;‘..\| hwy.“ . r 3.33 a... 3. .: ~ .v 1 1 1.). .50.}. unit... 2:3: ‘ 5v. .1 . IQ'ir‘ .. I}!a..:l-I 1rd. , . e x , ...;..v.z. 35$...» {2... P.” .. ....uflwfin. div :r mafia r E , on. .r. Huh: :2 $8.. . . ~ '35 1:. 2”}. 5:1. 9-... IID. LC 5' . \r}?.. I“. ‘1!!!\nt\ a 3,1»):25 ¢.‘: ; :1. :3. .4 (D A it )V it: \.03... D‘.\I\X§s\9b.l $sw._\-h~l\ (it. 5: r. {311: .v: IP»V 0.. t. is. ‘ 5:1! ‘51:.17 El r uumxwgéwjg _ 1!. 3'“... , E A .1. .JW.) .131: ,8 , , 3% .. a3: . , Ex. Exp .2. .3: . . .. 2...: . .. E , winnin. mi” llllllllllllllllllllIlllllllllllllllllllllllllll 3 1293 014172 This is to certify that the dissertation entitled "Physics of Single-Top Quark Production at Hadron Colliders" presented by Douglas Olaf Carlson has been accepted towards fulfillment of the requirements for Ph.D. degreein PhYSiCS Wm Major p essor Dme April 17, 1995 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State Unlverslty' RNBOXtoromu/ombchockmntromyoutrooord. PLACE II RETU More dd. duo. 1'0 AVOID FINES Mum on or DATE DUE DATE DUE DATE DUE PHYSICS OF SINGLE-TOP QUARK PRODUCTION AT HADRON COLLIDERS By Douglas Olaf Carlson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1995 ABSTRACT PHYSICS OF SINGLE—TOP QUARK PRODUCTION AT HADRON COLLIDERS By Douglas Olaf Carlson We discuss the physics of single-top quark production and decay at hadron col- liders, such as the Tevatron, the Di—TeV and the LHC. Our study includes how to measure the mass and the width of the top quark produced from a single-t or a single-f process. We also show how to probe new physics by studying the couplings of t-b—W and show what can be improved from measuring the production rate of single-top events. We also discuss how to probe CP properties of the top quark by measur- ing the single-top production rate. Finally, we present a Monte Carlo study on the detection of single-top events in hadron collisions. For Dawn. iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to the people who have contributed to me and to this work: To my thasis advisor, C .—P. Yuan, for his calm guidance over the past three years; To my mentor, Chip Brock, for sparking my interest in this field and introducing me to C.—P.; To the members of my Thesis Committee: Chip Brock, Wu—Ki Tung, S. D. Ma- hanti and Horace Smith, for their careful] reading of the manuscript; To my colleagues in the High Energy Theory group: Glenn Ladinsky and Pankaj Agrawal for many illuminating discussions; and my graduate student colleagues: Ehab Malkawi, Mike Wiest, Csaba Balasz, Liang—Hung Lai and Xiaoning Wong for our lunch gatherings; To my wife, Dawn, for her love and support; To my parents, Jerry and Justine, for everything. iv Contents LIST OF TABLES viii LIST OF FIGURES ix 1 Introduction to the Standard Model 1 1.1 The Matter Spectrum of Spin";- Fermions ............... 2 1.2 Force Mediators as Spin-1 Gauge Bosons ............... 4 1.3 Gauge Invariant Interactions of Fermion and Gauge Boson Fields . . 5 1.4 Gauge Field Self Interactions ...................... 7 1.5 The Higgs Mechanism .......................... 8 1.6 Review .................................. 11 2 Introduction to the Top Quark 12 3 The Single-Top Production Mechanism 16 4 Measuring the Tap Quark Mass 29 5 Measuring the Tap Quark Width 35 6 Top Quark Couplings to the W Gauge Boson 6.1 From the Decay of Top Quarks ..................... 6.2 From the Production of Top Quarks ................... 7 Probing CP Properties in Top Quarks 8 A Monte Carlo Study 8.1 Tevatron with \/S = 2TeV ....................... 8.2 Tevatron with J3” = 4 TeV ....................... 8.3 LHC with «'5‘ = 14 TeV ........................ 9 Discussions and Conclusions A Helicity Amplitude Method A.1 Helicity Amplitudes for ub -—> dt .................... A.2 Helicity Amplitudes for ub -—) dt in the CMS. ............. A.3 Helicity Amplitudes for u g —+ d t(—) bW+(—+ 8+ 14))5 ........ A.4 Helicity Amplitudes for u b —) d t(—) b W+(-> 6+ 14)) ......... A.5 Helicity Amplitudes for u (I —> W" —-> 5t(—) bW+(—) 8+ 113)) ..... A.6 Helicity Amplitudes for u (I —> 5bW+(—) 8“ V5) ............ B Event Rate of the (2 -> 3) Process ug —> th C Helicity Amplitudes of t -+ W+b and f —-> W‘E D The Total Rate for W—gluon Fusion vi 38 41 47 51 56 57 73 75 91 93 96 100 102 103 104 105 107 112 116 E The Eikonal Approximation for a(AB(ud) -> 5bW+ + jet) 118 F The Computer Program ONETOP 120 LIST OF REFERENCES 122 vii List of Tables 1.1 1.2 1.3 1.4 3.1 6.1 Lepton and Quark Masses ........................ Boson Masses ............................... Quantum numbers of the fermion spectrum ............... Quantum numbers of the Higgs doublet ................. Rates of the above processes for m, = 180(140) GeV. (Branching ratios are not included here.) For J? = 2TeV and 4TeV we include rates for a pp machine. At x/S = 14 TeV the rates are for a pp machine. For the single-top rates we only include single-t production ....... Results on the accuracy of measuring flL’R for various luminosities. (Only statistical errors are included at the 95% confidence level.) viii 0000th 21 46 List of Figures 2.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 6.1 6.2 6.3 6.4 Diagrams contributing to the QCD production of mi, 99 —> tt- . . . . 14 Diagrams for various single-top quark processes ............. 17 Rate in [pb] for qq, gg —-> tt, q’g(W+g) —) qtl), q’q -> W“ -> t5 and gb -—> W't at various energies of pp colliders ............... 18 Rate in [pb] for qq, 99 -—> tt, q'g(W+g) —) nt, (it? —> W" -—> t5 and gb -> W’t at various energies of pp colliders ............... 19 Feynman diagrams illustrating the subtraction procedure for calcu— lating the tot_al rate for W—gluon fusion: q’b -> qt $ q’ g(W+g) -) qtbe(g—>bb®q’b—)qt). ....................... 22 Rate in [pb] for single-t production: (b —> qt 2 —) ), q’g —1 qth (2 -+ 3) and the splitting piece 9 —> bb ® q’b -+ qt i which bb are collinear. The rates are for 13p colliders. ................ 23 Rate in [pb] for single-t production: {b -) qt (2 -) 2), (1’9 ‘2 qt; (2 —) 3) and the splitting piece 9 —} bb ® q’b -) qt in which bb are ”to collinear. The rates are for pp colliders. ................ 24 Rate of W—gluon fusion process versus scale Q for m, = 180 GeV and {S = 2 TeV. ............................... 25 The lepton+jet decay mode of tt production. ............. 30 The di-lepton decay mode of tt production. .............. 30 Distributions of mu (solid) and my (dash) in tt events for a 180 GeV top quark. ................................ 33 For a left-handed t-b—W vertex. .................... 42 For a right-handed t-b—W vertex. ................... 42 mu distribution for SM top quark (solid) and for pure right-handed t—b—W coupling of th(dash). .......................... 44 cos 0; distribution for SM top quark (solid) and for pure right-hand t-b—W coupling of th(dash). .......................... 45 ix 6.5 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 Constraint on lnLCCI and NRCC given 20% and 50% error in measure- ment of Standard Model rate for W-gluon fusion. Curves are identical for m, = 140 GeV and m, = 180 GeV. ................. Diagrams for ad, c? -> bbW+(-) (+11). ................. The rapidity distribution of the spectator quark q, after cuts in Equa- tion (8. 3), fo_r the signal q’b —> qt(—> bW+(—> (+11)), and of the spec- tator quark b for the major background q’q —-) bbW+(-> (+11) (dots), for m,— = 180 GeV (solid) and 140 GeV (dash), at the .Tevatron. (The vertical scale is arbitrary, but the relative size among these curves are absolute.) ................................. PT distribution of the b quark, after requiring P" > 15 GeV along with all the other cuts in (8. 3), for the signal q’b -) th(—> bW+(—> (+11)), and the major background q’q -) bbW+(—) (+11), at the Tevatron. The cos 9a; distribution prior to cut Equation (8.4) for the signal q’ b —+ qt(—) bW+(—-) (+11)) and the major background q’q —> bbW+(—> (+11), at the Tevatron ............................... The 171, distribution after the cuts Equation (8.3) and Equation (8.4) for m, = 180 GeV (solid) and 140 GeV (dash) at the Tevatron including both the signal and background events with W:h -> ei or 11*. . Same as Figure 8.5, but with detector resolution effects as described in Equation 8.9. ............................. mu distributions without (top) or with (bottom) smearing, after all the above analysis for the Tevatron. .................. The rapidity distribution, after Equation (8. 3), of the spectator jet (1'. e. ,b—jet if b—jet identified) 1n the W“ event for m1=180 GeV (solid) and 140 GeV (dash) against Figure 8. 2 (dots), at the Tevatron. The production rate for a left-handed (long dash) or a right-handed (short dash) top quark from the W“ process. The upper solid line is the total rate for the W—gluon fusion process, the lower solid line for the W“ process. ............................. The distribution of cos 019 in W“ event for m,=180 GeV (solid) and 140 GeV (dash) against Figure 8.4 (dots), at the Tevatron. ..... AR“; distributions in W + bb + jet (solid) and W + bb (dash) events after applying the cuts listed in (8.11). ................. Mb; distributions in W + bb + jet (solid) and W + bb (dash) events after applying the cuts listed in (8.11). ................. The rapidity distribution of the spectator quark q, after cuts in Equa- tion (8. 12), for the signal q’b -) qt(—> bW+(-) (+11)), and of the spec- tator quark b for the major background q’q —+ bbW+(—> 3" V) (dots), for m,- — 180 GeV (solid) and 140 GeV (dash), at the Di-TeV ...... PT distribution of the b quark, after requiring PT > 15 GeV along with all the other cuts in (8.12), for the signal q’b -> qt(—> bW+(—> (+11)), and the major background q’q' -+ bbW+(—) (+11), at the Di-TeV . X 49 57 59 61 62 64 67 68 70 77 78 79 80 81 82 8.15 cos 9a; distribution for the signal q’b -) qt(—> bW+(—> (+11)) and back- ground q’q —1 bbW+(—+ (”'11) at the Di-TeV. .............. 8.16 The m, distribution after the cuts Equation (8.12) and Equation (8.4) for mt = 180 GeV (solid) and 140 GeV (dash) at the Di-TeV, including both the signal and background events with W* -+ e35 or 11*. 8.17 Same as Figure 8.16 but with detector resolution effects as described in Equation (8.9). ............................ 8.18 P7- distribution of the b quark, after cuts in (8.13), for the signal q’ b -—) qt(-+ bW+(-> €+u)), and the major background q’q —1 bbW+(—> #11), at the LHC. ................................ 8.19 The rapidity distribution of the spectator quark q, after cuts in Equa- tion (8.13), f9r the signal q’b —> qt(—> bW+(—) [ft/D, and of the spec- tator quark b for the major background q’q -> bbW+(—) 3+1!) (dots), for m, = 180 GeV (solid) and 140 GeV (dash), at the LHC. ...... 8.20 cos 9a; distribution for the signal q’ b —> qt(—> bW+(-) (+11)) and back- ground q’q —> bbW+(—> (+12) at the LHC. ............... 8.21 The 111, distribution after the cuts Equation (8.13) and Equation (8.4) for m, = 180 GeV (solid) and 140 GeV (dash) at the LHC including both the signal and background events with Wi —) ei or 11*. 8.22 Same as Figure 8.21, but with detector resolution effects as described in Equation (8.9). ............................ A.1 Diagram for the (2 —-> 2) process ub —) dt. ............... A.2 Diagrams for ug —> dt(—) bW+(-) 6+ 14))5. ............. A.3 Diagram for ub —> dt(—) bW+(—+ 6+ 14)) ................. A.4 Diagram for uaI-+ bt(—) bW+(—> 6+ 111)) ................. A.5 Diagrams for 113—) bbW+(—+ 13“ V1). .................. B.1 Diagrams for 119 ——> dtb. ........................ D.1 n—body scattering ............................. 83 84 85 86 87 88 89 104 104 105 117 Chapter 1 Introduction to the Standard Model The Standard Model (SM) of elementary particle physics [1, 2, 3, 4] is a Yang—Mills gauge field theory with symmetry SU(3)C x SU(2)L x U(1)Y. (1.1) It has been very successful in explaining and predicting experimental data. The SU(3)C sector governs the strong force of the SM and is known as quantum chromo- dynamics (QCD). The SU(2) L x U(1)Y sector unifies the electromagnetic and weak forces, collectively known as the electroweak force. The unification of the electromag- netic and weak forces is accomplished in the SM via the mixing of the neutral SU(2) L gauge boson and the hypercharge gauge boson of U(1)Y. Masses are introduced in a gauge invariant way through spontaneous symmetry breaking which gives rise to the as yet undiscovered Higgs boson (H). This process of spontaneous symmetry breaking is known as the Higgs mechanism. The standard model does not incorporate the gravitational force. So far, no definitive quantum field theory of gravity exists. Gravity affects all massive particles, however, gravitational interactions with elementary particles are too weak and can be ignored. Table 1.1: Lepton and Quark Masses Particle Symbol Mass (GeV) Electron neutrino Ve 0 Electron e 0.00051 First Up quark 11 0.002 to 0.008 Generation Down quark d 0.005 to 0.015 Muon neutrino u“ 0 Muon p 0.106 Second Charm quark c 1.3 to 1.7 Generation Strange quark s 0.1 to 0.3 Tau neutrino 11,. 0 Tan T 1.78 Third Top quark t 174 Generation Bottom quark b 4.7 to 5.3 In the standard model there are three generations of leptons and quarks as listed in Table 1.1. Associated with each force is one or several gauge bosons as listed in Table 1.2. For reference we list the masses of each particle as found in Reference [5]. So far, the only particle for which a discovery is lacking is the scalar Higgs boson. In what follows, we briefly describe the particle spectrum, particle interactions and the 18 independent parameters which constitute the standard model. 