ni, n2, ... nnbin), which in our case would correspond to the bin content of the 1 ,, histogram. The modeling of the data contains one unknown parameter and we want to be able to make a probabilistic statement about that parameter once we look at the data. In other words we would like to obtain a posterior probability distribution for the parameter. We will call this parameter a, because in our particular case it corresponds to the signal crosssection times branching ratio. It is often the case that other parameters are involved, and their values are known with some uncertainty. We will assume their values are normally distributed with the uncert.,iiir being the standard deviation. We will denote these parameters v = (Vi, v2, ...) and call them nuisance parameters. We will formalize our prior knowledge of the nuisance parameters and a by introducing the prior probability density 7(a, v). In our case this can be factorized as a product of Gaussians for the nuisance parameters and a flat distribution for a. The Bayes theorem connects the likelihood of the measurement (prior probability) to the posterior density of a and v after the measurement: p(a, v n)= (n a, v) 7(a, v)/p(n) (81) where p(n) is the marginal probability density of the data p(n) f d f da(n a, v ( v) (82) In these equations p(a, vin) stands for the posterior density and (n a, v) stands for the prior density. We are not interested in the nuisance parameters so we integrate over them p(acn) = dup(Q, v\n) (83) to obtain the sought posterior probability density for the parameter of interest a. From this posterior p(aln) we can extract the information we need, like the most probable value, upper and lower limits at any confidence level, etc. 8.2 Application to This Analysis In our analysis the data n we observe is the binned iA ,, spectrum, the parameter of interest a is the resonant tt production cross section times branching ratio, axo BR(Xo  tt, and the nuisance parameters are: the integrated luminosity, acceptance, and crosssections. In order to build the likelihood (prior density) we need normalized A i,, templates for each process. We will use the notation Tj with j E {s, b} for the binned signal and background templates, and Tj for the ith bin of the jth template. Given the above definitions we can write the expected number of events in the ith bin of the spectrum as Pi = Ldt. a jcejTji = a8TAT + E NjT (84) jE{s,b} jE{b} where we separated the signal contribution from the backgrounds and we defined the auxiliary variables As f Ldt cs (also called effective acceptance) and Nj = f Ldt ajcj with j E {b}, the total expected number of events for each background, after event selection. The prior likelihood can be written: (nla, v) P( ) eATsii)i, NbjTji iC{nbins} iC{nbins} (85) As we already pointed out, we may not know exactly As and the expected number of events from background. It is customary to take as priors for these parameters a truncated (to positive values) Gaussian to represent our prior knowledge' For the signal cross section a( we use a flat prior. 8.2.1 Templates As pointed out in Eq. 85, in order to build the likelihood function we need to know the template distributions for the signal and for the backgrounds. Given the limited statistics available for the samples we decided to fit them and use the smoothed fit distributions as templates; this procedure removes 1il ii.1 i. .1l empty bins or bumps. As already mention in Chapter 5, we consider as possible background contributions the following processes: 1 Given that the total efficiency is often the product of several efficiencies, the lognormal prior is often used too. Standard Model tt W ev + 4 partons W Ltv + 4 partons W e + 2 partons 2b W i / + 2 partons 2b Dibosons WW, WZ, ZZ QCD (from data) Mean X0_600 X21/ndf Prob Constant 1 Mean1 Mean 300 Sigmal 2 Constant 1 Mean2 250 Sigma2 Constant 200 Mean3 4 Sigma3 1 150 100 50  45.17/43 0.3815 37.7 +20.5 603.9+ 2.4 22.85 +2.97 32.4+ 19.5 577.9 5.6 i7.79 6.44 86.6 11.0 165.7 +32.9 43.4+15.4 SMean SMtt I x2/ndf 48.09/56 Prob 0.7649 450 Constant 14.65 0.13 Slope 0.2152 0.0053 