Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0021188/00001
## Material Information- Title:
- Measurement of the Top Quark Mass in the All Hadronic Channel at the Tevatron
- Creator:
- Lungu, Gheorghe
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2007
- Language:
- english
- Physical Description:
- 1 online resource (181 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Physics
- Committee Chair:
- Konigsberg, Jacobo
- Committee Members:
- Field, Richard D.
Mitselmakher, Gena Ramond, Pierre Nishida, Toshikazu - Graduation Date:
- 8/11/2007
## Subjects- Subjects / Keywords:
- Antiprotons ( jstor )
Average linear density ( jstor ) Calorimeters ( jstor ) Electrons ( jstor ) Mass ( jstor ) Muons ( jstor ) Protons ( jstor ) Quarks ( jstor ) Signals ( jstor ) Vertices ( jstor ) Physics -- Dissertations, Academic -- UF - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Physics thesis, Ph.D.
## Notes- Abstract:
- This study presents a measurement of the top quark mass in the all hadronic channel of the top quark pair production mechanism, using 1 fb & #8722;1 of pp collisions at ps=1.96 TeV collected at the Collider Detector at Fermilab (CDF). Few novel techniques have been used in this measurement. A template technique was used to simultaneously determine the mass of the top quark and the energy scale of the jets. Two sets of distributions have been parameterized as a function of the top quark mass and jet energy scale. One set of distributions is built from the event-by-event reconstructed top masses, determined using the Standard Model matrix element for the tt all hadronic process. This set is sensitive to changes in the value of the top quark mass. The other set of distributions is sensitive to changes in the scale of jet energies and is built from the invariant mass of pairs of light flavor jets, providing an in situ calibration of the jet energy scale. The energy scale of the measured jets in the final state is expressed in units of its uncertainty, & #190;c. The measured mass of the top quark is 171.1 & #177;3.7(stat.unc.) & #177;2.1(syst.unc.) GeV/c2 and to the date represents the most precise mass measurement in the all hadronic channel and third best overall. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2007.
- Local:
- Adviser: Konigsberg, Jacobo.
- Statement of Responsibility:
- by Gheorghe Lungu.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Lungu, Gheorghe. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 660822582 ( OCLC )
- Classification:
- LD1780 2007 ( lcc )
## UFDC Membership |

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1-=- C-=a c-=- car e :-- -- :-= Figure D-2. Continued many interesting discussions we had and for being the friends I needed during difficult times. At last, yet most importantly, I want to thank my wife, Corina, whom I dedicate this work. Without her constant support, criticism and love I wouldn't have succeeded in finding the balance needed to reach this goal. Also I thank my father and my sisters for loving me, and my mother who will be ah-- 0-4 in my mind. Figure D-1. Continued 700 - 600 - 500 - 400 - 300 - 200 - 100 - 800 - 700 - 600 - 500 _ 400 - 300 - 200 - 100 - 600 500 - 400- 300- 200 - 100 - 400 - 300 - 200 - 100 - 700 600 - 500 - 400 - 300 - 200 - 100 - OE 50 100 150 200 25 E 700 - 600 - 500 - 400 - 300 - 200 - 100 - 9 50 100 150 200 250 MU 9E 600 - 500 - 400 - 300 - 200 - 100 - 0 500 - 400 - 300 - 200 . 100 - 0 700 - 600 - 500 - 400 - 300 - 200 - 100 - 01 50 100 150 200 25 mu - 600 - 500 - 400 - 300 - 200 - 100 - 01 50 100 150 200 250 U 600 - 500 - 400 - 300 - 200 - 100 - 0 500 - 400 - 300 - 200 - 100 - 0 Figure E-1. Continued interaction was not understood until later. However, the CP conservation by the strong force remains a mystery. Another thing that puzzled the physicists was why the decay of the muon into an electron and a photon was not observed. The solution adopted was the postulation of two types of neutrinos, the electron-neutrino and the muon-neutrino, along with the conservation of two new quantum numbers, the electron number and the muon number. The muon-neutrino was eventually discovered in 1962 by Lederman, Schwarts and Steinberger. Through the work of Feynman in 1947, the physicists were able to calculate the electromagnetic properties of the electron, positron and the photon using the Feynman diagrams. This constitutes the birth of quantum electrodynamics, or QED. The theory of weak interaction was first formulated by Fermi in 1933 and it was assuming a four-fermion interaction acting at a single point. The Fermi coupling constant GF=1.16639x10-s GeV-2 WaS giving the strength of the weak interaction. In 1956, Feynman and Gell-Mann incorporated the phenomenon of parity violation into this theory. The Fermi theory of weak interaction was able to explain the low-energy processes, but was making unacceptable prediction for high-energy weak interactions. The solution to this problem was to introduce a particle which mediates the weak interaction. This particle was thought to be a spin 1 boson, with three charge states, W-, Wo, W+ and was the result of work done by Schwinger, Bludman and Glashow in 1959. Later in 1967 Weinberg and Salam propose a theory that unifies the weak and the electromagnetic forces. In this theory the neutral boson carrying the weak force is called Zo. In addition to that a massive boson called the Higgs boson is predicted. The W and Z bosons will be eventually discovered in 1983 at CERN in according to the predictions. In 1964 the fundamental particles were: three quarks up (u), down (d) and strange (s), and two pairs of leptons the electron (e) with its neutrino (ve), and the muon (p-) with its neutrino (v,). Their corresponding antiparticles were also considered as ~ldf 231.3/11 6 "" pO 1.105+ 0.002257 .1.25 1 .2 1.15 4415r1 0 1.115+ 0.002278 1.2 1 1- 09 150 160 170 180 190 200 Input Top Mass (GeV/cS 150 160 170 180 190 200 Input Top Mass (GeV/ci Figure 9-5. Top mass pull width as a function of top mass for different treatment of the correlation between the event top mass and the dijet mass. On the left is the default case when the correlation is zero, while on the right is shown the situation with the full correlation. Figure 9-1 shows the event multiplicity single' I__- d events on the left, and for double I__- d events on the right. Figure 9-2 shows the histograms of the three parameters describing the gaussian fit for the single' I__- d events, while Figure 9-3 shows the equivalent plots in the case of the double I__ d events. The uncertainties on the background parameters as determined following the histogfram fluctuation are shown in Table 9-1. Varying the background parameters within these uncertainties results in a shift in top mass of 0.4 GeV. 9.7 Correlation Between Top Mass and Dijet Mass We investigate here the effect of the correlation between the event top mass and dijet mass has on the top mass pull widths and pull means. Our pseudo-experiments were formed by randomly selecting the event top masses from the top mass templates and by randomly selecting the dijet masses from the dijet mass templates. As a consequence the correlation between two masses is reduced to zero. Figure 9-4 shows on the left the top mass pull mean in the default case when the above correlation was reduced to zero, while on the right is shown the situation with full correlation. Figure 9-5 shows the equivalent comparison involving the top mass pull widths. On average over different top mass samples, the pull mean is consistent within the uncertainties between the two scenarios. However, the pull widths appear higher when the correlation between the event top mass and the dijet mass is zero. We conclude that there is no need for a systematic uncertainty, and we keep the default pull width as the correcting factor on the statistical error on the top mass since it represents the more conservative approach. 9.8 2D Calibration We have varied the parameters of Equations 8-5 and 8-6 within their uncertainties as listed in Table 8-2. We then re-calibrated the reconstructed values for the top mass. The change in top mass is 0.2 GeV. CHAPTER 3 EVENT RECONSTRUCTION In this chapter we will describe how we can identify the particles produced in a pp collision starting front the raw outputs of the different parts of the detector. First we will see how information from silicon detectors and COT are used to reconstruct charged particle trajectories. Then we will move to the reconstruction of jets of hadronic particles, based on calorinteters. A section will be devoted to the correction of jet energies for different error sources introduced by calorinteters and reconstruction algorithms. After a brief description of the identification of leptons and photons, we will end with the different methods used at CDF to identify a jet of particles originated front a b quark. 3.1 Tracks Track reconstruction is performed using data from silicon tracking system and COT. The reconstruction is based on the position of the hits left b charged particles on detector components. Combining these hits one can reconstruct particle trajectories. The whole tracking system is ininersed in a 1.4 T magnetic field. C'I Iaged particles moving in a homogeneous magnetic field follow a helix trajectory. The helix axis is parallel to the magnetic field. 1\easuring the radius of curvature of the helix, one can obtain the transverse montentunt of the particle, while the longitudinal montentunt is related to the helix pitch. To describe a helix five parameters are needed, three to paranieterize the circle in r projection and two to paranieterize the trajectory in x. At CDF, as shown by Equation :31, the helix of a charged particle is paranleterized. S= (cot0, C, xo, D, 00) (:31) The parameters used to describe the helix of a charged particle are: cot 8 is the cotangent of the polar angle at nmininiun approach to the origin; C is the half curvature, whose sign is given by the charge of the particle; xo is the position on x axis of the nmininiun approach to the helix origin; D is the signed impact parameter (i.e., the distance S= lisag 2tag JES 71 Both the single tag likelihood and the double tag likelihood are a product of four terms as shown in Equation 7-2. The top template term, top, iS Shown in Equation 7-3. The W template term, w, is shown in Equation 7-4. The constraint on total number of events, Onev, is shown in Equation 7-5. The constraint on the it number of events, L,,, is shown in Equation 7-6. 1,2tag = top .W .nev .n, (7-2) Both top and W template terms have the same structure: a weighted sum of the event signal probability at a given top mass and JES and the event background probability. The fraction of it events, us/(us + ab), is the weight of the signal probability and the fraction of background events, nb Es, + ab), is the weight of the background probability. Together with M~ and JES, the parameters as and nb are free in the likelihood fit. et n, Ptop(m,, | M, JE S) + nb tl- ) us + nb evt 1 Cw = r~v 11~U "et (7-4) n, + nb evt=1 The sum of signal and background events, as + ab, is constrained to the total number of observed events in the data, NVor gs, via a Poisson probability with a mean equal to Netot even~lts * Cnev = (eo enes/R 'b *p(NJnes/ (7-5) (us + nb) The number of signal events, us, is constrained to the expected number of it events, ae sp, ia a Gaussnian of mean equa~l to nsp and width equal to o-Ps. The width of the gaussian is simply the uncertainty on the expected number of it events. The epecnpted nu~mbers of sigrnal evepnts, ns, are 13 Single' I__- d and 14 double I__- d events, corresponding to a theoretical cross-section of 6.7'ji pb [55] and an energy. Moreover, the sum of transverse moment of all tracks pointing to the 0.4 cone should be less than 2 GeV/c. The line connecting the primary vertex to the shower max position of the photon candidate determines the photon direction. 3.6 Bottom Quark Tagging The hadrons produced by a b quark have two important properties: long lifetime allowing it to travel before decaying and the possibility of semi-leptonic decay b luIs. Typically, the lifetime is about 1.5 ps for a hadron with an energy of about 40 GeV, so the distance it travels if few millimeters. From these properties it is possible to construct algorithms to tag jets if they are produced by b quarks. At CDF there are used three such algorithms: the SecVtx algorithm, the JetProbability (JP) algorithm and the Soft Lepton T.--I ;::_ (SLT) algorithm. 3.6.1 SecVtx Algorithm This algorithm [47] exploits the fact that the B hadron travels before it decays and therefore the jet produced by it will contain a secondary vertex (Figure 3-6). The algorithm starts from COT and silicon tracks inside a cone and as a first step, using as discriminating variable their impact parameter, it removes tracks identified as Ks, A or y daughters, or consistent with primary vertex or too far from it. Then a three dimensional common vertex constrained fit is performed using two tracks: if X2 < 50 the two tracks are used as seed to find other tracks that point toward the same secondary vertex. If at least three tracks are found to be compatible with a secondary vertex, the jet containing them is considered a b-tag if it passes the following cuts: * |Le,| < 2.5 cm, where L,, is the decay length of the secondary vertex; this cut helps rejecting conversions from the first 1 ., -r of SVXII; ~Lz * if if is the invariant mass of the tracks, |mKs ii1T, I > 0.01 GeV and |mA i 0.006 GeV; * Lv-(i./Pr) m c -=~nii a I 0~uii I- 0ruii Figure D-1. Continued The inning in parton energy is defined such that each bin contains at least 3000 entries and it is wider than 5 GeV. This is done in each bin of pseudo-rapidity. Table 4-2 shows the definition of energy inning for the b-jets transfer functions, while Table 4-3 is for the W-jets transfer functions. In each bin the transfer function is represented by the distribution of the variable 1 Ejet/E,,rton. The shape of this distribution is fitted to the sum of two gaussians. Appendix C holds the fitted shapes. 4.5 Transverse Momentum of the it System The PT(p3 weight is written as dependent on the 4-vectors of the partons in the final state, generically represented by P'in the argument of the function. This dependence is difficult to parameterize. Therefore we will pick a more natural set of parameters to work with. In the next section we will detail the change of variables needed to accommodate this simplification. Until then we anticipate that the variables used for integration in Equa~~tion 4-6 are 6 and ,6 representing the projections of the transverse momentum of the it system along the x and y axes. The probability density related to the transverse momentum of the it system weight is shown in Equation 4-31. PT(p~ PT(p p) (4 31) The parameters we actually use are the magnitude of the transverse momentum of the it system, p), and the azimuthal angle, ~. The upper index means that these parameters are determined using the 6 partons in the final state. We expect to have a flat dependence on ## and therefore we can factorize the two dependence. The Equation 4-32 gives the normalization relation. dp d, PrX (p ~)= 1 = dpP~r (p~l )~ X d (4-32) The transverse momentum spectrum of the it events, represented by PT(p ) in the Equation 4-32, is obtained from a tt Monte Carlo sample with M~to = 178 GeV. The integrated luminosity of 943 pb-l. These numbers have been determined using a tt Monte Carlo sample with a cross-section equal to the theoretical value. The value of the top mass used in the it Monte Carlo sample just mentioned is 175 GeV and it also corresponds to the top mass value for which the theoretical cross-section has been calculated. Therefore we read the expected number of signal events from Table 5-4. The uncertainties on the numbers of signal events a,p are chosen to be the Poisson errors. This is a conservative approach since the Poisson errors are larger than the uncertainties derived based on the theoretical cross-section uncertainty. L,, = exp ( >" (7-6) The value of JES is constrained to the a priori determination of this parameter by the CDF Jet Energy Resolution group, JESemp. This constraint is a gaussian centered on JESemp and width equal to 1. The unit used is a, which represents the uncertainty on the jet energy scale. ~~ ~( (JES JESemp2(77 7.2 Top Templates 7.2.1 Definition of the Template As mentioned in section 7.1, we use the matrix element to build the top templates. The event probability defined in section 4 is plotted as a function of the top pole mass in the range 125 GeV and 225 GeV. In negative logarithmic scale this event probability will be minimized for a certain value of top mass which we'll use to form the top templates. The shape of these templates depends on the input top mass and JES for it events, but not for background events. 7.2.2 Parameterization of the Templates We form signal templates for the mass samples described in section 5 with 7 different JES values: -3, -2, -1, 0, 1, 2, 3, after all our selection cuts have been applied. In total there are 84 templates for signal used for parameterization. The function used to fit them -CH TopMass=175 sa -HW TopMass=130 slas HW TopMass=140 HW TopMass=150 sae HW TopMass=160 SHW TopMass=170 HW TopMass=180 HW TopMass=190 -HW TopMass=200 HW TopMass=210 - ~HW TopMass=220 -HW TopMass=230 --= Pvt TooMass=178 APPENDIX B TRANSVERSE MOMENTUM OF THE ft SYSTEM PtTTbar D I PtT.ba 0 20 40 60 80 100 means means 30 35 40 45 50 0 5 rms 10 15 20 25 20 25 0 5 10 15 Figure B-1. Transverse momentum of the it system for different generators and for different top masses. Upper plot: shapes of the transverse momentum of the it system for different generators (CompHep, Pythia and Herwig) and for different top masses. Middle plot: the Means of the distributions in the upper plot. Lower plot: the RMS of the distributions in the upper plot. Figure E-2. Continued zuu - 180 - 160 - 140 - 120 - 100 - 80 - 60 - 40 - 20 - 0 0 1 200 250 3 250 - 200 - 150 _ 100 - 50 - 0 0 1 200 250 3U 3 250 - 200 - 150 - 100 - 50 - 0 1 0 200 250 UU It 350 - 300 - 250 - 200 - 150 - 100 - 50 - 0 1 0 200 250 3UU It d dO dO 200 250 3UU 94 500 - 400 - 300 - 200 - 100 - 500 - 500 - 400 400 - 300 300 - 200 200 - 100 100 - 00 PtTThar 18000 u 2i Undelflow 16000 -C Overflow 197 Integral 4545e+05 14000 # \i~df 31 3e 92 12000C pO 5259e+05 i1445e+04 pl 3402 11018 10000t p2 2552e+04 i1083 p3 1658e+06 i2212e+04 8000 p4 -1183 i01924 p5 339219653 6000 p5 -1995e+06 i3403e+04 p7 -24 9110 05462 4000 p8 1514106078 2000- 0 Pt_ttbar [GeV] Figure 4-4. Transverse momentum of the it events. The fit is a sum of 3 gaussians. 22 Indf 1.269 I 3 Prob 0.7365 p0 177.210.03468 pl 0.9892 10.001879 -y=x - y =pO +(x -178)*pl 150 160 170 180 190 200 Input Top Mass [GeV] Graph F200 I<190 0180 170 160 150 X2 Indf 2.845 I 3 Prob 0.4162 pO 176.8 10.05232 pl 0.9866 10.002775 -y=x | = O + x -178)*pl 150 160 170 180 190 200 Input Top Mass [GeV] Figure 4-5. Reconstructed top mass versus input top mass at parton level. A) The energies of the partons have been smeared by 5' B) The energies of the partons have been smeared by 10'; C) The energies of the partons have been smeared by 211' D) The energies of the partons have been smeared using the transfer functions. 600 - 500 - 400 - 300 - 200 - 100 - OC 50 100 150 200 2 E 0 800 - 700 - 600 - 0 - 300 - 200 - 100 - OC 50 100 150 200 250 MU 9? 0 700 - 600 - 500 - 400 - 300 - 200 - 100 - OC E 0 mu - 600 - 500 - 0 - 200 . 100 - OC E 0 uu - 600 - 500 _ 400 - 300 - 200 - 100 - OE 50 100 150 200 25 700 - 600 - 500 - 400 - 300 - 200 - 100 - 50 100 150 200 250 U 700 - 600 - 500 - 400 - 300 - 200 - 100 - 0 700 - 600 - 500 - 0 - 200 - 100 - 0 800 - 700 - 600 - 500 - 400 - 300 . 200 - 100 - 01 50 100 150 200 25 700 - 600 - 500 - 400 - 300 - 200 - 100 - 01 50 100 150 200 250 U di 0 800 - 700 - 600 - 500 - 400 - 300 - 200 - 100 - 0 600 - 500 - 400 - 300 - 200 - 100 - 0 Figure E-1. Continued between the helix and the origin at minimum approach); 00 is the direction in r 4 of the helix at the point of minimum approach. If (.ro, Wo) is the center of the circle, then the impact parameter is calculated as in E~quation? :32, where p = 2~is the radius of the circle and Q2 the charge of th~e particle. D = Q ( r,~ x + YO2 p) (:32) Having described the parameterization of a particle trajectory, we'll turn on the main tracking algorithms developed for offline analysis, the Standalone and the Outside-In algorithms. Standalone tracking [:35] is a strategy to reconstruct tracks in the silicon detector. It consists in findings triplets of aligned :3D hits, extrapolating them and adding matching :3D hits on other 11s-c v ;. This technique is called standalone because it doesn't require any input from outside: it performs tracking completely inside the silicon detector. First the algorithm builds :3D hits from all possible couples of intersecting axial and stereo strips on each lI ... Once a list of such hits is available, the algorithm searches for triplets of aligned hits. This search is performed fixingf a 111-< v and doing a loop on all hits in the inner and outer 11s- and one in the outer 1.,-< c a straight line in the r x plane is drawn. Next step consists in examining the 1 .,-cc in the middle: each of its hits is used to build a helix together with the two hits of the inner and outer 111-c v ;. The triplets found so far are track candidates. Once the list of candidates is complete, each of them is extrapolated to all silicon 11s-< rs looking for new hits in the proximity of the intersection between candidate and 111-< v. If there is more than one hit, the candidate is cloned and a different hit is attached to each clone. Full helix fits are performed on all candidates. The best candidate in a clone group is kept, the others rejected. The Outside-In algorithm [:36] exploits information from both COT and silicon. The first step is tracking in the COT, which starts translating the measured drift times in APPENDIX D SIGNAL TOP TE1\PLATES I mn~uniin I I mnrunii I mn~uniaA Figure D-1. Top templates for it single' I__- d events for samples with different top masses: from 150 GeV to 200 GeV. A) Case of JES = -3. B) Case of JES = -2. C) Case of JES = -1. D) Case of JES = 0. E) Case of JES = 1. F) Case of JES = 2. G) Case of JES = 3. REFERENCES [1] F. Abe et [2] J.H. K~uhn, Lectures delivered at 2:3rd SLAC Suniner Institute, hep-ph/9707:321 (1997). [:3] V.M. Ahazov et [4] T. Affolder et [5] D. Acosta et [6] D. Acosta et [7] D. C'I I1:1 .I~orty, J. K~onigsherg and D.L. Rainwater, Ann. Rev. Nucl. Part. Sci. 53, :301 (200:3). [8] 31. Cacciari et [9] B. Abbott et [10] D. Acosta et [11] S.L. Glashow, J. Iliopoulos and L. Alaiani, Phys. Rev. D 2, 1285 (1970). [12] S. Eidelman et [1:3] F. Abe et [14] ALEPH, DELPHI, L:3 and OPAL Collaborations and The LEP Working Group for Higgs Boson Searches, hep-ex/06120:34 (2006). [15] P. Azzi et [16] ALEPH, DELPHI, L:3 and OPAL Collaborations and The LEP Working Group for Higgs Boson Searches, Phys. Lett. B 565, 61 (200:3). CHAPTER 10 CONCLUSION We have applied the method described in the previous chapters to the data sample corresponding to 943 ph l. In this sample, there are 48 single' I__- d and 24 double I---- d events after all the cuts have been applied. In the second column of Table 10-1, we show in the total number of events and the expected number of signal events used as input in the 2D likelihood of Equation 7-1. Note that in Equation 7-1 we need the uncertainty on the expected number of signal events and this is also shown in Table 10-1. The numbers of background events are shown as well, but they are not used as input values in the likelihood. In the third column we show the number of events as they result from the minimization of the 2D likelihood. Following the minimization of the 2D likelihood, we measured a top mass of 171.1 & 3.7 GeV, and a JES of 0.5 + 0.9 ec.. The value of the jet energy scale (JES) is therefore consistent with the previous determination of JES at CDF. The quoted uncertainty on the top mass represents the combination of the statistical uncertainty with the systematic uncertainty due to JES uncertainty. In order to obtain only the statistical uncertainty on the top mass, the minimization of the 2D likelihood is modified such that the JES parameter is fixed to 0.5 ce. (the result from 2D fit). Following this procedure the statistical uncertainty on the top mass is 2.8 GeV. Therefore the systematic uncertainty due to JES is 2.4 GeV. Figure 10-1 shows the distributions of event by event reconstructed top masses as the black points for data and as the orange histogfram for the combination of signal and background templates that best fitted the data. The blue histogram represents only the background template. The sample with single I__ d events is shown in the left plot, while the double' I__- d events are shown in the right plot. [36] W. Yao, K(. Bloom, "Outside-In silicon tracking at CDF", CDF Note 5991. [37] H. Stadie, W. Wagner, T. Muller, "VxPrim in Run II", CDF Note 6047. [38] J.F. Arguin, B. Heinemann, A. Yagil, "The z-Vertex Algorithm in Run II", CDF Note 0.25 [39] CDF collaboration, Jet Energy Group, "Jet Energy Corrections at CDF", CDF Note 7543. [40] A.A. Bhatti, K(. Hatakeyama, "Relative jet energy corrections using missing Et projection fraction and dijet I In 1al s! CDF Note 6854. [41] B. Cooper, M. D'Onofrio, G. Flanagan, \!.i11ll sle interaction corrections", CDF Note 7365. [42] A. Bhatti, F. Canelli, "Absolute corrections and their systematic uncert 1!il I. - CDF Note 5456. [43] J.F. Arguin, B. Heinemann, "Underlying event corrections for Run II", CDF Note 6293. [44] A. Bhatti, F. Canelli, L. Galtieri, B. Heinemann, "Out-of-Cone corrections and their Systematic Uncert .sist s. CDF Note 7449. [45] R. Wagner, "Electron Identification for Run II: algorithms", CDF Note 5456. [46] J. Bellinger, "A guide to muon reconstruction and software for Run 2", CDF Note 5870. [47] D. Glenzinski, "A detailed study of the SECVTX als.. )111 .1.. CDF Note 2925. [48] D. Acosta, "Introduction to Run II jet probability heavy flavor .- -I__-:, CDF Note 6315. [49] L. Cerrito, A. Taffard, "A soft muon' I---- 1- for Run II", CDF Note 6305. [50] P. Azzi, A. Castro, A. Gresele, J. K~onigsberg, G. Lungu and A. Sukhanov, 1 kinematical selection for All-hadronic tt events in the Run II multijet <1 II I-, I CDF Note 7717. [51] P. Azzi, A. Castro, A. Gresele, J. K~onigsberg, G. Lungu and A. Sukhanov, "B-' I__;l!