1.1 The Matter Spectrum of Spin--;- Fermions The matter spectrum consists of twelve fermions which are organized in Table 1.1 according to the symmetry structure of Equation (1.1). Each successive generation is a more massive copy of the previous generation, so only the quantum numbers of the first generation are shown in Table 1.3. Under the SU(2)L sector of the SM, Table 1.2: Boson Masses Particle Symbol Mass (GeV) Photon 7 0 Electromagnetic Force W Boson Wi 80.22 Charged Weak Force Z Boson Z0 91.187 Neutral Weak Force Gluon G 0 Strong Force Higgs H 60 < 1113 < 800 Spontaneous Symmetry Breaking Table 1.3: Quantum numbers of the fermion spectrum Chirality Q Ta, Y C VeL 0 1/2 -1 0 8L -1 -1/2 -I 0 11;, 2/3 1/2 1/3 r,g,b (it, -1/3 -1/2 1/3 r,g,b BR -1 0 -2 0 “R 2/3 0 4/3 r,g,b d}; -1/3 0 -2/3 r,g,b left-handed fermions transform as weak isospin (Tw) doublets, e. = (”)L at = (3); (1.2) whereas right-handed fermions transform as singlets, ' (6)12, (203. (dln. (13) Since neutrinos are massless Dirac fermions in the SM, there are no right-handed neutrinos. Once the third component of Weak isospin T3,, is assigned, the values of hypercharge Y can be determined to cancel the chiral anomalies [6]. With these quantum numbers in place, the charge quantization is determined by Q = T3, +32:. (1.4) Only quarks transform under the SU(3)C sector of the SM. Each quark flavor, (1'.e., q = u,d, s,c, b, t), carries red(r), green(g) or blue(b) color charge and therefore 11, = (3;). (1.5) 45 1.2 Force Mediators as Spin-1 Gauge Bosons transforms as a triplet: Associated with the group structure of the SM are twelve generators and each gen- erator is associated with a gauge boson. Therefore, U(1)Y has one generator, the neutral hypercharge gauge boson B”. SU(2)L has three (2 x 2 — 1 = 3) generators, two of which are charged SU(2)L gauge bosons W3: and one neutral SU(2)L gauge boson W3. Finally, SU(3)C has eight (3 x 3 — 1 = 8) generators and therefore, eight gluons, G;,a = 1,2, . . . ,8. As stated earlier, electroweak unification is accomplished by mixing the B” and W3 gauge bosons. Formally this is accomplished via the following rotation, (23) = (cost —sin9w) (W3), (16) A“ sm 9w cos 9w 3,, where our, called the weak mixing angle, is chosen such that A” only couples with charged particles. A” is then identified as the photon field in quantum electrodynam- its (QED) and additionally, a neutral weak force, Z2 , is obtained. At this point in the theory, all fermions and gauge bosons are massless to preserve gauge invariance. In the next two sections we describe how the force mediators inter- act with fermions and amongst themselves. The Higgs mechanism is also introduced to incorporate mass in a gauge invariant way. 1.3 Gauge Invariant Interactions of Fermion and Gauge Boson Fields We begin with the lagrangian for a massless free fermion field \II, cm =T1Iiaxp. (1.7) Equation (1.7) is the kinetic term for fermion fields.1 To make the lagrangian gauge invariant, we introduce the gauge covariant derivative ,1 . , ,\a a 2 W; — zg3—G”, (1.8) . Y . 8,, —> D” = 8,, — zgl—Bp -— 292 2 2 where j = 1,2,3 and a = 1,2,...,8. The T—l’s are the Pauli matrices defined in Appendix A and Tj = 35’- is the Weak isospin. The (3 x 3) matrices (generators) A“,a = 1, 2, . . . , 8, are the SU(3)C matrices. As a result, we obtain gauge interaction terms in the lagrangian. 1 = 0 “ where 6 is the Lorentz invariant s ace-time derivative and “ are the Dirac matrices. H7 1: P ’7 The Bu term acts on all fields with different Y’s, including leptons and quarks in Table 1.3 (same for the other two generations) and the Higgs doublet field (I) discussed in Section 1.5 below. The W; term acts only on the SU(2)L doublets with non-zero T62, and the field . In the process of obtaining electroweak unification the coupling constants, 91 and 92 become related through the weak mixing angle 0w via 93 = tanfiw, (1.9) 91 where 91 sin 0w = e. The value of e is related to the fine structure constant a by a = e2/47r. For electroweak interactions, the colors of the quarks have to be the same since SU(2)L x U(1)Y does not act on color space. Also, the A” and Z3 fields do not induce quark or lepton flavor changing at the Born level, (i.e. no u H c H t, d H s H b, V, H 11,, H V, or e H 11 H 1' transitions occur). However, charged current transitions via Wf do occur and for quarks, are not restricted by generation (2'.e. u,c,t H d,s,b transitions are allowed). Lepton flavor changes are restricted to generation due to massless neutrinos. Finally, the SM does not allow for direct lepton-quark transitions. The G2 term in Equation (1.8) only acts on colored fermions, i.e. quarks. The coupling strength 93 = 95 is universal for all colored quarks. Analogous to the fine structure constant a in electromagnetic theory is the strong coupling constant as = g§/47r in QCD. Here again, no flavor changes occur. 7 1.4 Gauge Field Self Interactions To complete the lagrangian for massless fermions and gauge bosons and their inter- actions, we must introduce the kinetic term for gauge bosons, 1 u 1 i 1' u 1 a a u LGK = 7&3" — Zwflmz ,. — Zg,,,,g # , (1.10) where 8,... = 6,3,, - 6.3,, (1.11) w;, -.= aw; -— aw; + gge‘ikwgwf, (1.12) g;,, = as: — as; + ggfmGzGfi. (1.13) The Lie group structure constants 6'“ and fab“ are defined through the following anti-commutation relations, 1'i 'rj .- T" _ _ =. 3.7k_ . . = [2,2] 26 2, 2,],k 1,2,3 (1.14) and "a Ab -1f "c ate-12 s (115) 212— abc21 11—11°'°1 ° for SU(2) and SU(3), respectively. We note that the pure Yang-Mills terms, —%W;,,W““’ and —§g;,gw contain factors that are trilinear and quadrilinear in W; and G2. These Yang-Mills terms expand out partially as 2 1" i ' u g 1” im ' mu _ g, e 1*(0PWV)W”‘W" — —4'~’- e 1" e’ W5WfW’“W (1.16) and 2 . . . __ gS fabc (apr’)Gprcu __ 9:5; fabc fade Gzaf’G-dpacu (117) Table 1.4: Quantum numbers of the Higgs doublet Q T3,, Y C vHio Lj-g—t— 0 1/2 -10 111- -1 -1/2 -1 o respectively, and correspond to self-couplings of non-abelian gauge fields. This is fundamentally different than in the abelian case where, in QED, photons do not directly couple with photons. It also accounts for the short distance interaction of the strong force, despite the fact that the gluon is massless. As can be seen in Chapter 2, the triple gluon interaction contributes to tt production via gg —) tt at hadron colliders. 1.5 The Higgs Mechanism The goal of the Higgs mechanism is to introduce mass to the particles in the SM in a gauge invariant way. We begin by defining a complex doublet scalar field (I) composed of four real scalar fields H, (to, ¢1 and (252 where - 0 v+H+i° <1>=i(”+H+'¢)= +2 . (1.18) fl i451 _ ¢2 to“ The quantum numbers of the Higgs field are as shown in Table 1.4. The fields 49° and 49* = ((151 1F i052” 1/5 are the unphysical would-be Goldstone bosons associated with spontaneous symmetry breaking. They give rise to the masses of the gauge bosons W3: and Z0. One physical field thus remains, which is the Higgs field H. The constant 12 2: 246 GeV is the scale characterizing the symmetry breaking scale and is called the vacuum expectation value of , where (<1)0 a (0)140) = (705) . (1.19) The lagrangian for the Higgs sector is c. = (D.>*(D“) — $0M)? -— Eon). (1.20) with ,1 . Y . Dp = [8,, - zgl—Bp — 292 2 2 as in Equation (1.8) without the gluon interaction. W3] «1 (1.21) If p < 0 and A > 0 then the minimum of the potential energy occurs at l—2l‘ The Goldstone bosons 45* and (25° are “eaten” by the vector bosons Wi and Z 0, respectively, where MW cos 0W ' MW = $9211 and M2 = (1.23) Therefore, Wi and Z 0 have three polarization states; two transverse and one longi- tudinal. The massless photon and gluon have only the two transverse polarization states. The Higgs mass is given by my = vx/X. (1.24) To introduce fermion mass in a gauge invariant way one introduces the Higgs mechanism through Yukawa coupling interactions. For the first generation 2m“ _ - 2m _ - . .. ‘/-v (UL dL)¢UR+ C d (uLdL)(—172 )dR LYulcawa = 72m. ‘0 + (171, EL) (—i1'2") e3 + hermitian conjugate (1.25) 10 where mu, m.) and me are up quark, down quark and electron masses, respectively (neutrinos are massless) and 0 —1 “5+. 0 —i72" = (1 0 ) <1; = 2%1319— . (1.26) As mentioned in Section 1.3, neutral currents coupled to 7, Z 0 and G do not change flavor, although G changes color, but charged currents coupled to W‘t do change flavor. For leptons, the flavor change does not exceed generational bounds, due to the massless neutrino. However, there is a chance that an up quark, for instance, can change to a down quark, a strange quark or even a bottom quark. This is called quark mixing, which is due to the weak eigenstates (indicated by the subscript “Weak”) of quarks being different than the mass eigenstates (indicated by the subscript “Mass”). By convention, the three charge 2 / 3 quarks u, c and t are unmixed: (2‘) =(l‘) . (1.27) t Weak t Mass All the mixing is therefore expressed in terms of a (3 x 3) unitary matrix V operating on the charge (-1/3) quarks d, s and b: d Vud Vus Vub d s 5 Va) Vc, V2,), 3 b Weak V“ W: Kb b Mass 1 - “/2 A AA3(p —in) d z -). 1 — /\2/2 A12 s . (1.28) AA3(1 — p — in) —AA2 1 b Mass The matrix V is known as the Cabibbo-Kobayashi—Maskawa matrix (CKM), which consists of three mixing angles and one phase. The second parameterization is due 11 to Wolfenstein [7], where A z 0.22, A z 1, n z 0.5 and —0.4 _<_ p _<_ 0.2. CP violation, which is the violation of combined charge conjugation C and the parity transformation P, is characterized by the CP violating phase in p — 2'17. 1.6 Review In this introduction to the SM, we have outlined how the SM is constructed based on gauge invariance and the Higgs mechanism. Although the SM has been successful in describing experimental data, there are 18 independent parameters which must be determined experimentally. These parameters are: 0 nine fermion masses: me,mp,m,,mu,md,mc,m,,m¢,mb (neutrinos are mass- less), 0 four CKM parameters: A,A,n, p, 0 four electroweak parameters: e,0w, MW, m”, e the strong coupling constant: as. Chapter 2 Introduction to the Top Quark If the SU(2) structure of the Standard Model (SM) holds, the top quark (t) has to exist as the weak isospin partner of the bottom quark (b) [8]. If the coupling of t—t-Z is as predicted in the SM, then from LEP and SLAC experiments the mass of the top quark (m,) has to be larger than half of the Z-boson mass (~ 45 GeV) independent of how the top quark decays. If the coupling of t-b-W is as predicted in the SM, then from the measurement of the total width of the W—boson, by measuring the ratio of the event rates of pp —-> W(—-) 81/) to pp —) Z (—> 13"?) 1, the mass of the top quark (mt) has to be larger than 62 GeV independent of how the top quark decays [9]. From examining the radiative corrections to low energy observables, such as the p parameter2 which is proportional to m? at the one loop level [10], m, has to be less than about 200 GeV. Based upon analysis of a broad range of Electroweak data, the mass of the SM top quark is expected to be in the vicinity of 150 to 200 GeV [11, 12, 13, 14]. Independently, from the direct search at the Tevatron, the top quark has been discovered and found to have mass of m, = 176 :l: 8 (stat.) :l: 10(sys.) GeV from CDF data [15], and m, = 19912;? (stat.) :h 22(sys.) GeV from DC data [16]. lA proton is denoted by p and an anti—proton by f) 2 2p = 173%, where Mw (or M2) is the mass of w=t (or Z) boson. 9w is the weak mixing angle. p has been measured to the accuracy of about 0.1%. 12 13 For a. heavy top quark, m, is of the order of the electroweak symmetry breaking scale 12 = (x/'2-G)t~)-l/2 = 246 GeV. In fact, recall the Yukawa coupling interaction, this time for third generation quarks, Jim: fimb LYukawa = v (5], 81,) T in + ‘0 (if, 8],) (—i7’2Q‘) 03. (2.1) We see that “in” ~ 1, (2.2) for m, = 175 GeV. Because the generation of fermion mass can be closely related to the electroweak symmetry breaking [17, 18], effects from new physics should be more apparent in the top quark sector than any other light sector of the electroweak theory. Thus, the top quark system may be used to probe the symmetry breaking sector. A few examples were discussed in Ref. [19] to illustrate that different models of electroweak symmetry breaking mechanism will induce different interactions among the top quark and the W— and Z—bosons. Therefore, hopefully through studying the top quark system one may eventually learn about the symmetry breaking sector of the electroweak theory. The most important consequence of a heavy top quark is that to a good approxi- mation it decays as a free quark because its lifetime is short and it does not have time to bind with light quarks before it decays [20]. Furthermore, because the heavy top quark has the weak two—body decay t —+ bW+, it will analyze its own polarization. Thus we can use the polarization properties of the top quark as additional observables to test the SM and to probe new physics. In the SM, the heavy top quark produced from the usual QCD process, at the Born level, is unpolarized. However, top quarks will have longitudinal polarization if weak effects are present in their production [21]. For instance, the top quark produced from the W—gluon fusion process is left-hand 14 gm——t g ——>——t g t 9 Alt Alt gm—e—f 9 L—«——{ 9 Z q t g a E Figure 2.1: Diagrams contributing to the QCD production of qq, 99 —> tt polarized. With a large number of top quark events, it will be possible to test the polarization effects of top quarks. How to detect a SM top quark pair produced via the QCD processes qq, gg —) tt-, as shown in Figure 2.1, has been extensively studied