400 Expo 0.6148 0.0033 350 300 1 250 200 150 100 50 "O 200 400 600 800 1000 1200 "0 200 400 600 800 1000 1200 Mt, GeV Mt, GeV Figure 81: Signal and background examples. The signal spectrum on the left (Mxo = 600 GeV/c2) has been fit with a triple Gaussian. The background spectrum from Standard Model tt has been fit with the exponentiallike function. Fit range starts at 400GeV/c2 The f., histograms are fit with an exponentiallike function f(x) = a e'" in the region above 400 GeV/c2. The signal histogram is fit with a double or triple Gaussian, or a truncated double Gaussian and a truncated exponential distribution2 An example is shown in Fig 81. All templates can be found at the end of the previous chapter. 2 This set of the fitting functions guarantees a fit with good x2 probability. III _L We discussed the backgrounds in Chapter 5, and we will remind the reader that we decided it is safe to absorb the small W + 2 partons + 2 b contributions into the W + 4 partons templates. Similarly, the WZ and ZZ contributions are absorbed in the ZZ template by increasing by 20% the nominal WW cross section. 8.2.2 Template Weighting Equation 84 shows that in order to build the likelihood we need to know the number of background events Nj for each background type. Table 81: Acceptances for background samples. Sample Event Selection Reconstruction and 400GeV/c2 cut Total acceptance '\! tt 0.045 0.72 0.032 WW 0.0014 0.60 0.0008 W(ev) 0.0076 0.66 0.0050 W(ep,) 0.0072 0.65 0.0047 QCD 0.0070 0.71 0.0050 In general we estimate the crosssection, acceptance and integrated luminosity in order to get this number, but since the cross sections for the processes pp  W + nj and multijets (QCD) are not known with good precision we decided to estimate the number of events from these backgrounds based on the total number of events seen in the data: N dt (1sAs + A t A+ + wwAww) + NWe4p + N 4p NQCD (86) with the constraints NWe4p/AWe4p NW 4p/AW 4p, ii i,. = 10. NQCD (87) The relative weights for We4p, Wk4p backgrounds have been set such that they have the same number of events before the event selection and reconstruction because the (unknown) cross sections are considered to be the same. The relative weight between QCD and W+4p has been set to 10% as discussed in Chapter 5 and established in this analysis [29]. Acceptances used in calculations are listed in Tables 81 and 82. Crosssections are listed in section 5.4, Table 53. Table 82: Acceptances for resonance samples. Mxo (GeV/c2) Event Selection Reconstruction and 400 GeV/c2 cut Total 450 0.047 0.86 0.040 500 0.051 0.93 0.048 550 0.055 0.94 0.051 600 0.057 0.97 0.055 650 0.059 0.97 0.057 700 0.062 0.97 0.060 750 0.062 0.98 0.060 800 0.063 0.98 0.061 850 0.063 0.97 0.061 900 0.061 0.98 0.059 8.2.3 Implementation After building the likelihood for a given observation n according to Eq. 85 we need to calculate the posterior density for ( according to Equations 81, 82 and 83. In practice we do not divide by p(n) in Eq. 81 since that is only a global normalization factor we can apply at the end. In this way we do not need Eq. 82 any more and we can rewrite Eq. 81 in a simplified and more explicit form: p(a(; A,, Nb n) = (nl.(; A,, Nb) 7(s; A,, Nb) (88) To obtain the posterior probability density for ( only we carry out the integration on the nuisance parameters As and Nb using a Monte Carlo method. Following the ;.. i ..i, in [28] on page 20, we implement the "Sample & Scan" method. We repeatedly (1000 times) sample the priors 7(As) and 7j(Nj), which are truncated Gaussians with respective widths of 6As and 6Nj. Then we scan (400 bins) the ( up to some value where the posterior is negligible. At each scan point we add to the corresponding bin in a histogram of os a weight equal to (n os, As, Nb) 7r(o, As, Nb). This yields the posterior density for as. 8.2.4 Cross Section Measurement and Limits Calculation Having calculated the signal cross section posterior density we can extract limits and ,in ..i ire" the cross section. We define as our estimator for the cross section and therefore as our measurement the most probable