_; efficiency and background estimate in the Run II multijet <1 It I-- I CDF Note 7723. [52] Roger Barlow, "Application of the Bootstrap resampling technique to Particle Physics exp.~ Hin,! Il- MAN/HEP/99/4 April 14 2000. [53] J.F. Arguin, P. Sinervo, "b-jets Energy Scale Uncertainty From Existing Experimental Constraints", CDF Note 7252. Figure 2-4 shows the initial instantaneous luminosity and total integrated luminosity as a function of year. The initial instantaneous luminosity increased with running time due to intprovenients such as using the Recycler to store antiprotons. Total integrated luminosity is separated according to that delivered hv the Tevatron and that recorded to tape by the CDF detector. 2.2.2 Silicon Tracking The innermost component of CDF is a tracking system composed of silicon micro-strip arrays. Its main function is to provide precise position measurements near collision vertices, and it is essential for identification of secondary vertices. Constructed in three separate components, LOO [28], SVXII [29] and ISL [:30], the silicon tracking system covers detector |vy| < 2. LOO is a single 1... -r mounted directly on the beam pipe, r = 1.6 cm, and is a single-sided array with a pitch of 50 ftn providing solely axial measurements. SVXII is mounted outside of LOO, 2.4 < r < 10.7 cm, and is composed of 5 concentric 1.,-< cms in 4 and :3 segments, or barrels, in x. Each lI.-c c is further subdivided into 12 segments in ~, or wedges. Double-sided arrays provide axial (r 4) measurements on one side and stereo (x) measurements on the other. The stereo position of li n-c- c 1 and :3 is perpendicular to the x-axis, and that of lI .-cc 2 and 4 is is -1.2" and +1.2", respectively. The SVXII detector spatial resolution for axial measurements is 12 pn1. ISL surrounds SVXII, 20 < r < 29 cm, and is composed of three l o,-c rs of double-sided arrays. As with SVXII, one side provides axial measurements and the other stereo measurements at 1.2" relative to the x-axis. The ISL detector resolution for axial measurements is 16 pni (Figure 2-5). 2.2.3 Central Outer Tracker The Central Outer Tracker (COT) [:31] comprises the bulk of CDF's tracking volume, located between 40 < r < 1:32 cm and detector |vy| < 1. The COT provides the best measurements of charged particle montentunt, but does not measure position as precisely as the silicon tracking system. It is a 96-1 .,-cc open-cell drift chamber subdivided into 8 the optimization can be found in [50]. Table 5-1 shows the number of events in the data sample. Table 5-2 shows the number of events in a tt Monte Carlo sample with My = 170 GeV. The SVX b-' I__-- used has a higher efficiency in the Monte Carlo than in the data. Therefore we need to degrade the number of' I__- d events according to the appropriate scale factor which is SF = 0.91. Taking this scale factor into account, and converting to the luminosity of the data, we show in Table 5-3 the signal to background ratios, S/B, for different top masses after the kinematical cuts for single and double I__ d events separately. The conversion to the observed luminosity is done by using the theoretical it cross section. The number of background events is the difference between the observed number of events in the data shown in Table 5-1 and the signal expectation. An additional cut is introduced to further cut down the background. This new variable we cut on is the minimum of the event probability given in Equation 4-6 of section 4. Figure 5-1 shows the distribution of the minimum of the negative log event probability for a signal sample versus the background shape. Note that the top mass value for which this event probability is minimized will be used in the final top mass reconstruction, and the value of the probability in negative log scale is used as a discriminating variable between it and background. We denote this value as minLKL, and the cut definition is requiring this variable to be less than 10. The value of this last cut has been obtained by minimizing the statistical uncertainty on the top mass value as reconstructed in section 4, that is using only the matrix element calculation. Table 5-4 shows the efficiency of this cut relative to the number of events after' I__h;~! and after the kinematical cuts, for signal at different top masses and for background. The table also shows the number of signal events corresponding to 943 pb-l and the appropriate signal to background ratio. Comparing the signal-to-background ratios S/B between Table 5-3 and Table 5-4 there is an improvement of about a factor of 3 for samples with one I__ d heavy The 2D likelihood is shown in Figure 10-2. The central point corresponds to the nmininiun of the likelihood, while the contours represent the 1-signia, 2-signia, and :$-sigma levels, respectively. Using a tt 1\onte Carlo sample with a top mass equal to 170 GeV and the number of signal and background events as resulted front the data fit, we formed pseudo-experintents and determined the expected uncertainty on the top mass due to statistical effects and JES. About 41 of the pseudo-experintents had such combined uncertainty on the top mass lower than the measured value of :3.7 GeV. This can he seen in Figure 10-3, where the histogram shows the results of the pseudo-experintents and the blue line represents the measured uncertainty. In conclusion, the measured combined statistical and JES uncertainties on the top mass agrees with the expectation. The total uncertainty on the top mass in this analysis is 4.3 GeV. The previous best mass measurement in this channel had an equivalent total uncertainty of 5.3 GeV [56] which is 2 :' more. The source for this intprovenient is the uncertainty due to jet energy scale (JES) on the top mass. In this analysis this uncertainty amounts to 2.4 GeV compared to 4.5 GeV in the previous best result which is M' more. Some of this gain in precision is lost due to the somewhat higher systematic uncertainties front other sources and due to a slightly worse statistical uncertainty in this analysis compared with the previous best mass result in this channel. A more careful estimation of the other sources of systematic uncertainties on the top mass as well as a more efficient it event selection will help further reduce the total uncertainty on the top mass. Compared to mass measurements in other it decay channels, the mass measurement front this analysis ranked third in the top mass world average [57] with a 11 weight. The two better measurements were performed in the lepton+jets channel as it can he seen in Figure 10-4. This measurement promotes the all hadronic channel as the second best channel for the top quark mass analyses in Run II at the Tevatron. In conclusion, it is for the first time in the it all hadronic channel to have a simultaneous measurement of the top mass and of the jet energy scale. It is also the first mass measurement in this channel that involved the use of the it matrix element either in the event selection or in the mass measurement itself. All of the above were successfully mixed together resulting in the best top mass measurement in the all hadronic channel. r Table 10-1. Number of events for the it expectation and for the observed total for a luminosity of 943 pb-l passing all the cuts. The input values for signal have the uncertainties next to them in parenthesis. The background expectation being the difference between total and signal is also shown. For the output values, the numbers in the parenthesis are the uncertainties. Number of Events Input Reconstructed Total Observed(1tag) 48 47.8 Expected Signal (1tag) 13 + 3.6 13.2 + 3.7 Background (1tag) 35 34.6 + 7.2 Total Observed(2tags) 24 23.3 Expected Signal (2tags) 14 & 3.7 14.1 & 3.4 Background (2tags) 10 9.2 & 4.3 CDF Runil preliminary L=943pbl CDF Runil preliminary L=943pbl 16 Single Tags 1 --Double Tags $1v Data 14 -- Dnal+Bakground M Signal+Bakground 5 2 Background I Background Event Top Mass (GeV/cz) Figure 10-1. Event reconstructed top mass for data (black points), signal+background (orange) and only background events (blue). Single I__ d sample is on the left, while the double' I__- d sample is on the right. xM103 600 _ 500 - 400 - 300- 200 - 100 - 00 50 100 150 200 hMW1c Entries 6570 Mean 89.85 RMS 33.51 Underfow 0 Overfow 0 Integral 4.82e+06 X'Indf 4.569e+04126 Prob 0 pO 6.297e+06+i18356 pl 80.17+0.01 p2 7.005+0.01g p3 1.568e+07+41300 p4 99.65+0.06 p5 29.81+ 0.03 p6 1.152e+07+i36575 p7 0.04026+i0.00000 pa 10.4+0.0 pg1.886+0.007 250 300 350 hMtN2c 7080 84.68 32.44 0 0 13e+05 946130 0 +5379 7+ 0.05 +gage +9212 4+0.32 8 +0.18 +8169 0.0003 .4+ 0.2 +0.028 350 RMS 40000F Underfow Overfow 35000 -Integral 2.7 X lndf 4( 30000:Prb pO 6.582e+05 : pl 80.1i 25000 -p2 gagg5 p3 6.69e+05 20000t p4 94.6 p5 33.51 15000F l ) p6 5.412e+05 p7 0.0408+ 10000~ p8 19 pg1.579 5000 00 50 100 150 200 250 300 Entries Mean Figure 7-4. Dijet mass templates for background events. Single tags in the double tags in the right plot. left plot, and Table 7-4. Values of the parameters describing the dijet mass templates shapes in the case of the background events. Parameter Values (1Tag) Uncertainties (1Tag) 1 1.88e-01 9.52e-02 2 8.02e 01 4. 29e-02 3 7.01e 00 1.70e-02 4 4.68e-01 9.52e-02 5 9.97e 01 4. 29e-02 6 2.98e 01 1.70e-02 7 3.44e-01 9.52e-02 8 4.03e-02 4.29e-02 9 1.04e 01 1.70e-02 10 1.89e 00 9.52e-02 Values (2Tags) 3.53e-01 8.02e 01 9.13e 00 3.59e-01 9.46e 01 3.36e 01 2.90e-01 4.08e-02 1.04e 01 1.58e 00 Uncertainties (2Tags) 2.39e-01 1.12e-01 4.41e-02 2.39e-01 1.12e-01 4.41e-02 2.39e-01 1.12e-01 4.41e-02 2.39e-01 CDF RunlI preliminary L=943pb' 2 A In L=4.5 A In L=2 1C A In L=0.5 -2- 165 170 175 180 Top Mass (GeV/c2) Figure 10-2. Contours for 1-sigma (red), the 2-sigma (green) and the 3-sigma (blue) levels of the mass and JES reconstruction in the data. Figure 10-3. Histogram shows the expected statistical uncertainty from 1\onte Carlo using pseudo-experiments, while the line shows the measured one. About 41 of all pseudo-experiments have a lower uncertainty. CDF RunlI preliminary L=943pbl 250 - 200 - 150 . 100 - 50 - OC 50 100 150 200 250 MU 900 350 - 300 - 250 - 200 - 150 - 100 - 50 - OC 50 100 150 200 250 3UU dt0 900 - 800 - 700 - 600 - 500 - 400 - 300 . 200 - 100 - OC 0 500 - 400 - 300 - 200 . 100 - OC 0 300 - 250 - 200 - 150 - 100 - 50 - OE 50 100 150 200 250 E0 400 - 350 - 300 . 250 - 200 - 150 - 100 - 50 - E 50 100 150 200 250 3UU di0 450 - 400 - 350 - 300 - 250 - 200 - 150 - 100 - 50 - OE E 0 500 - 400 - 300 - 200 - 100 - OE E 0 350 - 300 - 250 - 200 - 150 - 100 - 50 - 01 50 100 150 200 250 U di0 400 - 350 - 300 - 250 - 200 - 150 - 100 . 50 - 01 50 100 150 200 250 3UU di0 500 - 400 - 300 - 200 - 100 - 0 500 - 400 - 300 - 200 - 100 - 0 Figure E-1. Continued computer network to a storage facility using a robotic tape library. This data is then processed with offline reconstruction software for use in analyses. Fermilab's ACCE L ER TOR CHAIN fci~ ~\MAIN INJECTOR .RECYCLER TEVATRON 11 D:ERI Deetr TAGET HACL Experiment ANTIPROTON : (Colllder Deictp Infrmlb .7 :- -p Iner~l BOOSTER COCKCROFT-WALTON PROTON Dumps Figure 2-1. Diagram of the Tevatron accelerator complex CENTRAL DR IT CHAMBER ---~~C*O -- -- iAET IC x, EM SHAMER HADRONIC CALORIMETER MUON DR IT CHAMBERS --- INTILL TOR ISL 3 LAER S svX I (3 BARRELS) z I-SOLENolD COlL PRESHowER DETECTOR Figure 2-2. Elevation view of the East hall of the CDF detector. The West half is nearly symmetric. I ml~liiiI II ml~liiiI II ml~liiiID Fi ur D-2 Cotiue 17 - 6 17e JES =-1 oa JES=1 26 =3 8 JES =-1 JES=1 a JES=3 Figure 7-1. Top templates for it events, single tags in the left plot, double tags in the right plot. The upper plots show the parameterized curves, while the bottom plots show the original histograms. The left column shows the templates variation with top mass at JES = 0. The right column shows their variation with JES at top mass lHop = 170 GeV. minLKLmassV1 minLKLmassV1 minLKLmassV1 Mann 1851 ovdernow a Integrol 4165 a 7ine J814*<4/22 minLKLmassV1 Emneso losi Mean 1899 Rus 26os Integral 4231.*04 findr a44/22 proa po 1o'ii"oo looeso372e" 112 2940+6417 35000 30000 _ 25000 - 20000 - 15000- 10000- 5000- Figure 7-2. 140 160 180 200 220 Scanned Top Mass (GeVlc 2) 0 140 160 180 200 220 Scanned Top Mass (GeVlc 2) Top templates for background events. Single tags in the left plot, and double tags in the right plot. O O O B rCIJ ,r I Figure 7-3. Dijet mass templates for it events, single tags in the left plot, double tags in the right plot. The upper plots show the parameterized curves, while the bottom plots show the original histograms. The left column shows the templates variation with top mass at JES = 0. The right column shows their variation with JES at top mass lHop = 170 GeV. -JES=-3 JES=-1 JES=1 JES= hits positions: once all COT hit candidates in the event are known, the eight super-l} ... rs are scanned looking for line segments. A line segment is defined as a triplet of aligned hits which belong to consecutive l o,-;- s. A list of candidate segments is formed and ordered by increasing slope of the segment with respect to the radial direction so that high momentum tracks will be given precedence. Once segments are available, the tracking algorithm tries to assemble them into tracks. At first, axial segments are joined in a 2D track and then stereo segments and individual stereo hits are attached to each axial track. Outside-In algorithm takes COT tracks and extrapolates them into the silicon detectors, adding hits vi a progressive fit. As each lI .-cc of silicon is encountered (going from the outside in), a road size is established based on the error matrix of the track: currently, it is four standard deviations hig. Hits that are within the road are added to the track, and the track parameters and error matrix are refit with this new information. A new track candidate is generated for each hit in the road, and each of these new candidates are then extrapolated to the next 1.,-c c in, where the process is repeated. At the end of this process, there may be many track candidates associated with the original COT track. The candidate that has hits in the largest number of silicon 1 .,-c rs is chosen as the real track: if more than one candidate has the same number of hits, the X2 of the fit in the silicon is used to choose the best track. 3.2 Vertex Reconstruction The position of the interaction point of the pp collision (primary vertex) is of fundamental importance for event reconstruction. At CDF two algorithms can he use for primary vertex reconstruction. One is called PrimVtx [37] and starts by using the beam line x-position (seed vertex) measured during collisions. Then the following cuts (with respect to the seed vertex position) are applied to the tracks: |Itrk Xertezr| < 1.0 cm, |do| < 1.0 cm, where do is track impact parameter, and ( < 3.0, where o- is error on do. C E C E~liiV C E~miI II ml~miV I mn~uniin I mr~unia I mr~uniaB Figure D-2. Continued In Equation 3-10, R is the clustering cone radius, PT is the raw energy measured in the cone and if the pseudo-rapidity of the jet: f,7, Af,, fabs, UE and 000 are respectively relative, multiple interactions, absolute, underlying event and out-of-cone correction factors . 3.4 Leptons Reconstruction 3.4.1 Electrons Being a charged particle, an electron traversingf the detector first leaves a track in the tracking system and then loses its energy in the electromagnetic calorinteter. So a good electron candidate is made of a cluster in the electromagnetic calorinteter (central or plug) and one or more associated tracks; if available, shower nmax cluster and preshower clusters can help electron identification. The shower has to be narrow and well defined in shape, both longitudinally and transversely. The ratio between hadronic and electromagnetic energies has to be small and track montentunt has to match electromagnetic cluster energy [45]. 3.4.2 Muons Muons can leave a track in the tracking system and in the nmuon system, with little energy deposition in the calorinteter. Aluons are reconstructed using the information coming front nuon chambers (CMET, CM~P, CM~X, BIfET) and nmuon scintillators (CSP, CSX, BSU, TSU). The first provide measurements of drift time, which is then converted to a drift distance (i.e., a distance front the wire to a location that the nmuon has occupied in its flight, in the plane perpendicular to the chamber sense wire). Scintillators, on the other hand, only produce timing information. The output of chambers and scintillators produce nmuon hits. A nmuon track segment (a stub) is obtained by fitting the nmuon hits. Finally, COT tracks are extrapolated to the nmuon chambers and matched to nmuon stubs in the r plane [46]. TABLE OF CONTENTS page ACK(NOWLEDGMENTS ....._.__ .. .. 4 LIST OF TABLES ....._.. ... 9 LIST OF FIGURES ......... .... .. 10 ABSTRACT ......_ .._ ._ .. 14 CHAPTER 1 INTRODUCTION ..... ... 15 1.1 History of Particle Physics ......... .. .. 15 1.2 The Standard Model ......... . .. 21 1.3 Top Quark Physics ......... .. .. 2:3 1.4 Highlights of Mass Measurement . ..... .. :32 2 EXPERIMENTAL APPARATUS ....._ .. :38 2.1 Tevatron Overview ........ . .. :38 2.2 CDF Overview and Design ........ ... .. 40 2.2.1 Cl.,~ i. al:0,v Luminosity Counters ..... ... .. 41 2.2.2 Silicon Tracking ........ .. .. 42 2.2.3 Central Outer Tracker . .. .. .. 42 2.2.4 Calorinteters ........ ... .. 4:3 2.2.5 The bluon System ....... ... .. 44 2.2.6 The Ti1;__ 1- System ....... .. .. 44 :3 EVENT RECONSTRUCTION ........ .. 51 :3.1 Tracks ........ . .. 51 :3.2 Vertex Reconstruction ........ . .. 5:3 :3.3 Jets Reconstruction ........ .. .. .. 54 :3.3.1 Relative Energy Scale Correction .... .. .. 56 :3.3.2 Multiple Interactions Correction .... .... 57 :3.:3.3 Absolute Energy Scale Correction .... .. .. 57 :3.3.4 Underlying Event Correction . ... .. 58 :3.3.5 Out of Cone Correction ..... .. .. 58 :3.4 Leptons Reconstruction ........ .. 59 :3.4.1 Electrons ........ . .. 59 :3.4.2 Aluons ........ . .. 59 :3.4.3 Tau Leptons ........ .. .. 60 :3.4.4 Neutrinos ........ . .. 60 :3.5 Photon Reconstruction ........ .. .. 60 :3.6 Bottom Quark TI_-! .. it,-; ... .. .. .. 61 quark separates itself from all other quarks. For example, it is the most massive fermion by a factor of nearly 40 (the bottom being the closest competitor). Interestingly, even though the top quark is the most recent quark observed, its mass is the best known of all quarks. This is because it has such a short lifetime that it decays before any hadronization effects can occur. We should not be satisfied with this relative success and a more accurate determination of M~to is Strongly motivated inside and beyond the SM. The top quark is the weak isospin partner of the b-quark in the Standard Model. As such, it carries the following quantum numbers: an electric charge +2/3, an intrinsic spin of 1/2 and a color charge associated with the strong force. Due to the relatively small data sample collected in Run I of the Tevatron, none of these assignments have been measured directly. However, strong indirect evidence exists. First, the precision electroweak data of Z boson decay properties requires the existence of an isospin partner of the b-quark with electric charge +2/3 and a large mass. Furthermore, the predicted rate of top quark pair production, which is very sensitive to the spin and strong coupling of the top quark, is in good agreement with the data [3] [4] [5] [6]. Therefore, current observations lead us to believe that the particle observed at the Tevatron is indeed the top quark. However, direct measurements are still desirable and will be attempted in the case of the electric charge and spin using data from the Run II of the Tevatron or the LHC [7]. The other intrinsic properties of an elementary particle are its mass and lifetime. The most precise knowledge of the mass comes from direct measurements. The current world average containing only measurements performed during Run I at the Tevatron is 178 + 4.3 GeV/c2. In quantum mechanics, the lifetime of a particle is related to its natural width through the relationship -r = &/0. The branching ratio for the electroweak top quark decay t Wb is far larger than any other decay mode and thus its full width can be approximately calculated from the partial width F(t Wb). Assuming Myw = M1,. = 0, the lowest order calculation of the partial width has the expression shown in Equation 1-1, 4.7 Checks of the Matrix Element Calculation The event probability described in the section 4.6 depends on the top quark pole mass and is expected to be minimized in negative log scale around the true masses in the event. Multiplying all the event probabilities we obtain a likelihood function that depends on the top pole mass. Equation 4-48 shows the expression of the likelihood. L(Mtop) = (jMtp)8) events In negative log scale this likelihood is expected to have a minimum around the true pole mass, and so the top mass reconstruction can be performed. This reconstruction is the traditional matrix element top mass reconstruction. However, we only use this reconstruction to check the matrix element calculation. We use Monte Carlo samples generated at various input top masses. Only signal events are used. For each sample, the reconstructed top mass done by using only the matrix element calculation can be plotted against the input top mass. This can be done at various levels of complexity. Ideally, we'd see a linear dependence with no bias and a unitary slope. The first check to do is at the parton level. We take the final state partons moment from our Monte Carlo, smear their energies and use them as jets moment. Figure 4-5 shows a good linearity in the case of a 5' uniform smearing. There is a small bias of about 0.8 GeV, but the slope is consistent with 1. As the smearing is increased the bias becomes more evident, and slope degrades slightly. This can be also seen in Figure 4-5 for 101' smearing and for 211' smearing, respectively. In all of these situations a gaussian centered on 0 and with width equal to the amount of smearing used has been emploi-x I as a transfer function in the event probability computation. The partons can also be smeared using the functions described in section 4.4, in which case the same functions are used as transfer functions in the event probability computation. This test makes the transition between the parton level to the jets level, '~ ~'7 exp ( n (ment W S)) exx to (mi az) c03 \"ev "4 2 exp -(-2 NV(M, JES) from Equation 7-12 is given in eve N(M~, JES) x (mentW a~s)t9 The expression for normalization term Equation 7-13. 1: 3k+1 JES + p3k+2 J~ES2) Mk~' (713) The dependence of the parameters asi from Equation 7-12 as a function of the top mass M~ and jet energy scale JES is given by Equation 7-14. as = p3i+6 + 3i+7 M~ + p3i+8 JES, i = 0, 9 (7-14) The X2 per degree of freedom is 3551/2636 = 1.35 for the single' I__- d sample and 2972/2524 = 1.18 for the double' I__- d sample. The X2 has the same definition as in Equation 7-11. In each sample, the values of the 36 parameters, p, are given in Table 7-3. The shapes of few of the signal templates as well as the parameterized curves are shown in Figure 7-3. The background template shape is build in the same way as the signal templates. The top contamination is removed in the same way as in the case of the top templates (see section 7.2). The background template is fitted to a normalized sum of two gaussians and a gamma integfrand. For both the single' I__- d and the double' .,---- d samples, we show the values of the parameters in Table 7-4. CHAPTER 5 DATA SAMPLE AND EVENT SELECTION 5.1 Data and Monte Carlo Samples The data events are the Run2 CDF multi-jet events selected with the TOP_M~ULTIJET trigger, and it amounts to approximately 943 pb-l. This trigger selects about M' of the it all hadronic events. The Monte Carlo samples are the official CDF samples. We use 12 different samples generated with the Herwigf package to parameterize the mass dependence of our templates. The mass takes values from 150 GeV to 200 GeV in 5 GeV increments. There are also samples with a top mass of 178 GeV used to determine various systematic uncertainties: different choice of generator (in this case we used the Pythia package), different modeling of the initial state radiation (ISR) and of the final state radiation (FSR), different choice of proton parton distribution function (PDF). The background model described in section 6 is validated with the help of two Monte Carlo samples generated with the Alpgfen package: one with events having bb+4 light partons in the final state and another with events having 6 light partons in the final state. 5.2 Event Selection Before describing and listing the selection cuts, we need to mention the sample composition. The multi-jet events contain beside our signal events, a multitude of backgrounds: * QCD multi-jets * hadronic W,Z production * single top production * pair production in other channels The QCD multi-jet production has the N----- -1 contribution, while the others can be neglected since they involve electroweak couplings. M~w and lMop. Indeed, the correction to the W boson mass Ar given in Equation 1-4 contains additional terms due to Higgs boson loops. These corrections depend only logarithmically on M~H and have thus weaker dependence on M~H than on lMop. Still, precise determination of lMop and M~w can be used to obtain meaningful constraints on M~H aS illuStrated in Figure 1-5. Numerically, the constraints are [14] made explicit in Equations 1-7 and 1-8. M~H = 126+47 GeV/c2 (7 M~H < 280 GeV/c2 at 95' C.L., (1-8) Only the top quark mass measurements from Run I have been used. Such constraints on M~H can help direct future searches at the Tevatron and LHC and constitutes another stringent test of the Standard Model when compared to limits from direct searches or mass measurements from an eventual discovery. Even though the Standard Model successfully describes experimental data up to a few hundred GeV, it is believed that new physics must come into pIIl w at some greater energy scale. At the very least, gravity effects are expected at the Planck scale (a 1019 GeV) that the SM ignores in its current form. The SM can thus be thought of as an effective theory with some unknown new physics existing at higher energy scale. A link exists between the new physics and the SM that manifests itself through radiative corrections to SM particles. The Higgs boson sector is the most sensitive to loops of new physics. For example the Higgs boson mass corrections from fermion loops shown in diagram (a) of Figure 1-6 are given by Equation 1-9, where mf is the fermion mass and A is the "cut-off" scale used to regulate the loop integral. AM -i 2A +6f ln(A~ i /my + f ..., (1-9) The parameter A can be interpreted as the scale for new physics that typically corresponds to the scale of the Grand Unified Theory (GUT) near 1016 GeV. This is a X2 /ndf 0.5199 /6 Prob 0.9976 pO 0.05343 & 0.09964 X2 / ndf 36.54 / 6 Prob 2.168e-06 pO 1.058 &0.002828 -3 -2 0. 