in the literature [22]. In this paper we will concentrate on how to detect and study the top quark produced from the single-top quark processes q’g(W+g) —> qtb, q’b -—> qt, gb —> W‘t, and q’q —> W“ —-) tb. For the single-top production we will only consider the decay mode of t -+ bW+ —> b€+1x, with 6+ = e’r or 11+. (The branching ratio for this decay mode is Br = g.) The rest of this paper is organized as follows. In Chapter 3 we discuss the pro- duction rates of top quarks at hadron colliders. Following that, we will discuss in Chapters 4 and 5, respectively, how to measure the mass and the width of the top quark. In Chapter 6 we discuss what we have learned about the couplings of the top quark to the weak gauge bosons and show what can be improved from measuring the production rate of single-top quark events. We will also discuss in Chapter 7 the potential of the Tevatron as a pp collider to probe CP properties of the top quark by simply measuring the single—top quark production rate. Finally, in Chapter 8 we present a Monte Carlo study on the detection of a single-top quark event in hadron 15 collisions. Various unique features of the kinematics of the single-top quark signal will be discussed. Chapter 9 contains our conclusions. Based upon the results of the FNAL CDF and DO , the mass of the SM top quark m, is not likely to be lighter than, say, 140 GeV. Throughout this paper we will use 111, = 180 GeV (or 140 GeV) as an example of a heavy (or a light) top quark for our studies. Chapter 3 The Single-Top Production Mechanism In this chapter we consider the production rate of a single-top quark at the Tevatron, the Di-TeV (the upgraded Tevatron) and the LHC (Large Hadron Collider) colliders. In referring to single-top production, unless stated otherwise, we will concentrate only on the positive charge mode (i.e., only including single—t, but not single-t). The colliders we consider are the Tevatron (a pp collider) with the Main Injector at \/S = 2 TeV, the Di-TeV (a pp collider) at 4 TeV and the LHC (a pp collider) at \/— = 14 TeV with an integrated luminosity of 1fb'1, 10fb'1, and 100 fb—l, respectively.1 A single-top quark signal can be produced from either the W—gluon fusion process q’g(W+g) —+ qtb (or q’b -> qt) [23, 24], the Drell-Yan type process q'q —+ W" —-> tb (also known as “W'” production) [25], or Wt production via gb -) W’t [26]. The corresponding Feynman diagrams for these processes are shown in Figure 3.1. In Figures 3.2 and 3.3 we show the total cross sections of these processes for the Tevatron, the Di-TeV and the LHC energies referred to above. For reference we include plots of the cross sections of top quarks as a function of m, in both 1In reality, the integrated luminosity can be higher than the ones used here. For instance, with a couple of years of running a 2 TeV Tevatron can accumulate, say, 10 lb“1 luminosity. Similarly, it is not out of question to have a 4TeV Di-TeV to deliver an integrated luminosity of about 100 lb"1 . 16 17 W ‘ (C) ((1) Figure 3.1: Diagrams for various single-top quark processes. 18 ‘ «Is = 2 TeV, pp ‘02 s” seam->6 : - - - - Wg->t5.Wb->t °°°°°° (IQ->6 .............. o. w)” 103_ Q 77" V Cross-Section 0(pb) 3 10° r ........................ 10" r ........... l L 4L M 1 I 1 1 l l 100 150 200 250 300 TopMassNGeV) 104 5 ils =14 TeV. pp = swam->6 - - - - Wg->Ib.Wb->t 1- >15 3 103 tr 0 P g 102 FF- _____________ 0 ‘0‘ E ................................................... t ........ 10° g .................. >1 M L 1 4 1M 1 1 ‘1 100 150 200 250 $0 Top Mass m,(GeV) Figure 3.2: Rate in [pb] for qq, gg —-) tt, q’g(W+g) -) qtb, q’q -) W" —) tb and gb —> W‘t at various energies of pp colliders. 19 102 wls = 2 TeV, pp Men->8 ' - - - - Wg->16.Wb->1 10‘ E- -------- oil->11: E . ~ ~ .............. w,” Cross-Section o(pb) 100 150 200 250 300 Top Mass m.(GaV) 103 _ 5 wls = 4 TeV. pp was»! . - - - - W9:§1.Wb—>t )— ........ m- g 102 ............... 9w”, 0 g 101 r 8 E .. _ - ~ - g --------- 5 10" .r ““131? ................... 1°" :- 1 . . . 1 . . . ....... 1 100 150 200 250 300 Top Mass New) ‘04 F Vs =14 TeV. pp E ee->fi.qq->tf - - ‘- - Wg->lb,Wb->t >6 Cross-Section 0(pb) 1 n 1 n A; l 1 n i 1 l 1 4 1 1 l 1 1 1 n l 100 150 200 250 300 Top Mass NGeV) Figure 3.3: Rate in [pb] for qq, gg -) tt, qg(W+g) —> qtb, q’q -) W' -> tb and gb —-) W‘t at various energies of pp colliders. 20 the pp collisions, shown in Figure 3.2, and pp collisions, shown in Figure 3.3. The parton distribution function (PDF) used in our calculation is the leading order set CT EQ2L [27]. We note that taking the AQCD value given in CTEQ2L PDF we obtain a,(Mz) = 0.127 which is about 15% larger than the value of 0.110 in CTEQ2M PDF [27]. We found that if we rescale the tt production rates obtained from CTEQ2L PDF with born level amplitudes by the ratio of af(Q,AQCD) from CT EQ2M and that from CTEQ2L, which yields 0.7 for Q = M 2, then our total rates are in good agreement with those obtained using N LO PDF and NLO amplitudes [28], see, for example, Reference [29]. Hereafter we shall use the scaled results for our rates. The constituent cross sections are all calculated at tree level for simplicity to study the kinematics of the top quark and its decay products. To include the production rates for both single-t and single-t events at pp colliders, a factor of 2 should be multiplied to the single-t rates shown in Figures 3.2 and 3.3 because the parton luminosity for single-t production is the same as that for single-t. Similarly, at pp colliders the rates should be multiplied by about 1.5 for the center- of-mass energy (x/S) of the collider up to approximately 4 TeV, but almost a factor of two at higher energies (say, x/S Z 8 TeV up to about 14 TeV) because the relevant parton luminosities for producing a singlet and a single-t event in pp collisions are difl'erent. As shown in Figures 3.2 and 3.3 the total rate for singletop production is about the same at pp and pp colliders for VS 2 8 TeV because the relevant valence and sea quark parton distributions are about equal for 100 GeV < m, < 300 GeV. For smaller VS, up to about 4 TeV, a pp collider is preferred over a pp collider for heavy top quark production because of its larger parton luminosities. Similarly, for tt pair productions at small \/S, the quark initiated process qq H tt is more important than the gluon fusion process 99 H tt. At x/S from 8 to 14 TeV the tt- rate is about the same in pp and pp collisions because the 99 H tt subprocess becomes dominant. 21 Table 3.1: Rates of the above processes for m, = 180(140) GeV. (Branching ratios are not included here.) For E = 2TeV and 4TeV we include rates for a pp machine. At \/S = 14 TeV the rates are for a pp machine. For the single-top rates we only include single-t production. Cross Section (pb) JflTeV) qq, gg H tt’ q'g -+ qtb (or q’b H qt) q’q -> W‘ —> tb gb H W“t 2 4.5(16) 1(2) 0.3(0.8) 0.1(0.3) 4 26(88) 7(11) 0.8(2.1) 1.3(2.9) 14 430(1300) 100(140) 4.6(11) 3.6(8.8) For later reference in this paper, we show the rates of the above processes in Table 3.1 for m, = 180(140) GeV. (Branching ratios are not included here.) For J? = 2 and 4 TeV we include only the rates for a pp machine, whereas at x/S = 14 TeV the rates are for a pp machine. Again, for the single-top rates we only include t production. Both in Figures 3.2 and 3.3 and Table 3.1, we have given the cross section of single-top quarks produced from either the q’g(W+g) H qtb or q’b H qt processes. From now on, we will refer to this production rate as the rate of the W—gluon fusion process. The single-top quark produced from the W-gluon fusion process involves a very important and not yet well-developed technique of handling the kinematics of a heavy b parton inside a hadron. Thus the kinematics of the top quark produced from this process can not be accurately calculated yet. However, the total event rate for single-top quark production via this process can be estimated using the method proposed in Reference [30]. The total rate for the W—gluon fusion process involves the 0(02) (2 H 2) process q’b H qt plus the 0(a2cz,) (2 H 3) process q’g(W+g) H qtb (where the gluon splits to bb) minus the splitting piece 9 H bb (S) q’ b H qt in which bb are nearly collinear. These processes are shown diagrammatically in Figure 3.4. The 22 q: q 9' q W+ W+ 69 b t b t _ 9 b It i 9 3w b b 9M b Figure 3.4: Feynman diagrams illustrating the subtraction procedure for calculating the total rate for W-gluon fusion: q’b H qt €B q’g(W+g) H qtb e (g H bb ® q’b H qt). helicity amplitudes and the cross sections for these processes are given in Appendices A and B respectively. The splitting piece is subtracted to avoid double counting the regime in which the b propagator in the (2 H 3) process is close to on-shell.2 The procedure is to resum the large logarithm a, ln(m?/m§) in the W—gluon fusion process to all orders in a, and include part of the higher order 0(020,) corrections to its production rate. (m5 is the mass of the bottom quark.) We note that to obtain the complete 0(a2a,) corrections beyond just the leading log contributions one should also include virtual corrections to the (2 H 2) process, but we shall ignore these non-leading contributions in this work. Using the prescription described as above we found that the total rate of the W—gluon fusion process is about 25% less as compared to the (2 H 2) event rate for m, = 180 (140) GeV regardless of the energy or the type (i.e., pp or pp) of the machine. In Figures 3.5 and 3.6 we show the total rate of W—gluon fusion versus 111, with scale Q = m, as well as a breakdown of the contributing processes at the Tevatron, the Di-TeV and the LHC. 2The total rate of the (2 H 3) process is extensively discussed in the Appendix C. 23 10a : : proton-antiproton : —— Total _ ‘45 — 14 TeV _____ 2,,2 102 g E 3. C c .5 10‘ .~ g E o . 10° 5- ; 10'1 5' 100 Top Mass mt(GeV) Figure 3.5: Rate in [pb] for single-t production: q b H qt (2 H 2), q’ g H qtb (2 H 3) and the splitting piece 9 H bb <8) q’ b H qt in which bb are collinear. The rates are for pp colliders. 24 103 : : proton-proton * -— Total [ \Is—14TeV _____ 2.>2 t 102 L a r- e u- o . 8 g 10‘ g 5 F 10° 3 10“ 100 Top Mass m((GeV) Figure 3.6: Rate in [pb] for single-t production: q’b -_—> qt (2 H 2), q’g H qtb (2 H 3) and the splitting piece g H bb 81 q’ b H qt in which bb are collinear. The rates are for pp colliders. 101 Cross-Section 0(pb) 10“ Figure 3.7: Rate of W—gluon fusion process versus scale Q for m, \/§ = 2TeV. 25 111111111111111111111111 \ls=2TeV rm: 180 GeV —— Total ----- 2->2 --------- 2->3 ................ _- splitting .................................................................................... o ‘. ‘§ . 5-- ..... --- -- --- -- u--- - ---—----------. 1 l 60 80 100 120 140 160 180 Scale Q(GeV) = 180 GeV and 26 To estimate the uncertainty in the production rate due to the choice of the scale Q in evaluating the strong coupling constant a, and the parton distributions, we show in Figure 3.7 the scale dependence of the W—gluon fusion rate. As shown in the figure, although the individual rate from either (2 H 2), (2 H 3) or the splitting piece is relatively sensitive to the choice of the scale, the total rate as defined by (2 H 2) + (2 H 3) — (splitting piece) only varies by about 30% for MW / 2 < Q < 2m, at the Tevatron. At the Di-TeV and the LHC, it varies by about 30% and 10%, respectively. Based upon the results shown in Figure 3.7, we argue that Q < MW / 2 probably is not a good choice as the relevant scale for the production of the top quark from the W-gluon fusion process because the total rate rapidly increases by about a factor of 2 in the low Q regime. In view of the prescription adopted in calculating the total rate, the only relevant scales are the t0p quark mass m, and the virtuality of the W—line in the scattering amplitudes. Since the typical transverse momentum of the quark (q), which comes from the initial quark (q’) after emitting the W-line, is about half of the W-boson mass, the typical virtuality of the W- line is about Mw/2 2 40 GeV. m5 :2 5GeV is thus not an appropriate scale to be used in calculating the W-gluon fusion rate using our prescription. We note that in the (2 H 2) process the b quark distribution efl'ectively contains sums to order [01, ln(Q/mb)]" from n-fold collinear gluon emission, whereas the subtraction term (namely, the splitting piece) contains only first order in a, 1n(Q/mb). Therefore, as Q H m), the (2 H 2) process picks up only the leading order in a,ln(Q/mb) and so gets largely cancelled in calculating the total rate. Consequently, as shown in Figure 3.7, the total rate is about the same as the (2 H 3) rate for Q H m5. We also note that at Q s: Mw/2, the (2 H 2) and (2 H 3) processes have about the same rate. As Q increases the (2 H 2) rate gradually increases while the (2 H 3) rate decreases such that the total rate is not sensitive to the scale Q. It is easy to see 27 also that the total rates calculated via this prescription will not be sensitive to the choice of PDF although each individual piece can have different results from different PDF’s, based upon the factorization of the QCD theory [30]. Another single-top quark production mechanism is the Drell-Yan type process q’ q H W" H tb. As shown in Figures 3.2 and 3.3, for top quarks with mass on the order of 180 GeV the rate for W‘ production is about one fifth that of W-gluon fusion at 3 = 2 TeV. The W" process becomes much less important for a heavier top quark. This is because at higher invariant masses 5‘ (for producing a heavier top quark) of the tb system, W“ production suffers the usual 1/§ suppression in the constituent cross section. However, in the W—gluon fusion process the constituent cross section does not fall ofi as 1/5 but flattens out asymptotically to 1/M3V. (The analytical results of these amplitudes are given in Appendix A for reference.) For colliders with higher energies, therefore with large range of 3, the W" production mechanism for heavy top quarks becomes much less important. However, the kinematics of the top quarks produced from this process are diflerent from those in the W—gluon fusion events. Moreover, possible new physics may introduce a high mass state (say, particle V) to couple strongly with the tb system such that the production rate from q’q H W" H V H tb can largely deviate from the SM W" rate.3 We will however not discuss it in detail here because its rate is highly model dependent. The W-gluon fusion process becomes more important for a heavier top quark. Why? Effectively, the W—gluon fusion process can be viewed as the scattering of a longitudinal W—boson (WL) with gluon to produce a top quark and a bottom anti- quark (ng H tb) after applying the effective-W approximation [32]. For large 5 this scattering process is equivalent to (49+ g H tb) where 45“ is the corresponding ’This is similar to the speculations made in Reference [31] for having some high mass resonance in t? production. 