value of the distribution. This choice is supported by many linearity tests we run both with fake signal templates (simple Gaussians) and with real Xo templates. Lum=1000pb1 Lum=1000pb1 XoMass450 5 Xo Mass 600 o fpb] ( O pb] XoXo Masasss 890000 psLum=1000pb1 e a i Lum=1t f pb1 s Xo Mass 800 0 1 2 3 4 opb] 0o s 1 2 2s .es pb %pb] d t io Pb] Figure 82: Linearity tests on fake (left) and real (right) templates. As test fake signal templates we used Gaussians with 60 GeV/c2 widths and means of 800 and 900 GeV/c2. We used also real templates with masses from 450 to 900 GeV/c2 The top plots show the input versus the reconstructed cross section after 1000 pseudoexperiments at integrated luminosity J L = 1000pb1. Bottom plots show the deviation from linearity in expanded scale, with reddotted lines representing a 2% deviation Figure 82 shows the results of the tests with fake Gaussian signal templates of 800 and 900 GeV/c2 masses and 60 GeV/c2 width and with real 1 1, templates for Xo masses from 450 to 900 GeV/c2 at an integrated luminosity equal to f 1000lpb1. The reconstructed cross section agrees very well with the input value, showing only a small relative shift of about . However our measurement is meaningless as long as it is consistent with the null hypothesis, being only a statistical fluctuation. Therefore the key quantities to extract are the upper and lower limits (UL, LL) on the crosssection at a given confidence level. This is done by finding an interval defined by limits LL and UL, which satisfy: f UL JLL p(T(ln) f L aC (89) jo P(a n) and p(LLIn) = p(ULln) (810) with a the desired confidence level, for example 0.95 for 95' Crosssection posterior p.d.f. hxsec x10 5 Entn.rs 400000 x10 Man 2598 0.2 RMS 03435 0.18 Undflo 0 0.16 Ig 0.14 0.12 oa < 3.225 at 95% CL 0.1 0.08 0.06 0.04 0.02 S 1 2 3 4 5 6 7 8 9 10 S,pb Figure 83: Example posterior probability function for the signal cross section for a pseudoexperiment with input signal of 2 pb and resonance mass of 900 GeV/c2 The most probable value estimates the cross section, and 95% confidence level (CL) upper and lower limits are extracted. The red arrow and the quoted value correspond to the 95% CL upper limit In this way we can extract LL and UL for each pseudoexperiment or for data. Figure 83 shows an example of posterior for a pseudoexperiment with input signal of 2 pb, Mxo = 900 GeV/c2 and total integrated luminosity f 1000pb1. Before looking at the data we need to know what are the expected limits without any signal present and what are their fluctuations for certain integrated luminosities. For these purposes we ran many (1000) pseudoexperiments for each Mxo and integrated luminosity and we filled histograms with the most likely value, LL and UL from each pseudoexperiment. The median of the UL histogram is considered the expected upper limit in the absence of any signal. We also define 68% and 95% CL intervals around the central value in order to get a feeling of the expected fluctuations in the upper limits. We also ran similar series of pseudoexperiments with signal in order to see what are our chances of observing a nonzero LL in a given scenario. More specifically, we computed the probability of observing a nonzero LL for a given resonance mass, integrated luminosity and signal crosssection. This quantity is very useful in assessing the power of the algorithm and what signal crosssections are realistically possible to observe at any integrated luminosity. 8.2.5 Expected Sensitivity and Discovery Potential Figure 84 shows the distribution of the expected upper limit (UL) at 95% CL for various masses and two integrated luminosity scenarios, f = 319, 1000pb1. Figure 85 shows the power of the algorithm in distinguishing signal from background. On the x axis we have input signal crosssection and on the y axis the fraction is the probability of observing a nonzero LL at 95% CL for f L = 1000pb1. This plots do not include shape systematics, or systematic effects that lead to change in the shape of the templates. We will explore the treatment of shape systematics in the next chapter. 