3 -2 -1 0 - Input JES 2 3 Input JES Figure 8-12. JES pull means versus input top mass, for input top mass equal to 170 GeV. Figure 8-13. JES pull widths versus input top mass, for input top mass equal to 170 GeV. X2 / ndf 3.468 /11 Prob 0.983 pO 0.05026 10.02838 X2 / ndf 550.7 /111 Prob 0 pO 1.0441 0.0008059 150 160 170 180 190 200 Input Mass 150 160 170 180 190 200 Input Mass Figure 8-15. Average of JES pull widths versus input top mass. Figure 8-14. Average of JES pull means versus input top mass. 0.4 - ,0.3 0- I- mn-==0iV I [--=-VI II n 0liiV Figur D-2 Cotiue I mnlKlmiiiVI I id_ I mnrKuniiin I I mnrKuniiin I I mnrKuniiin I I mnlKlmiiiVI I Ikl I mnrKuniia I I mnrKuniia I ~LI I mnrKuniia I I mnlKlmiiiVI I I mnrKuniia I I mnrKuniia I I mnrKuniia I M Figure D-2. Top templates for it double' I__- d events for samples with different top masses: from 150 GeV to 200 GeV. A) Case of JES = JES = -2. C) Case of JES = -1. D) Case of JES = JES = 1. F) Case of JES = 2. G) Case of JES = 3. -3. B) Case of 0. E) Case of 80.5- (D CD S80.4- E 80.3- Figure 1-5. Constraint on the Higgfs hoson mass as a function of the top quark and W hoson measured masses as of winter 2007. The full red curve shows the constraints (0.1' C.L.) conting from studies at the Z hoson pole. The dashed blue curve shows constraints (0.*' C.L.) front precise nicasurenient of M~w and f S I \ I ( I H\ I (2) (b) Figure 1-6. Loop contributions to the Higgs hoson propagator front (a) fernlionic and (b) scalar particles. 175 mt [GeV] background shapes one corrected for top of 160 GeV and the other corrected for top of 180 GeV. The change in the value of the reconstructed top mass is 0.9 GeV. 9.6 Background Statistics Another effect we address here is the effect of the limited statistics of the sample used to generate the background sample. To estimate this effect is enough to vary the parameters describing the background shapes. First we notice that the dijet mass template histograms for background are quite smooth, so only the event top mass template histograms will be modified. One has to reniember that the background model is based on about 2600 pretag data events passing the kinentatical selection. Then using the nxistag matrix we artificially increased the size of this sample by calling i.; 10 any distinct I__ d configuration. Therefore any of the original 2600 events will generate a number of these artificial .; .. 1 '. This number will be referred to as the multiplicity of the real event. In order to find the uncertainties on the background parameters, we need to fluctuate the content of the template histograms. Given the fact that entries of these histograms are not real events, but artificial i... at ', we have to somehow fluctuate the number of real events front each hin. The procedure is described below: * assume the event multiplicity the same for all real events and equal to the average multiplicity for the whole sample: 735 for single tags and 41 for double tags * before the it contamination removal and based on the constants above, we scale down the template histograms * fluctuate the content of the scaled histograms using the Poisson probability * after the Poisson fluctuation, scale back up the histogframs, remove the it contamination and fit with a gaussian to obtain the new template function * repeat the above steps 10,000 times, and histogfram the parameters of the new templates * extract the uncertainties on the background parameters front these last histograms S2007 Gheorghe Lungu 0.7 0.8 0.9 APPENDIX A PARTON DISTRIBUTION FUNCTION OF THE PROTON pdfs _u pdfs_u ntes 9800 sen 0.2078 us .1ss terl 2072 Oup Odown Ogluon Oubar Ot*3r strange charm Obottom sum Fntrier 9800 Mean 0.00094 RMS 0.1239 Underfow 0 0.9954 Integral=0.995420 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure A-1. Upper plot section 4.3. PDFs. shows the PDF shapes used in the matrix element calculation of Bottom plot shows a cross check of the normalization of these 1.5 0.5 00 0.1 0.2 0.3 0.4 0.5 0.6 The term aro is due to the running of a~. The term Ap is due to the one-loop top quark correction to W-boson propagators shown in Figure 1-4, and is given by Equation 1-6. 3 GF Mt2, Ap = (1-6) 80K2~r The uncertainty on the Fermi constant GF is completely negligible with respect to the one on the top quark mass in the computation of Ap. The term aro and the Weinberg angle in Equation 1-5 are known to a precision of 0."' The uncertainty on the top quark mass is currently about an order of magnitude larger than the other uncertainties and moreover it contributes quadratically to Ar. Thus the precision on Met, is currently the limiting factor in the theoretical prediction of the W boson mass. The parameter Ap is qualified as "universal" in the literature because it enters in the calculation of many other electroweak observable like sinew and the ratio of the production of b-quark hadrons of all types (usually denoted Rb), to name a few. Therefore, the top quark mass pha7i~ a central role in the interplay between theoretical predictions and experimental observables that aims to test consistency of the SM. One consistency check is to compare the measured value of M~t, with the predicted value from SM precision observables (excluding of course direct measurements of Meo). The indirect constraints, inferred from the effect of top quark radiative corrections, yields M~t, = 181'82 GeV/c2 [14]. The relatively small uncertainty is achieved because of the large dependence of M~t, on many electroweak observables. This is in remarkable agreement with the Run I world average of M~t, = 178 + 4.3 GeV/c2 [15], and is considered a success of the SM. A similar procedure can be used to constrain the Higgs boson mass (M~H), the last particle in the SM that has yet to be observed. The only direct information on M~H is a lower bound obtained from searches at LEP-II: M~H > 114 GeV/c2 at 95' confidence level [16]. Indirect constraints on M~H can be obtained with precise measurements of fundamental. Observing the pattern of the leptons many physicists started believing in the existence of a fourth quark, called charm (c). In 1970 Glashow, Iliopoulos and Alaiani proposed a mechanism through which the weak theory will allow flavor-conserving Zo-mediated weak interactions. This mechanism was requiring the existence of a fourth quark. Later in 197:3, at CERN, Perkins found evidence of weak interactions with no charge exchange. The existence of charm was confirmed in 1974 by Richter and Ting who found a charm-anticharm meson called J/W, and then reconfirmed in 1976 by Goldhaber and Pierre who found a charm-antiup meson called DO. A quantum field theory of strong interaction is formulated in 197:3 hv Fritzsch and Gell-Mann. They introduce the gluon (g) as a massless quanta of the strong force. This theory of quarks and gluons is similar in structure to QED, but since strong interaction deals with color charge this theory is called quantum chromodynamics, or QCD. The color charge was a concept introduced earlier in 196:3 by Greenberg, Han and Nambu. The hadrons made of quarks were considered color neutral. In 197:3, Politzer, Gross and Wilczek discover that at short distances the strong force was vanishing. This special property was called.movi-nlll Ie freedom. In 1979, a strong evidence for a gluon radiated by a quark is found at DESY, in Hamburg, Germany. In 1976, another unexpected particle is discovered. This new particle seen by Perl at SLAC was the tau lepton, denoted 7r, and it was the first particle of the third generation. In 1977, the existence of a third generation was confirmed by Lederman at Fermilah by discovering a new quark, called bottom (b). In 1989, the experiments at SLAC and CERN strongly supported the hypothesis of only three generations of fundamental particles by measuring the lifetime of Zo-hoson. Later in 1995 at Fermilah the remaining quark of the third generation is discovered. This is called the top quark (t) and it has mass much larger than the other quarks. Also at Fermilah the third generation is completed by the discovery of the tau neutrino (v-) in 2000. 9.11 Summary of the Systematic Uncertainties The total systematic uncertainty on the top mass combining all the effects listed above is 2.1 GeV. Table 9-3 suninarizes all sources of systematic uncertainties with their individual contribution as well as the combined effect. EntrgdMt63 EntbgM~t12 45 Mean 7348 600 -Mean 4063 RMS 4969 RMS 388 40 Undemfow 0Undemfow0 35 ovemow o 500 -ovelfow Integral 633 Integral 1120 30 400- 25 ,, n,300- 0500 1000 1500 2000 2500 3000 050 100 150 200 250 300 350 400 Figure 9-1. Event multiplicity for background events. On the left is shown the plot for single' I__- d events, while on the right the plot for double I__ d events is shown. Table 9-1. Uncertainties on the parameters of the top mass templates for background. Parameter 1 tag 2 tags Constant 10.2e-04 7.0e-04 Mean 2.59 :3.35 Sigma 272.1 711.9 Table 9-2. Residual jet energy scale uncertainty on the top mass. Level Uncertainty (GeV/c2) L1 0.2 L4 0.1 L5 0.5 L6 0.0 L7 0.5 L8 0.1 Total JES Residual 0.7 ACKENOWLED GMENTS The first person I want to acknowledge is my advisor, Prof. Jacobo K~onigsberg, for guiding and supporting me during my graduate student years in ner IlrJ rlis. His dedication, his commitment to his work and his students, and his savviness in the high-energy experimental field serve as an example to which I aspire as a physicist and as a scientist. Also I will be forever grateful to Dr. Valentin Necula in many aspects. He made possible many things for me starting with lending me money to pI li the tests needed for admission in the graduate school at the University of Florida. Moreover, he contributed greatly to the success of this analysis, from the writing the C++ code for main tools and ending with rich and enlightening discussions on the topic. His great skills and his excellence represent a standard for me. I would like to mention the great influence I received in my first years at the University of Florida from Prof. K~evin Ingersent and Prof. Richard Woodard. With or without their awareness, they helped me deepen my knowledge in theoretical physics. Also I take this opportunity to thank the members of the committee supervising this thesis: Dr. Toshikazu Nishida, Dr. Richard Field, Dr. Pierre Ramond and Dr. Guenakh Mitselmakher. I will be inspired by their tremendous work and by their extraordinary achievements in physics. Despite our rather brief interaction, I want to mention that my experience during my Oral Examination helped redefine me as a physicist and as a person. At CDF I drew much knowledge from interacting with many people such as Dr. Roberto Rossin, Dr. Andrea Castro, Dr. Patrizia Azzi, Dr. Fabrizio Margfaroli, Dr. Florencia Canelli, Dr. Daniel Whiteson, Dr. Nathan Goldschmidt, Dr. Unki Yang, Dr. Erik Brubaker, Dr. Douglas Glenzinski, Dr. Alexandre Pronko, Dr. Mircea Coca, Dr. Gavril Giurgiu. Special thanks to Dr. Dmitri Tsybychev, Dr. Alexander Sukhanov and Dr. Song Ming Wang who helped me greatly getting up to the speed of the experimental physics at CDF. Also I want to mention and thank Yuri Oksuzian and Lester Pinera for P~jsn = ) j dzedzbfx) x b a b I 6 (2r32Edi (2x,)4 (4)(Efi, Ei,i) 4EEblV U a 2)3E tot (m)e(m) Ncombi combi i= 1 |A|2 C E|2,, x 6(2) 36) i=1 i 2x (p )2 As mentioned previously, we will not use any constant that can be factored out in the expression of the probability density. From now on we will omit all such constants except for the number of combinations, Ncombi. Also in the- argument- of Prw ilu just p6 but it should be understood ()2 2Which in turn should be understood as a function of the 4-vectors of the final state partons. We will move to spherical coordinates in the integration over the partons moment. Due to the assumption that the angles of the partons are known as the measured angles of the jets, made explicit by the delta functions, 6(2)(R04 04p), all the integrals after the angles will be dropped together with the aforementioned delta functions. Also we use (4 instead of ((ji) in the argument of T. One should notice that |E|2 1S divided out by the energy factors in the denominator as seen in Equation 4-37. P~j n)= j Va -, Itot(m)e(m) Ncombi ji p iF~ p)] combi i= 1 x t -I' Py -t P (|Mag|R2 + | |17~,2 6(4)(Efi,, Ez,i) (4-37) p6 To reduce the number of integrals we will work in the narrow width approximation for the W-bosons. This translates in two more delta functions arising from the square of the W-boson propagators as shown by Equation 4-38. Pw =1 rw~l MW 6 2 -14 M ) (4-38) (P& M)2+ W M~Wry is a normalized product of a Breit-Wigfner function and an exponential. The parameters of this function depend linearly on top mass and JES. The Equation 7-8 d~;-1i ph the fit function and the dependence of its parameters on top mass and JES. 11 x (7-8) The expression for normalization term NV(M, JES) from Equation 7-8 is given in Equation 7-9. N(MJES)= ( 3k 3+1 JES + p3k+2 JES2) Mk~ _79) The dependence of the parameters asi from Equation 7-8 as a function of the top mass M~ and jet energy scale JES is given by Equation 7-10. asc = p1s i = (7-10) p3i+13 + 3i+14 M~ + p3i+15 JES i = 2, 3 The X2 per degree of freedom is 1554/1384 = 1.12 for the single' I__- d sample and 1469/1140 = 1.29 for the double' I__- d sample. The expression for X2 1S given ill Equation 7-11. p12 p7 Nl~bins hbin -fbin2 tm= 1 j= 1 bin=] hi <@ )) (E2 p b ihns 1) 25 where hbin is the bin content of the template histogram and fbin is the value of the function from Equation 7-8 at the center of the bin. The summation in Equation 7-11 is done for all templates and for all the bins for which Abin has more than 5 entries. The denominator of Equation 7-11 is the number of degrees of freedom. For each sample, the values of the 25 parameters, p, are given in Table 7-1. The shapes of few of the signal templates as well as the parameterized curves are shown in Figure 7-1. [17] H. Haber and R. Hempfling, Phys. Rev. Lett. 66, 1815 (1991); Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys., 85, 1 (1991); J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 257, 83 (1991); J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 262, 477 (1991); R. Barbieri and M. Fr-igeni, Phys. Lett. B 258, 395 (1991). [18] S. Heinemeyer, W. Hollik and G. Weinglein, Eur. Phys. J. C 9, 343 (1999); G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weinglein, Eur. Phys. J. C 28, 133 (2003). [19] S. Heinemeyer and G. Weinglein, hep-ph/0412214 (2004). [20] A review of dynamical electroweak symmetry breaking models can be found in: C.T. Hill and E.H. Simmons, Phys. Rept. 381 235 (2003); Eratum-ibid. 390, 553 (2004). [21] S. Weinberg, Phys. Rev. D 13 974 (1976); L. Susskind, Phys. Rev. D 20 2619 (1979). [22] C.T. Hill, Phys. Lett. B 266, 419 (1991). [23] D. Cronin-Hennessy, A. Beretvas, P.F. Derwent, Nucl. Instrum. Meth. A 443, 37-50 (2000). [24] S. Van Der Meer et al., Phys. Rep. 58, 73 (1980). [25] R. Blair et al. (CDF Collaboration), Fermilab Report No. FERMILAB-Pub-96-390-E, Section 12 (1996). [26] D. Acosta et al. (CDF Collaboration), Phys. Rev. D 71 032001 (2005). [27] D. Acosta et al. (CDF Collaboration), Nucl. Instrum. Meth. A 461 540-544 (2001). [28] C.S. Hill et al. (CDF Collaboration), Nucl. Instrum. Meth. A 530 1 (2004). [29] A. Sill et al. (CDF Collaboration), Nucl. Instrum. Meth. A 447 1-8 (2000). [30] T. Affolder et al. (CDF Collaboration), Nucl. Instrum. Meth. A 453 84 (2000). [31] T. Affolder et al. (CDF Collaboration), Nucl. Instrum. Meth. A 526 249-299 (2004). [32] L. Balka et al. (CDF Collaboration), Nucl. Instrum. Meth. A 267 272-279 (1998); S. Bertolucci et al. (CDF Collaboration), Nucl. Instrum. Meth. A 267 301-314 (1998). [33] M. Albrow et al. (CDF Collaboration), Nucl. Instrum. Meth. A 480 524-545 (2002); R. Blair et al. (CDF Collaboration), Fermilab Report No. FERMILAB-Pub-96-390-E, Section 9 (1996); G. Apollnari et al. (CDF Collaboration), Nucl. Instrum. Meth. A 412 515-526 (1998). [34] A. Artikov et al. (CDF Collaboration), Nucl. Instrum. Meth. A 538 358-371 (2005). [35] P. Gatti, "Performance of the new tracking system at CDF II", CDF Note 5561. leading six jets, and of the sub-leading four jets, aplanarity and centrality as defined in section 5. 6.2.2 Validation in Control Region 2 We compare shapes between our background model for this region and a Monte Carlo background. The background model for this region is formed by taking the pretag data sample in this kinematical region and by using the mistag matrix to obtain the tag rates. The Monte Carlo sample used has bb + 4 light partons in its final state. One variable we can look at is the sum of the event probabilities as defined in section 4 using the matrix element. The sum is between a top mass equal to 125 GeV up to 225 GeV in steps of 1 GeV. Figure 6-3 shows the shapes of Monte Carlo background and of the data-driven background. Another interesting variable is the invariant mass of all the untl I_ d pairs of jets in the event. Figure 6-4 shows this variable for the I__ d events before the minLKL cut, while Figure 6-5 shows the case of' I__- d events after the minLKL cut. 6.2.3 Validation in the Signal Region The top mass value for which the event probability is minimized represents another interesting variable. Figure 6-6 shows this variable for events after the minLKL cut. The event by event most probable top mass and the dijet mass variables are of particular interest since they will be used in the reconstruction of the top mass and of the JES variable to be described in section 7. All these comparisons show good agreement between our data-driven background model and the Alpgfen bb + 4 light partons. 6.2.4 Effects on the Statistical Uncertainty Using a top mass reconstruction technique based solely on the matrix element, we can vary the background fraction of our mixture of signal and background events and observe the effects on the statistical uncertainty of the top mass. The goodness of the mass reconstruction is related to the parameters of the reconstructed versus the input top mass. The statistical uncertainty is affected by the Once collected into a beam, the antiproton are sent to the Debuncher, a triangular synchrotron with a radius of 90 m, where their spread in energy is reduced using a synchronized oscillating potential in the RF cavities. This potential is designed to accelerate slower particles and decelerate faster particles. Uniform velocities of antiprotons enables more efficient beam manipulation and increases instantaneous luminosity by reducing bunch widths. Thus prepared, the antiprotons are collected and stored until they are needed for acceleration and collisions. One storage unit, the Accumulator, is a synchrotron in the same tunnel as the Debuncher, labeled .1.1sI n-IIn~~ source" in Figure 2-1. The Accumulator reduces the longitudinal momentum of the antiprotons using a synchronized potential and stochastic cooling [24]. Stochastic cooling was developed at CERN in the 1970s and dampens unwanted momentum phase-space components of the particle beam using a feedback loop. Essentially, the beam orbit is measured with a pickup and corrected with a kicker. The other antiproton storage unit is the Recycler, a synchrotron in the same ring as the Main Injector. The Recycler was originally designed to collect antiprotons from the Tevatron once collisions for a given store were finished, but attempts to use it for this purpose have not been worthwhile. As an additional storage unit, the Recycler has allowed increased instantaneous luminosity since 2004. The Recycler takes advantage of electron c....11nlr in which a 4.3 MeV beam of electrons over 20 m is used to reduce longitudinal momentum. When a store is ready to begin, antiprotons are transferred from either or both the Accumulator and the Recycler to the Tevatron for final acceleration. 2.2 CDF Overview and Design The Collider Detector at FNAL (CDF) is a general purpose charged and neutral particle detector [25] [26]. It surrounds one of the beam crossing points described in section 2.1. The detector observes particles or their decay remnants via charged tracks bending in a 1.4 T solenoidal field, electromagnetic and hadronic showers in calorimeters, Tracks surviving the cuts are ordered in decreasing pr and used in a fit to a common vertex. Tracks with X2 TelatiVe tO the vertex greater than 10 are removed and the remaining ones are fit again to a common point. This procedure is iterated until no tracks have X2 > 10 relative to the vertex. The second vertex finding algorithm developed at CDF is ZVertex~oll [38]. This algorithm starts from pre-tracking vertices (i.e., vertices obtained from tracks passing minimal quality requirements). Among these, a lot of fake vertices are present: ZVertex~oll cleans up these vertices requiring a certain number tracks with pT > 300 MeV be associated to them. A track is associated to a vertex if it is within 1 cm from silicon standalone vertex (or 5 cm from COT standalone vertex). Vertex position z is calculated from tracks positions zo weighed by their error 6 according to Equation 3-3. z = (3-3) Vertices found by ZVertex~oll are classified by quality flags according to the number of tracks with silicon/COT tracks associated to the vertex. Associated COT tracks have shown to reduce the fake rate of vertices thus higher quality is given to vertices with COT tracks associated: * Quality 0: all vertices * Quality 4: at least one track with COT hits * Quality 7: at least one track with COT hits, at least 6 tracks with silicon hits * Quality 12: at least 2 tracks with COT hits * Quality 28: at least 4 tracks with COT hits * Quality 60: at least 6 tracks with COT hits 3.3 Jets Reconstruction Jets are reconstructed by applying a clustering algorithm to calorimeter data. This algorithm determines the number of jets in an event, their energies and directions. future prospects (e.g., black curve for Tevatron/LHC and red curve for the International Linear Collider (ILC)) demonstrates very good discriminating power. The radiative corrections from MSSM particles to the SM precision observables are discussed in more detail in [19]. Other alternatives to replace the SM at energies near the TeV scale are theories involving dynamical breaking of the electroweak symmetry [20]. These models, one well-known example being Technicolor [21], do not include an elementary Higgs boson, but rather give mass to the SM particles by introducing a new strong gauge interaction that produce condensates of fermions that act as Higgs bosons. In some versions of these models, denoted "topl In i the new gauge interaction acts only on the third generation, and the fermion condensates are made of top quarks [22]. Such a model could be discovered by looking for evidence of new particles in the it invariant mass at the Tevatron or LHC. 1.4 Highlights of Mass Measurement Now that the top quark was placed in the context of particle physics and of the Standard Model, the most successful theory describing it, we stop to outline the remaining of the study. In the following chapters a detailed analysis of the measurement of the mass of the quark will be presented. The experimental apparatus used to produce and collect the data is described in broad details. This description is divided into a section dedicated to the accelerator of particles, Tevatron, and another for detailing the particle detector, the Collider Detector at Fermilab (CDF). M1 I.ny techniques are used for the identification of particles separately for leptons, photons, quarks and gluons. A more sophisticated tool involves the calculation of the matrix element for the process us i tt bbundd used in the computation of a probability to observe such process. This probability will be later used in the event selection and the in the mass bbTFE_0_ bTFE 1 1 E'ntri son8 Entum 02 men oomo mea on. a.. B 300~~~,,. us 091 Rs 01 undemow 1I 300C undemow "ealow o1 omow 25 -Iteoria so27 rI Inteoral Iml xinar 3a26/34 250C -I Ix inr loeoiso Pronl o2e22 I Pronl o42o 20 -consti 2713mas I consl 1307+2s3 emisl oose22ioooae 200C -I meani oloostooloo slumal oosalifoooml I slumal oo11matooose 10-const2 2m82r8ozes cona2 1863+262 Meml2 oI991soa242 150C mean. 2 oo4374tooem sigmn omo2rosi2o I slma2 oo282otooo41o 100 100- -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 EntiebTF_ 2 41 500os menoose ans o1632 undemow ouealow 400 Itoera 4168 ,irnr 4564/2s Prone oolme8 conni 2193+307 300- umsl o14oosiool11 slumal ominooo2e conn2 2Bli29e 200 oslum2 oo232i00ooas 100mi -016~0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure C-2. Continue 147 Graph S 4 - 3- O 2 -2- -3- 150 160 170 180 190 200 output Mass Figure 8-1. JES versus Top Mass plane. The mass. points represent the reconstructed JES and X2 /ndf 4.116 /10 020 Prob 0.942 SpO 175.3 i0.2763 8190 pl 0.9674 10.01897 - 150 160 170 180 190 200 Input Top Mass [GeV] Figure 8-2. Reconstructed top mass versus input top mass, for input JES equal to 0. 