28 Goldstone boson of the gauge boson W+ due to the Goldstone Equivalence Theorem [33, 34]. Since the coupling of t-b-¢ is proportional to the mass of the top quark, the constituent cross section of the W—gluon fusion process grows like m3/Mv2v when m, increases. This explains why the W—gluon fusion rate only decreases slightly as the mass of the top quark increases even though both the parton luminosity and the available phase space decrease for a heavier top quark. In contrast, the tt pair produc- tion rate from the QCD processes decreases more rapidly as m, increases because the constituent cross section of qq, gg H tt goes as 1/§ and the phase space for producing a ti pair is smaller than that for producing a single-t. Therefore, the W—gluon fusion process becomes more important for the production of a heavy top quark. Before closing this chapter, we note that the Effective-W approximation has been the essential tool used in studying the strongly interacting longitudinal W system to probe the symmetry breaking sector at the supercolliders such as the LHC [35]. By studying single—top production from the W—gluon fusion process at the Teva- tron, one can learn about the validity of the Effective-W approximation prior to the supercolliders. Chapter 4 Measuring the Top Quark Mass By the year 2000, we expect results from the Tevatron (with Nb“) and results from LEP-200, giving an error of about 50 MeV on MW. Due to Veltman’s screening theorem, the low energy data are not sensitive to the mass of the Higgs boson [36]. For a heavy Higgs boson, the low energy data can at most depend on m H logarithmically up to the one loop level. Therefore, within the SM one needs to also know the mass of the top quark to within about 5GeV to start getting useful information on mg with an uncertainty less than a few hundred GeV. This can be done by studying radiative corrections to the low energy data which include LEP, SLC, and neutrino experiments [11, 12, 13, 14]. (Of course, m 3 will be measured to better precision if it is detected from direct production at colliders.) How accurate can the mass of the top quark be measured at hadron colliders? At hadron colliders, m, can be measured in the tt events by several methods [22, 37]. The first method is to use the lepton+jet decay mode of the it pair, as shown in Figure 4.1. This is done by reconstructing the invariant mass of the three jets in the opposite hemisphere from the isolated lepton I (= eor p) in t H bW(H £11), and requiring that two of the three jets reconstruct to a W and the third be tagged as a b—jet. The second method is to use the di-lepton decay mode of the tt pair, 29 30 b 1+ t 1 W V w- J i ‘ , - J b Figure 4.1: The lepton+jet decay mode of ti production. b IT t + W V w- ’2' t i _ .. 1/ b Figure 4.2: The di-lepton decay mode of ti production. 31 as shown in Figure 4.2. Here it is required that both W’s decay leptonically. In addition, one of the b’s must decay semileptonically to measure the mass distribution of the non-isolated lepton 3;, (from b decay) and one of the two isolated leptons (£1 and (2 from Wi decay) which is closer to (1,. The third method is to measure the cross section of the di-lepton decay mode of the tt- pair. At the LHC, there will be about 108 tt— pairs produced in one year of running for m, < 200 GeV. With such a large number of events, the ATLAS and CMS collaborations concluded that m, can be measured with a precision of 5 5 GeV using the first method described above and with about a factor of 2 improvement using the second method [38, 39]. A similar conclusion was also drawn by the CDF and the D0 collaborations for the Tevatron with Main Injector after the upgrade of their detectors [40]. This is remarkable given that the tt cross section at the Tevatron is smaller by about two orders of magnitude as compared with that at the LHC, as shown in Figures 3.2 and 3.3. Next, we would like to discuss how to measure the mass of the top quark in the W—gluon fusion process. Since m, has been measured by the FNAL CDF and D0 groups in the tt events [15, 16], why do we care? To check whether it is a SM top quark, we should verify its production rate predicted by the SM for other production processes such as the single-top quark process. Suppose the coupling of t-b—W is not of the SM nature, then we would find that the single-top quark production rate of the W—gluon fusion process is different from the SM prediction because its production rate is directly proportional to the square of this coupling. (We will discuss more on this point in Chapter 6.) Hence, without knowing the nature of the t-b—W interactions one can not use the production rates of the single—top quark events to measure m,. Alternatively, we propose two methods to measure m, in the single-top quark events. We will refer to them as the fourth and the fifth method. The fourth method is a slight variation of the second method. Instead 32 of measuring the invariant mass of the leptons, we propose to directly measure the invariant mass (mu) of the f and b in t H bW(H [11). We expect that the efficiency of b tagging using the displaced vertex is higher for detecting a heavier top quark, and the b jet energy measurement is better for b having larger transverse momentum from a heavy top quark decay. Thus mu can be used to measure the mass of a SM top quark. The details of our Monte Carlo study are given in Chapter 8 for a single-top quark event. In the tt event there are two b’s, therefore this method may not work as well as in the single-top event which only contains one b. However it is not entirely impossible to use this method because, as shown in Figure 4.3, the sum of the invariant mass distributions of b! and bi for a 180 GeV t0p quark still show a bump near the region that the distribution of mu peaks. (With a larger sample of t? events one might be able to afford using the electric charge of the soft-lepton from b—decay to separate b from b on an event-by-event basis at the cost of the small branching ratio of b H p + X, of about 10%.) We will explain in more detail how to use flag (the fraction of longitudinal W—bosons from top quark decay), derived from the distribution of mu, to measure m, in Chapter 6. The fifth method is to reconstruct the invariant mass of the top quark in the t H bW(H (V) decay mode by measuring the missing transverse momentum and choosing a two-fold solution of the longitudinal momentum of the neutrino from the mass constraint of the W boson. In Chapter 8 we conclude that it is possible to measure m, using either of these last two methods to a precision of 5GeV at the Tevatron (J? = 2TeV) with 1fb'l integrated luminosity. We also find that after applying all the kinematical cuts to suppress the dominant background W + bb, at most 10% of W“ events contribute to single-top production for a 180 (140) GeV top quark. The SM W" production rate is already much smaller than the W—gluon fusion rate for a heavier top quark, therefore the contribution from the W“ is not important 33 6 I I t t I I l t I I t 1 I t l l I I r ‘l I r I C l 6_ —( v- -( :3 . . o 4_ _ o )- .1 \ . e t - 3— — x p d 5 l- «I: \ L _‘ b 2, - 'u _ . 1;— LL“ "2 I )- ~‘_,.( 0 l _L l I l d 0 50 100 150 200 Figure 4.3: Distributions of mu (solid) and my (dash) in tt events for a 180 GeV top quark. 34 in our study although we do include its small effects in our analysis as described in Chapter 8. Chapter 5 Measuring the Top Quark Width As shown in Reference [41] the intrinsic width of the top quark can not be measured at a high energy hadron collider such as the LHC through the usual QCD processes.l For instance, the intrinsic width of a 150 GeV Standard Model top quark is about 1 GeV, and the full width at half maximum of the reconstructed top quark invariant mass (from t H bW(H jets) decay mode) is about 10 GeV after including the detector resolution effects by smearing the final state parton momenta. Here, the ratio of the measured width and the intrinsic width for a 150 GeV top quark is about a factor of 10. For a heavier top quark, this ratio may be slightly improved because the jet energy can be better measured. (The detector resolution AE / E for a QCD jet with energy E is proportional to l/x/E.) A similar conclusion was also given from a hadron level analysis presented in the SDC Technical Design Report which concluded that reconstructing the top quark invariant mass gave a width of 9 GeV for a 150 GeV top quark [42]. Is there a way to measure the top quark width I‘(t H bW+), say, within a factor of 2 or better, at hadron colliders? Yes, it can in principle be measured in single-top events. lIn Reference [41], the effects of QCD radiation in top quark decay (at one loop level) to the measurement of mg in t? events produced in hadron collisions was studied. It was concluded that the peak position of the m. distribution remains about the same as the tree level result, but the shape is difl'erent. It was also found that the ma distribution is not sensitive to QCD radiation in top decay. 35 36 The width I‘(t H bW+) can be measured by counting the production rate of top quarks from the W-b fusion process which is equivalent to the W—gluon fusion process by a proper treatment of the bottom quark and the W boson as partons inside the hadron. The W-boson which interacts with the b—quark to produce the top quark can be treated as an on-shell boson in the leading log approximation [32, 43]. The result is that even under the approximations considered, a factor of 2 uncertainty in the production rate for this process gives a factor of 2 uncertainty in the measurement of I‘(t H bW+). This is already much better than what can be measured from the invariant mass distribution of the jets from the decay of top quarks in the tt events produced via the usual QCD processes. More precisely, as argued in Chapter 3, the production rate of single-top events at the Tevatron can probably be known within about 30%, thus it implies F(t H bW+) can be measured to about the same accuracy.2 Therefore, this is an extremely important measurement because it directly tests the couplings of t-b-W . W—gluon fusion can also tell us about the CKM matrix element IthI. Assuming only three generations of quarks, the constraints from low energy data together with unitarity of the CKM matrix require [th] to be in 0.9988 to 0.9995 at the 90% confidence level [5]. As noted in Reference [5] the low energy data do not preclude there being more than three generations of quarks (assuming the same interactions as described by the SM). Moreover, the entries deduced from unitarity might be altered when the CKM matrix is expanded to accommodate more generations. When there are more than three generations the allowed ranges (at 90% CL) of the matrix element |V,¢,| can be anywhere between 0 and 0.9995 [5]. Since |V¢b| is directly involved zstrictly speaking, from the production rate of single-top events, one measures the sum of all the possible partial decay widths, such as I‘(t H bW+) + I‘(t H sW+) + I‘(t H dW+) + - - -, therefore, this measurement is really measuring the width of I‘(t H X W+) where X can be more than one particle state as long as it originates from the partons inside the proton (or anti-proton). In the SM, I‘(t H bW+) is about equal to the total width of the top quark. 37 in single-top production via W-gluon fusion, any deviation from SM value in thbl will either enhance or suppress the production rate of single-top events. It can therefore be measured by simply counting the single-top event rates. For instance, if the single—top production rate is measured to within 30%, then [141,] is determined to within 15%. In conclusion, after the top quark is found, the branching ratio of t H bW+(H 8+1!) can be measured from the ratio of (23 + jets) and (1! + jets) rates in tt- events. The measured single-top quark event rate is equal to the single-top production rate multiplied by the branching ratio of t H bW+(H 8+1!) for the (1! + jets) mode and the same t-b—W couplings appearing in the decay of t in this process appear also in the production of t. Thus, a model independent measurement of the decay width I‘(t H bW+) can be made by simply counting the production rate of t in the W—gluon fusion process. Should the top quark width be found to be different from the SM expectations, we would then have to look for non-standard decay modes of the top quark. We note that it is important to measure at least one partial width (say, I‘(t H bW+)) precisely in order to discriminate between different models of new physics, if any. In the SM, the partial width I‘(t H bW+) is about the same as the total width of the top quark at the tree level because of the small CKM matrix element |V,,|, thus measuring the single—top quark production rate measures the lifetime of the top quark. Chapter 6 Top Quark Couplings to the W Gauge Boson It is equally important to ask what kind of interactions the t-b-W vertex might involve [44]. For instance, one should examine the form factors of t-b—W which result from higher order corrections due to SM strong and / or electroweak interactions. It is even more interesting to examine these form factors to test the plausibility of having nonuniversal gauge couplings of t-b—W due to some dynamical symmetry breaking scenario-[45, 19]. The QCD [46] and the electroweak [47] corrections to the decay process t H bW+ in the SM have been done in the literature. The most general operators for this coupling are described by the interaction lagrangian -— 1 _— L = i=2— [w, b7”(f1"P_ + may - $6qu ba’“’( {10. + ffP+)t] — ‘ . 1 - ‘ ‘ +3}; [Warm P. + 1. Pub — Wampum. P. + f2 Fob] , (6.1) where Pi = %(1 :l: '75), i0” = —%['y“,7"] and the superscript =1: denotes the complex conjugate. In general, the form factors f1“! and sz’R can be complex. Note that in Equation (6.1), if there is a relative phase between f1” and ff or between f1“ and ff, 38 39 then CP is violated. For instance, in the limit of m5 = 0 , a CP-violating observable will have a coeflicient proportional to Im(f{’ .f') for a left-handed bottom quark, and Im(f13f2’") for a right-handed bottom quark [44]. (We will discuss more on CP violation in Chapter 7.) If the W—boson can be off—shell then there are additional form factors such as afiw;t(f3’~P_ + f3"P+)t + 6‘“ij(f3 ‘P. + f3 ‘P,.)b, (6.2) which vanish for an on—shell W-boson or when the off—shell W-boson couples to massless on-shell fermions. Here, we only consider on—shell W—bosons for m, > Mw+mb. At tree level in the SM the form factors are ff = 1 and fl” : f2” = ff = 0. These form factors will in general affect the experimental observables related to the top quark, such as the fraction of longitudinal W’s produced in top quark decays. The fraction (fans) of longitudinally polarized W-bosons, produced in the rest frame of the decaying top quark, strongly depends on the form factors flL’R and 21"", as shown in Appendix C. Hence, fLong is a useful observable for measuring these form factors. The definition of fLong is simply the ratio of the number of longitudinally polarized W—bosons produced with respect to the total number of W— bosons produced in top quark decays: f _ POW =0) Long — I‘(AW = 0) +I-‘(AW = —) +P(AW = +). (6.3) We use I‘(Aw) to refer to the decay rate for a top quark to decay into a W—boson with polarization Aw. (Aw = -, +, 0 denotes a left-handed, right-handed, and longitudinal W—boson.) Clearly, the polarization of the W—boson depends on the form factors f1 and f2.l Therefore, one can measure the polarization of the W-boson to measure these form factors. As shown in Appendix C, the polarization of the W—boson can be 1The fraction of long'tudinal W’s in top quark decays contributed from the form factor f,“ is the same as that from ff [44]. 40 determined by the angular distribution of the lepton, say, e+ in the rest frame of W+ in the decay mode t H bW+(H e+u). However, the reconstruction of the W—boson rest frame (to measure its polarization) could be a non-trivial matter due to the missing longitudinal momentum (P2) (with a two-fold ambiguity) of the neutrino (V) from W decay. Fortunately, as shown in Equation (6.4), one can determine the polarization of the W—boson without reconstructing its rest frame by using the Lorentz-invariant observable mix, the invariant mass of b and e from t decay. The polar angle 0;. distribution of the e+ in the rest frame of the W+ boson, whose z-axis is defined to be the moving direction of the W+ boson in the rest frame of the top quark, can be written in terms of m5, through the following derivation: cosB‘+ EeEb — Po . pb c lfiellfibl ~ 1 “'pb—1 2mg, (64) _ E,Eb — m? — M3,, ' The energies E, and E), are evaluated in the rest frame of the W+ boson from the top quark decay and are given by M3,, + m3 - m3, 1 F—‘— Ee = 2MW 1 lpel : E3 _ m3: 2 m2—M2 —m ,, E5 = 1 2M: b1 lpbl : VEE-mg (65) m, (m,) denotes the mass of e+ (11,) for the sake of bookkeeping. The first line in Equation (6.4) is exact when using Equation (6.5), while the second line of Equa- tion (6.4) holds in the limit of m), = 0. It is now trivial to find fLong by first calculating the cos 0;, distribution then fitting it according to the decay amplitudes of the W- boson from top quark decay, as given in the Appendix C. In what follows we will show how to use the distribution of 1m, to measure the mass of the top quark and its couplings to the W—boson. 41 In Reference [19], we considered the effective couplings 1+ 1:00 W—tL—bLI 2% 2 L 7110—75) (6-6) and g REC W — tn - bat @7711“ + 75) (6-7) derived from an electroweak chiral lagrangian with the symmetry SU(2)L x U(1)Y broken down to U(1)EM. (Here, KEG = ff — l, and n20 = flR.) At the Tevatron and the LHC, heavy top quarks are predominantly produced from the QCD process gg, qq H ti and the W—gluon fusion process qg(Wg) H tb, tb. In the former process, one can probe KLCC and KRCC from the decay of the top quark to a bottom quark and a W—boson. In the latter process, these non-standard couplings can also be measured by simply counting the production rates of signal events with a single t or 5. Let us C discuss KL C and K300 in more detail as follows. 6.1 From the Decay of Top Quarks To probe 141,00 and nRCC from the decay of the top quark to a bottom quark and a W- boson, one needs to measure the polarization of the W-boson, which can be measured from the distribution of the invariant mass mu. For a massless b, the W—boson from top quark decay can only be either longitudinally or left-hand polarized for a left- hand charged current (5300 = 0). For a right—hand charged current (KLCC = —1) the W—boson can only be either longitudinally or right-hand polarized. (Note that the handedness of the W—boson is reversed for a massless b from t decays.) This is a consequence of helicity conservation, as diagrammatically shown in Figures 6.1 and 6.2 for a polarized top quark. In these figures we show the preferred moving direction of the lepton from a polarized W—boson in the rest frame of a polarized top quark for either a left-handed or a. right-handed t-b—W vertex. As indicated in 42 (a) left-handed top (b) right-handed top 1+ 1+ ‘ \ / ‘ b c: = ¢-——-—.’VWWVVVVV\P AMANVVVVVV.-—————O = a =' = “7+ W"? + (left-hand) (left—hand) 1 1+ \ t t / WVVVWVVVV'T ‘——=>———.WVVVV\NVV\P WZong W'lfong t boost direction t boost direction Figure 6.1: For a left-handed t—b—W vertex. a - an top ri t- an e top ()lefth ded (b) gh h dd 1* 1+ W. = TOW + °+ W(right—hand) H[(riglu-haml) 1+ 1+ t / \ t b = = b TOW/v MMANWWVO = W3... W3,” t boost direction t boost direction Figure 6.2: For a right-handed t-b—W vertex. 43 these figures, the invariant mass mu depends on the polarization of the W-boson from the decay of a polarized top quark. Also, mu is preferentially larger for a pure right-handed t-b—W vertex than a pure left-handed one. This is clearly shown in Figure 6.3, in which the peak of the mu distribution is shifted to the right and the distribution falls off sharply at the upper mass limit for a pure right-handed t-b-W vertex. In terms of cos 0;, their difference is shown in Figure 6.4. However, in both cases the fraction (fans) of longitudinal W’s from top quark decay is enhanced by m¢2/2Mw2 as compared to the fraction of transversely polarized W’s [44], namely, f Long = —’,x,,2— (6-8) 1 + m- Therefore, for a heavier top quark, it is more difficult to untangle the KLCC and K300 contributions. On the other hand, because of the very same reason, the mass of a heavy top quark can be accurately measured from fLong (discussed below) irrespective of the nature of the t-b—W couplings (either left-handed or right-handed). The QCD production rate of ti is obviously independent of “CC and 19300. (Here we assume the electroweak production rate of ch H A, Z H tt remains small as in the SM.) Let us estimate how well the couplings 111,00 and K300 can be measured at the Tevatron, the Di-TeV, and the LHC. First, we need to know the production rates of the top quark pairs from the QCD processes. As shown in Table 3.1, the QCD production rate of gg,qq H tt for a 180 GeV t0p quark is about 4.5 pb, 26 pb and 430 pb at the Tevatron, the Di-TeV, and the LHC, respectively. For simplicity, let’s 8 57-, where consider the 3* + Z 3 jet decay mode whose branching ratio is Br = 2% = the charged lepton (i can be either ei or p*. We assume the experimental detection efficiency (6), which includes both the kinematic acceptance and the efficiency of b- tagging, to be 15% for the signal event [48]. Let’s further assume that there is no 44 IOOTIIT 60 4O da/dmg. pb/O GeV 20 II'IIUTITIUITTIIIUVWIII'I Figure 6.3: mu distribution for SM top quark (solid) and for pure right-handed t-b-W coupling of th(dash). 45 da(pb)/dcoso Figure 6.4: cos 9; distribution for SM top quark (solid) and for pure right-hand t-b—W coupling of th(dash). 46 Table 6.1: Results on the accuracy of measuring ff’R for various luminosities. (Only statistical errors are included at the 95% confidence level.) Integrated Number of Luminosity reconstructed 97’;- A f,“ A731 fb—l tt events. 1 1 200 8% :l:0.5 4% 3 600 4% i0.3 2% 10 2000 2% :l:0.2 1% ambiguity in picking up the right b (b) to combine with the charged lepton 6+ (6‘) to reconstruct t (or t), then in total there are 4.5 pb x 103 pb"1 x ~28—7- x 0.15 = 200 reconstructed tt— events to be used in measuring KLCC and [£300 at \/S' = 2 TeV. The same calculation at the Di-TeV and the LHC yields 1100 and 19000 reconstructed tf events, respectively. Given the number of reconstructed top quark events, one can fit 00 and KRCC. For example we have done a study the mu distribution to measure KL for the Tevatron. Let us assume the effects of new physics only modify the SM results (ff = 1 and f1” = 0 at Born level) slightly and the form factors fgL’R are as small as expected from the usual dimensional analysis [49, 50].2 We summarize our results on the accuracy of measuring flL’R for various luminosities in Table 6.1 [51]. (Only statistical errors are included at the 95% confidence level.) In the same table (i.e.,Table 6.1) we also show our estimate on how well the mass of the top quark m. can be measured from fung. By definition of fLonS, for a SM top quark (i.e., ff' = 1 and f1” = O), the distribution of cos 0; has the functional form ,_ 1 — c0302 2 sinfi; 2 . F(cosl9,) ~ (1 - fLong) (T) + fLong (W) - (6-9) 2The coeficients of the form factors 1‘21“”, assumed to be induced through loop effects, will be a factor of 111;;- smaller than that of the form factors ff’R. 47 Therefore, fun; can be calculated by fitting with the distribution of cos 0;, or equiv- alently with the distribution of mu. We prefer to measure KLCC and KRCC using the distributions of mu than of cos 6; because the former can be directly calculated from the measured momenta of b and 8. However, to convert from the distributions of mu to cos 0;, as given in Equation (6.4), the effects from the width of the W—boson and the top quark might slightly distort the distribution of cos 9;. (Notice that in the full calculation of the scattering amplitudes the widths of the W-boson and the top quark have to be included in the Breit-Wigner form to generate a finite event rate.) However, in reality, the momenta of the bottom quark and the charged lepton will be smeared by detector effects and another problem in this analysis is the identifica- tion of the right b to reconstruct t. There are three possible strategies to improve the efficiency of identifying the right b. One is to demand a large invariant mass of the tt- system so that t is boosted and its decay products are collimated. Namely, the right b will be moving closer to the lepton from t decay. This can be easily enforced by demanding leptons with a larger transverse momentum. Another is to identify the soft (non-isolated) lepton from b decay (with a branching ratio Br(b -) u+X) ~ 10%). The other is to statistically determine the electric charge of the b-jet (or b—jet) to be 1/3 (or —1/3) [52]. All of these methods may further reduce the reconstructed signal rate by an order of magnitude. How will these affect our conclusion on the determi- nation of the non-universal couplings KLCC and KRCC? It can only be answered by detailed Monte Carlo studies which are yet to be done. 6.2 From the Production of Top Quarks Here we propose another method to measure the couplings mac and K. 360 from the production rate of the single-top quark process. 