2 / ndf 0.6064 /5 Prob 0.9877 pO 0.0459310.0849 pl 0.98910.04234 // -3 -2 -1 0 1 2 3 Input JES Figure 8-3. Reconstructed JES versus input JES, for input top mass equal to 170 GeV. corresponds to the measured mass Myw a 80.4 GeV/c2), given that Mt, > M~w + Ml, This is an important characteristic of it events that is exploited in this analysis in the reconstruction of the top quark mass and the W boson mass. The W boson will in trn decay to two quarks about 2/3 of the time and a charged lepton associated with a neutrino about 1/3 of the time. The experimental signature of top quarks thus emerge. They are produced as it pairs, each one decaying immediately to a real W boson and a b-quark, the latter hadronizing to form a b-jet. The resulting W decay defines the it final state: There can be two hadronic decays (all-hadronic channel), one leptonic and one hadronic decay (lepton + jets channel), and two leptonic decays (dilepton channel), where the leptonic decays considered are usually only to electrons and muons (with their associated neutrinos) due to the experimental difficulty of identifying tau leptons. The approximate branching ratios for each channel are given in Table 1-3. The top quark pIIl us a central role in the predictions of many SM observables by contributing to their radiative corrections. Good examples are the W and Z boson propagators, in which loops involving top quarks are expected to strongly contribute, as illustrated in Figure 1-4. These diagrams can exist for any type of quark or lepton, but the very large value of M~t, makes the top quark contribution dominant. To illustrate the effect of the top quark, we consider in Equation 1-4 the theoretical calculation of the W boson mass [12]. xacl 1 z/Gysin28w 1 ar ) a~ is the fine structure constant, Ow is the Weinberg angle and Ar contains the radiative corrections and is approximately given by Equation 1-5. In Figure 3-3 the absolute jet energy scale corrections for jets cone size of 0.4 as a function of the jet momentum (blue). The uncertainty for this correction is also shown as a function of the jet momentum (black). 3.3.4 Underlying Event Correction In a hadron-hadron collision, in addition to the hard interaction that produces the jets in the final state, there is also activity in the detector originating from soft spectator interactions. In some event, the spectator interaction may be hard enough to produce soft jets. Energy from the underlying event can fall in the jet cones of the hard scattering process thus biasing jet energy measurements. A correction factor for such effect has been calculated using a sample of minimum bias events as for multiple interaction correction, but selecting only those events with one vertex [43]. For each event, transverse energy Er inside cones of different radii (0.4, 0.7 and 1.0) is measured in a region far away from cracks (0.1 < |9|l < 0.7). The correction factor is extracted from the mean values of ET distribution (Figure 3-4). 3.3.5 Out of Cone Correction The jet clustering may not include all the energy from the initiating partons. Some of the partons generated during fragmentation may fall outside the cone chosen for the clustering algorithm. This energy must be added to the jet to get the parton level energy. A correction factor is obtained using MC events [44]: hadron-level jets are matched to partons if their distance in the rl plane is less than 0.1. Then the difference in energy between hadron and parton jet is parameterized using the same method as for absolute correction (Figure 3-5). We have seen different corrections that account for different sources of jet energy mis-measurement. Depending on the physics analysis, all of them or just a subset can be applied. The corrections are applied to the raw measured jet momentum. PT (R, PT, r) = (P}"(R)- f,(R, PT r)-M,~(R))- fabs(R PT)- UE(R)+ OOC(R, PT) (3-10) shape of this distribution is normalized to unity and therefore we have in Equation 4-33 the value for #1. dp r, (p ) = 1, O (4-33) As mentionedl before wei needT tnpnpo ex rpres vrythingin terms of p6 andl p6 This can be done just by changing the variables from the polar to the Cartesian coordinates as shown in Equation 4-34. dp d~ PT(p t) = 1= dpn6dp T("'tp = (p6)2 Fiue 4-2 o riei quto 3 the shapepesio of the transverse momentum of tei vnsi hw itdt u of ~p 3 gaussians.43 section 4.3,t 4.4 an 4.5~ offeredig detailsnc on the exreso ms b i uns of eea iprat piece 4 Ipeenteringd vlutono the probability density.UigEutos42,43an435wecnrten Equation 4-36 the new expression for the probability density. 9.9 B-jet Energy Scale We study the effect of the uncertainty on the modeling of heavy flavor jets due to the uncertainty in the senli-leptonic branching ratio, the modeling of the heavy flavor fragmentation and due to the color connection effects. To determine this we reconstruct the top mass in a Monte Carlo sample where the b-quarks could be geometrically matched to a jet, and the energy of such jets was modified by 1 As it turns out in [53], 0.10' of the jet energy uncertainty on the b-jets is coming front the effects listed above. Therefore the final shift on the top mass following our 1 shift in b-jets energies needs to be scaled down by a factor of 0.6. The systematic uncertainty on the top mass due to the b-jet energy scale is 0.4 GeV. 9.10 Residual Jet Energy Scale Fr-on the hi-dintensional fit for top mass and JES, we extract an uncertainty on the top mass that includes a statistical component as well as a systematic uncertainty due to the uncertainty on the JES parameter. However, the JES parameter is defined as the sunt of six independent effects, and therefore the systematic uncertainty on the top mass included in the 2D fit is only a leading order uncertainty due to our limited understanding of the jet energy scale. Second order components of this uncertainty arise front the limited understanding of the six individual contributions to JES. Additional details on this source of uncertainty can he found in [54]. For this we have to study the effect on the top mass reconstruction front each of these six sources: level 1, 4, 5, 6, 7 and 8. A Monte Carlo sample has been used where the energies of the jets have been shifted up or down by the uncertainty at each level separately, so a total of 12 samples have been obtained. We reconstruct the top mass in each of them, without applying any constrain on the value of JES. In Table 9-2 we present the average shift on the top mass at each level, and their sunt in quadrature. We conclude front this that the residual jet energy uncertainty on top mass is 0.7 GeV. CDF Tracking Volume ,30 n = 2.0 *n =3.0 -30 5 10o 15 SVX II INTERMEDIATE 5 LAYERS SILICON LAYERS Figure 2-3. Schematic of trackingf volume and plugf calorimeters of the upper east quadrant of the CDF detector. Year2002 2003 2004 2005 2006 2007 Year2002 2003 2004 2005 2006 2007 Ms nth 4 7 10 1 4 7 101 4 7 1 47101 7 0 Ms nth 4 7 10 1 4 7 101 4 7 1 47101 7 0 150 12500 50 ls i~3500 eird 0 u 1000 1500 2000 2500 3000 3500 4000 4500 5000 1000 1500 2000 2500 3000 3500 4000 4500 5000 Store Number Store Number Figure 2-4. Initial instantaneous luminosity (left) and total integrated luminosity (right) as a function of year since the start of Run II. Figure 3-5. Jet corrections due to out-of-cone effect for jets with cone size of 0.4 as a function of the jet momentum (red). The uncertainty for this correction is also shown as a function of the jet momentum (black). 20 40 60 80 100 120 140 160 180 200 Corrected jet PT (GeV) t 0.14 0.12 C~0.08 U)0. 06 LI - 0.04 Underlying Event Systematic Uncertainty SCon 0 -oe0 Figure 3-4. Fractional transverse systematic uncertainty due to underlying event as a function of jet momentum for different jet cone sizes. **** Correction for Cone 0 4 jets - Uncertaintyio 20 40 60 80 100 120 140 160 180 200 PT particle-jet (GeV) mass is concentrated in a nucleus with the electrons orbiting around it. Several years later in 1918, following a different scattering experiment with ac-particles, he will conclude that the hydrogen nucleus is an elementary particle and it is present inside the nucleus of every atom. This new particle was later called proton. While the proton was able to explain the charge of the nucleus, it couldn't explain the mass of heavier atoms. Rutherford believed that a neutral particle he called neutron exists, but this was confirmed experimentally only in 1932 by ChI I [wick. Rutherford's atom was not a satisfactory model. The electron going around the nucleus would be accelerated centripetally and therefore should emit electromagnetic radiation according to the classical theory of electromagnetism. The loss of energy through radiation should make the electron collapse on the nucleus rendering Rutherford's atom unstable. In 1913, Bohr will propose a different model for the atom in which the electrons sit on orbits with discrete values of the orbital angular momentum. The electron can move from one orbit to another by releasing or receiving a photon with an energy equal to the energy difference between the orbits. This model will receive support from the Franck-Hertz experiment where it was observed that the atoms can absorb only specific amounts of photons. Bohr's atom was still not explaining several experimental observations like the splitting of the atomic spectral lines (Zeeman effect) or the splitting of a beam of electrons when passing a magnetic field (Stern-Gerlach experiment). To explain this, in 1925, Uhlenbeck and Goudsmit proposed that the electron spins on its axis as it orbits around the nucleus. Soon Pauli introduced the exclusion principle stating that two particles can occupy a state defined by the same quantum numbers explaining why the electrons were spread overall several orbits. In 1924, De Broglie extended the particle-wave duality from photons to any particle such as the electron. The wavelike character of the electron was observed in 1927 in a diffractive experiment hv Davisson and Germer. Based on this idea, Schrodinger Tree level Feynman diagram for the process us i tt bbundd ...... Cross section for it production versus the top mass, from CompHep ... Transverse momentum of the it events ...... Mass reconstruction using smeared parton energies ...... Mass reconstruction using jets matched to partons ...... Reconstructed top mass versus input top mass using realistic jets. .... Minimum of the negative log event probability ....... Background validation in control region 1 for single I__ d events .... Background validation in control region 1 for double I__ d events ... Sum of event probabilities calculated for for background samples. .... .... 86i . .. 87 . 88 . 88 . 89 . 90 . 95 .... 100 .... 101 .... 101 6-4 Dijet invariant mass of the ulrnt I_ d jets for background before the cut on the signal-like probability. 6-5 Dijet invariant mass of the ulrnt I_ d jets for background on the signal-like probability ...... 6-6 Event by event most probable top mass distributions for after the signal-like probability cut ..... 6-7 Effect of the background contamination in the top mass r only the matrix element technique. ..... 7-1 Top templates for it events. ..... 7-2 Top templates for background events ..... 7-3 Dijet mass templates for it events. ..... 7-4 Dijet mass templates for background events ..... 8-1 Raw reconstruction in the JES versus Top Mass plane . 8-2 Reconstructed top mass versus input top mass, for input 8-3 Reconstructed JES versus input JES, for input top mass 8-4 Slope of the mass calibration curve versus input JES. .. 8-5 Constant of the mass calibration curve versus input JES. 8-6 Slope of the JES calibration curve versus input JES. .. 8-7 Constant of the JES calibration curve versus input JES. samples after the cut .. 102 background samples .. 103 reconstruction using .. 103 . 111 . 111 . 111 .. 113 . 120 JES equal to 0. .. 120 equal to 170 GeV. .. 120 .. 121 .. 121 .. 121 . . 121 is also determined with the help of a Monte Carlo sample and we'll offer more details in section 4.5. Therefore the new expression for the probability density is shown in Equation 4-5. 1 dzdz a b P~jIm= totmt(m>em 4 Xdb(,fX~EaEb Ug t, | i= (2xr)32Ei ,,,, Even though a tt event in the all hadronic final state is fully reconstructed, there is an ambiguity in assigning the jets to the partons. Therefore all the possible combinations are considered and their contributions averaged. The number of possible assignments depends on the topology of the event and this will be discussed in section 4.2. Until then the Equation 4-6 gives the most general expression of the probability density. 1 d a z a b P(j Im)=x atot (m)e(m) Ncombi ddxf,)xb4EEblV i, r, | I (2xr)32Ei1 x |Mz~(m, p)|12(2xT)4 b(4) (Efi, Ei,i )TF(j13 |p P(pi3 (4-6) 4.2 Combinatorics In general, there are 6! = 720 r-wsi~ to assign the observed jets to the six partons of the final state in an all hadronic tt process. This number can be reduced by making few observations and assumptions. First, one has to notice that the matrix element is symmetric to t +-4 t. Let's write down in Equation 4-7 the spin averaged matrix element squared for the process us i t. 4 M 288(p,g +' ps)m)v ~~T YL(~-m-y(j~ tl( spons Assuming that the masses of the up quarks are zero and omitting the constant and the gluon propagator term, we can write Equation 4-8 spons APPENDIX C TRANSFER FUNCTIONS A Figure C-1. Transfer functions for the W-boson decay partons. A) For partons with the value for pseudo-rapidity |9| < 0.7. B) For partons with pseudo-rapidity 0.7 < |9| < 1.3. C) For partons with pseudo-rapidity 1.3 < |9|l < 2. Graph S3 -2 -2 -3 150 160 170 180 190 200 Corrected output Mass Figure 8-16. JES versus Top Mass plane. The points represent the reconstructed JES and mass after the 2D correction. for which data is stored, CDF uses information from some detector components to make a decision to save an event, called a tri ~-r. Data is stored in buffers until trim. r--i decisions cause some of the events to be read out and stored on computer disk or the buffer to be emptied. The trigger is divided into three levels of increasing sophistication in object identification (Figure 2-9). Data is stored in synchronous buffers awaiting an initial trigger decision. The first trigger level returns a decision with a latency of 5.5 ps and a maximum accept rate of 50 kHz and will ak- -l-s occur in time to read out the event. Level one uses solely custom hardware operating in three parallel streams. One stream, the extremely Fast Tracker (XFT), reconstructs transverse COT tracks and extrapolates them to calorimeters and muon chambers. Another stream detects possible electron, photon or jet candidates, along with total and missing transverse energy. The final stream searches for tracks in muon chambers. These streams are combined in the final level one decision. After a level one accept, the event information is read out into .l-inchronous buffers. Since events remain in these buffers until a level two decision is made, it is possible some events passing level one will be lost when these buffers are full. The level two tr~i ;r returns a decision with a latency of 25 ps and a maximum accept rate of 300 Hz. Level two used custom hardware and modified commercial microprocessors to cluster energy in calorimeters and reconstruct tracks in the silicon detector using the Silicon Vertex Tracker (SVT). Calorimeter clusters estimate the total jet energy and help to identify electrons and photons. The SVT measures the impact parameters of tracks, part of locating displaced vertices. The third trigger level runs on a commercial dual microprocessor farm and returns a decision with a maximum accept rate of 150 Hz. The farm runs a version of CDF offline reconstruction merging information from many detector systems to identify physical objects in the event. Data passing level three tr~i ;r requirements is transferred via Aplanarity Figure 6-1. Jet Eta CDF Runil preliminary L=943pb' 0.2 0.1 U 5 'l'15 20 Number of Z's CDF Runil preliminary L=943pb' 0.12 0.15 - 0. 8 - 0. 6 - OU 100 O 0 SumEt3 (GeV/c2) CDF Runil preliminary L=943pb' O.02C -~ U 0.2 ~~040.6 0.8 Centrality 0 2 4 6 Jet Phi CDF Runil preliminary L=943pb 0.0 0. . slope of the calibration curve. The bias in the mass reconstruction is related to the intercept of the calibration curve. In the upper plot, Figure 6-7 shows how the slope decreases with the background fraction, while the lower plot shows how the intercept changes with the background fraction. The slope decrease indicates a decrease in the sensitivity, in other words an increase in the statistical uncertainty on the top mass. For the calibration curves studied in these plots the intercept should be 178 GeV, and it can he seen that as the background fraction increases the intercept gets further from the 178 GeV value, that is the bias mecreases. The reason for the background fraction to have such a big effect on the mass reconstruction using the matrix element technique of section 4 is because the background is completely ignored in the matrix element calculation or in assessing a background event probability. In this analysis we still won't calculate a background matrix element, but we will use a background probability instead, which will be described in the next sections. CDF Runil preliminary L=943pb' CDF Runil preliminary L=943pb' "U 5 100 15 o 20 -O1 Jet Et (GeV/c2) CDF Runil preliminary L=943pb Background validation in control region 1 for single I__ d events. The red points are the data points, while the black points are from the background model. C E C E~miV mlliI II n~miV ~W ~TI -E I -0~uii I- -0 cn~mii -= a~mi c =r=-ia c c ===iin IImnamii Figure D-2. Continued The Jacobian is obtained by solving a system of equations for pb and pg. The relations entering the system of equations are shown in Equation 4-46. (4-46) We can then write in Equation 4-47 the expression of the probability density in its final form which is used inside a C++ code. P(jm) =/ x ator (m) a(m) Nvcombi (12 2 Lo34 2p294 combi x W((4|p) 6 t Pi (|Man|$ + |Mc~L|2 (47 fi= 1 F~p)]Dpn6 The integration is performed by simply giving values to the 4 integration variables and then by adding up the integrand obtained at each step. The limits of the integration are -60 GeVi 60 GeV~ for ,6 and 10 GeV 300 GeV for p24. The step of integration is 2 GeV. Given these limits, at each step of integration we have to check the physicality of the components entering Equation 4-47. The probability density is evaluated for top mass values going in 1 GeV increments from 125 GeV 225 GeV. The dependence on mass of the it cross-section is obtained from values calculated by CompHep Monte Carlo generator for the processes us i t, dd i t and gg i t. The absolute values for these cross sections are not as important as their top mass dependence. Figure 4-1 shows this dependence. For the proton andl antiproton PDF, f ( ) f (p3i), we~ wVill use the~ CTE5LU dUistibutions with the scale corresponding to 175 GeV. The shapes are given in Appendix A. The it acceptance, e(m), depends on the top mass and will be described later when the event selection is addressed. The final expression of the probability density has been given and its implementation has been detailed. The following section is dedicated to the checks we performed in order to assure the proper functionality of the matrix element technique. Figure C-2. Transfer functions for the b-quark partons. A) For partons with the value for pseudo-rapidity |vy| < 0.7. B) For partons with pseudo-rapidity 0.7 < |vy| < 1.3. C) For partons with pseudo-rapidity 1.3 < |vy| < 2. Equation 4-6, we will need to sum over all the possible spin configurations of the initial state. We find two non-zero contributions corresponding to the situations when the incoming partons have the same handedness. Therefore for the term I from Equation 4-12 is expressed in Equation 4-20. IgR = ZEd (0, 1, i,0) I = (p- o (s) =(4-20) If, = ~E~(0,1,: -i, 0) In principle, we need to average over all the possible spin configurations of the final state. The Equations 4-18 and 4-19 represent the non-zero contributions. Using Equations 4-18, 4-19 and 4-20, the product of the terms I, T, W1 and W2 is giVeH in Equation 4-21. I T W1 W2 = Ex ManR,LL (4-21) Fr-om Equation 4-21, the term E proportional to the product of the energies of all particles, incoming or outgoing, is shown in Equation 4-22. May,LL ( f ) (7 .; o ) 0(h b naL ), m2 bRR,LL) 0 (4-23) The terms ManR and MrsL, shown in Equation 4-23, are calculated in a C++ code using Equation 4-15 and the matrix algebra. Therefore we can write down the expression of the matrix element squared from Equation 4-6 in the form of Equation 4-24. IM"-1 |A|=~2 -C || |Ad|2~~~~ 2622 -P P P (|Man|$ + |McL|2) (4-24) spons colors 7.50 in the wall, and 0.16 x 7.50 or 0.2-0.6 x 150 in the plug. The energy resolution of the C1EM is o-(E)/E = 0.135/ Er(TGeV) 0.015. Figures 2-'7 shows a c~ross-sectional vie~w of the plug calorimeter. 2.2.5 The Muon System Whereas electrons create showers confined to the calorimeters, the mass of muons makes them nearly minimum ionizing particles (jl\l's), and high momentum pass through the calorimeters. The calorimeters (and in some cases additional steel shielding) block the 1 in .0 lRy of hadronic particles from reaching the outer detector. Drift chambers placed on the outside of the detector identify charged tracks from muons and measure their position. There are three muon detection systems: C \l U, C \lP' and CijlS [34]. CijlU and CijlP cover detector |9|l < 0.6, with CijlP located outside CijlU, and CijlS covers detector 0.6 The C \LU chambers surround the central calorimeter in ~. They are composed of 4 concentric 1... ris of cells containing argon-ethane gas and high-voltage sense wires parallel to the beam pipe (Figure 2-8). The CijlP chambers are separated from the C11lU chambers by 60 cm of steel shielding. They are similar in construction to the C \!U chambers, but the lIn-;-rs are successively offset by half of a cell. The C \! X chambers are nearly identical to the C \LU chambers. They are arranged in four logical 1 ... rs successively offset by half of a cell. Each logical 111-;-r consists of two partially overlapping physical 1... ris of cells. On average, a particle will traverse six cells. Sense wires are independent in the CijlP chambers, but are shared between 4 neighbors in CijlU and C'j lS The single-hit r resolution is 0.25 mm. Measurements in z with a resolution of 1.2 mm are also possible by using differences in arrival times and amplitudes of pulses measured at either end of each wire in neighboring cells. 2.2.6 The Trigger System Collisions occur every 396 ns (2.5 MHz), far too quickly even for CDF's custom hardware to process and read out detector information. To reduce the number of collisions ------ i ----- i i /I 00 R=29 cm Port Cards 90crn Layer 00) SVX 11 64cm Figure 2-5. Schematic with the r-< and the y-z views of the Run II CDF system. silicon tracking I II ' I + Potential wires *Sensewires X Shaper wires BareMylar Gold on Mylar (Field Panel) j6 58 60 62 64R SL2 Layer # 1 2 3 4 5 6 7 8 Cells 188 192 240 288 336 384 432 480 Figure 2-6. East end-plate slots Sense and field planes are at the clock-wise edge of each slot (left). Nominal cell layout (right). super-111--c rs. Each super-11s-c r is further divided with gold covered Mylar field sheets into cells containing 25 wires alternating between potential and sense wires, see Figure 2-6. In half of the super-11s-c r~s, the wires are parallel to the beam line and provide axial measurements, while in the other half, the wires are alternately at +2" and provide stereo measurements. The innermost super-l} ... r provides a stereo measurement and subsequent 1 .,;- comprised of 50'; argon and 50'; ethane (and lately, some oxygen was added to prevent corrosion). This results in a maximum drift time of 100 ns, far shorter than the time between hunch collisions. The single hit resolution of the COT is 140 pm, and the track momentum resolution using muon cosmic ray,~s is o-,g,;~ M 0.001 (GeV/c)j- 2.2.4 Calorimeters Calorimeters provide energy and position measurements of electron, photon and hadron showers. They are divided into electromagnetic (EM) and hadronic (HA) segments, with EM positioned closer to the interaction region than the HA. Both regions are sampling calorimeters with alternating 11s- generate photons in the scintillators which are collected and carried to PMTs with wavelength shifting optical fibers. Lead is used as the absorber in EM segments and iron in HA segments. The EM segment closest to the interaction region acts as a pre-shower detector useful for photon and ~ro discrimination. A shower-maximum detector, placed at about 6 radiation lengths in the EM calorimeter, measures the shower profile and obtains a position measurement with a resolution on the order of a few mm. Due to detector geometry, calorimeters are divided into a barrel shaped region surrounding the solenoid, the central calorimeters (CPR, CES, CEM and CHA) [:32]; and calorimeters capping the barrel, the plug calorimeters (PPR, PES, PEM and PHA) [:33]. A wall hadronic calorimeter (WHA) fills the gap between the two. The central region covers detector |vy| < 1, the wall 0.6 < |vy| < 1.3, and the plug 1.1 < |vy| < :3.6. Each of these regions is further segmented in ty and 4 into towers covering 0.1 x 15" in the central, 0.1 x Starting from the quantities in Equation 3-4, the jet transverse energy, transverse momentum and pseudo-rapidity are calculated in Equations 3-5, 3-6 and 3-7. PT = (3 5) Er = PT, (3-6) E pz The jet 4-momenta measured in the calorimeter suffer from intrinsic limits of both calorimeter and jet algorithm. Different particles produce different responses in calorimeters and some of them can fall in uninstrumented regions of the detector. Moreover, calorimeter response to particle energies is non-linear. The jet clustering algorithm, on the other hand, doesn't take into account multiple interactions and energy that can be radiated outside the fixed radius cone. For all these reasons, a set of corrections has been developed in order to scale measured jet energy back to the energy of the particle [39]. 3.3.1 Relative Energy Scale Correction Relative (or rl-dependent) jet energy corrections [40] are applied to raw jet energies to correct for non-uniformities in calorimeter response along rl. Calorimeter response in each rl bin is normalized to the response in the region with 0.2 < |9|l < 0.6, because this region is far away from detector cracks and it is expected to have a stable response. The correction factor is obtained using the dijet balancing method applied to dijet events. This method starts selecting events with one out of two jets in the region 0.2 < |9|l < 0.6. This jet is defined as tr~i ;r jet. The other jet is defined as probe jet. If both jets are in the region of 0.2 < |9|l < 0.6, tr~i -;r and probe jet are assigned randomly. The transverse momentum of two jets in a 2 2 process should be equal and this property is used to calculate first a pT balancing fraction Apr f as shown in Equation 3-8 ,~~,prPTobe t riggerj (8 are fixed in the likelihood. However the JES is constrained via a gaussian centered on the true JES and with a width of 1. Figure 8-1 shows the reconstructed JES and the reconstructed top mass represented by the points, versus the true JES and true top mass represented by the grid. Ideally the points should match the grid crossings. Figure 8-2 shows reconstructed top mass versus the true top mass for a true JES of 0. Ideally, this curve should have a slope of 1, and an intercept of 175 consistent with no hias. Figure 8-3 shows reconstructed JES versus the true JES for a true top mass of 170 GeV, and again, ideally, this curve should have a slope of 1, and an intercept of 0 consistent with no hias. Figure 8-4 shows how the slope of Figure 8-2 changes with the true JES, while Figure 8-5 shows how the intercept of Figure 8-2 changes with the true JES. Figure 8-6 shows how the slope of Figure 8-3 changes with the true top mass, while Figure 8-7 shows how the intercept of Figure 8-3 changes with the true top mass. Figure 8-8 shows the mass pull means versus true top mass, while Figure 8-9 shows the mass pull widths versus true top mass. In both plots the true JES is 0. Based on these figures it results that the uncertainty on top mass has to be inflated by 10.5' The average mass pull mean as a function of true JES is shown in Figure 8-10, while the average mass pull width as a function of true JES is shown in Figure 8-11. For a given true JES value, the average is over all the mass samples. Figure 8-12 shows the JES pull means versus true JES, while Figure 8-13 shows the JES pull widths versus true JES. In both plots the true top mass is 170 GeV. Based on these plots it results that the uncertainty on the JES has to be inflated by 5.>' The average JES pull mean as a function of true top mass is shown in Figure 8-14, while the average JES pull width as a function of true top mass is shown in Figure 8-15. For a given true top mass value, the average is over all the JES samples. As it can he seen in Figure 8-1, there seems to be a slight hias in the reconstruction of JES and top mass. We can extract the slope and the intercept of the dependence of the reconstructed mass on the true mass. This can he done for different JES values. 120 - 100 - 80 - 60 - 40 - 20 - 0 0 dO 200 250 3 00 220 - 200 - 180 - 160 - 140 - 120 - 100 - 80 - 60 - 40 - 20 - 0 0 150 200 250 3UU 00 0 - 350 - 300 - 250 - 200 - O - 50 - OC 0 400 - 350 - 300 - 250 - 200 - 150 . 100 . 50 - OC 0 160 - 140 - 120 - 100 - 80 - 60 - 40 - 20 - oC 0 dO 200 250 3 If 0 250 - 200 - 150 - 100 . 50 - 0 1 200 250 3UU If 0 300 - 250 - 200 - 150 - 100 - 50 - OE E 0 450 - 400 - 350 - 300 - 250 - 200 _ 150 - 100 . 50 - 0 E 0 180 - 160 - 140 - 120 - 100 - 80 - 60 - 40 - 20 - 0 dO 1 200 250 3UU ?0 250 . 200 - 150 - 100 - 50 - 0 1 1 200 250 3UU ?0 350 - 300 - 250 - 200 . 150 . 100 - 50 - 0 450 - 400 - 350 - 300 - 250 - 200 - 150 - 100 - 50 - 0 Figure E-2. Continued CHAPTER 7 DESCRIPTION OF THE MASS MEASUREMENT METHOD The N----- -r contribution to the uncertainty on the top quark mass is the jet energy scale uncertainty. The jet energy scale and its uncertainty is measured independently at CDF by the Jet Energy Resolution working group. It takes into account the differences between the energy scale of the jets in our Monte Carlo samples and the scale observed in the data. Its value depends on the transverse energy, pseudo-rapidity and the electromagnetic fraction of the total energy of a jet. So the jet energy uncertainty is different from jet to jet, but we will generically denote that with ac. The environment in which this scale and uncertainty is determined is quite different than that of the it events, and additional corrections might be needed at this level. We define a variable, JES, called Jet Energy Scale, measured in units of ac. There is a correlation between the top mass and the value of JES, and that's why we plan to measure them simultaneously to avoid any double counting in the final uncertainty on the mass. Our technique starts by modeling the data using a mixture of Monte Carlo signal and Monte Carlo background events. The events will be represented by two variables: dijet invariant mass and an event-by-event reconstructed top mass. The latter is obtained using the matrix element technique described in section 4. For signal, the shapes obtained in these two variables are parameterized as a function of top quark pole mass and JES. For background no such parameterization is needed. Hence our model will depend on the top mass and the JES. The measured values for the top quark mass and for the JES are determined using a likelihood technique described in this section. 7.1 Likelihood Definitions The likelihood function used to reconstruct the top mass, shown in Equation 7-1, is product of 3 terms: the single tag likelihood used for single I__ d events, ~Lte,, the double tag likelihood used for double I__ d events, 2tag and the JES constraint, JES, whose expression is shown in Equation 7-7. 350 - 300 - 250 - 200 - 150 - 100 - 50 - 0 50 0 1 200 250 UU ? 0 400 - 350 - 300 - 250 - 200 - 150 - 100 . 50 - 0 0 1 200 250 UU 7? 0 450 400 - 350 - 300 - 250 - 200 - 150 - 1 OC E 0 600 - 500 - 400 - 300 - 200 - 100 _ OC E 0 400 - 350 - 300 - 250 - 200 - 150 - 100 - 50 - oC 50 100 150 200 250 3UU 3 500 - 400 - 300 - 200 - 100 - 0 1 200 250 3UU 500 - 400 - 300 - 200 - 100 - 0 500 - 400 - 300 - 200 - 100 - 0 400 - 350 - 300 - 250 - 200 - 150 - 100 - 50 - 01 50 100 150 200 250 3UU di 0 500 - 400 - 300 - 200 - 100 - 0 1 1 200 250 3UU ? 0 600 - 500 - 400 - 300 - 200 - 100 - 0 500 - 400 - 300 - 200 - 100 - 0 Figure E-2. Continued All the discoveries described above led to the formulation of a theory that suninarizes the current knowledge of the fundamental particles and the interactions between them. This theory is called the Standard Model of particle physics and it will be described in more detail in the next section. 1.2 The Standard Model The Standard Model of particle physics is a theory which describes three of the four known fundamental interactions between the elementary particles that make up all matter. It is a quantum field theory which is consistent with both quantum niechanics and special relativity. To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions. However, the Standard Model falls short of being a complete theory of fundamental interactions, primarily because of its lack of inclusion of gravity, the fourth known fundamental interaction, but also because of the large number of numerical parameters (such as masses and coupling constants) that must he put "by hand" into the theory (rather than being derived front first principles). The matter particles described by the Standard Model all have an intrinsic spin whose value is determined to be 1/2, making them fernxions. For this reason, they follow the Pauli exclusion principle in accordance with the spin-statistics theorem giving them their material quality. Apart front their antiparticle partners, a total of twelve different types of matter particles are known and accounted for by the Standard Model. Six of these are classified as quarks (up, down, strange, charm, top and bottom), and the other six as leptons (electron, nmuon, tau, and their corresponding neutrinos). Each quark carries any one of three color charges red, green or blue, enabling them to participate in strong interactions. The up-type quarks (up, charm, and top quarks) carry an electric charge of +2/3, and the down-type quarks (down, strange, and bottom) carry an electric charge of -1/3, enabling both types to participate in electromagnetic interactions . 72 / ndf 15.62 /4 pO 0.0058941 0.0007298 pl 0.35631 0.0006464 e 3.5 o (u3. E o "2.5 A Lu .5 v 1 0.5 1 23 45 67 8 Nurnber of primary vertices Figure 3-2. Average transverse energy as a function of the number of primary vertices in the event: a correction factor is extracted from the slope of the fittingf line. u 16 S15 S14 12 ***--- CorrectionforCone04Jets -Uncertainty to - -.~ 50 100 150 200 250 300 350 400 450 500 PT jet (GeV) Figure 3-3. Absolute jet energy scale corrections for jets with cone size of 0.4 as a function of the jet momentum (blue). The uncertainty for this correction is also shown as a function of the jet momentum (black). flavor jets and about a factor of 6 for samples with two I__ d heavy flavor jets. This improvement in the signal-to-background ratio will result in a better resolution in the top mass reconstruction. Table 5-1. Number of events in the multi-jet data after the clean-up cuts, kinematical cuts and' I__;h! The integrated luminosity is L 943 pb-l Cut Events Fr-action ( .) Initial |z| < 60cm |z z,|1 < 5cm Lepton Veto fr/CE < 3 Netightets = 6 K~inematic Cuts 1 tag > 2 ta f 12274958 3555054 3397341 3392551 3333451 380676 4172 782 148 100 28.9 27.7 27.6 27.2 3.1 0.034 6.37e-5 1.21e-5 Table 5-2. Number of events in the it Monte Cut Events Fraction ( .) Initial 233233 100 |z| < 60cm 128169 55.0 |z z,|1 < 5cm 128045 54.9 Tigfht Lepton Veto 113970 48.9 fr/CE < 3 88027 37.7 Neightjets = 6 29485 12.6 K~inematic Cuts 5999 2.6 1 tagf 2603 1.1 > 2 taf 1599 0.69 Carlo sample with M~top 170 GeV. Table 5-4. Number of events, minLKL cut efficiency (e) relative to the kinentatical cuts and the signal to background ratios for the it 1\onte Carlo samples with top masses between 150 GeV and 200 GeV for a luminosity of 94:3 ph l. These events pass all the cuts. The efficiency for background events is also shown. M~,,, (GeV/c2) Single Tag S/B Double Tag S/B 150 18 0.25 1/2 14 0.32 :3/1 155 17 0.2:3 1/2 15 0.:33 4/1 160 16 0.21 1/2 14 0.31 :3/1 165 16 0.22 1/2 14 0.3 4/1 170 15 0.2 1/2 14 0.29 4/1 175 1:3 0.19 1/:3 14 0.29 :3/1 178 14 0.18 1/:3 14 0.28 4/1 180 12 0.18 1/:3 1:3 0.27 :3/1 185 11 0.16 1/:3 11 0.26 :3/1 190 9 0.15 1/4 11 0.25 :3/1 195 9 0.15 1/4 10 0.25 2/1 200 7 0.12 1/5 8 0.22 2/1 Background -0.05 --0.04 Data Events 48 -24 Table 1-1. Classification of the fundamental fermions arranged in three generations. Generation Flavor Mass (GeV/c2) ! U~p (u) 0.003 I Down (d) 0.006 e-Neutrino (ve) < 2 x10-6 Electron (e) 0.0005 ('1. ) is (c)1.5 II Strange (s) 0.1 p--Neutrino (v,) < 2 x10-6 Muon (p) 0.1 Top (t) 171 III Bottom (b) 4.2 -rNeutrino (v,) < 2 x10-6 Tau (-r) 1.7 in Standard Model. They are Weak Isospin - - 1 - - Table 1-2. Force carriers described in Standard Model. Boson Force Mass (GeV/c2) l ie Photon (y) EM 0 0 W* weak 80.4 +1 Zo weak 91.2 0 Gluon (g) strong 0 0 Figure 1-1. Leadingf order diagram for it production via quark-antiquark annihilation. In this figure the incident quarks are the up-quarks. O I O I p Figure 1-2. Leading order diagrams for it production via gluon-gluon fusion. I I P s Table 5-:3. Number of events and expected signal to background ratios for the it Monte Carlo samples with top masses between 150 GeV and 200 GeV for a luminosity of L 94:3 ph l. The number of data events is shown too. These events are passing the kinentatical selection, but not the nxininiun likelihood cut. M~t<, (GeV/c2) Single Tag S/B Double Tag S/B 150 7:3 1/10 45 1/2 155 72 1/10 46 1/2 160 74 1/10 45 1/2 165 74 1/10 48 1/2 170 74 1/10 49 1/2 175 71 1/10 47 1/2 178 75 1/9 50 1/2 180 69 1/10 47 1/2 185 67 1/11 44 1/2 190 61 1/12 4:3 1/2 195 59 1/12 :39 1/:3 200 56 1/1:3 :38 1/:3 Data Events 782 -148 minLKLminLKLb 0.14 -Ma s 0.12 -Uddo 0.08- 0.06- Figure 5-1. Mininiun of the negative log event probability. In blue it's shown the curve for it sample of M~t<> = 175 GeV, while in red it's shown the background shape. 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 MEASITREMENT OF THE TOP QUARK( MASS IN THE ALL HADRONIC CHANNEL AT THE TEVATRON By Gheorghe Lungu August 2007 C'I I!r: Jacoho K~onigfshergf Major: Physics This study presents a measurement of the top quark mass in the all hadronic channel of the top quark pair production mechanism, using 1 fb- of p collisions at 2@=1.96 TeV collected at the Collider Detector at Fermilah (CDF). Few novel techniques have been used in this measurement. A template technique was used to simultaneously determine the mass of the top quark and the energy scale of the jets. Two sets of distributions have been parameterized as a function of the top quark mass and jet energy scale. One set of distributions is built from the event-by-event reconstructed top masses, determined using the Standard Model matrix element for the it all hadronic process. This set is sensitive to changes in the value of the top quark mass. The other set of distributions is sensitive to changes in the scale of jet energies and is built from the invariant mass of pairs of light flavor jets, providing an in situ calibration of the jet energy scale. The energy scale of the measured jets in the final state is expressed in units of its uncertainty, oy.. The measured mass of the top quark is 171.1+3.7(stat.unc.)+2.1(syst.unc.) GeV/c2 and to the date represents the most precise mass measurement in the all hadronic channel and third best overall. CDF RunlI preliminary L=943pb ~ 3C Bckg Data --BB4P 0130 140 150 160 170 180 190 200 210 22U Event Top Mass (GeV/c2) 08 06 01 02 02 01 06 08 Bb*BrOUnll~rd0110n 7 '82 180 178 176 171 172 170 CDF RunlI preliminary L=943pb 1 Figure 6-6. Event by event most probable top masses. These are the events after the minLKL cut for Alpgfen bb + 4 light partons in blue, and for the background model in red. The plot to the left shows the single' I__- d events, while the plot to the right shows the double' I__- d events. Figure 6-7. Effect of the background contamination in the top mass reconstruction using only the matrix element technique. The upper plot: slope of the calibration curve versus the background fraction. The lower plot: intercept of the calibration curve versus the background fraction. The calibration curves are built using only the matrix element reconstruction technique described in section 4. Grapl ~200 / ndf 3.627 / 3 Prob 0.3047 SpO 178.510.1308 pl0.939410.0071 ,190- 8180- 170- 160- y=x y= p + x -178)*pl 150- 150 160 170 180 190 200 Input Top Mass [GeV] Figue 47. econstructed top mass versus input top mass using realistic jets. C mnr~iia C Er~liiin C Er~liiin c== c== c=== Figure D-2. Continued Considering the high-energy limit, we have that the invariant mass of the W-boson decay products is given by Equation 4-39. 01,2 is a geneTic HOtation for the polar, 01~,2 and the azimuthal, ~1,2, angles of the two decay products. Arl2a is the difference in pseudo-rapidities of the two decay partons and a#12 = 1 2-. P( = 2lp~snip28 ( 882COSha 012 COS 12~) = 2plp2 a(12 12) (4-39) Making the change of variables P i pi, the Equation 4-38 can be written as a delta function depending on the energy of one of the W-boson decay partons as shown in Equation 4-40, where-- pO =VW MS/(2p2 12 Pw wtw 1- p) (4-40) The mass of the W-boson is 80.4 GeV and its width is 2.1 GeV. Without these new constants and using the expression from Equation 4-40 for both W-boson squared propagators, we can write in Equation 4-41 the probability density. P~~j mj C i~~~ dzedzb (a f( b) ip dPbR3 PT~p) 00mb -v i, |"Ttot (m) e(m) Ncombi p294 pT x ((4|p) t b'" 4(Ef,,, E,?,) (4-41) i= 1 When we calculated the matrix element in section 4.3 we assumed that the incoming partons were traveling along the z-axis. This means their transverse momentum is zero. Therefore the energy conservation is violated in the transverse coordinates since based on Figure 4-2 we considered non-zero transverse momentum for the it system. However, we expect this to be a small effect covered by the uncertainty on the parton distribution functions of the proton and of the antiproton. Anyway, we need ignore the delta functions requiring energy conservation along the x and y axes as shown in Equation 4-42. BIOGRAPHICAL SKETCH Gheorghe Lungu was born in Galati, Galati County, Romania, on December 16th 1977. After graduating from high school in 1996 he was accepted in the Physics Department of the University of Bucharest. He graduated with a B.Sc. in physics in 2000, entered the Physics Graduate Department at University of Florida in 2001 and moved to Fermilab in 2003 for research within the CDF collaboration under the supervision of Prof. Jacobo K~onigfsberg. CDEE mlliiI C mlEEECiI I CnlEEECiV I Figure D-1. Continued |

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TherstpersonIwanttoacknowledgeismyadvisor,Prof.JacoboKonigsberg,forguidingandsupportingmeduringmygraduatestudentyearsinmanyways.Hisdedication,hiscommitmenttohisworkandhisstudents,andhissavvinessinthehigh-energyexperimentaleldserveasanexampletowhichIaspireasaphysicistandasascientist.AlsoIwillbeforevergratefultoDr.ValentinNeculainmanyaspects.HemadepossiblemanythingsformestartingwithlendingmemoneytopaythetestsneededforadmissioninthegraduateschoolattheUniversityofFlorida.Moreover,hecontributedgreatlytothesuccessofthisanalysis,fromthewritingtheC++codeformaintoolsandendingwithrichandenlighteningdiscussionsonthetopic.Hisgreatskillsandhisexcellencerepresentastandardforme.IwouldliketomentionthegreatinuenceIreceivedinmyrstyearsattheUniversityofFloridafromProf.KevinIngersentandProf.RichardWoodard.Withorwithouttheirawareness,theyhelpedmedeepenmyknowledgeintheoreticalphysics.AlsoItakethisopportunitytothankthemembersofthecommitteesupervisingthisthesis:Dr.ToshikazuNishida,Dr.RichardField,Dr.PierreRamondandDr.GuenakhMitselmakher.Iwillbeinspiredbytheirtremendousworkandbytheirextraordinaryachievementsinphysics.Despiteourratherbriefinteraction,IwanttomentionthatmyexperienceduringmyOralExaminationhelpedredenemeasaphysicistandasaperson.AtCDFIdrewmuchknowledgefrominteractingwithmanypeoplesuchasDr.RobertoRossin,Dr.AndreaCastro,Dr.PatriziaAzzi,Dr.FabrizioMargaroli,Dr.FlorenciaCanelli,Dr.DanielWhiteson,Dr.NathanGoldschmidt,Dr.UnkiYang,Dr.ErikBrubaker,Dr.DouglasGlenzinski,Dr.AlexandrePronko,Dr.MirceaCoca,Dr.GavrilGiurgiu.SpecialthankstoDr.DmitriTsybychev,Dr.AlexanderSukhanovandDr.SongMingWangwhohelpedmegreatlygettinguptothespeedoftheexperimentalphysicsatCDF.AlsoIwanttomentionandthankYuriOksuzianandLesterPinerafor 4 PAGE 5 5 PAGE 6 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 15 1.1HistoryofParticlePhysics ........................... 15 1.2TheStandardModel .............................. 21 1.3TopQuarkPhysics ............................... 23 1.4HighlightsofMassMeasurement ........................ 32 2EXPERIMENTALAPPARATUS .......................... 38 2.1TevatronOverview ............................... 38 2.2CDFOverviewandDesign ........................... 40 2.2.1CherenkovLuminosityCounters .................... 41 2.2.2SiliconTracking ............................. 42 2.2.3CentralOuterTracker ......................... 42 2.2.4Calorimeters ............................... 43 2.2.5TheMuonSystem ............................ 44 2.2.6TheTriggerSystem ........................... 44 3EVENTRECONSTRUCTION ........................... 51 3.1Tracks ...................................... 51 3.2VertexReconstruction ............................. 53 3.3JetsReconstruction ............................... 54 3.3.1RelativeEnergyScaleCorrection ................... 56 3.3.2MultipleInteractionsCorrection .................... 57 3.3.3AbsoluteEnergyScaleCorrection ................... 57 3.3.4UnderlyingEventCorrection ...................... 58 3.3.5OutofConeCorrection ......................... 58 3.4LeptonsReconstruction ............................. 59 3.4.1Electrons ................................. 59 3.4.2Muons .................................. 59 3.4.3TauLeptons ............................... 60 3.4.4Neutrinos ................................ 60 3.5PhotonReconstruction ............................. 60 3.6BottomQuarkTagging ............................. 61 6 PAGE 7 ............................ 61 3.6.2JetProbabilityAlgorithm ....................... 62 3.6.3SoftLeptonTagAlgorithm ....................... 62 4DESCRIPTIONOFTHEMATRIXELEMENTMACHINERY ......... 68 4.1ProbabilityDensityDenition ......................... 68 4.2Combinatorics .................................. 70 4.3CalculationoftheMatrixElement ...................... 71 4.4TransferFunctions ............................... 76 4.5TransverseMomentumofthettSystem .................... 78 4.6ImplementationandEvaluationoftheProbabilityDensity ......... 79 4.7ChecksoftheMatrixElementCalculation .................. 84 5DATASAMPLEANDEVENTSELECTION ................... 91 5.1DataandMonteCarloSamples ........................ 91 5.2EventSelection ................................. 91 6BACKGROUNDMODEL .............................. 97 6.1Denition .................................... 97 6.2ValidationoftheBackgroundModel ..................... 98 6.2.1ValidationinControlRegion1 ..................... 98 6.2.2ValidationinControlRegion2 ..................... 99 6.2.3ValidationintheSignalRegion .................... 99 6.2.4EectsontheStatisticalUncertainty ................. 99 7DESCRIPTIONOFTHEMASSMEASUREMENTMETHOD ......... 104 7.1LikelihoodDenitions ............................. 104 7.2TopTemplates ................................. 106 7.2.1DenitionoftheTemplate ....................... 106 7.2.2ParameterizationoftheTemplates ................... 106 7.3DijetMassTemplates .............................. 108 7.3.1DenitionoftheTemplate ....................... 108 7.3.2ParameterizationoftheTemplates ................... 108 8MODELVALIDATIONANDSENSITIVITYSTUDIES ............. 114 8.1Pseudo-experimentsSetup ........................... 114 8.2ValidationoftheModel ............................ 115 8.3ExpectedStatisticalUncertainty ........................ 118 9SYSTEMATICUNCERTAINTIES ......................... 127 9.1JetFragmentation ............................... 127 9.2InitialStateRadiation ............................. 127 9.3FinalStateRadiation .............................. 128 7 PAGE 8 ......................... 128 9.5BackgroundShape ............................... 128 9.6BackgroundStatistics .............................. 129 9.7CorrelationBetweenTopMassandDijetMass ................ 130 9.82DCalibration ................................. 130 9.9B-jetEnergyScale ............................... 131 9.10ResidualJetEnergyScale ........................... 131 9.11SummaryoftheSystematicUncertainties ................... 132 10CONCLUSION .................................... 136 APPENDIX APARTONDISTRIBUTIONFUNCTIONOFTHEPROTON .......... 141 BTRANSVERSEMOMENTUMOFTHETTSYSTEM .............. 142 CTRANSFERFUNCTIONS ............................. 143 DSIGNALTOPTEMPLATES ............................ 149 ESIGNALDIJETMASSTEMPLATES ....................... 163 REFERENCES ....................................... 177 BIOGRAPHICALSKETCH ................................ 181 8 PAGE 9 Table page 1-1ClassicationofthefundamentalfermionsinStandardModel. .......... 34 1-2ForcecarriersdescribedinStandardModel. .................... 34 1-3Branchingratiosofthettdecaychannels. ..................... 35 4-1Denitionofthebinningofthepartonpseudo-rapidity .............. 85 4-2Denitionofthebinningofthepartonenergyforb-jets .............. 86 4-3DenitionofthebinningofthepartonenergyforW-jets ............. 87 5-1Numberofeventsinthemulti-jetdata ....................... 94 5-2Numberofeventsinthet tMonteCarlosample .................. 94 5-3Expectedsignaltobackgroundratiosforthet tMonteCarlosamples. ...... 95 5-4EciencyoftheminLKLcutforthet tMonteCarlosamples. .......... 96 7-1Valuesoftheparametersdescribingtheshapesofthetoptemplatesforthettsamples. ........................................ 110 7-2Valuesoftheparametersdescribingtheshapesofthetoptemplatesinthecaseofthebackgroundevents. .............................. 110 7-3Valuesoftheparametersdescribingthedijetmasstemplatesshapesforthettsamples. ........................................ 112 7-4Valuesoftheparametersdescribingthedijetmasstemplatesshapesinthecaseofthebackgroundevents. .............................. 113 8-1Valueofthecorrelationfactorbetweenanytwopseudo-experiments ....... 119 8-2LinearitycheckoftheMtopandJESreconstruction ................ 119 9-1Uncertaintiesontheparametersofthetopmasstemplatesforbackground. ... 132 9-2Residualjetenergyscaleuncertaintyonthetopmass. .............. 132 9-3Summaryofthesystematicsourcesofuncertaintyonthetopmass. ....... 133 10-1Expectedandobservednumberofeventsforthet tevents ............. 138 9 PAGE 10 Figure page 1-1Leadingorderdiagramforttproductionviaquark-antiquarkannihilation .... 34 1-2Leadingorderdiagramsforttproductionviagluon-gluonfusion. ......... 34 1-3Cross-sectionofttpairproductionasafunctionofcenter-of-massenergy .... 35 1-4Diagramsfortheself-energiesofW-bosonandZ-boson .............. 35 1-5ConstraintontheHiggsbosonmass ......................... 36 1-6LoopcontributionstotheHiggsbosonpropagator ................. 36 1-7ExperimentalconstraintsonMWandMtop. .................... 37 2-1DiagramoftheTevatronacceleratorcomplex ................... 46 2-2ElevationviewoftheEasthalloftheCDFdetector ................ 46 2-3Schematicoftrackingvolumeandplugcalorimeters ................ 47 2-4InitialinstantaneousluminosityandtotalintegratedluminosityinRunII .... 47 2-5SchematicviewoftheRunIICDFsilicontrackingsystem. ............ 48 2-6Eastend-plateslotsSenseandeldplanesinCOT ................ 48 2-7Crosssectionofupperpartofnewendplugcalorimeter. ............. 49 2-8Congurationofsteel,chambersandcountersfortheCMUdetector ....... 49 2-9ReadoutfunctionalblockdiagraminRunII. .................... 50 3-1Jetscorrectionfactorasafunctionof. ...................... 63 3-2Averagetransverseenergyasafunctionofthenumberofprimaryverticesintheevent ....................................... 64 3-3Absolutejetenergyscalecorrectionsforjetswithconesizeof0.4 ........ 64 3-4Fractionalsystematicuncertaintyduetounderlyingevent ............ 65 3-5Jetcorrectionsduetoout-of-coneeectforjets .................. 65 3-6Schematicviewofaneventcontainingajetwithasecondaryvertex. ...... 66 3-7Jetprobabilitydistributionforprompt,charmandbottomjets. ......... 66 3-8Signedimpactparameterdistribution ........................ 67 4-1TreelevelFeynmandiagramfortheprocessuu!tt 85 10 PAGE 11 86 4-3Crosssectionfort tproductionversusthetopmass,fromCompHep ....... 87 4-4Transversemomentumofthet tevents ....................... 88 4-5Massreconstructionusingsmearedpartonenergies ................ 88 4-6Massreconstructionusingjetsmatchedtopartons ................. 89 4-7Reconstructedtopmassversusinputtopmassusingrealisticjets. ........ 90 5-1Minimumofthenegativelogeventprobability ................... 95 6-1Backgroundvalidationincontrolregion1forsingletaggedevents ........ 100 6-2Backgroundvalidationincontrolregion1fordoubletaggedevents ........ 101 6-3Sumofeventprobabilitiescalculatedforforbackgroundsamples. ........ 101 6-4Dijetinvariantmassoftheuntaggedjetsforbackgroundbeforethecutonthesignal-likeprobability ................................. 