48 For m. = 180 GeV, the sum of the production rates of single-t and single-f events is about 2 pb and 14 pb for \/§ = 2 TeV and \/§ = 4 TeV respectively. The branching ratio of interest is Br = g. The kinematic acceptance of this event at \fS = 2TeV is about 0.55, as shown in Chapter 8. Assuming the efficiency of b—tagging is about 30%, then there will be 2 pb x 103 pb"1 x g x 0.55 x 0.3 = 75 events reconstructed for a 1 fb’l integrated luminosity. At x/S = 4TeV, as shown in Chapter 8, the kinematic acceptance of this event is about 0.40 which, from the above calculation, yields about 3700 reconstructed events for 10 1b“1 integrated luminosity. Based on statistical error alone, this corresponds to a 12% and 2% measurement on the single- top cross section. A factor of 10 increase in the luminosity of the collider can improve the measurement by a factor of 3 statistically. Taking into account the theoretical uncertainties, as discussed in Chapter 3, we examine two scenarios: 20% and 50% error on the measurement of the cross section for single-top production. The rasults, which are not sensitive to the energies of the colliders considered here (either 2TeV or 4TeV), are shown in Figure 6.5 for a 180 GeV top quark at the Tevatron. We found that KLCC and K300 are well constrained inside the region bounded by two (approximate) ellipses (cf. Appendix A). To further determine the sizes of KLCC and K300 one needs to study the kinematics of the decay products, such as the charged lepton I, of the top quark. Since the top quark produced from the W—gluon fusion process is almost one hundred percent left-hand (right-hand) polarized for a left- hand (right-hand) t-b—W vertex, the charged lepton 3+ from t decay has a harder momentum for a right-handed t-b—W coupling than for a left-handed coupling. (Note that the couplings of light-fermions to W—boson have been well tested from the low energy data to be left-handed as described in the SM.) As shown in Figures 6.1 and 6.2, this difference becomes smaller when the top quark is much heavier because the W—boson from the top quark decay tends to be more longitudinally polarized. 49 0.4 .' ~15 = 2.0 TeV : m, = 180 GeV 03 ' —— 50% ; ----- 20% o ””””””””” 82' -02 0.4 - -O.6 - y. -0.8 — -1 4 1 1 l o o to .0 b o a: o G d Incfi‘l Figure 6.5: Constraint on InLCC| and KR“: given 20% and 50% error in measurement of Stande Model rate for W—gluon fusion. Curves are identical for m, = 140 GeV and m, = 180 GeV. 50 A right-hand charged current is absent in a linearly SU(2) L invariant gauge theory with massless bottom quark. In this case, KRCC = 0, then KLCC can be constrained to within about -0.08 < m,“ < 0.03 (—0.20 < am < 0.08) with a 20% (50%) measurement on the production rate of single-top quark at the Tevatron [19]. (Here we assume the experimental data agrees with the SM prediction within 20% (50%).) This means that if we interpret (1 + KLCC) as the CKM matrix element [141,], then |V¢b| can be bounded as [thl > 0.9 (or 0.75) for a 20% (or 50%) measurement on the single-top production rate. Before closing this chapter, we remark that in the Refs. [19] and [53] some bounds on the couplings of KLCC and x300 were obtained by studying the low energy data with the assumption that the effects of new physics at low energy can only modify the couplings of KLCC and KRCC but not introduce any other light fields in the ef- fective theory. However, nature might not behave exactly in this way. It is possible that some light fields may exist just below the TeV scale, then the bounds obtained from Refs. [19] and [53] may no longer hold. Thus, it is important to have direct measurements on all the form factors listed in Equation (6.1) from the production of top quarks, in spite of the present bounds on It’s derived from radiative corrections to low energy data. Chapter 7 Probing CP Properties in Top Quarks It is known that explicit CP violation requires the presence of both the CP non- conserving vertex and the complex structure of the physical amplitude. Due to the origin of this complex structure, the possible CP-violating observables can be sepa- rated into two categories. In the first category, this complex structure comes from the absorptive part of the amplitude due to the final state interactions. In the sec- ond category, this complex structure does not arise from the absorptive phase but from the correlations in the kinematics of the initial and final state particles involved in the physical process. Hence', it must involve a triple product correlation (i.e., a Levi-Civita tensor). To distinguish the symmetry properties between these two cases, we introduce the transformation T, as defined in Reference [54], which is simply the application of time reversal to all momenta and spins without interchanging initial and final states. The CP-violating observables in the first category are CP-odd and CPT—odd, while those in the second category are CP-odd and CPT-even. Of course, both of them are CPT-even. To illustrate the above two categories, we consider CP-violating observables for 51 52 the decay of the top quark. Consider the partial rate asymmetry I‘(t —) bW+) — I‘(f—) Bw-) AW I‘(t -> bW+) + P(f—> bW-)' (7.1) A non-vanishing Abw clearly violates CP and CPT, therefore this observable belongs to the first category. We note that because of CPT invariance, the total decay width of the top quark I"(t) has to equal the total decay width of the top anti-quark I‘fl. Thus, any non-zero Abw implies that there exists a state (or perhaps more than one state) X such that t can decay into X and i- into X. The absorptive phase of t —> bW+ is therefore generated by re-scattering through state X, i.e., t —) X —) bW+, where X 75 bW+ because the final state interaction should be off-diagonal [55]. Next, let’s consider the observable of the second category. In the decay of t -—> bW+(-) #111), for a polarized t quark, time reversal invariance (T) is violated if the expectation value of 6‘, X 171, ' 135+ (7.2) is not zero [44]. Assuming CPT invariance, this implies CP is violated. Therefore, this observable is CP-odd but C PT-even. A non-vanishing triple product observable, such as that in Equation (7.2), from the decay of the top quark violates T. However, it may be entirely due to final state interaction effects without involving any CP-violating vertex. To construct a truly CP-violating observable, one must combine information from both the t and t- quarks. For instance, the difference in the expectation values of 6’, x 1'27, - 152+ and 6’,- x 15'; - 13',- would be a true measure of an intrinsic CP violation. There have been many studies on how to measure the CP-violating effects in the ti- system produced in either electron or hadron collisions. (For a review, see a recent paper in Reference [56].) At hadron colliders, the number of ti- events needed to measure a CP-violating effect of the order of 10‘3 — 10"2 is about 107 — 108. To 53 examine the potential of various current and future hadron colliders in measuring the CP-violating asymmetries, we estimate the total event rates of tt- pairs for a 180 GeV SM top quark produced at these colliders. At the Tevatron, the Di-TeV and the LHC, an integrated luminosity of 10, 100 and 100 fb“ will produce about 4.5 x 104, 2.6 x 106 and 4.3 x 107 tt- pairs, respectively, as given in Table 3.1. Therefore, the LHC would be able to probe the CP asymmetry of the top quark at the level of a few percent. A similar number of the ti- pairs is required in electron collisions to probe the CP asymmetry at the same level. Thus, for a t/S = 500 GeV e‘e+ collider, an integrated luminosity of about 104 — 105 fb‘1 has to be delivered. This luminosity is at least a factor of 100 higher than the planned next linear colliders. We note that although the initial state in a pp collision (such as at the LHC) is not an eigenstate of a CP transformation, these CP-odd observables can still be defined as long as the production mechanism is dominated by 99 fusion. This is indeed the case for tt-. pair production at the LHC. In the SM, the top quark produced via the W-gluon fusion process is about one hundred percent left-hand (longitudinally) polarized, see Appendix A. Given a polar- ized top quark, one can use the triple product correlation, as defined in Equation (7.2), to detect CP violation of the top quark. For a polarized top quark, one can either use b" x 15}, or 15%“ x ii}, to define the decay plane of t —-> bW(—) (+11). Obviously, the latter one is easier to implement experimentally. Define the asymmetry to be N (8* out of the decay plane) — N (3+ into the decay plane) Aio N(€+ out of the decay plane) + N (2+ into the decay plane) ° (7.3) If Ago is not zero, then the time-reversal T is not conserved, therefore CP is violated for a CPT invariant theory. Due to the missing momentum of the neutrino from the decay of the W-boson, it is difficult to reconstruct the azimuthal angle (W) of the W—boson from the decay of the top quark. Once the angle w is integrated over, 54 the transverse polarization of the top quark averages out and only the longitudinal polarization of the top quark contributes to the asymmetry Ago. Thus, the asymmetry A50 can be used to study the effects of CP violation in the top quark, which in the SM is about one hundred percent left-hand (longitudinally) polarized as produced from the W—gluon fusion process. To apply the CP-violating observable Ago, one needs to reconstruct the directions of both the t and b quarks. It has been shown in Reference [57] that it takes about 107 — 108 single-top events to detect CP violation at the order of ~ 10'3 — 10*. For m, = 180 GeV at the Tevatron, the Di-TeV and the LHC, an integrated luminosity of 10, 100 and 100 fb‘1 will produce about 2 x 10“, 1.4 x 106 and 2 x 107 single-t or single-t- events, respectively, from Table 3.1. At the NLC, the single top quark production rate is much smaller. For a 2 TeV electron collider, the cross sections for €_8+ -i efi'zetb and 6+7 -> Detb are 8 fb and 60 fb, respectively [58]. Hence, it will be extremely difficult to detect CP violation effects at the order of _<_ 10‘2 in the single-top events produced in electron collisions. A few comments are in order. First, to extract the genuine CP-violating effects, we need to study the difference in the asymmetry A50 measured in the single-t and single-t- events because the time-reversal violation in Age of the t (or D alone could be generated by final state interactions without CP-violating interactions. Second, the detection eficiency for this method is not close to one, so a good understanding of the kinematics of the decay products and how the detector works are needed to make this method useful. The asymmetry Ago belongs to the second category of CP-violating observables and is CP-odd and CPT-even. Consider another asymmetry A, which belongs to the first category of CP-violating observables and is CP-odd and CPT-odd. Using A, for 55 detecting CP-violating effects is to make use of the fact that pp is a CP eigenstate; therefore, the difference in the production rates for pp -+ tX and pp -) {X is a signal of CP violation. This asymmetry is defined to be 0(139 -+ tX) - 0031) -+ t7f) 0(pp -) tX) + 0(pp —) fX) At (7.4) As discussed in Chapter 6, the production rate of pp —> tX is proportional to the decay rate of t -9 bW+ and the rate of pp —+ t-X is proportional to the rate of t. -—) bW’. This implies that A, = Abw, cf. Equation (7.1). There have been quite a few models studied in the literature about the asymmetry in Abw. For instance, in the Supersymmetric Standard Model where a CP-violating phase may occur in the left-handed and right-handed top-squark, Aow can be as large as a few percent depending on the details of the parameters in the model [59]. Next, let’s examine how many top quark events are needed to detect a few percent effect in the CP-violating asymmetry A). Consider t -) bW+ —> b€+u, where E = cor [1. Define the branching ratio Bw as the product of Br(t —-) bW+) and Br(W+ -) (+11), where Br(W+ —> (“‘11) is 2/9. (Br(t —> bW+) depends on the details of a model and is almost 1 in the SM.) Let us assume that the efficiency of b-tagging (chug) is about 30% and the kinematic acceptance (6):) of reconstructing the single- top event, pp —> tX -) bW+X -> b€+uX, is about 50%. (See, a Monte Carlo study in Chapter 8.) The number of single-t and single-f events needed to measure A, is 1 1 2 ’V‘ = m (It) - (7‘5) Thus, to measure A, of a few percent, M has to be as large as ~ 106, which corre- sponds to an integrated luminosity of 100 fb‘1 at the Di-TeV. Chapter 8 A Monte Carlo Study It was shown in Reference [24] that due to the characteristic features of the transverse momentum and rapidity distributions of the spectator quark which emitted the virtual W an almost perfect efficiency for “kinematic b tagging” can be achieved. In addition, the ability of performing b-tagging using a vertex detector increases the detection efficiency of a heavy top quark produced via the W—gluon fusion process. In this chapter we show that this process is useful at the Tevatron with the Main Injector. We also estimate results for t/S = 4 TeV at the Di-TeV and for x/S = 14 TeV at the LHC in separate subsections. To show that a heavy top quark produced from the W—gluon fusion process can be detected at the Tevatron, we performed a Monte Carlo study on the W + 2 jets mode of the (2 —-) 2) process q’b —> qt(-) bW+(—> EU» (8.1) with 3+ = e+ or M”. More specifically, we assume that the b—quark jet from the top quark decay can be tagged so that the decay mode of interest is identified to be W + b + jet. Throughout this study, we assume that the efficiency of the b—quark tagging is 30% for P," > 30 GeV with no misidentifications of a b-jet from other QCD jets. For clarity we only give rates for top quark (not including top-antiquark) 56 plot pm the Sill del 997 sin Th me 0111 At fusi 57 production in this chapter, unless specified otherwise. To include the top-antiquark production one can refer to Chapter 3 for its production rate as compared with that of a top quark. For simplicity we only consider the intrinsic backgrounds (i.e., those present at the parton level) for the W + b + jet final state, and will not invoke any detailed study on effects due to hadronization of the partons or the imperfectness of the detectors used in experiments. The intrinsic backgrounds in the SM for the mode W + b + jet are the electroweak-QCD process q’cj —) W + b + b and QCD process ch, gg —> tt —) W + b + jet. We will show that the dominant backgrounds for the single-top signal come from the electroweak-QCD processes (as shown in Figure 8.1) 113, CS" —> bbW+(—) 8+V). (8.2) The other backgrounds such as cg —) bW+ are suppressed due to the small CKM matrix element [Vcbl 2 0.03 to 0.048 [5]. As done in the previous sections, we will give our numerical results in this section for either a 140 GeV or a 180 GeV top quark. 