102 6-5Dijetinvariantmassoftheuntaggedjetsforbackgroundsamplesafterthecutonthesignal-likeprobability ............................. 102 6-6Eventbyeventmostprobabletopmassdistributionsforbackgroundsamplesafterthesignal-likeprobabilitycut ......................... 103 6-7Eectofthebackgroundcontaminationinthetopmassreconstructionusingonlythematrixelementtechnique. ......................... 103 7-1Toptemplatesforttevents. ............................. 111 7-2Toptemplatesforbackgroundevents ........................ 111 7-3Dijetmasstemplatesforttevents. ......................... 111 7-4Dijetmasstemplatesforbackgroundevents .................... 113 8-1RawreconstructionintheJESversusTopMassplane ............... 120 8-2Reconstructedtopmassversusinputtopmass,forinputJESequalto0. .... 120 8-3ReconstructedJESversusinputJES,forinputtopmassequalto170GeV. ... 120 8-4SlopeofthemasscalibrationcurveversusinputJES. ............... 121 8-5ConstantofthemasscalibrationcurveversusinputJES. ............. 121 8-6SlopeoftheJEScalibrationcurveversusinputJES. ............... 121 8-7ConstantoftheJEScalibrationcurveversusinputJES. ............. 121 11 PAGE 12 ........ 122 8-9Masspullwidthsversusinputtopmass,forinputJESequalto0. ........ 122 8-10AverageofmasspullmeansversusinputJES. ................... 122 8-11AverageofmasspullwidthsversusinputJES. ................... 122 8-12JESpullmeansversusinputtopmass,forinputtopmassequalto170GeV. .. 123 8-13JESpullwidthsversusinputtopmass,forinputtopmassequalto170GeV. .. 123 8-14AverageofJESpullmeansversusinputtopmass. ................. 123 8-15AverageofJESpullwidthsversusinputtopmass. ................ 123 8-16CorrectedreconstructionintheJESversusTopMassplane ............ 124 8-17SlopeoftheMtopcalibrationcurveversustrueJESafterthe2Dcorrection. ... 125 8-18InterceptoftheMtopcalibrationcurveversustrueJESafterthe2Dcorrection. 125 8-19SlopeoftheJEScalibrationcurveversustrueMtopafterthe2Dcorrection. ... 125 8-20InterceptoftheJEScalibrationcurveversustrueMtopafterthe2Dcorrection. 125 8-21Massreconstructionusingblindmasssamples ................... 125 8-22JESreconstructionusingblindJESsamples .................... 125 8-23Expecteduncertaintyontopmassversusinputtopmass ............. 126 8-24ExpecteduncertaintyonJESversusinputJES .................. 126 9-1Eventmultiplicityforbackgroundevents ...................... 132 9-2Parametersofthetopmasstemplateforsingletaggedbackgroundevents .... 133 9-3Parametersofthetopmasstemplatefordoubletaggedbackgroundevents ... 134 9-4TopmasspullmeanasafunctionofMtopconsideringthecorrelationbetweentheeventtopmassandthedijetmass ....................... 134 9-5TopmasspullwidthasafunctionofMtopconsideringthecorrelationbetweentheeventtopmassandthedijetmass. ....................... 135 10-1Eventreconstructedtopmassinthedata ...................... 138 10-2ContoursofthemassandJESreconstructioninthedata ............. 139 10-3ExpectedstatisticaluncertaintyfromMonteCarlo ................. 139 10-4MostprecisetopmassresultsatFermilab ..................... 140 12 PAGE 13 .... 141 B-1Transversemomentumofthettsystemfordierentgeneratorsandtopmasses. 142 C-1TransferfunctionsfortheW-bosondecaypartons ................. 143 C-2Transferfunctionsfortheb-quarkpartons ..................... 146 D-1Toptemplatesforttsingletaggedevents ...................... 149 D-2Toptemplatesforttdoubletaggedevents ..................... 156 E-1Dijetmasstemplatesforttsingletaggedevents .................. 163 E-2Dijetmasstemplatesforttdoubletaggedevents .................. 170 13 PAGE 14 pcollisionsatp 14 PAGE 15 15 PAGE 16 16 PAGE 17 17 PAGE 18 18 PAGE 19 19 PAGE 20 20 PAGE 21 21 PAGE 22 1-1 ).Theforce-mediatingparticlesdescribedbytheStandardModelallhaveanintrinsicspinwhosevalueis1,makingthembosons(Table 1-2 ).Asaresult,theydonotfollowthePauliExclusionPrinciple.Thephotonsmediatethefamiliarelectromagneticforcebetweenelectricallychargedparticles(thesearethequarks,electrons,muons,tau,W-boson).Theyaremasslessandaredescribedbythetheoryofquantumelectrodynamics.TheWandZgaugebosonsmediatetheweaknuclearinteractionsbetweenparticlesofdierentavors(allquarksandleptons).Theyaremassive,withtheZ-bosonbeingmoremassivethantheW-boson.AninterestingfeatureoftheweakforceisthatinteractionsinvolvingtheWgaugebosonsactonexclusivelyleft-handedparticles.Theright-handedparticlesarecompletelyneutraltotheWbosons.Furthermore,theW-bosonscarryanelectricchargeof+1and-1makingthosesusceptibletoelectromagneticinteractions.TheelectricallyneutralZ-bosonactsonparticlesofbothchiralities,butpreferentiallyonleft-handedones.Theweaknuclearinteractionisuniqueinthatitistheonlyonethatselectivelyactsonparticlesofdierentchiralities;thephotonsofelectromagnetismandthegluonsofthestrongforceactonparticleswithoutsuchprejudice.Thesethreegaugebosonsalongwiththephotonsaregroupedtogetherwhichcollectivelymediatetheelectroweakinteractions. 22 PAGE 23 1 ].Thediscoveryofthetopquarkwasnotasurprise.Indeed,theexistenceofanisospinpartnerfortheb-quarkisstronglymotivatedbyargumentsoftheoreticalconsistencyoftheStandardModel,absenceofavorchangingneutralcurrentinBmesondecaysandstudiesofZbosondecays[ 2 ].However,thelargemassofthetopquark,nearly175GeV/c2,wasinitselfasurpriseatthetime.Inthisregard,thetop 23 PAGE 24 3 ][ 4 ][ 5 ][ 6 ].Therefore,currentobservationsleadustobelievethattheparticleobservedattheTevatronisindeedthetopquark.However,directmeasurementsarestilldesirableandwillbeattemptedinthecaseoftheelectricchargeandspinusingdatafromtheRunIIoftheTevatronortheLHC[ 7 ].Theotherintrinsicpropertiesofanelementaryparticleareitsmassandlifetime.Themostpreciseknowledgeofthemasscomesfromdirectmeasurements.ThecurrentworldaveragecontainingonlymeasurementsperformedduringRunIattheTevatronis1784.3GeV/c2.Inquantummechanics,thelifetimeofaparticleisrelatedtoitsnaturalwidththroughtherelationship=~=.Thebranchingratiofortheelectroweaktopquarkdecayt!Wbisfarlargerthananyotherdecaymodeandthusitsfullwidthcanbeapproximatelycalculatedfromthepartialwidth(t!Wb).AssumingMW=Mb=0,thelowestordercalculationofthepartialwidthhastheexpressionshowninEquation 1{1 24 PAGE 25 0(t!Wb)=GFM2topjVtbj2 tpairsattheTevatronviathestronginteraction.Atacenter-of-massenergyof1.96TeV,theprocessq q!t tandgg!t toccurapproximately85%and15%ofthetime,respectively.TheleadingorderdiagramsforthetwoprocessesareshowninFigure 1-1 andinFigure 1-2 .Calculationsofthetotalttcross-sections(tt)havebeenperformeduptothenext-to-leadingorder(NLO)inthecouplingconstantofthestrongforce(s).Thetheoreticalvalueatacenter-of-massenergyof1.96TeV[ 8 ]isshowninEquation 1{2 forMtop=175GeV/c2. 1-3 whereweshow(tt)asafunctionofthecenter-of-mass 25 PAGE 26 3 ][ 4 ]and1.96TeV(RunII)[ 5 ][ 6 ].Figure 1-3 illustratesonemotivationtomeasureaccuratelyMtop:theknowledgeofthetopquarkmassisnecessarytocompareaspreciselyaspossiblethetheoreticalpredictionsandmeasurementsofthettcross-section.Aneventualdiscrepancycouldbeasignofnewphysicsasdiscussedinmoredetailin[ 7 ].TheelectroweakproductionofsingletopquarksisalsopredictedbytheStandardModelbuthasnotbeenobservedtodate[ 9 ][ 10 ].Theproductioncross-sectionispredictedtobesmallerthanfortt(2.4pb)andtheexperimentalsignaturesuersfrommuchlargerbackgroundcontamination.Thetopquarkdecayismediatedbytheelectroweakinteraction.SinceavorchangingneutralcurrentsareforbiddenintheStandardModelduetotheGIMmechanism[ 11 ],thedecaysofthetopquarkinvolvingZorbosonsinthenalstate(e.g.,t!Zc)arehighlysuppressedandcanonlyoccurthroughhigherorderdiagrams.Therefore,thetopquarkdecayvertexmustincludeaWboson.Threepossiblenalstatesexist:t!Wb,t!Wsandt!Wd.AsillustratedinEquation 1{1 ,thepartialwidthofchargedcurrenttopdecaysisproportionaltothesquareofthecorrespondingCKMmatrixelement.AssumingaStandardModelwiththreefamilies,therelevantCKMmatrixelementshavetheconstraints[ 12 ]giveninEquation 1{3 0:0048 PAGE 27 1-3 .ThetopquarkplaysacentralroleinthepredictionsofmanySMobservablesbycontributingtotheirradiativecorrections.GoodexamplesaretheWandZbosonpropagators,inwhichloopsinvolvingtopquarksareexpectedtostronglycontribute,asillustratedinFigure 1-4 .Thesediagramscanexistforanytypeofquarkorlepton,buttheverylargevalueofMtopmakesthetopquarkcontributiondominant.Toillustratetheeectofthetopquark,weconsiderinEquation 1{4 thetheoreticalcalculationoftheWbosonmass[ 12 ]. 1r;(1{4)isthenestructureconstant,WistheWeinbergangleandrcontainstheradiativecorrectionsandisapproximatelygivenbyEquation 1{5 rr0 tan2W(1{5) 27 PAGE 28 1-4 ,andisgivenbyEquation 1{6 =3GFM2top 1{5 areknowntoaprecisionof0.2%.Theuncertaintyonthetopquarkmassiscurrentlyaboutanorderofmagnitudelargerthantheotheruncertaintiesandmoreoveritcontributesquadraticallytor.ThustheprecisiononMtopiscurrentlythelimitingfactorinthetheoreticalpredictionoftheWbosonmass.Theparameterisqualiedas\universal"intheliteraturebecauseitentersinthecalculationofmanyotherelectroweakobservablelikesinWandtheratiooftheproductionofb-quarkhadronsofalltypes(usuallydenotedRb),tonameafew.Therefore,thetopquarkmassplaysacentralroleintheinterplaybetweentheoreticalpredictionsandexperimentalobservablesthataimstotestconsistencyoftheSM.OneconsistencycheckistocomparethemeasuredvalueofMtopwiththepredictedvaluefromSMprecisionobservables(excludingofcoursedirectmeasurementsofMtop).Theindirectconstraints,inferredfromtheeectoftopquarkradiativecorrections,yieldsMtop=181+129GeV/c2[ 14 ].TherelativelysmalluncertaintyisachievedbecauseofthelargedependenceofMtoponmanyelectroweakobservables.ThisisinremarkableagreementwiththeRunIworldaverageofMtop=1784.3GeV/c2[ 15 ],andisconsideredasuccessoftheSM.AsimilarprocedurecanbeusedtoconstraintheHiggsbosonmass(MH),thelastparticleintheSMthathasyettobeobserved.TheonlydirectinformationonMHisalowerboundobtainedfromsearchesatLEP-II:MH>114GeV/c2at95%condencelevel[ 16 ].IndirectconstraintsonMHcanbeobtainedwithprecisemeasurementsof 28 PAGE 29 1{4 containsadditionaltermsduetoHiggsbosonloops.ThesecorrectionsdependonlylogarithmicallyonMHandhavethusweakerdependenceonMHthanonMtop.Still,precisedeterminationofMtopandMWcanbeusedtoobtainmeaningfulconstraintsonMHasillustratedinFigure 1-5 .Numerically,theconstraintsare[ 14 ]madeexplicitinEquations 1{7 and 1{8 OnlythetopquarkmassmeasurementsfromRunIhavebeenused.SuchconstraintsonMHcanhelpdirectfuturesearchesattheTevatronandLHCandconstitutesanotherstringenttestoftheStandardModelwhencomparedtolimitsfromdirectsearchesormassmeasurementsfromaneventualdiscovery.EventhoughtheStandardModelsuccessfullydescribesexperimentaldatauptoafewhundredGeV,itisbelievedthatnewphysicsmustcomeintoplayatsomegreaterenergyscale.Attheveryleast,gravityeectsareexpectedatthePlanckscale(1019GeV)thattheSMignoresinitscurrentform.TheSMcanthusbethoughtofasaneectivetheorywithsomeunknownnewphysicsexistingathigherenergyscale.AlinkexistsbetweenthenewphysicsandtheSMthatmanifestsitselfthroughradiativecorrectionstoSMparticles.TheHiggsbosonsectoristhemostsensitivetoloopsofnewphysics.ForexampletheHiggsbosonmasscorrectionsfromfermionloopsshownindiagram(a)ofFigure 1-6 aregivenbyEquation 1{9 ,wheremfisthefermionmassandisthe\cut-o"scaleusedtoregulatetheloopintegral. MH22+6m2fln(=mf)+:::;(1{9)TheparametercanbeinterpretedasthescalefornewphysicsthattypicallycorrespondstothescaleoftheGrandUniedTheory(GUT)near1016GeV.Thisisa 29 PAGE 30 1{10 ,wherevisthevacuumexpectationvalueoftheHiggseldthatisknownfrompropertiesoftheweakinteractiontobeapproximately171GeV. 1{9 forfermionicparticles).Moreover, 30 PAGE 31 17 ]asshowninEquation 1{11 ,whereM~t1andM~t2arethemassesofthelightestandtheheavieststopquarks,respectively. M2hGFM4toplogM~t1M~t2 18 ].Usingthecurrentmeasurementsofprecisionobservables,itisalreadypossibletosetmeaningfulconstraintsonSUSY.Forexample,Figure 1-7 showsthecurrentmeasurementsofMtopandMWaswellastheregionallowedexclusivelyinsidetheMSSM(green),theSM(red)aswellasanoverlapregionbetweentheMSSMandSM(blue).Ascanbeseen,theadditionalradiativecorrectionsfromSUSYparticlesarelargeenoughsuchthattheoverlapregionbetweenSMandMSSMissmallintheMtopMWplane.Thecurrentexperimentalaccuraciesarenotgoodenoughtodistinguishbetweenthetwotheories,but 31 PAGE 32 19 ].OtheralternativestoreplacetheSMatenergiesneartheTeVscalearetheoriesinvolvingdynamicalbreakingoftheelectroweaksymmetry[ 20 ].Thesemodels,onewell-knownexamplebeingTechnicolor[ 21 ],donotincludeanelementaryHiggsboson,butrathergivemasstotheSMparticlesbyintroducinganewstronggaugeinteractionthatproducecondensatesoffermionsthatactasHiggsbosons.Insomeversionsofthesemodels,denoted\topcolor",thenewgaugeinteractionactsonlyonthethirdgeneration,andthefermioncondensatesaremadeoftopquarks[ 22 ].SuchamodelcouldbediscoveredbylookingforevidenceofnewparticlesinthettinvariantmassattheTevatronorLHC. 32 PAGE 33 33 PAGE 34 ClassicationofthefundamentalfermionsinStandardModel.Theyarearrangedinthreegenerations. GenerationFlavorMass(GeV/c2)ChargeWeakIsospin Up(u)0.0032 31 2IDown(d)0.006-1 3-1 2e-Neutrino(e)<210601 2Electron(e)0.0005-1-1 2 31 2IIStrange(s)0.1-1 3-1 2-Neutrino()<210601 2Muon()0.1-1-1 2 31 2IIIBottom(b)4.2-1 3-1 2-Neutrino()<210601 2Tau()1.7-1-1 2 ForcecarriersdescribedinStandardModel. BosonForceMass(GeV/c2)Charge Photon()EM00Wweak80.41Z0weak91.20Gluon(g)strong00 Leadingorderdiagramforttproductionviaquark-antiquarkannihilation.Inthisguretheincidentquarksaretheup-quarks. Leadingorderdiagramsforttproductionviagluon-gluonfusion. 34 PAGE 35 Cross-sectionofttpairproductionasafunctionofcenter-of-massenergyforthetheorypredictionandCDFmeasurements. Table1-3. Branchingratiosofthettdecaychannels. ChannelBranchingRatio all-hadronic44%lepton+jets30%dilepton5%taulepton+X21% Diagramsfortheself-energiesofW-bosonandZ-bosonwherealoopinvolvingthetopquarkiscontributing. 35 PAGE 36 ConstraintontheHiggsbosonmassasafunctionofthetopquarkandWbosonmeasuredmassesasofwinter2007.Thefullredcurveshowstheconstraints(68%C.L.)comingfromstudiesattheZbosonpole.Thedashedbluecurveshowsconstraints(68%C.L.)fromprecisemeasurementofMWandMtop. LoopcontributionstotheHiggsbosonpropagatorfrom(a)fermionicand(b)scalarparticles. 36 PAGE 37 ExperimentalconstraintsonMWandMtop(outerblueellipse),theprojectedconstraintsattheendoftheTevatronandLHC(middleblackellipse)andattheILC(redinnerellipse).AlsoshownaretheallowedregionforMSSM(greenhatched),theSM(redcross-hatched)andtheoverlapregionbetweentheSMandMSSM(blueverticallines). 37 PAGE 38 psynchrotronacceleratorsupportsseveralexperiments,includingtwocolliderdetectors,oneofwhich,theColliderDetectoratFermilab(CDF),collecteddataforthisanalysis.Theacceleratoralsoprovidesprotonstoxedtargetexperiments.CDFisageneralpurposehardscatteringdetectorsupportingawidevarietyofphysicsanalyses.OneoftheprioritiesofFNALisaprecisemeasurementofthetopquarkmass.SeveralhundredpeoplesupporttheoperationoftheacceleratorandanotherseveralhundredareresponsibleforthecommissioningandoperationoftheCDFdetector.Acompetingcollaboration,D0,independentlymeasuressimilarphysicsquantities.Combinedresultsfromthesetwocollaborationshaveresultedinincreasinglyprecisemeasurementsofthetopquarkmassandotherinterestingphysicalphenomena.Thischapteroutlinesthebasicoperationandstructureoftheacceleratorandofthedetector. 2-1 schematicallydescribestheTevatroncomplex.ProtonscollidingintheTevatronstartoutashydrogengas.ThehydrogenisionizedbyaddinganelectronandthenfedtoaCockroft-Waltondirectcurrentelectrostaticaccelerator.ExitingtheCockroft-Waltonwith750keV,thehydrogenionsarefedintoaRFlinearaccelerator,theLinac,andrampedto400MeV.Thehydrogenionsthenstrikeastationarytargetofcarbonfoil,strippingthetwoelectronsfromtheionsandleavingbareprotons. 38 PAGE 39 23 ]asshowninEquation 2{1 (p+ 39 PAGE 40 2-1 .TheAccumulatorreducesthelongitudinalmomentumoftheantiprotonsusingasynchronizedpotentialandstochasticcooling[ 24 ].StochasticcoolingwasdevelopedatCERNinthe1970sanddampensunwantedmomentumphase-spacecomponentsoftheparticlebeamusingafeedbackloop.Essentially,thebeamorbitismeasuredwithapickupandcorrectedwithakicker.TheotherantiprotonstorageunitistheRecycler,asynchrotroninthesameringastheMainInjector.TheRecyclerwasoriginallydesignedtocollectantiprotonsfromtheTevatrononcecollisionsforagivenstorewerenished,butattemptstouseitforthispurposehavenotbeenworthwhile.Asanadditionalstorageunit,theRecyclerhasallowedincreasedinstantaneousluminositysince2004.TheRecyclertakesadvantageofelectroncooling,inwhicha4.3MeVbeamofelectronsover20misusedtoreducelongitudinalmomentum.Whenastoreisreadytobegin,antiprotonsaretransferredfromeitherorboththeAccumulatorandtheRecyclertotheTevatronfornalacceleration. 25 ][ 26 ].Itsurroundsoneofthebeamcrossingpointsdescribedinsection 2.1 .Thedetectorobservesparticlesortheirdecayremnantsviachargedtracksbendingina1.4Tsolenoidaleld,electromagneticandhadronicshowersincalorimeters, 40 PAGE 41 2-2 .CDFiscylindricalinconstruction,withthebeamlinedeningthez-axisorientedwiththedirectionofprotontravel,whichisalsothedirectionofthesolenoidaleldlines.Thex-axisisdenedaspointingawayfromtheTevatronring,andthey-axisisdenedaspointingdirectlyupward.Transversecomponentsaredenedtobeperpendiculartothebeamline,inotherwordsthepolarrdimensionasgiveninEquation 2{2 .AnotherusefulcoordinatevariableistherapidityshowninEquation 2{3 .Thepseudo-rapidity,,isthemasslesslimitofrapidityandisgiveninEquation 2{4 2lnE+pz 2ln(tan):(2{4)Pseudo-rapidityisalwaysdenedwithrespecttothedetectorcoordinatesunlessexplicitlyspecied.ManyofthecomponentsofCDFaresegmentedinpseudo-rapidity.Figure 2-3 showsthecoordinatesrelativetothetrackingvolumeandplugcalorimeter. 27 ]arepositionednearthebeamline,3.7 PAGE 42 2-4 showstheinitialinstantaneousluminosityandtotalintegratedluminosityasafunctionofyear.TheinitialinstantaneousluminosityincreasedwithrunningtimeduetoimprovementssuchasusingtheRecyclertostoreantiprotons.TotalintegratedluminosityisseparatedaccordingtothatdeliveredbytheTevatronandthatrecordedtotapebytheCDFdetector. 28 ],SVXII[ 29 ]andISL[ 30 ],thesilicontrackingsystemcoversdetectorjj<2.L00isasinglelayermounteddirectlyonthebeampipe,r=1.6cm,andisasingle-sidedarraywithapitchof50mprovidingsolelyaxialmeasurements.SVXIIismountedoutsideofL00,2.4 PAGE 43 2-6 .Inhalfofthesuper-layers,thewiresareparalleltothebeamlineandprovideaxialmeasurements,whileintheotherhalf,thewiresarealternatelyat2oandprovidestereomeasurements.Theinnermostsuper-layerprovidesastereomeasurementandsubsequentlayersalternatebetweenaxialandstereomeasurements.Thegasllingthechamberiscomprisedof50%argonand50%ethane(andlately,someoxygenwasaddedtopreventcorrosion).Thisresultsinamaximumdrifttimeof100ns,farshorterthanthetimebetweenbunchcollisions.ThesinglehitresolutionoftheCOTis140m,andthetrackmomentumresolutionusingmuoncosmicraysispT=p2T0.001(GeV/c)1. 32 ];andcalorimeterscappingthebarrel,theplugcalorimeters(PPR,PES,PEMandPHA)[ 33 ].Awallhadroniccalorimeter(WHA)llsthegapbetweenthetwo.Thecentralregioncoversdetectorjj<1,thewall0.6 PAGE 44 2-7 showsacross-sectionalviewoftheplugcalorimeter. 34 ].CMUandCMPcoverdetectorjj<0.6,withCMPlocatedoutsideCMU,andCMXcoversdetector0.6 PAGE 45 2-9 ).Dataisstoredinsynchronousbuersawaitinganinitialtriggerdecision.Thersttriggerlevelreturnsadecisionwithalatencyof5.5sandamaximumacceptrateof50kHzandwillalwaysoccurintimetoreadouttheevent.Leveloneusessolelycustomhardwareoperatinginthreeparallelstreams.Onestream,theextremelyFastTracker(XFT),reconstructstransverseCOTtracksandextrapolatesthemtocalorimetersandmuonchambers.Anotherstreamdetectspossibleelectron,photonorjetcandidates,alongwithtotalandmissingtransverseenergy.Thenalstreamsearchesfortracksinmuonchambers.Thesestreamsarecombinedinthenallevelonedecision.Afteraleveloneaccept,theeventinformationisreadoutintoasynchronousbuers.Sinceeventsremaininthesebuersuntilaleveltwodecisionismade,itispossiblesomeeventspassinglevelonewillbelostwhenthesebuersarefull.Theleveltwotriggerreturnsadecisionwithalatencyof25sandamaximumacceptrateof300Hz.LeveltwousedcustomhardwareandmodiedcommercialmicroprocessorstoclusterenergyincalorimetersandreconstructtracksinthesilicondetectorusingtheSiliconVertexTracker(SVT).Calorimeterclustersestimatethetotaljetenergyandhelptoidentifyelectronsandphotons.TheSVTmeasurestheimpactparametersoftracks,partoflocatingdisplacedvertices.Thethirdtriggerlevelrunsonacommercialdualmicroprocessorfarmandreturnsadecisionwithamaximumacceptrateof150Hz.ThefarmrunsaversionofCDFoinereconstructionmerginginformationfrommanydetectorsystemstoidentifyphysicalobjectsintheevent.Datapassinglevelthreetriggerrequirementsistransferredvia 45 PAGE 46 DiagramoftheTevatronacceleratorcomplex ElevationviewoftheEasthalloftheCDFdetector.TheWesthalfisnearlysymmetric. 46 PAGE 47 SchematicoftrackingvolumeandplugcalorimetersoftheuppereastquadrantoftheCDFdetector. Figure2-4. Initialinstantaneousluminosity(left)andtotalintegratedluminosity(right)asafunctionofyearsincethestartofRunII. 47 PAGE 48 Schematicwithther-andthey-zviewsoftheRunIICDFsilicontrackingsystem. Figure2-6. Eastend-plateslotsSenseandeldplanesareattheclock-wiseedgeofeachslot(left).Nominalcelllayout(right). 48 PAGE 49 Crosssectionofupperpartofnewendplugcalorimeter. Detailshowingthecongurationofsteel,chambersandcountersfortheCentralMuonUpgradewalls.Amuontrackisdrawntoestablishtheinteractionpoint.Counterreadoutislocatedatz=0.CounterslayersareosetfromthechambersandfromeachotherinxtoallowoverlappinglightguidesandPMTs,minimizingthespacerequired. 49 PAGE 50 ReadoutfunctionalblockdiagraminRunII. 50 PAGE 51 pcollisionstartingfromtherawoutputsofthedierentpartsofthedetector.FirstwewillseehowinformationfromsilicondetectorsandCOTareusedtoreconstructchargedparticletrajectories.Thenwewillmovetothereconstructionofjetsofhadronicparticles,basedoncalorimeters.Asectionwillbedevotedtothecorrectionofjetenergiesfordierenterrorsourcesintroducedbycalorimetersandreconstructionalgorithms.Afterabriefdescriptionoftheidenticationofleptonsandphotons,wewillendwiththedierentmethodsusedatCDFtoidentifyajetofparticlesoriginatedfromabquark. 3{1 ,thehelixofachargedparticleisparameterized. 51 PAGE 52 3{2 ,where=1 2CQistheradiusofthecircleandQthechargeoftheparticle. 35 ]isastrategytoreconstructtracksinthesilicondetector.Itconsistsinndingtripletsofaligned3Dhits,extrapolatingthemandaddingmatching3Dhitsonotherlayers.Thistechniqueiscalledstandalonebecauseitdoesn'trequireanyinputfromoutside:itperformstrackingcompletelyinsidethesilicondetector.Firstthealgorithmbuilds3Dhitsfromallpossiblecouplesofintersectingaxialandstereostripsoneachlayer.Oncealistofsuchhitsisavailable,thealgorithmsearchesfortripletsofalignedhits.Thissearchisperformedxingalayeranddoingalooponallhitsintheinnerandouterlayerswithrespecttothexedone.Foreachhitpair-oneintheinnerandoneintheouterlayer-astraightlineintherzplaneisdrawn.Nextstepconsistsinexaminingthelayerinthemiddle:eachofitshitsisusedtobuildahelixtogetherwiththetwohitsoftheinnerandouterlayers.Thetripletsfoundsofararetrackcandidates.Oncethelistofcandidatesiscomplete,eachofthemisextrapolatedtoallsiliconlayerslookingfornewhitsintheproximityoftheintersectionbetweencandidateandlayer.Ifthereismorethanonehit,thecandidateisclonedandadierenthitisattachedtoeachclone.Fullhelixtsareperformedonallcandidates.Thebestcandidateinaclonegroupiskept,theothersrejected.TheOutside-Inalgorithm[ 36 ]exploitsinformationfrombothCOTandsilicon.TherststepistrackingintheCOT,whichstartstranslatingthemeasureddrifttimesin 52 PAGE 53 pcollision(primaryvertex)isoffundamentalimportanceforeventreconstruction.AtCDFtwoalgorithmscanbeuseforprimaryvertexreconstruction.OneiscalledPrimVtx[ 37 ]andstartsbyusingthebeamlinez-position(seedvertex)measuredduringcollisions.Thenthefollowingcuts(withrespecttotheseedvertexposition)areappliedtothetracks:jztrkzvertexj<1.0cm,jd0j<1.0cm,whered0istrackimpactparameter,andd0 53 PAGE 54 38 ].Thisalgorithmstartsfrompre-trackingvertices(i.e.,verticesobtainedfromtrackspassingminimalqualityrequirements).Amongthese,alotoffakeverticesarepresent:ZVertexCollcleansuptheseverticesrequiringacertainnumbertrackswithpT>300MeVbeassociatedtothem.Atrackisassociatedtoavertexifitiswithin1cmfromsiliconstandalonevertex(or5cmfromCOTstandalonevertex).Vertexpositionziscalculatedfromtrackspositionsz0weighedbytheirerroraccordingtoEquation 3{3 54 PAGE 55 3{4 assumingthateachvectorcorrespondstoamasslessparticlethatdepositedallitsenergyinthetowerbarycenter. (3{4) 55 PAGE 56 3{4 ,thejettransverseenergy,transversemomentumandpseudo-rapidityarecalculatedinEquations 3{5 3{6 and 3{7 P(3{6) 39 ]. 40 ]areappliedtorawjetenergiestocorrectfornon-uniformitiesincalorimeterresponsealong.Calorimeterresponseineachbinisnormalizedtotheresponseintheregionwith0.2jj0.6,becausethisregionisfarawayfromdetectorcracksanditisexpectedtohaveastableresponse.Thecorrectionfactorisobtainedusingthedijetbalancingmethodappliedtodijetevents.Thismethodstartsselectingeventswithoneoutoftwojetsintheregion0.2jj0.6.Thisjetisdenedastriggerjet.Theotherjetisdenedasprobejet.Ifbothjetsareintheregionof0.2jj0.6,triggerandprobejetareassignedrandomly.Thetransversemomentumoftwojetsina2!2processshouldbeequalandthispropertyisusedtocalculaterstapTbalancingfractionpTfasshowninEquation 3{8 pTf=pT 56 PAGE 57 3{9 3-1 weshowthecorrectionfactorasafunctionoffordijetdata(black)andfordijetMonteCarlousingPythiaasgenerator(red). pinteractioncanoccur.Energyfromthesenonoverlappingminimumbiaseventsmayfallintothejetclusteringconeofthehardinteractioncausingthusamis-measurementofjetenergy.Acorrectionforthiseectisextractedusingasampleofminimumbiasevents[ 41 ]:foreachevent,transverseenergyETinsideconesofdierentradii(0.4,0.7and1.0)ismeasuredinaregionfarawayfromcracks(0.1jj0.7):then,thedistributionofaverageETasafunctionofthenumberofquality12verticesisttedwithastraightlineandtheslopeofthettinglinesaretakenascorrectionfactors(Figure 3-2 ). 42 ].Theproceduretoextractacalorimeter-to-hadroncorrectionfactorisbasedonthefollowingsteps:usefullysimulatedCDFsampleswhereparticleshavepTrangingfrom0to600GeV,clusterthecalorimetertowersandtheHEPGparticles,associatecalorimeter-leveljetswithhadron-leveljets,parameterizethemappingbetweencalorimeterandhadron-leveljetsasafunctionofhadron-leveljets,asacorrectionfactor,extracttheprobabilitiesofmeasuringajetwithpcalTgivenajetwithxedvalueofphadT. 57 PAGE 58 3-3 theabsolutejetenergyscalecorrectionsforjetsconesizeof0.4asafunctionofthejetmomentum(blue).Theuncertaintyforthiscorrectionisalsoshownasafunctionofthejetmomentum(black). 43 ].Foreachevent,transverseenergyETinsideconesofdierentradii(0.4,0.7and1.0)ismeasuredinaregionfarawayfromcracks(0.1jj0.7).ThecorrectionfactorisextractedfromthemeanvaluesofETdistribution(Figure 3-4 ). 44 ]:hadron-leveljetsarematchedtopartonsiftheirdistanceintheplaneislessthan0.1.Thenthedierenceinenergybetweenhadronandpartonjetisparameterizedusingthesamemethodasforabsolutecorrection(Figure 3-5 ).Wehaveseendierentcorrectionsthataccountfordierentsourcesofjetenergymis-measurement.Dependingonthephysicsanalysis,allofthemorjustasubsetcanbeapplied.Thecorrectionsareappliedtotherawmeasuredjetmomentum. 