8.1 Tevatron with x/S = 2 TeV At the Tevatron (pp, \/S = 2 TeV) the single-top production rate from the W—gluon fusion process is 1(2) pb, as shown in Table 3.1. For the final state of Equation (8.1), b b M g “,0 b U,C _ b u, C l) d, 50 — 1+ - 1+ d, .‘s' d, 5 w"!- IV" V V Figure 8.1: Diagrams for ml, c§ —> bbW+(—) (+11). the h (hem insit abm the den The ledl be a for I with rapil Sign; Tera- Droba mow": the ll PDF . 58 the branching ratio of t —) bW+(—i (+1!) for 8+ = 6+ or M“ is about 2 / 9 in the SM. As discussed in Chapter 3, we found that after properly treating the b quark as a parton inside the proton (or antiproton) the total rate for the W—gluon fusion process is about 30% smaller than that of the (2 -) 2) process. Hereafter, we shall rescale all the numerical results of our analysis for the (2 -> 2) process to the total event rate of the W—gluon fusion process by multiplying them by a factor of 0.7. To show that a 180 (140) GeV top quark produced from this process can be detected at the Tevatron, we first impose the following kinematic cuts: P; > 15 GeV, Inql < 3.5, P; > 15 GeV, M < 2, P; > 35 GeV, [17"] < 2, ET > 15 GeV, Aqu > 0.7 . (8.3) The efficiency of these cuts for a 180 (140) GeV top quark is 32%(53%). Including the reduction factor from the assumed b-tagging efficiency, 30%, the signal rate is found to be about 0.045 (0.063) pb. In Equation (8.3), PT stands for transverse momentum, 17 for pseudo-rapidity, ET for missing transverse momentum, and AR = ([0517)2 + ((545))2 with 03 being the azimuthal angle. It is important to note in Figure 8.2 that the typical rapidity of the spectator jet in the signal event is about 1.6 although almost all the signal events have [17" | < 3.5 [24]. The distribution of 17" is asymmetric becauselthe Tevatron is a pp collider. To produce a heavy top quark, which decays to a positively charged lepton, the valence quark from the proton is most important, implying a large probability for 17" to be positive. (We define the positive z-direction to be the proton moving direction in the laboratory frame.) Similarly, a top-antiquark produced from the W—gluon fusion process would prefer a negative 17" due to the large up-antiquark PDF inside the antiproton. 59 15.0 12.5 10.0 7.5 5.0 [VIII[IUITIIUUW1I'II1111j lllllllllllllllllllllllllllll .5 Figure 8.2: The rapidity distribution of the spectator quark q, after cuts in Equa- tion (8.3), for the signal q’b -> qt(—> bW+(—> €+u)), and of the spectator quark b for the major background q’ ('1’ -) bbW+(—> (+11) (dots), for m, = 180 GeV (solid) and 140 GeV (dash), at the Tevatron. (The vertical scale is arbitrary, but the relative size among these curves are absolute.) 60 In the W + bb background process, the bb pair comes from a virtual gluon con- version, therefore its rate is highly suppressed if the invariant mass of the bb pair is large. Since both b and b have about the same transverse momentum (PT) in the background event, the requirement of P} > 35 GeV efi‘ectively forces a similar PT cut on b. This generates a large invariant mass of b and b (i.e., the virtuality of the gluon), strongly suppressing the background rate. In contrast, in the signal event the final parton g (from 9’, after emitting a virtual W) typically has a smaller PT than the b-quark (from the decay of a heavy top quark). Typically, in the signal event, P7“. :2 m, / 3. In Figure 8.3 we show the PT distribution of the tagged b from t in both the signal and the background events. Hence, demanding an asymmetric cut on PT (i.e., P} > 15 GeV and P} > 35 GeV) will suppress background effectively and keep most of the signal events. This is why an asymmetric cut on PT was used in our analysis to suppress the major background process W + bb. To compare the efficiency of this asymmetric cut in PT, we note that using P} > 15 GeV along with all the other cuts in (8.3) yields a signal-to—background ratio (S/ B) of about 1 /3(2/ 3). Re- ' quiring P} > 35 GeV excludes about 60% of the background events sacrificing about 10%(30%) of the signal. After imposing the kinematic cuts in Equation (8.3), we found that S / B 2 0.9(1.3). However, the signal-to-background ratio can be further improved by imposing cos 014 > —0.4. (8.4) Because the top quark produced from the W—gluon fusion process is left-hand polar- ized, 2+ tends to move against the moving direction of the top quark in the center- of-mass frame of q and t, cf. Figure 6.1. However, in the background event, the distribution of cos 0a,, as shown in Figure 8.4, is almost flat after imposing the cuts of (8.3). (qu = 7r — 0,, where 61 is the the polar angle of 6+ in the rest frame of t 61 UFIIITTWIIIIII‘II[ITIIIIIIUIIT 125 100 75 50 25 'llllllll'lllllLllllllllll Figure 8.3: PT distribution of the b quark, after requiring P7". > 15 GeV along with all the other cuts in (8.3), for the signal q’b —> qt(—9 bW+(—+ (+11)), and the major background q’ (j —-) bbW+(—> fill), at the Tevatron. 62 10 I I I I I T I I I l I I I I I I T I—I 8;— 'l 1 _ l' .- d t I ' . __ r-—J I _ l— (I ll: I. 6,_ IllJ I-: b '- b '- P '1 4.— '— 2— o J 1 J 1 J 1 1 l 1 I l l l l l 1 1 l 1 --10 -0.5 0.0 05 10 “3911 Figure 8.4: The cos 01,, distribution prior to cut Equation (8.4) for the signal q’ b —> qt(-) bW+ (-) (+11)) and the major background q’q —> bbW+(-> flu), at the Tevatron. 63 defined in the center-of-mass frame of q and t.) To calculate cos 9&1, the P; information must be constructed. Since both 6+ and u come from a real W+ boson, we can use the W—boson mass constraint 2 2 MW 2 (Pl + Pu) (8-5) and the ET information to specify the longitudinal momentum P; of the neutrino. There are two solutions for P; and typically, both of them are physical solutions for a signal event. Therefore, one has to fix a prescription to choose the one which will most likely give the correct distribution of the invariant mass of (”Cu and b. We choose the solution which has the smaller |P;|. Here we exploit the fact that the W has finite width. If a physical solution for P; is not found with MW = 80 GeV, we generate a resonant mass of the W—boson using a Breit-Wigner distribution. We use a full half-maximum width of the W—boson, where I‘w = 2.1 GeV, and solve for P; , repeating the algorithm for up to three trials if necessary. We found that the survival probability for finding a solution using this algorithm is about 90%, and the difference between this solution and the value actually generated by the Monte Carlo generator is a Gaussian distribution peaking at 0 with a width about the order of My. After the additional cut imposed on cos 0&1, we obtained S/ B 2 1.2(l.8). About 55%(40%) of the total signal event rate remains after applying the cuts (8.3) and (8.4) to the process (8.1). We conclude that for an integrated luminosity of lfb‘1 at a 2TeV pp collider, there will be about 75(105) signal events detected with a significance S/fB of about 10(14), including both the single-t and single-t events as defined in (8.1). To measure the mass of the top quark, we calculate the reconstructed invariant mass (m,) of the top quark using m? = (Pb + p: + p.92- (8.6) The distribution of m,, including both the signal and the backgrounds, is shown in 64 I I F I I I W T T I I r I I I I I I I I I I I T I I I I I 40 —- f: __ .5 t I I 1 on p I I d > I I 0 .- ., o _ : : .. g I- I | d On " I | m - ‘6 " I I 'l .0 ' 20 *— I I '— 5 l' I | ' a l : ' I t0 I 5 b I ' u > 10 _ I' “I i a E I I o- I q 0 I. I a I I I I I I I I4 I I I 1.3754 I I I I d 0 50 100 150 200 250 300 :11, [GeV] Figure 8.5: The m, distribution after the cuts Equation (8.3) and Equation (8.4) for m, = 180 GeV (solid) and 140 GeV (dash) at the Tevatron including both the signal and background events with Wi -+ ei or pi. 65 Figure 8.5, in which a clear mass peak appears unmistakably. Therefore, we conclude that the top quark can be detected and studied via this process at the Tevatron. In Figure 8.5, other less important backgrounds, such as tt- events, were also included. The tt background is not important after vetoing the events with more than 2 jets [24]. To support this we did a study for the two decay modes of tt: tt‘ —> bW+(—> (MSW—H q'q) (8.7) and tt—> bW+(—) (fix) bW’(—> (2'17), 82 = eoru. (8.8) For both modes we require one of the jets to be a b or 5. Consider first the t —) bq’q decay mode. We require at least one of the q’- and quets and b (if b is tagged) or b (if b is tagged) to be within rapidity 3.5, otherwise we reject the event. If any one (and only one) of the untagged jets is within rapidity 3.5 we call it the spectator jet and then apply our cuts. If two of the untagged jets are within rapidity 3.5, then we require their AR separation to be less than 0.7 to classify them as one spectator jet. When all three untagged jets are within rapidity 3.5 we choose the jet with the largest PT and check its AR separation with the other two. If the lower PT jets are within AR = 0.7 of the high PT jet, we call this the spectator jet and apply our cuts, otherwise, reject the event. Recall from Table 3.1 for a 180 GeV top quark, the tt rate is about 4.5 pb. After applying the cuts (8.3) and (8.4) and including the branching ratio for this mode, %3 = 1,173 the event rate, of approximately 3 x 10'4 pb, is very small as compared with the signal rate of 0.075 pb (including t and t) This is because for most of the t —> Mp decay modes all three jets are within rapidity 3.5 and Aij and Ajo are in general large . For the t --I 56517 decay mode we require that (2 be undetected. Specifically, if (2 is within rapidity 2 with P? > 15 GeV we reject the event. If 2 < [17"] < 3.5 we 66 require P? to be less than the minimum PT for detecting leptons, i.e., < 15 GeV in accordance with Equation (8.3). After the cuts in (8.3), this mode (with branching ratio %% = 54;) already suffers, being about 5% of the signal rate. This mode suffers another factor of about two loss to the failure of reconstructing P; due to the presence of two neutrinos in the final state. After imposing the cos 01,, cut the rate for tt -> bW+(—) (+11) bW"(—) €517) is about 3 x 10'3 pb which is about a factor of 25 smaller than the signal rate. Hence the dominant background (of the same order as the signal rate) comes from the electroweak-QCD processes as given in Equation (8.2). As summarized in Figure 8.5, even with the very minimum kinematic cuts of (8.3) and (8.4) the single-top signal can already be detected, assuming a perfect detector with b-tagging efficiency of 30%. To incorporate the effects of detector efficiencies, we smear the final state parton momenta using a Gaussian distribution with (AE/E); = 15%/7E, and (AB/E)“, = 50%/7E. (8.9) The m, distribution becomes slightly broader as shown in Figure 8.6; however, both the signal and the background rates are almost the same as those obtained with a perfect detector. As discussed in the previous chapters, the distribution of the invariant mass mu is extremely useful in either determining the mass of the top quark or measuring the form factors of t-b—W. For completeness we show in Figure 8.7 the distribution of mu with or without smearing, after all the above analysis. Because the b-jet is required to have large PT, cf. Equation (8.3), so AE/E for the b—jet is small, therefore the two mu distributions do not differ much and the difference becomes smaller for larger mu. Thus, the position of the bump at large mu in the signal events remain a good signature for detecting the single-top signals and determining the mass of the top quark (discussed in Chapter 4) or the couplings of t-b—W (discussed in Chapter 6). 67 _w I Ifi I Ifi Ifjfi I T I Ti I I I I I I I l I I I I I _‘ 40— _. 5 I I .0 P ., r. h - 0 30— — K) I- .. 3 F ‘ DI l" .. I- P’ d g 20— — G ,. -. fl _- d 5 I'll - .. F' c: L . OFJ l l l d O 300 Figure 8.6: Same as Figure 8.5, but with detector resolution effects as described in Equation 8.9. 68 15 '- _— I- ., a L ' 2'! Id I_ ‘ I- I 'l I I d 10 — I II _ I- ld ' q . ‘ I . ‘_ [- . 5 _ ~|d eeeee — P '_ : 0 At I I I I 4 I I I I‘l—A—J4l A__|'l_1-(l_.l 0 50 100 150 200 250 mu [GeV] Figure 8.7: mu distributions without (top) or with (bottom) smearing, after all the above analysis for the Tevatron. 