58 PAGE 59 3{10 ,Ristheclusteringconeradius,PTistherawenergymeasuredintheconeandthepseudo-rapidityofthejet:f;MI;fabs;UEandOOCarerespectivelyrelative,multipleinteractions,absolute,underlyingeventandout-of-conecorrectionfactors. 3.4.1ElectronsBeingachargedparticle,anelectrontraversingthedetectorrstleavesatrackinthetrackingsystemandthenlosesitsenergyintheelectromagneticcalorimeter.Soagoodelectroncandidateismadeofaclusterintheelectromagneticcalorimeter(centralorplug)andoneormoreassociatedtracks;ifavailable,showermaxclusterandpreshowerclusterscanhelpelectronidentication.Theshowerhastobenarrowandwelldenedinshape,bothlongitudinallyandtransversely.Theratiobetweenhadronicandelectromagneticenergieshastobesmallandtrackmomentumhastomatchelectromagneticclusterenergy[ 45 ]. 46 ]. 59 PAGE 60 3{11 ,Eiistheenergyoftheithtower,iisthepolarangleofthelinepointingfromtheinteractionpointtotheithtowerand~niisthetransverseunitvectorpointingfromtheinteractionpointtothecenterofthetower. 60 PAGE 61 47 ]exploitsthefactthattheBhadrontravelsbeforeitdecaysandthereforethejetproducedbyitwillcontainasecondaryvertex(Figure 3-6 ).ThealgorithmstartsfromCOTandsilicontracksinsideaconeandasarststep,usingasdiscriminatingvariabletheirimpactparameter,itremovestracksidentiedasKS;ordaughters,orconsistentwithprimaryvertexortoofarfromit.Thenathreedimensionalcommonvertexconstrainedtisperformedusingtwotracks:if2<50thetwotracksareusedasseedtondothertracksthatpointtowardthesamesecondaryvertex.Ifatleastthreetracksarefoundtobecompatiblewithasecondaryvertex,thejetcontainingthemisconsideredab-tagifitpassesthefollowingcuts:jLxyj<2.5cm,whereLxyisthedecaylengthofthesecondaryvertex;thiscuthelpsrejectingconversionsfromtherstlayerofSVXII;Lxy 61 PAGE 62 48 ].Theprobabilitydistributionisuniformlydistributedforajetarisingfromtheprimaryvertex,whileitshowsapeakatzeroforalong-livedjet(Figure 3-7 ).Theprobabilityisbasedontrackimpactparametersandontheiruncertainties.Alltracksassociatedtotheprimaryvertexhaveequalprobabilitytobeeitherpositivelyornegativelysignedasfarastheirimpactparameterisconcerned.Thewidthoftheimpactparameterdistributionfromthesetracksissolelyduetothetrackingdetectorresolutionandmultiplescattering.Along-livedparticlewillproducemoretrackswithpositiveimpactparameter(Figure 3-8 ).Tominimizethecontributionofmis-measuredtracks,thenalprobabilityiscomputedusingthesignedimpactparametersignicance(ratiooftheimpactparametertoitsmeasurederror)insteadoftheparameteritself.GivenatrackwithimpactparametersignicanceSd0,theprobabilitythatatrackfromalightquarkhasalargervalueofSd0iscalculated.Combiningprobabilitiesforalltracksinajet,oneobtainsthejetprobability.Byconstruction,thisprobabilityisatforjetscomingfromlightquarksorpeakedatzeroforthosecomingfromheavyquarks. 62 PAGE 63 49 ].First,thetaggabletracksarefound(i.e.,tracksthatcouldhavebeenleftbymuons).Totakeintoaccountthefactthatthemuonmightnothavehadenoughenergytoreachthemuonchambers,trackswhosemomentumislowerthan2.8GeVarerejected.Moreover,ithastopointtoavolumelimitedbythephysicaledgesofthemuonchambers,oradistanceof3MSinside/outsidethephysicaledges.HereMSisthestandarddeviationofthemaximumdeectionexpectedfrommultiplescatteringthroughthematerialofthedetector.Ifatrackistaggableandhasastubassociatedtoit,thealgorithmcomputesalikelihoodcomparingalltheavailableinformationaboutthemuoncandidatewiththeexpectedvalues.Besidesvariablesfrommuondetectors,forthelikelihoodonecanusealsosometrackqualityinformation,likethenumberofCOThits,thebeamline-correctedimpactparameterandthetrackz0position. Correctionfactorasafunctionoffordijetdata(black)andfordijetMonteCarlousingPythiaasgenerator(red).Thejetswerereconstructedwithaconeof0.4. 63 PAGE 64 Averagetransverseenergyasafunctionofthenumberofprimaryverticesintheevent:acorrectionfactorisextractedfromtheslopeofthettingline. Absolutejetenergyscalecorrectionsforjetswithconesizeof0.4asafunctionofthejetmomentum(blue).Theuncertaintyforthiscorrectionisalsoshownasafunctionofthejetmomentum(black). 64 PAGE 65 Fractionalsystematicuncertaintyduetounderlyingeventasafunctionofjettransversemomentumfordierentjetconesizes. Jetcorrectionsduetoout-of-coneeectforjetswithconesizeof0.4asafunctionofthejetmomentum(red).Theuncertaintyforthiscorrectionisalsoshownasafunctionofthejetmomentum(black). 65 PAGE 66 Schematicviewofaneventcontainingajetwithasecondaryvertex. Jetprobabilitydistributionforprompt,charmandbottomjets. 66 PAGE 67 Signedimpactparameterdistributionfortracksfromprimaryvertex(left)andfromsecondaryvertex(right). 67 PAGE 68 4{1 4EaEbjvavbjjM(m;j)j2(2)4(4)(EfinEini)6Yi=1d3~ji 4{1 ,jisagenericnotationbywhichweunderstandallsix4-momentadescribingthenalstate;za(zb)isthefractionoftheproton(anti-proton)momentumcarriedbythecollidingpartons;f(za)andf(zb)standforthepartondistributionfunctionsforprotonandforanti-protonrespectively;M(m;j)isthematrixelementcorrespondingtotheallhadronictt;Efinisagenericnotationforthe4-vectorofthenalstate,andsimilarlyfortheinitialstateweuseEini.Iftheelementarycross-sectionsfromagroupofeventsareaddedupweshouldobtainafractionoftotalttcross-section,tot(m),fortopmassmasshowninEquation 4{2 68 PAGE 69 4{3 4EaEbjvavbjjM(m;j)j2(2)4(4)(EfinEini) (2)32Ei(4{3)ThequantityP(jjm)Q6i=1d3~jiwillbetheprobabilityforaneventdenedbythesetofsixjets(i.e.,six4-momenta)tobetheresultofttproductionfollowedbyanallhadronicdecayfortopmassm.Sofarwedidn'tworryabouthowaccuratelywecandeterminethesix4-momenta.Inreality,thenalstatepartonswhichareobservedasjetsinthedetector,canbemis-measured.WecanaccountforthisusingourttMonteCarlosamplesanddetermineaprobabilityforapartonwith4-momentumptobeobservedasajetwith4-momentumj.ThisnewprobabilityiscalledTransferFunctionTF(~jj~p)andallthetechnicaldetailsonhowwedeterminethemwillbepresentedinsection 4.4 .Sincewedon'tknowwhatistheparton4-momentumthatgeneratedagivenjet4-momentumwehavetoconsiderallpossibilitiesandintegrateoverthemweighedbythetransferfunctions.TheEquation 4{3 canberewrittenasinEquation 4{4 4EaEbjvavbjZ6Yi=1d3~pi (4{4) ThepartoncongurationsintegratedoverinEquation 4{4 areweighedbythetransferfunctionssothatthosemorelikelytoproduceagiven6-jetseventareenhanced.Ideallythettphasespaceshouldbeenhancedaswellandnotdiminished.Inordertoenforcethislastaspectoftheintegration,weintroduceanadditionalweight,PT(~p),thatfollowstheshapeofthetransversemomentumofthettsystem.Thislastweight 69 PAGE 70 4.5 .ThereforethenewexpressionfortheprobabilitydensityisshowninEquation 4{5 4EaEbjvavbjZ6Yi=1d3~pi (4{5) Eventhoughatteventintheallhadronicnalstateisfullyreconstructed,thereisanambiguityinassigningthejetstothepartons.Thereforeallthepossiblecombinationsareconsideredandtheircontributionsaveraged.Thenumberofpossibleassignmentsdependsonthetopologyoftheeventandthiswillbediscussedinsection 4.2 .UntilthentheEquation 4{6 givesthemostgeneralexpressionoftheprobabilitydensity. 4EaEbjvavbjZ6Yi=1d3~pi (4{6) 4{7 thespinaveragedmatrixelementsquaredfortheprocessuu!tt. 1 4XspinsjMj2=g4s 4{8 1 4XspinsjMj2Tr[6pu6p 70 PAGE 71 4{9 1 4XspinsjMj232(pup 4{9 thet$ t=(b2;W2)g;ft=(b1;W2); t=(b2;W1)g.Itisobviousthatswappingtheb'sisequivalentwithswappingthetopquarks.Inconclusion,duetothet$ 4{10 summarizesthepossiblevaluesforNcombi. 71 PAGE 72 4{6 .Theinvariantamplitudefortheprocessuu!tt!bbuuddisgivenbelowasaproductofseveralfactorsasshowninEquation 4{11 4{11 aredetailedbytheEquation 4{12 (pu+p 72 PAGE 73 bWverticeswiththenumeratorsofthetopquarkandtheantitopquarkpropagators.ThetermsPtandP dandWd uvertices.ThetermsPW1andPW2arethedenominatorsoftheW+andWpropagators.WehaveusedtheFeynmangaugefortheWbosonpropagator.TheDiracgammamatricesaredenedintheDiracrepresentationasshowninEquation 4{13 ,where=(1;~)and 4{14 73 PAGE 74 4{15 2(1^p~)1 2(1+^p~)1CA;v(p)=p 2(1^p~)1 2(1+^p~)1CA(4{15)Thepresenceoftheoperator^p~willprojectthespinstatesalongthedirectionofmovementdenedby^p.Foraparticletravelinginthedirectiondenedbythepolarangleandbytheazimuthalangle,thespinstatesalongthisdirectionareshowninEquation 4{16 4{17 4{15 and 4{17 ,wecanrewriteinEquation 4{18 the4-vectorsW1andW2fromEquation 4{12 AlsothetensorintermTfromEquation 4{12 canberewrittenintheformgivenbyEquation 4{19 74 PAGE 75 4{6 ,wewillneedtosumoverallthepossiblespincongurationsoftheinitialstate.Wendtwonon-zerocontributionscorrespondingtothesituationswhentheincomingpartonshavethesamehandedness.ThereforeforthetermIfromEquation 4{12 isexpressedinEquation 4{20 u(0;1;i;0)ILL=p u(0;1;i;0)(4{20)Inprinciple,weneedtoaverageoverallthepossiblespincongurationsofthenalstate.TheEquations 4{18 and 4{19 representthenon-zerocontributions.UsingEquations 4{18 4{19 and 4{20 ,theproductofthetermsI,T,W1andW2isgiveninEquation 4{21 4{21 ,thetermEproportionaltotheproductoftheenergiesofallparticles,incomingoroutgoing,isshowninEquation 4{22 u(4{22) 4{23 ,arecalculatedinaC++codeusingEquation 4{15 andthematrixalgebra.ThereforewecanwritedowntheexpressionofthematrixelementsquaredfromEquation 4{6 intheformofEquation 4{24 26XspinscolorsjMj2=jAj2CjEj2 75 PAGE 76 4{24 aredetailedinEquation 4{25 93Xi;j;k;l=1aijakl36=234fPg=jPgj2=1 (pu+p (p2tm2)2+m22teP (p2 (P2W+M2W)2+M2W2WgPW2=jPW2j2=1 (P2WM2W)2+M2W2W 4{6 isinfactaproductofsixterms,oneforeachofthenalstatequarks:twofortheb-quarksandfourforthedecayproductsoftheW-boson.TheprobabilitydensityforthetransferfunctionsisgiveninEquation 4{26 4{27 toexpressthetransferfunctionsinamoregeneralway. 76 PAGE 77 4{28 4{29 givestheirnormalization. 4{26 againwiththefullexpressionenteringEquation 4{6 holdingtheprobabilitydensityforthettallhadronicprocess. tMonteCarlosamples.Moreexactly,ajetisassociatedtoapartonifitsdirectioniswithinaconeofR=0:4aroundthepartondirection.Wesaythatajetismatchedtothepartonifnootherjetshouldsatisfythisgeometricalrequirement.Wecallaneventasbeingamatchedeventifeachofthesixpartonsinthenalstatehasadierentjetmatchedtoit.Ofallthet tMonteCarloeventspassingthekinematicalselectiondenedlaterinsection 5 ,about50%arematchedevents.ThejetsformedbythedecaypartonsoftheW-bosonshaveadierentenergyspectrumthanthejetsoriginatingfromtheb-quarks.Thusweformdierentsetsoftransferfunctionsdependingontheavorofthepartonthejethasbeenmatchedto.Thetransferfunctionsaredescribedusingaparameterizationinbinsofthepartonenergiesandofthepartonpseudo-rapidities.Table 4-1 showsthedenitionofthebinninginpseudo-rapidity.Thesamedenitionholdsforb-jettransferfunctionandforW-jetstransferfunctions. 77 PAGE 78 4-2 showsthedenitionofenergybinningfortheb-jetstransferfunctions,whileTable 4-3 isfortheW-jetstransferfunctions.Ineachbinthetransferfunctionisrepresentedbythedistributionofthevariable1Ejet=Eparton.Theshapeofthisdistributionisttedtothesumoftwogaussians.Appendix C holdsthettedshapes. 4{6 arep6xandp6y,representingtheprojectionsofthetransversemomentumofthettsystemalongthexandyaxes.TheprobabilitydensityrelatedtothetransversemomentumofthettsystemweightisshowninEquation 4{31 4{32 givesthenormalizationrelation. 4{32 ,isobtainedfromattMonteCarlosamplewithMtop=178GeV.The 78 PAGE 79 4{33 thevaluefor6T. 2(4{33)Asmentionedbeforeweneedtoexpresseverythingintermsofp6xandp6y.ThiscanbedonejustbychangingthevariablesfromthepolartotheCartesiancoordinatesasshowninEquation 4{34 2=1=Zdp6xdp6yePTp6T=q q 2==Zdp6xdp6yPT(p6x;p6y) (4{34) WecannowwriteinEquation 4{35 thefullexpressionofthetransversemomentumofthettsystemweight. q 2(4{35)TheshapeofePT(p6T)hasaslightdependenceonthetopmass,butitturnsoutthatchoosingtheshapeobtainedwithMtop=178GeVdoesn'tintroduceasignicantbiasinthenalmassreconstruction.SeeAppendix B forthemassdependenceofthisshape.InFigure 4-2 theshapeofthetransversemomentumofthetteventsisshownttedtoasumof3gaussians. 4{6 .Thesections 4.3 4.4 and 4.5 oereddetailsontheexpressionsofseveralimportantpiecesenteringtheprobabilitydensity.UsingEquations 4{24 4{30 and 4{35 ,wecanwriteinEquation 4{36 thenewexpressionfortheprobabilitydensity. 79 PAGE 80 4EaEbjvavbjZ6Yi=1d3~pi Asmentionedpreviously,wewillnotuseanyconstantthatcanbefactoredoutintheexpressionoftheprobabilitydensity.Fromnowonwewillomitallsuchconstantsexceptforthenumberofcombinations,Ncombi.AlsointheargumentofePTwewillputjustp6T,butitshouldbeunderstoodq 4{37 (4{37) ToreducethenumberofintegralswewillworkinthenarrowwidthapproximationfortheW-bosons.ThistranslatesintwomoredeltafunctionsarisingfromthesquareoftheW-bosonpropagatorsasshownbyEquation 4{38 (P2WM2W)2+M2W2WWMW!(P2WM2W) MWW(4{38) 80 PAGE 81 4{39 .1;2isagenericnotationforthepolar,1;2,andtheazimuthal,1;2,anglesofthetwodecayproducts.12isthedierenceinpseudo-rapiditiesofthetwodecaypartonsand12=12. 4{38 canbewrittenasadeltafunctiondependingontheenergyofoneoftheW-bosondecaypartonsasshowninEquation 4{40 ,wherep01=M2W=(2p2!12). MWW1 2p2!12(1;2)(p1p01)(4{40)ThemassoftheW-bosonis80.4GeVanditswidthis2.1GeV.WithoutthesenewconstantsandusingtheexpressionfromEquation 4{40 forbothW-bosonsquaredpropagators,wecanwriteinEquation 4{41 theprobabilitydensity. (!12)2(!34)2(4)(EfinEini) (4{41) Whenwecalculatedthematrixelementinsection 4.3 weassumedthattheincomingpartonsweretravelingalongthez-axis.Thismeanstheirtransversemomentumiszero.ThereforetheenergyconservationisviolatedinthetransversecoordinatessincebasedonFigure 4-2 weconsiderednon-zerotransversemomentumforthettsystem.However,weexpectthistobeasmalleectcoveredbytheuncertaintyonthepartondistributionfunctionsoftheprotonandoftheantiproton.Anyway,weneedignorethedeltafunctionsrequiringenergyconservationalongthexandyaxesasshowninEquation 4{42 81 PAGE 82 InEquation 4{41 ,wemadethechangeofvariablesza!puandzb!p 4{43 theexpressionfortheenergy-conservingdeltafunction,wherep0u=P6i=1pi(1+cosi)=2andp0 2(pup0u)(p (4{43) Usingalloftheabove,theexpressionfortheprobabilitydensityisgivenbyEquation 4{44 inanalmostnalform. (!12)2(!34)2p2p46Yi=1gTF(ijpi)ePT(p6T) (4{44) Insection 4.5 ,weannouncedourpreferencetointegrateoverthexandycomponentsofthemomentumofthettsystem.Thatisaccomplishedbyalastchangeofvariablesfpb;p 4{45 82 PAGE 83 4{46 4{47 theexpressionoftheprobabilitydensityinitsnalformwhichisusedinsideaC++code. (!12)2(!34)2p2p46Yi=1gTF(ijpi)ePT(p6T) (4{47) Theintegrationisperformedbysimplygivingvaluestothe4integrationvariablesandthenbyaddinguptheintegrandobtainedateachstep.Thelimitsoftheintegrationare-60GeV!60GeVforp6x;yand10GeV!300GeVforp2;4.Thestepofintegrationis2GeV.Giventheselimits,ateachstepofintegrationwehavetocheckthephysicalityofthecomponentsenteringEquation 4{47 .Theprobabilitydensityisevaluatedfortopmassvaluesgoingin1GeVincrementsfrom125GeV!225GeV.Thedependenceonmassofthet tcross-sectionisobtainedfromvaluescalculatedbyCompHepMonteCarlogeneratorfortheprocessesu u!t t,d d!t tandgg!t t.Theabsolutevaluesforthesecrosssectionsarenotasimportantastheirtopmassdependence.Figure 4-1 showsthisdependence.FortheprotonandantiprotonPDF,f(p0u)f(p0 A .Thet tacceptance,(m),dependsonthetopmassandwillbedescribedlaterwhentheeventselectionisaddressed.Thenalexpressionoftheprobabilitydensityhasbeengivenanditsimplementationhasbeendetailed.Thefollowingsectionisdedicatedtothechecksweperformedinordertoassuretheproperfunctionalityofthematrixelementtechnique. 83 PAGE 84 4.6 dependsonthetopquarkpolemassandisexpectedtobeminimizedinnegativelogscalearoundthetruemassesintheevent.Multiplyingalltheeventprobabilitiesweobtainalikelihoodfunctionthatdependsonthetoppolemass.Equation 4{48 showstheexpressionofthelikelihood. 4-5 showsagoodlinearityinthecaseofa5%uniformsmearing.Thereisasmallbiasofabout0.8GeV,buttheslopeisconsistentwith1.Asthesmearingisincreasedthebiasbecomesmoreevident,andslopedegradesslightly.ThiscanbealsoseeninFigure 4-5 for10%smearingandfor20%smearing,respectively.Inallofthesesituationsagaussiancenteredon0andwithwidthequaltotheamountofsmearingusedhasbeenemployedasatransferfunctionintheeventprobabilitycomputation.Thepartonscanalsobesmearedusingthefunctionsdescribedinsection 4.4 ,inwhichcasethesamefunctionsareusedastransferfunctionsintheeventprobabilitycomputation.Thistestmakesthetransitionbetweenthepartonleveltothejetslevel, 84 PAGE 85 4-5 showsthelinearitycheckinthiscaseaswell.Thenextcheckismovingclosertorealitybyusinginthereconstructionthejetsthathavebeenmatchedtothepartons.Thisisalreadyacheckatthejetslevelandthefunctionsdenedinsection 4.4 havetobeused.Figure 4-6 showsthelinearitycheck.Thenalcheckisthemostrealisticwecangetusingonlysignalevents,andthatisweusealltheeventswehavewithdisregardtowhetherthejetshavebeenmatchedornottothepartons.Figure 4-7 showsthelinearitycheckinthiscase.Allthecheckswehavelistedaboveshowthegoodperformanceofourmatrixelementcalculation.Ingeneral,thetraditionalmatrixelementapproachisexpectedtoprovideabetterstatisticaluncertaintyonthetopmassthanthetemplateanalyses.Inthecaseofthepresentanalysis,thetraditionalmatrixelementmethoddoesbetteronlythereconstructionisperformedonsignalsamples.Whenthebackgroundismixedin,thetemplatemethodweusehasagreatersensitivity. TreelevelFeynmandiagramfortheprocessuu!tt Denitionofthebinningofthepartonpseudo-rapidityfortheparameterizationofthetransferfunctions. Binjj 85 PAGE 86 Denitionofthebinningofthepartonenergyfortheb-jetstransferfunctionsparameterization. Bin0jj<0:70:7jj<1:31:3jj2:0 110!5310!8310!1253!6483!111364!74111!1474!85585!97697!1147114!1 TreelevelFeynmandiagramfortheprocessuu!tt!bbuudd PAGE 87 DenitionofthebinningofthepartonenergyfortheW-jetstransferfunctionsparameterization. Bin0jj<0:70:7jj<1:31:3jj2:0 110!3210!5010!98232!3850!6398!1338!4463!76444!4976!90549!5490!108654!59108!1759!64864!69969!751075!811181!891289!991399!11314113!1 Crosssectionfort tproductionasafunctionofthetopmass,asobtainedfromCompHep.Thelineisnotat. 87 PAGE 88 Transversemomentumofthet tevents.Thetisasumof3gaussians. A BFigure4-5. Reconstructedtopmassversusinputtopmassatpartonlevel.A)Theenergiesofthepartonshavebeensmearedby5%.B)Theenergiesofthepartonshavebeensmearedby10%.C)Theenergiesofthepartonshavebeensmearedby20%.D)Theenergiesofthepartonshavebeensmearedusingthetransferfunctions. 88 PAGE 89 DFigure4-5. Continued Reconstructedtopmassversusinputtopmassusingjetsthatwereuniquelymatchedtopartons. 89 PAGE 90 Reconstructedtopmassversusinputtopmassusingrealisticjets. 90 PAGE 91 MULTIJETtrigger,anditamountstoapproximately943pb1.Thistriggerselectsabout88%ofthettallhadronicevents.TheMonteCarlosamplesaretheocialCDFsamples.Weuse12dierentsamplesgeneratedwiththeHerwigpackagetoparameterizethemassdependenceofourtemplates.Themasstakesvaluesfrom150GeVto200GeVin5GeVincrements.Therearealsosampleswithatopmassof178GeVusedtodeterminevarioussystematicuncertainties:dierentchoiceofgenerator(inthiscaseweusedthePythiapackage),dierentmodelingoftheinitialstateradiation(ISR)andofthenalstateradiation(FSR),dierentchoiceofprotonpartondistributionfunction(PDF).Thebackgroundmodeldescribedinsection 6 isvalidatedwiththehelpoftwoMonteCarlosamplesgeneratedwiththeAlpgenpackage:onewitheventshavingb b+4lightpartonsinthenalstateandanotherwitheventshaving6lightpartonsinthenalstate. 91 PAGE 92 PET<3(GeV)1=2removeeventshavingmuonsorelectronsTheseclean-upcutsselectabout37%ofthet tMonteCarlosamplesoutofwhichabout84%areall-hadronicevents.Inthedataonly27%oftheeventspassthesecuts,mostoftheeventsfailingthegoodrunlistandthetriggercuts.Next,thekinematicalandtopologicalcutsareappliedinordertoenhancethet teventsoverthebackground:requireeventswithexactly6jetswithjj<2andET>15GeVAplanarity+0:005PET3>0:96centrality>0:78PET>280GeV1SVXtagswhereETissumofallthetransverseenergiesofallthesixjetsintheevent,3ETisthesumofallthesixjetsminusthetwomostenergeticones,CentralityisdenedinEquation 5{1 andtheAplanarityisdenedas3=2ofthesmallesteigenvalueofthesphericitymatrix^Sij.Thesphericitymatrix^SijisdenedinEquation 5{2 92 PAGE 93 50 ].Table 5-1 showsthenumberofeventsinthedatasample.Table 5-2 showsthenumberofeventsinat tMonteCarlosamplewithMtop=170GeV.TheSVXb-taggerusedhasahighereciencyintheMonteCarlothaninthedata.ThereforeweneedtodegradethenumberoftaggedeventsaccordingtotheappropriatescalefactorwhichisSF=0:91.Takingthisscalefactorintoaccount,andconvertingtotheluminosityofthedata,weshowinTable 5-3 thesignaltobackgroundratios,S=B,fordierenttopmassesafterthekinematicalcutsforsingleanddoubletaggedeventsseparately.Theconversiontotheobservedluminosityisdonebyusingthetheoreticalt tcrosssection.ThenumberofbackgroundeventsisthedierencebetweentheobservednumberofeventsinthedatashowninTable 5-1 andthesignalexpectation.Anadditionalcutisintroducedtofurthercutdownthebackground.ThisnewvariablewecutonistheminimumoftheeventprobabilitygiveninEquation 4{6 ofsection 4 .Figure 5-1 showsthedistributionoftheminimumofthenegativelogeventprobabilityforasignalsampleversusthebackgroundshape.Notethatthetopmassvalueforwhichthiseventprobabilityisminimizedwillbeusedinthenaltopmassreconstruction,andthevalueoftheprobabilityinnegativelogscaleisusedasadiscriminatingvariablebetweenttandbackground.WedenotethisvalueasminLKL,andthecutdenitionisrequiringthisvariabletobelessthan10.Thevalueofthislastcuthasbeenobtainedbyminimizingthestatisticaluncertaintyonthetopmassvalueasreconstructedinsection 4 ,thatisusingonlythematrixelementcalculation.Table 5-4 showstheeciencyofthiscutrelativetothenumberofeventsaftertaggingandafterthekinematicalcuts,forsignalatdierenttopmassesandforbackground.Thetablealsoshowsthenumberofsignaleventscorrespondingto943pb1andtheappropriatesignaltobackgroundratio.Comparingthesignal-to-backgroundratiosS=BbetweenTable 5-3 andTable 5-4 thereisanimprovementofaboutafactorof3forsampleswithonetaggedheavy 93 PAGE 94 Table5-1. Numberofeventsinthemulti-jetdataaftertheclean-upcuts,kinematicalcutsandtagging.TheintegratedluminosityisL=943pb1. CutEventsFraction(%) Initial12274958100jzj<60cm355505428.9jzzpj<5cm339734127.7LeptonVeto339255127.66ET=p PET<3333345127.2Ntightjets=63806763.1KinematicCuts41720.0341tag7826.37e-52tag1481.21e-5 Table5-2. Numberofeventsinthet tMonteCarlosamplewithMtop=170GeV. CutEventsFraction(%) Initial233233100jzj<60cm12816955.0jzzpj<5cm12804554.9TightLeptonVeto11397048.96ET=p PET<38802737.7Ntightjets=62948512.6KinematicCuts59992.61tag26031.12tag15990.69 94 PAGE 95 Numberofeventsandexpectedsignaltobackgroundratiosforthet tMonteCarlosampleswithtopmassesbetween150GeVand200GeVforaluminosityofL=943pb1.Thenumberofdataeventsisshowntoo.Theseeventsarepassingthekinematicalselection,butnottheminimumlikelihoodcut. Minimumofthenegativelogeventprobability.Inblueit'sshownthecurvefort tsampleofMtop=175GeV,whileinredit'sshownthebackgroundshape. 95 PAGE 96 Numberofevents,minLKLcuteciency()relativetothekinematicalcutsandthesignaltobackgroundratiosforthet tMonteCarlosampleswithtopmassesbetween150GeVand200GeVforaluminosityof943pb1.Theseeventspassallthecuts.Theeciencyforbackgroundeventsisalsoshown. 96 PAGE 97 teventsbasedontheStandardModel.TheshapeofthebackgroundeventscanbedeterminedwiththehelpofourMonteCarlosamples.However,duethesmallstatisticsofthissamples,wewillbeforcedtore-sampleheavilywhenwewillperformthesensitivitystudiesofourtechnique.Inordertoovercomethat,wewillformasampleofbackground-likeeventsusingdataeventsfromasamplequasi-dominatedbybackground.Thenwe'llmakesurethattheshapeofthisdata-drivenbackgroundmodelcorrespondstotheshapefromMonteCarlobackgroundevents.Toformthedata-drivenbackgroundevents,westartwithourpretagdataeventsbeforetheminimumlikelihoodcut,butafteralltheclean-upandkinematicalcuts.Inthissamplethesignaltobackgroundratioisabout1=25.Thenwestarttorandomlyb-tagthejetsoftheseeventsbyusingtheb-tagratesofthemistagmatrixdenedintheallhadroniccross-sectionanalysis[ 51 ].Eacheventcanendupinanyofthepossibletaggedcongurationsbyhavinganumberoftaggedjetsbetween0and6.Weiteratethisarticialb-taggingproceduremanytimeskeepingallthecongurationsthathaveatleastoneb-taggedjet.Somecongurationswillappearmultipletimesinthisprocess,andwewilluseitthatofteninourstudiesasifitwereadistinctconguration.The 97 PAGE 98 6-1 showsthecomparisonintheexclusivesingletaggedsample,whileFigure 6-2 showsthecomparisonintheinclusivedoubletaggedsample.Thevariableschosenforthiscomparisonarethetransverseenergies,pseudo-rapidityandthepolarangleofthejets,andthenumberofvertices,sumofthetransverseenergiesofthe 98 PAGE 99 5 b+4lightpartonsinitsnalstate.Onevariablewecanlookatisthesumoftheeventprobabilitiesasdenedinsection 4 usingthematrixelement.Thesumisbetweenatopmassequalto125GeVupto225GeVinstepsof1GeV.Figure 6-3 showstheshapesofMonteCarlobackgroundandofthedata-drivenbackground.Anotherinterestingvariableistheinvariantmassofalltheuntaggedpairsofjetsintheevent.Figure 6-4 showsthisvariableforthetaggedeventsbeforetheminLKLcut,whileFigure 6-5 showsthecaseoftaggedeventsaftertheminLKLcut. 6-6 showsthisvariableforeventsaftertheminLKLcut.TheeventbyeventmostprobabletopmassandthedijetmassvariablesareofparticularinterestsincetheywillbeusedinthereconstructionofthetopmassandoftheJESvariabletobedescribedinsection 7 .Allthesecomparisonsshowgoodagreementbetweenourdata-drivenbackgroundmodelandtheAlpgenb b+4lightpartons. 99 PAGE 100 6-7 showshowtheslopedecreaseswiththebackgroundfraction,whilethelowerplotshowshowtheinterceptchangeswiththebackgroundfraction.Theslopedecreaseindicatesadecreaseinthesensitivity,inotherwordsanincreaseinthestatisticaluncertaintyonthetopmass.Forthecalibrationcurvesstudiedintheseplotstheinterceptshouldbe178GeV,anditcanbeseenthatasthebackgroundfractionincreasestheinterceptgetsfurtherfromthe178GeVvalue,thatisthebiasincreases.Thereasonforthebackgroundfractiontohavesuchabigeectonthemassreconstructionusingthematrixelementtechniqueofsection 4 isbecausethebackgroundiscompletelyignoredinthematrixelementcalculationorinassessingabackgroundeventprobability.Inthisanalysiswestillwon'tcalculateabackgroundmatrixelement,butwewilluseabackgroundprobabilityinstead,whichwillbedescribedinthenextsections. Figure6-1. Backgroundvalidationincontrolregion1forsingletaggedevents.Theredpointsarethedatapoints,whiletheblackpointsarefromthebackgroundmodel. 100 PAGE 101 Backgroundvalidationincontrolregion1fordoubletaggedevents.Theredpointsarethedatapoints,whiletheblackpointsarefromthebackgroundmodel. Figure6-3. SumofeventprobabilitiescalculatedforMtop=125GeVuptoMtop=225GeVinstepsof1GeV.ThesearetheeventsbeforetheminLKLcutforAlpgenb b+4lightpartonsinblue,andforthebackgroundmodelinblack.Theplottotheleftshowsthesingletaggedevents(Kolmogorov-Smirnovprobabilityis1%),whiletheplottotherightshowsthedoubletaggedevents(Kolmogorov-Smirnovprobabilityis13%). 101 PAGE 102 Dijetinvariantmassoftheuntaggedjets.ThesearetheeventsbeforetheminLKLcutforAlpgenb b+4lightpartonsinblue,andforthebackgroundmodelinblack.Theplottotheleftshowsthesingletaggedevents(Kolmogorov-Smirnovprobabilityis25%),whiletheplottotherightshowsthedoubletaggedevents(Kolmogorov-Smirnovprobabilityis43%). Figure6-5. Dijetinvariantmassoftheuntaggedjets.ThesearetheeventsaftertheminLKLcutforAlpgenb b+4lightpartonsinblue,andforthebackgroundmodelinblack.Theplottotheleftshowsthesingletaggedevents(Kolmogorov-Smirnovprobabilityis90%),whiletheplottotherightshowsthedoubletaggedevents(Kolmogorov-Smirnovprobabilityis70%). 102 PAGE 103 Eventbyeventmostprobabletopmasses.ThesearetheeventsaftertheminLKLcutforAlpgenb b+4lightpartonsinblue,andforthebackgroundmodelinred.Theplottotheleftshowsthesingletaggedevents,whiletheplottotherightshowsthedoubletaggedevents. Eectofthebackgroundcontaminationinthetopmassreconstructionusingonlythematrixelementtechnique.Theupperplot:slopeofthecalibrationcurveversusthebackgroundfraction.Thelowerplot:interceptofthecalibrationcurveversusthebackgroundfraction.Thecalibrationcurvesarebuiltusingonlythematrixelementreconstructiontechniquedescribedinsection 4 103 PAGE 104 tevents,andadditionalcorrectionsmightbeneededatthislevel.Wedeneavariable,JES,calledJetEnergyScale,measuredinunitsofc.ThereisacorrelationbetweenthetopmassandthevalueofJES,andthat'swhyweplantomeasurethemsimultaneouslytoavoidanydoublecountinginthenaluncertaintyonthemass.OurtechniquestartsbymodelingthedatausingamixtureofMonteCarlosignalandMonteCarlobackgroundevents.Theeventswillberepresentedbytwovariables:dijetinvariantmassandanevent-by-eventreconstructedtopmass.Thelatterisobtainedusingthematrixelementtechniquedescribedinsection 4 .Forsignal,theshapesobtainedinthesetwovariablesareparameterizedasafunctionoftopquarkpolemassandJES.Forbackgroundnosuchparameterizationisneeded.HenceourmodelwilldependonthetopmassandtheJES.ThemeasuredvaluesforthetopquarkmassandfortheJESaredeterminedusingalikelihoodtechniquedescribedinthissection. 7{1 ,isproductof3terms:thesingletaglikelihoodusedforsingletaggedevents,L1tag,thedoubletaglikelihoodusedfordoubletaggedevents,L2tagandtheJESconstraint,LJES,whoseexpressionisshowninEquation 7{7 104 PAGE 105 7{2 .Thetoptemplateterm,Ltop,isshowninEquation 7{3 .TheWtemplateterm,LW,isshowninEquation 7{4 .Theconstraintontotalnumberofevents,Lnev,isshowninEquation 7{5 .Theconstraintonthet tnumberofevents,Lns,isshowninEquation 7{6 tevents,ns=(ns+nb),istheweightofthesignalprobabilityandthefractionofbackgroundevents,nb=(ns+nb),istheweightofthebackgroundprobability.TogetherwithMandJES,theparametersnsandnbarefreeinthelikelihoodt. (ns+nb)!(7{5)Thenumberofsignalevents,ns,isconstrainedtotheexpectednumberoft tevents,nexps,viaaGaussianofmeanequaltonexpsandwidthequaltonexps.Thewidthofthegaussianissimplytheuncertaintyontheexpectednumberoft tevents.Theexpectednumbersofsignalevents,nexps,are13singletaggedand14doubletaggedevents,correspondingtoatheoreticalcross-sectionof6:7+0:70:9pb[ 55 ]andan 105 PAGE 106 5-4 .TheuncertaintiesonthenumbersofsignaleventsnexpsarechosentobethePoissonerrors.ThisisaconservativeapproachsincethePoissonerrorsarelargerthantheuncertaintiesderivedbasedonthetheoreticalcross-sectionuncertainty. 7.2.1DenitionoftheTemplateAsmentionedinsection 7.1 ,weusethematrixelementtobuildthetoptemplates.Theeventprobabilitydenedinsection 4 isplottedasafunctionofthetoppolemassintherange125GeVand225GeV.Innegativelogarithmicscalethiseventprobabilitywillbeminimizedforacertainvalueoftopmasswhichwe'llusetoformthetoptemplates.TheshapeofthesetemplatesdependsontheinputtopmassandJESfort tevents,butnotforbackgroundevents. 5 with7dierentJESvalues:3;2;1;0;1;2;3,afterallourselectioncutshavebeenapplied.Intotalthereare84templatesforsignalusedforparameterization.Thefunctionusedtotthem 106 PAGE 107 7{8 displaysthetfunctionandthedependenceofitsparametersontopmassandJES. (mtopevt1)2+224 (7{8) TheexpressionfornormalizationtermN(M;JES)fromEquation 7{8 isgiveninEquation 7{9 7{8 asafunctionofthetopmassMandjetenergyscaleJESisgivenbyEquation 7{10 7{11 7{8 atthecenterofthebin.ThesummationinEquation 7{11 isdoneforalltemplatesandforallthebinsforwhichhbinhasmorethan5entries.ThedenominatorofEquation 7{11 isthenumberofdegreesoffreedom.Foreachsample,thevaluesofthe25parameters,p,aregiveninTable 7-1 .TheshapesoffewofthesignaltemplatesaswellastheparameterizedcurvesareshowninFigure 7-1 107 PAGE 108 7-2 .Figure 7-2 showstheshapesofthebackgroundtemplatesaswellastheparameterizedcurves,forsingleanddoubletaggedevents.InAppendix D ,allthetoptemplatescorrespondingtosignaleventsaredisplayed. 7.3.1DenitionoftheTemplateThedijetmasstemplatesareformedbyconsideringtheinvariantmassofallpossiblepairsofuntaggedjetsinthesample.TheshapeofthesetemplatesdependsontheinputtopmassandJESfort tevents,butnotforbackgroundevents. 7{12 showsthetfunctionandthedependenceofitsparametersontopmassandJES. 108 PAGE 109 TheexpressionfornormalizationtermN(M;JES)fromEquation 7{12 isgiveninEquation 7{13 7{12 asafunctionofthetopmassMandjetenergyscaleJESisgivenbyEquation 7{14 7{11 .Ineachsample,thevaluesofthe36parameters,p,aregiveninTable 7-3 .TheshapesoffewofthesignaltemplatesaswellastheparameterizedcurvesareshowninFigure 7-3 .Thebackgroundtemplateshapeisbuildinthesamewayasthesignaltemplates.Thetopcontaminationisremovedinthesamewayasinthecaseofthetoptemplates(seesection 7.2 ).Thebackgroundtemplateisttedtoanormalizedsumoftwogaussiansandagammaintegrand.Forboththesingletaggedandthedoubletaggedsamples,weshowthevaluesoftheparametersinTable 7-4 109 PAGE 110 7-4 showstheshapesofthebackgroundtemplatesaswellastheparameterizedcurves,forsingleanddoubletaggedevents.InAppendix E ,allthedijetmasstemplatescorrespondingtosignaleventsaredisplayed. Table7-1. Valuesoftheparametersdescribingtheshapesofthetoptemplatesforthettsamples. ParameterValues(1Tag)Uncertainties(1Tag)Values(2Tags)Uncertainties(2Tags) Table7-2. Valuesoftheparametersdescribingtheshapesofthetoptemplatesinthecaseofthebackgroundevents. ParameterValues(1Tag)Uncertainties(1Tag)Values(2Tags)Uncertainties(2Tags) 11.53e-023.09e-051.28e-029.08e-0521.59e+027.68e-021.63e+023.73e-0131.79e+037.17e+003.28e+036.42e+01 110 PAGE 111 Toptemplatesforttevents,singletagsintheleftplot,doubletagsintherightplot.Theupperplotsshowtheparameterizedcurves,whilethebottomplotsshowtheoriginalhistograms.TheleftcolumnshowsthetemplatesvariationwithtopmassatJES=0.TherightcolumnshowstheirvariationwithJESattopmassMtop=170GeV. Figure7-2. Toptemplatesforbackgroundevents.Singletagsintheleftplot,anddoubletagsintherightplot. Figure7-3. Dijetmasstemplatesforttevents,singletagsintheleftplot,doubletagsintherightplot.Theupperplotsshowtheparameterizedcurves,whilethebottomplotsshowtheoriginalhistograms.TheleftcolumnshowsthetemplatesvariationwithtopmassatJES=0.TherightcolumnshowstheirvariationwithJESattopmassMtop=170GeV. 111 PAGE 112 Valuesoftheparametersdescribingthedijetmasstemplatesshapesforthettsamples. ParameterValues(1Tag)Uncertainties(1Tag)Values(2Tags)Uncertainties(2Tags) 112 PAGE 113 Dijetmasstemplatesforbackgroundevents.Singletagsintheleftplot,anddoubletagsintherightplot. Table7-4. Valuesoftheparametersdescribingthedijetmasstemplatesshapesinthecaseofthebackgroundevents. ParameterValues(1Tag)Uncertainties(1Tag)Values(2Tags)Uncertainties(2Tags) 11.88e-019.52e-023.53e-012.39e-0128.02e+014.29e-028.02e+011.12e-0137.01e+001.70e-029.13e+004.41e-0244.68e-019.52e-023.59e-012.39e-0159.97e+014.29e-029.46e+011.12e-0162.98e+011.70e-023.36e+014.41e-0273.44e-019.52e-022.90e-012.39e-0184.03e-024.29e-024.08e-021.12e-0191.04e+011.70e-021.04e+014.41e-02101.89e+009.52e-021.58e+002.39e-01 113 PAGE 114 52 ],wefoundthatforanydistributionthestatisticaluncertaintyonthemeanshouldbeexpressedasinEquation 8{1 ,thewidthshouldbeexpressedasinEquation 8{2 andthestatisticaluncertaintyonthewidthshouldbeexpressedasinEquation 8{3 (NPE1)(1)+ 114 PAGE 115 8{1 8{2 and 8{3 ,NPEisthenumberofpseudo-experiments,rawistheuncorrectedwidthofadistribution,andistheaveragecorrelationbetweenanytwopseudo-experiments.Thevalueofthecorrelationfactorsdependsonthesizeofthenumberofeventsperpseudo-experimentandonthetotalnumberofeventsavailable.Sincethelasttwonumbersdependonthetopmass(seeTable 5-4 )thentheaveragecorrelationbetweenanytwopseudo-experimentswilldependonthetopmass.ThevaluesforthesecorrelationtermsaregiveninTable 8-1 .WhentheJESpriorisapplied,thevalueoftheJESeachpseudo-experimentisconstrainedtoisrandomlyselectedbasedonagaussiancenteredonthetrueJESofthesampleandofwidthequalto1.Thevariablesextractedfromeachpseudo-experimentarethevaluesofmass,Mout,andJES,JESout,thatminimizethelikelihoodsdenedinsection 7 ;thestatisticaluncertaintiesontheabovevariables,MoutandJESoutandthepullsasdenedbyEquation 8{4 7 .Neitherthetopmass,northeJES 115 PAGE 116 8-1 showsthereconstructedJESandthereconstructedtopmassrepresentedbythepoints,versusthetrueJESandtruetopmassrepresentedbythegrid.Ideallythepointsshouldmatchthegridcrossings.Figure 8-2 showsreconstructedtopmassversusthetruetopmassforatrueJESof0.Ideally,thiscurveshouldhaveaslopeof1,andaninterceptof175consistentwithnobias.Figure 8-3 showsreconstructedJESversusthetrueJESforatruetopmassof170GeV,andagain,ideally,thiscurveshouldhaveaslopeof1,andaninterceptof0consistentwithnobias.Figure 8-4 showshowtheslopeofFigure 8-2 changeswiththetrueJES,whileFigure 8-5 showshowtheinterceptofFigure 8-2 changeswiththetrueJES.Figure 8-6 showshowtheslopeofFigure 8-3 changeswiththetruetopmass,whileFigure 8-7 showshowtheinterceptofFigure 8-3 changeswiththetruetopmass.Figure 8-8 showsthemasspullmeansversustruetopmass,whileFigure 8-9 showsthemasspullwidthsversustruetopmass.InbothplotsthetrueJESis0.Basedontheseguresitresultsthattheuncertaintyontopmasshastobeinatedby10:5%.TheaveragemasspullmeanasafunctionoftrueJESisshowninFigure 8-10 ,whiletheaveragemasspullwidthasafunctionoftrueJESisshowninFigure 8-11 .ForagiventrueJESvalue,theaverageisoverallthemasssamples.Figure 8-12 showstheJESpullmeansversustrueJES,whileFigure 8-13 showstheJESpullwidthsversustrueJES.Inbothplotsthetruetopmassis170GeV.BasedontheseplotsitresultsthattheuncertaintyontheJEShastobeinatedby5:8%.TheaverageJESpullmeanasafunctionoftruetopmassisshowninFigure 8-14 ,whiletheaverageJESpullwidthasafunctionoftruetopmassisshowninFigure 8-15 .Foragiventruetopmassvalue,theaverageisoveralltheJESsamples.AsitcanbeseeninFigure 8-1 ,thereseemstobeaslightbiasinthereconstructionofJESandtopmass.Wecanextracttheslopeandtheinterceptofthedependenceofthereconstructedmassonthetruemass.ThiscanbedonefordierentJESvalues. 116 PAGE 117 8-4 and 8-5 showthedependencesontheJESoftheslopesand,respectively,oftheintercepts.Similarly,inthecaseofJESreconstructionweobtainFigures 8-6 and 8-7 .BasedonthetsfromFigures 8-4 and 8-5 ,wecanexpressanalyticallyhowthereconstructedmassdependsonthetruetopmassandonthetrueJES.ThisisshowninEquation 8{5 .UsingthetsfromFigures 8-6 and 8-7 ,wecanwritesimilarexpressionsforthereconstructedJES.ThisisshowninEquation 8{6 (8{5) TheparametersCm,Cj,Sm,andSjfromEquations 8{5 and 8{6 dependonthetruevaluesoftopmassandjetenergyscaleasshowninEquation 8{7 .ThevaluesoftheparametersoftheseequationscorrespondtothetparametersofFigures 8-4 8-5 8-6 and 8-7 .TheyarelistedinTable 8-2 8{5 and 8{6 asasystemofequationsandsolvethemforthetruetopmass,Mtrue,andthetrueJES,JEStrue.AfterthesecorrectionsareappliedthenewreconstructedvaluesforJESandtopmassareconsistentwiththetruevaluewithintheuncertainties,asitcanbeseeninFigures 8-16 8-17 8-18 8-19 and 8-20 117 PAGE 118 8-21 showstheresidualofthetopmassreconstructionusingsamplesforwhichtheinputtopmasswasunknowntous,andFigure 8-22 showstheJESresidualsforsampleswithunknowntrueJES.Thetopmassgroupconvenersprovidedthesamplesandtheyweretheonlyonesabletocalculatetheseresiduals.TheplotsindicatethatwithintheuncertaintiesthetopmassandJESreconstructionisunbiased. 8{5 and 8{6 ,weobtainanothersystemofequationstobesolvedfortherealuncertainties.SolvingEquations 8{8 and 8{9 willprovidethecorrectuncertaintiesontopmassandonJES. 8-23 showstheexpecteduncertaintyontopmassversusinputtopmass,usinganinputJESof0.Figure 8-24 showstheexpecteduncertaintyontheJESversusinputJESforaninputtopmassof170GeV.TheexpecteduncertaintiesshowninFigure 8-23 containboththepurestatisticaluncertaintyonthetopmassandtheuncertaintyduetoJES.Thisuncertaintydependsonthetopmassbecausetheexpectednumberoft teventsdependsonthetopmass.InordertodisentanglethestatisticalcontributionfromtheJEScomponentofthisuncertainty,weperformedadierentreconstructionofthetopmassbyxingtheJEStothetruevalueinthe2Dt.Followingthisreconstruction,theuncertaintyonthetopmassispurelyofstatisticalnature.Foratopmassof170GeVtheexpectedstatisticaluncertaintyis2.5GeV,whereasthecombinedstatisticalandJES-systematicuncertainty,asperFigure 8-23 ,is3.2GeV.ThatmeansthesystematicuncertaintyduetoJESontopmassis2.0GeV.Thissystematicuncertaintyshowsanimprovementof10%overthe1DJESsystematicuncertaintyontopmassof2.2GeV. 118 PAGE 119 Table8-1. Valueoftheaveragecorrelationfactorbetweenanytwopseudo-experiments.Thedependenceonthevalueofthetopmassisduetothettcross-sectiondependenceontopmass. Table8-2. ValuesoftheparametersdescribingthelineardependenceonthetrueJESandonthetrueMtop,oftheinterceptandslopeoftheMtopcalibrationcurveandoftheJEScalibrationcurverespectively. ParameterValueUncertainty 119 PAGE 120 JESversusTopMassplane.ThepointsrepresentthereconstructedJESandmass. Figure8-2. Reconstructedtopmassversusinputtopmass,forinputJESequalto0. Figure8-3. ReconstructedJESversusinputJES,forinputtopmassequalto170GeV. 120 PAGE 121 SlopeofthemasscalibrationcurveversusinputJES. Figure8-5. ConstantofthemasscalibrationcurveversusinputJES. Figure8-6. SlopeoftheJEScalibrationcurveversusinputJES. Figure8-7. ConstantoftheJEScalibrationcurveversusinputJES. 121 PAGE 122 Masspullmeansversusinputtopmass,forinputJESequalto0. Figure8-9. Masspullwidthsversusinputtopmass,forinputJESequalto0. Figure8-10. AverageofmasspullmeansversusinputJES. Figure8-11. AverageofmasspullwidthsversusinputJES. 122 PAGE 123 JESpullmeansversusinputtopmass,forinputtopmassequalto170GeV. Figure8-13. JESpullwidthsversusinputtopmass,forinputtopmassequalto170GeV. Figure8-14. AverageofJESpullmeansversusinputtopmass. Figure8-15. AverageofJESpullwidthsversusinputtopmass. 123 PAGE 124 JESversusTopMassplane.ThepointsrepresentthereconstructedJESandmassafterthe2Dcorrection. 124 PAGE 125 SlopeoftheMtopcalibrationcurveversustrueJESafterthe2Dcorrection. InterceptoftheMtopcalibrationcurveversustrueJESafterthe2Dcorrection. SlopeoftheJEScalibrationcurveversustrueMtopafterthe2Dcorrection. InterceptoftheJEScalibrationcurveversustrueMtopafterthe2Dcorrection. Figure8-21. Dierencebetweenthereconstructedmassandthetruemassforblindmasssamples. Figure8-22. DierencebetweenthereconstructedandthetrueJESforblindJESsamples. 125 PAGE 126 Expecteduncertaintyontopmassversusinputtopmass,forinputJESequalto0.ThisuncertaintyincludesthepurestatisticaluncertaintyandthesystematicuncertaintyduetoJES. Figure8-24. ExpecteduncertaintyonJESversusinputJES,forinputtopmassequalto170GeV. 126 PAGE 127 teventsisexclusivelybasedonthesimulationwhichdoesn'tdescribethephysicsofsucheventsveryprecisely.Themajorsourcesofuncertaintiesappearfromourunderstandingofjetfragmentation,ourmodelingoftheradiationotheinitialornalpartons,andourunderstandingoftheprotonandantiprotoninternalstructure.Apartfromthesegenericuncertainties,wealsoaddressotherissuesspecictothepresentmethodsuchastheshapeofthebackgroundtoptemplatesfollowingthet tdecontamination,thecorrelationbetweenthedijetmassesandthetopmassdeterminedforeachevent,andthelevelofimprecisioninthedeterminationofthebi-dimensionalcorrectionofthereconstructedtopmassandJES. 127 PAGE 128 tcontamination.Toremovethetopcontamination,weassumedatopmassof170GeV,andnowwehavetoestimateeectofthisassumption.Wehavemodifyourassumptiononthetopmassofthetopcontaminationby10GeV,thatiswegottwo 128 PAGE 129 tcontaminationremovalandbasedontheconstantsabove,wescaledownthetemplatehistogramsuctuatethecontentofthescaledhistogramsusingthePoissonprobabilityafterthePoissonuctuation,scalebackupthehistograms,removethet tcontaminationandtwithagaussiantoobtainthenewtemplatefunctionrepeattheabovesteps10,000times,andhistogramtheparametersofthenewtemplatesextracttheuncertaintiesonthebackgroundparametersfromtheselasthistograms 129 PAGE 130 9-1 showstheeventmultiplicitysingletaggedeventsontheleft,andfordoubletaggedeventsontheright.Figure 9-2 showsthehistogramsofthethreeparametersdescribingthegaussiantforthesingletaggedevents,whileFigure 9-3 showstheequivalentplotsinthecaseofthedoubletaggedevents.TheuncertaintiesonthebackgroundparametersasdeterminedfollowingthehistogramuctuationareshowninTable 9-1 .Varyingthebackgroundparameterswithintheseuncertaintiesresultsinashiftintopmassof0.4GeV. 9-4 showsontheleftthetopmasspullmeaninthedefaultcasewhentheabovecorrelationwasreducedtozero,whileontherightisshownthesituationwithfullcorrelation.Figure 9-5 showstheequivalentcomparisoninvolvingthetopmasspullwidths.Onaverageoverdierenttopmasssamples,thepullmeanisconsistentwithintheuncertaintiesbetweenthetwoscenarios.However,thepullwidthsappearhigherwhenthecorrelationbetweentheeventtopmassandthedijetmassiszero.Weconcludethatthereisnoneedforasystematicuncertainty,andwekeepthedefaultpullwidthasthecorrectingfactoronthestatisticalerroronthetopmasssinceitrepresentsthemoreconservativeapproach. 8{5 and 8{6 withintheiruncertaintiesaslistedinTable 8-2 .Wethenre-calibratedthereconstructedvaluesforthetopmass.Thechangeintopmassis0.2GeV. 130 PAGE 131 53 ],0:6%ofthejetenergyuncertaintyontheb-jetsiscomingfromtheeectslistedabove.Thereforethenalshiftonthetopmassfollowingour1%shiftinb-jetsenergiesneedstobescaleddownbyafactorof0:6.Thesystematicuncertaintyonthetopmassduetotheb-jetenergyscaleis0.4GeV. 54 ].Forthiswehavetostudytheeectonthetopmassreconstructionfromeachofthesesixsources:level1,4,5,6,7and8.AMonteCarlosamplehasbeenusedwheretheenergiesofthejetshavebeenshiftedupordownbytheuncertaintyateachlevelseparately,soatotalof12sampleshavebeenobtained.Wereconstructthetopmassineachofthem,withoutapplyinganyconstrainonthevalueofJES.InTable 9-2 wepresenttheaverageshiftonthetopmassateachlevel,andtheirsuminquadrature.Weconcludefromthisthattheresidualjetenergyuncertaintyontopmassis0.7GeV. 131 PAGE 132 9-3 summarizesallsourcesofsystematicuncertaintieswiththeirindividualcontributionaswellasthecombinedeect. Figure9-1. Eventmultiplicityforbackgroundevents.Ontheleftisshowntheplotforsingletaggedevents,whileontherighttheplotfordoubletaggedeventsisshown. Table9-1. Uncertaintiesontheparametersofthetopmasstemplatesforbackground. Parameter1tag2tags Constant10.2e-047.0e-04Mean2.593.35Sigma272.1711.9 Table9-2. Residualjetenergyscaleuncertaintyonthetopmass. LevelUncertainty(GeV/c2) L10.2L40.1L50.5L60.0L70.5L80.1TotalJESResidual0.7 132 PAGE 133 Histogramsoftheparametersofthegaussiantofthebackgroundeventtopmasstemplateforsingletaggedevents.Upperleftplotshowstheconstantofthegaussian,upperrightshowsthemeanofthegaussian,lowerleftshowsthewidthofthegaussian,andlowerrightplotshowsthenormalizationofthegaussian. Table9-3. Summaryofthesystematicsourcesofuncertaintyonthetopmass. SourceUncertainty(GeV/c2) InitialStateRadiation0.3FinalStateRadiation1.2PDFchoice0.5Pythiavs.Herwig1.0MethodCalibration0.2BackgroundShape0.9BackgroundStatistics0.4SampleComposition0.1HeavyFlavorJES0.4ResidualJES0.7Total2.1 133 PAGE 134 Histogramsoftheparametersofthegaussiantofthebackgroundeventtopmasstemplatefordoubletaggedevents.Upperleftplotshowstheconstantofthegaussian,upperrightshowsthemeanofthegaussian,lowerleftshowsthewidthofthegaussian,andlowerrightplotshowsthenormalizationofthegaussian. Figure9-4. Topmasspullmeanasafunctionoftopmassfordierenttreatmentofthecorrelationbetweentheeventtopmassandthedijetmass.Ontheleftisthedefaultcasewhenthecorrelationiszero,whileontherightisshownthesituationwiththefullcorrelation. 134 PAGE 135 Topmasspullwidthasafunctionoftopmassfordierenttreatmentofthecorrelationbetweentheeventtopmassandthedijetmass.Ontheleftisthedefaultcasewhenthecorrelationiszero,whileontherightisshownthesituationwiththefullcorrelation. 135 PAGE 136 10-1 ,weshowinthetotalnumberofeventsandtheexpectednumberofsignaleventsusedasinputinthe2DlikelihoodofEquation 7{1 .NotethatinEquation 7{1 weneedtheuncertaintyontheexpectednumberofsignaleventsandthisisalsoshowninTable 10-1 .Thenumbersofbackgroundeventsareshownaswell,buttheyarenotusedasinputvaluesinthelikelihood.Inthethirdcolumnweshowthenumberofeventsastheyresultfromtheminimizationofthe2Dlikelihood.Followingtheminimizationofthe2Dlikelihood,wemeasuredatopmassof171.13.7GeV,andaJESof0.50.9c.Thevalueofthejetenergyscale(JES)isthereforeconsistentwiththepreviousdeterminationofJESatCDF.ThequoteduncertaintyonthetopmassrepresentsthecombinationofthestatisticaluncertaintywiththesystematicuncertaintyduetoJESuncertainty.Inordertoobtainonlythestatisticaluncertaintyonthetopmass,theminimizationofthe2DlikelihoodismodiedsuchthattheJESparameterisxedto0.5c(theresultfrom2Dt).Followingthisprocedurethestatisticaluncertaintyonthetopmassis2.8GeV.ThereforethesystematicuncertaintyduetoJESis2.4GeV.Figure 10-1 showsthedistributionsofeventbyeventreconstructedtopmassesastheblackpointsfordataandastheorangehistogramforthecombinationofsignalandbackgroundtemplatesthatbestttedthedata.Thebluehistogramrepresentsonlythebackgroundtemplate.Thesamplewithsingletaggedeventsisshownintheleftplot,whilethedoubletaggedeventsareshownintherightplot. 136 PAGE 137 10-2 .Thecentralpointcorrespondstotheminimumofthelikelihood,whilethecontoursrepresentthe1-sigma,2-sigma,and3-sigmalevels,respectively.UsingattMonteCarlosamplewithatopmassequalto170GeVandthenumberofsignalandbackgroundeventsasresultedfromthedatat,weformedpseudo-experimentsanddeterminedtheexpecteduncertaintyonthetopmassduetostatisticaleectsandJES.About41%ofthepseudo-experimentshadsuchcombineduncertaintyonthetopmasslowerthanthemeasuredvalueof3.7GeV.ThiscanbeseeninFigure 10-3 ,wherethehistogramshowstheresultsofthepseudo-experimentsandthebluelinerepresentsthemeasureduncertainty.Inconclusion,themeasuredcombinedstatisticalandJESuncertaintiesonthetopmassagreeswiththeexpectation.Thetotaluncertaintyonthetopmassinthisanalysisis4.3GeV.Thepreviousbestmassmeasurementinthischannelhadanequivalenttotaluncertaintyof5.3GeV[ 56 ]whichis23%more.Thesourceforthisimprovementistheuncertaintyduetojetenergyscale(JES)onthetopmass.Inthisanalysisthisuncertaintyamountsto2.4GeVcomparedto4.5GeVinthepreviousbestresultwhichis88%more.Someofthisgaininprecisionislostduetothesomewhathighersystematicuncertaintiesfromothersourcesandduetoaslightlyworsestatisticaluncertaintyinthisanalysiscomparedwiththepreviousbestmassresultinthischannel.Amorecarefulestimationoftheothersourcesofsystematicuncertaintiesonthetopmassaswellasamoreecienttteventselectionwillhelpfurtherreducethetotaluncertaintyonthetopmass.Comparedtomassmeasurementsinotherttdecaychannels,themassmeasurementfromthisanalysisrankedthirdinthetopmassworldaverage[ 57 ]witha11%weight.Thetwobettermeasurementswereperformedinthelepton+jetschannelasitcanbeseeninFigure 10-4 .ThismeasurementpromotestheallhadronicchannelasthesecondbestchannelforthetopquarkmassanalysesinRunIIattheTevatron. 137 PAGE 138 Table10-1. Numberofeventsforthet texpectationandfortheobservedtotalforaluminosityof943pb1passingallthecuts.Theinputvaluesforsignalhavetheuncertaintiesnexttotheminparenthesis.Thebackgroundexpectationbeingthedierencebetweentotalandsignalisalsoshown.Fortheoutputvalues,thenumbersintheparenthesisaretheuncertainties. NumberofEventsInputReconstructed TotalObserved(1tag)4847.8ExpectedSignal(1tag)133.613.23.7Background(1tag)3534.67.2TotalObserved(2tags)2423.3ExpectedSignal(2tags)143.714.13.4Background(2tags)109.24.3 Figure10-1. Eventreconstructedtopmassfordata(blackpoints),signal+background(orange)andonlybackgroundevents(blue).Singletaggedsampleisontheleft,whilethedoubletaggedsampleisontheright. 138 PAGE 139 Contoursfor1-sigma(red),the2-sigma(green)andthe3-sigma(blue)levelsofthemassandJESreconstructioninthedata. Figure10-3. HistogramshowstheexpectedstatisticaluncertaintyfromMonteCarlousingpseudo-experiments,whilethelineshowsthemeasuredone.About41%ofallpseudo-experimentshavealoweruncertainty. 139 PAGE 140 MostpreciseresultsfromeachchannelfromtheD0andCDFexperimentatFermilabbyMarch2007.Takingcorrelateduncertaintiesproperlyintoaccounttheresultingpreliminaryworldaveragemassofthetopquarkis170.91.1(stat)1.5(syst)GeV/c2whichcorrespondstoatotaluncertaintyof1.8GeV/c2.Thetopquarkmassisnowknownwithaprecisionof1.1%. 140 PAGE 141 UpperplotshowsthePDFshapesusedinthematrixelementcalculationofsection 4.3 .BottomplotshowsacrosscheckofthenormalizationofthesePDFs. 141 PAGE 142 Transversemomentumofthettsystemfordierentgeneratorsandfordierenttopmasses.Upperplot:shapesofthetransversemomentumofthettsystemfordierentgenerators(CompHep,PythiaandHerwig)andfordierenttopmasses.Middleplot:theMeansofthedistributionsintheupperplot.Lowerplot:theRMSofthedistributionsintheupperplot. 142 PAGE 143 AFigureC-1. TransferfunctionsfortheW-bosondecaypartons.A)Forpartonswiththevalueforpseudo-rapidityjj<0:7.B)Forpartonswithpseudo-rapidity0:7jj<1:3.C)Forpartonswithpseudo-rapidity1:3jj2. 143 PAGE 144 Continued 144 PAGE 145 Continued 145 PAGE 146 Transferfunctionsfortheb-quarkpartons.A)Forpartonswiththevalueforpseudo-rapidityjj<0:7.B)Forpartonswithpseudo-rapidity0:7jj<1:3.C)Forpartonswithpseudo-rapidity1:3jj2. 146 PAGE 147 Continued 147 PAGE 148 Continued 148 PAGE 149 AFigureD-1. Toptemplatesforttsingletaggedeventsforsampleswithdierenttopmasses:from150GeVto200GeV.A)CaseofJES=3.B)CaseofJES=2.C)CaseofJES=1.D)CaseofJES=0.E)CaseofJES=1.F)CaseofJES=2.G)CaseofJES=3. 149 PAGE 150 Continued 150 PAGE 151 Continued 151 PAGE 152 Continued 152 PAGE 153 Continued 153 PAGE 154 Continued 154 PAGE 155 Continued 155 PAGE 156 Toptemplatesforttdoubletaggedeventsforsampleswithdierenttopmasses:from150GeVto200GeV.A)CaseofJES=3.B)CaseofJES=2.C)CaseofJES=1.D)CaseofJES=0.E)CaseofJES=1.F)CaseofJES=2.G)CaseofJES=3. 156 PAGE 157 Continued 157 PAGE 158 Continued 158 PAGE 159 Continued 159 PAGE 160 Continued 160 PAGE 161 Continued 161 PAGE 162 Continued 162 PAGE 163 AFigureE-1. Dijetmasstemplatesforttsingletaggedeventsforsampleswithdierenttopmasses:from150GeVto200GeV.A)CaseofJES=3.B)CaseofJES=2.C)CaseofJES=1.D)CaseofJES=0.E)CaseofJES=1.F)CaseofJES=2.G)CaseofJES=3. 163 PAGE 164 Continued 164 PAGE 165 Continued 165 PAGE 166 Continued 166 PAGE 167 Continued 167 PAGE 168 Continued 168 PAGE 169 Continued 169 PAGE 170 Dijetmasstemplatesforttdoubletaggedeventsforsampleswithdierenttopmasses:from150GeVto200GeV.A)CaseofJES=3.B)CaseofJES=2.C)CaseofJES=1.D)CaseofJES=0.E)CaseofJES=1.F)CaseofJES=2.G)CaseofJES=3. 170 PAGE 171 Continued 171 PAGE 172 Continued 172 PAGE 173 Continued 173 PAGE 174 Continued 174 PAGE 175 Continued 175 PAGE 176 Continued 176 PAGE 177 [1] F.Abeetal.,Phys.Rev.Lett.74,2626(1995);S.Abachietal.,Phys.Rev.Lett.74,2632(1995). [2] J.H.Kuhn,Lecturesdeliveredat23rdSLACSummerInstitute,hep-ph/9707321(1997). [3] V.M.Abazovetal.(D0Collaboration),Phys.Rev.D67,012004(2003). 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