69 We note that the data sample obtained after all the above analysis can be further purified at the cost of somewhat reducing the signal rates. This can be easily done, for instance, by noting the distinct differences between the signal events and the background events from the distributions of rapidity of the spectator jet (Figure 8.2), transverse momentum of the bottom quark (Figure 8.3), and the angular correlation cos0¢q (Figure 8.4) due to the polarization of the top quark in signal events. However, for a more realistic simulation, one should also consider the possibility of having a charm-jet (or even an ordinary QCD-jet) faking a bottom jet in b—tagging so that the actual background rate measured by the detector would be larger than that given here. This is outside the scope of our parton level study. As discussed in Chapter 3, another process which produces a single-top is the W“ production q’rj —> W“ —> tb. (8.10) At 2 TeV for a 180 (140) GeV top quark, the W“ production rate is about 1/5(1/3) of the W-gluon fusion rate. Applying the kinematic cuts defined in Equation (8.3) we find that the W“ process passes with about the same efficiency as the W—gluon fusion process. However, there are a few obvious differences in the kinematics of their final state partons. First, in the W" event, there are two b-jets (one for b from t decay and another for b from production), therefore there is a 50% chance of tagging the wrong b and giving the wrong reconstructed top quark invariant mass, as defined in Equation (8.6). To improve the invariant mass distribution of the top quark, one has to be able to distinguish a b-jet from a b—jet by making further selections at the cost of reducing the single-top rate from this process. (Some of the techniques have been discussed in Chapter 4.) Second, the rapidity distribution of the spectator jet (i.e., b-jet if b-jet identified) in the W" event peaks around zero (i.e., central, as 70 15.0 I I I I I I I I I I :I I I II. I I I I I 12.5 10.0 7.5 5.0 2.5 IIIIIIIIIIITIIIIIIIIIIIIIITIT IllIIIIIIIIIIIILIIIIIIIIIIIIL 0.0 a. Figure 8.8: The rapidity distribution, after Equation (8.3), of the spectator jet (i.e., b—jet if b—jet identified) in the W" event for m¢=180 GeV (solid) and 140 GeV (dash) against Figure 8.2 (dots), at the Tevatron. 71 shown in Figure 8.8) because tb is produced through the s—channel process.1 This is in contrast to that in the W—gluon fusion event where the rapidity distribution (17‘?) of the spectator jet (labeled as q-jet) is asymmetric and less likely to be around zero. Third, the polarization of the top quark produced from the W" process is not purely left-hand polarized as in the case of the q’b -+ qt process. For a 180 (140) GeV top quark, the ratio of the event rates for producing a left-handed top versus a right- handed top in the W‘ event is about 3.5(3.4). In Figure 8.9, we show the production rate for a left-handed or a right-handed top quark from the W" process and compare them with that from the W—gluon fusion process as a function of m,. Because the top quark is not 100% polarized in the W“ process, the angular correlation of t and the spectator jet will not be as strong as that in the W—gluon fusion process. In Figure 8.10 we show the distribution of cos 0“ in W“ events for a 140 and 180 GeV top quark. (Here, q-jet denotes the spectator jet.) Following through the previous analysis done for the W-gluon fusion events we found that W’ production compliments the W—gluon fusion process by increasing the single-top production rate by about 10%.2 Therefore, its contribution to our final results of various distributions is small. In conclusion we found that at the Tevatron (pp, x/S = 2TeV) the single-top production rate from the W—gluon fusion process after including the branching ratio for t -) bW+(—> (+11) is about 0.22(0.44) pb for a 180 (140) GeV top quark, where 8+ = e+ or p+. The kinematic acceptance after the kinematic cuts (8.3) and (8.4) is about 55% (40%). Assuming a 30% b-tagging efficiency we concluded that the single— top event rate from the W—gluon fusion process is about 0.036(0.052) pb. For an 1We note that the rapidity distribution of the b-quark in the W‘ —) tb event is slightly asymmetric around zero. It favors a slightly negative rapidity. (Recall that in the W—gluon events for producing single-t, the rapidity of the spectator quark q favors positive values.) This is similar to the lepton rapidity asymmetry observed in the pp -) W+ —-) (*1! events due to the ratio of the down-quark and the up—quark parton distributions inside the proton and the anti-proton. 2The W‘ production rate is about one fifth of the W-gluon fusion rate for a 180 GeV top quark, and the kinematic acceptance of the W‘ event is about half of that of the W-gluon fusion event. 72 integrated luminosity of 1 fb'l, this yields 36(52) reconstructed single-top events. (To include top-antiquark production, a factor 2 should be included.) The dominant backgron process is the electroweak-QCD process W + bb whose rate is about 80%(60%) of the signal rate in the end of the analysis. The tt events are not as important to our study. The tt rate for a 180 GeV top quark is only 0.4% and 4% of the signal rate for its lepton+jet and di-lepton mode, respectively. In both Figs. 8.5 and 8.6 for the distribution of m,, we have also included another single-top production process (a single-top produced from W‘) which increases the single-top rate by about 10%. Let us make a side remark about the dominant background 9% -> W + bb before we close this section. In the above analysis we did not include the possibility of having an additional QCD jet from either the radiation or the conversion of the incoming quark jet ( q’ or q). The concern is that this jet may be identified as a forward jet which would fake the single-top signal event. In this case, b and b in the W + bb + jet background event have to both fall into a cone of AR = 0.7 in order to fake the tagged b-jet (only one b) in the single-t event (cf. Equation (8.1)). To examine the possibility for this to happen, we have applied the eikonal approximation [60] to calculate the rate of W+ bb+ jet from the square of the W+ bb amplitude. (The results are shown in Appendix E.) After the basic kinematic cuts: P?" > 15GeV, |nid| < 3.5, P; > 15GeV, ln‘l < 2, P41" > 15GeV, Inb"| < 3.5, (8.11) the rate for W + bb + jet is already about a factor of 5 smaller than that for W + bb. Naively, one might expect a factor of a,(Mw) (~ 0.1) suppression factor for emitting 73 an additional QCD jet (quark or gluon) in the hard scattering process. However, as compared to the large invariant mass (Mng) of the W + bb system a jet PT of 15 GeV may be small enough to generate large logs, such as ln(Mwu-,/PT), in the amplitudes. Hence, because a, ln(MW,,5/PT) is not negligible, the rate for W + bb + jet is not suppressed by a factor of 10 relative to the rate of W + bb, but only a factor of 5. To see how often b and b will fall into a cone of AR = 0.7 we show in Figs. 8.11 and 8.12 the A125; and the Mg, distributions in W + bb + jet events after applying the kinematic cuts listed in (8.11). The same distribution in W + bb events is also shown for comparison. The AR“; distributions look alike, and the Mg, distribution falls slowly as M); increases. Also, the W +bb+ jet event prefers a larger Mg, because the P7» of b and b are larger in this process than that in the W + bb process. We find that only about 20% of the W + bb + jet events can possibly fake the single-t event by having b and 5 inside the same jet-cone and thus fake a tagged b-jet. Hence, the additional background rate from W+bb+ jet events is about a factor of i— x 20% = 4% of the electroweak-QCD background rate. Although our estimate is not precise, we believe our conclusion for this additional background should hold within a factor of 2. Hence, this additional background is negligible at the Tevatron. However, it can be important at the LHC. Because the energy of the LHC collider is much higher, it is more likely to have additional radiation in the event and to boost the bb system to make them closer and thus fall into the same jet-cone. 8.2 Tevatron with \/S = 4 TeV Here we present our results for a possible upgrade of the Tevatron with x/S = 4 TeV. After the following kinematic cuts: P} > 15 GeV, [an < 3.5, 74 P} > 15 GeV, [17‘] < 2, P} > 30 GeV, lan < 2, ET > 15 GeV, Aqu > 0.7 , (8.12) the signal rate is about 0.28(0.37) pb. (The efficiency of these cuts is 45%(56%).) In Figure 8.13 the typical rapidity of the spectator jet in the signal event is about 2, and almost all the signal events have Inql < 3.5. An asymmetric cut on P7 was used once again to suppress the major background process W + b + jet. Demanding P} > 15 GeV along with the other cuts in Equation (8.12), the signal-to-background ratio (S/B) is about 1.1(1.7). We show in Figure 8.14 the PT distribution of the tagged b from t. Requiring P} > 30 GeV excludes about half of the background events sacrificing about 6% (20%) of the signal. After all the cuts listed in Equation (8.12), S/B‘o.’ 2.3(3.0), with the signal rate at 0.28(0.37) pb. We show in Figure 8.15 the distribution of cos 6a,. After applying the cut (8.4), the ratio S/B 2' 2.9(3.8) with the signal rate of 0.22(0.29‘) pb. In the end of the analysis there are about 2200(2900) single-t events for an integrated luminosity of 10 lb”1 at x/S = 4 TeV (a pp collider) with a significance S/x/B of about 80(105). The kinematic acceptance of the signal event is about 43%(34%). Note that in all the above rates we have included the reduction factor from a 30% b-tagging efficiency. In Figure 8.16, we show the reconstructed invariant mass (m,) of the top quark. Once again, to incorporate the effects of detector efficiencies, we smear the final state parton momenta as in (8.9). The m, distribution becomes slightly broader as shown in Figure 8.17; however, both the signal and the background rates are almost the same as those obtained with a perfect detector. 75 8.3 LHC with \/S = 14 TeV Here we present our results for the LHC with t/S = 14 TeV. After the following kinematic cuts P} > 40 GeV, 1< Inql < 4, P} > 40 GeV, In‘l < 2, P} > 40 GeV, Inbl < 2, ET > 40 GeV, Aqu > 0.7 , (8.13) the signal rate is about 0.44(0.24) pb. (The efficiency of the cuts is 94%(98%). We still assume a 30% efficiency for the b—tagging at the LHC.) Here we did not impose a smaller P} out because a lower PT jet will be more difficult to be identified at the LHC. (A typical QCD event at the supercollider will be engulfed by soft gluon radiation.) Since the signal event yield is large at the LHC, we decided to purify our data simply by requiring a large P} cut. Notice that the rate for m, = 180 GeV‘ is larger at the LHC than that for m, = 140 GeV after our cuts, opposite to the behavior of the rate at 2 TeV and 4 TeV. This is due in part to less sensitivity to m. at higher energies, but mainly because the b from top decay is much harder for larger top mass and thus is less sensitive to the cut of P} > 40 GeV. The typical P} for m, = 180 (140) GeV is 60(40) GeV. We show in Figure (8.18) the PT distribution of the tagged b from t. In Figure 8.19 the typical rapidity of the spectator jet in the signal event is about 3, but a cut on Inql < 4 keeps almost all the signal events. Excluding the [17‘1 | > 1 cut in (8.13) the signal-to-background ratio (S/ B) is about 10(7). Requiring Inql > 1 excludes about 40% of the background events sacrificing about 15%(20%) of the signal. After the kinematic cuts in (8.13), the ratio S/ B 2 25(14). 76 We show in Figure 8.20 the distribution of cos 919 at the LHC. After the cos 03,, cut, S/ B 2 40(20). In the end of the analysis, there are about 30, 000( 15, 000) single—top events for an integrated luminosity of 100 fb"1 at x/S = 14 TeV (a pp collider) with a significance S/x/B of about 32(16). Hence, about 4%(2%) of the total signal event rate remains. This is thus the kinematic acceptance for the signal process (8.1). In Figure 8.21, we show the reconstructed invariant mass (m,) of the top quark for a perfect detector. Once again, to incorporate the effects of detector efficiencies, we smear the final state parton momenta as in (8.9) and reconstruct the m, distribution for the LHC is shown in Figure 8.22. Notice that the above analysis for the LHC is less reliable because the energy of the collider is much higher and therefore it is more likely to have additional soft- jets accompanying the signal and the background events. As discussed at the end of section 7.1, it would be more reliable to use a full event generator such as ISAJ ET [61], PYTHIA [62] or HERWIG [63] for this study because these generators contain radiation from either the initial or final states. However, these generators currently do not have the correct angular correlations in Z and jets, as discussed in this analysis. It would therefore be important in the future to improve these generators to incorporate the polarization effects of the top quark and the W-boson for studying physics of the top quark in hadron collisions. 77 101 : ' \ls = 2 TeV, pp Wg->tb,Wb_->1 total q’q->1b - - - - (hlvl'bk('1lzv1’2) 3 100 r“ -------- (ththUIZJ/Zl a \ V [ ‘s D l ‘s c “s, 2 t g .1 l (I) 10 l ID 9 0 ~, V. 10'2 — .......... " I I I I I I I I I I I I I I L I I I I I _I 100 150 200 250 300 Top Mass m,(GeV) Figure 8.9: The production rate for a left-handed (long dash) or a right-handed (short dash) top quark from the W" process. The upper solid line is the total rate for the W—gluon fusion process, the lower solid line for the W" process. 78 10 I I Ifi I I I I I I I I I If] I I I I P '1 I- III I- -I .. J 8 II— _ P C C . " "l 8 I—— —I I- -I I- -I I- -l I. -4 4 — ——I h ..d -3 - .. :- I- ..... I OI: Eu F- I1 u. :d 0 I 2— III"J"“--I~ f- ‘I II}. o-“ . § E _- ’---'— .i :: p“.."b — l' -- '5 b.- i o l I l J 1 l l l i J 4% I l l l l l l -10 -05 0.0 05 10 c080,,i Figure 8.10: The distribution of cos 9&1 in W' event for mt=180 GeV (solid) and 140 GeV (dash) against Figure 8.4 (dots), at the Tevatron. 79 80IIIIIIIIIIIIIIIjjTrIIIIIITIII l l L L 60 4O - I 20— 1 l l l I l l l J I l l l l l O H N ca .5 Q Figure 8.11: AR“; distributions in W + b5 + jet (solid) and W + b5 (dash) events after applying the cuts listed in (8.11). 80 100 j—I I I I I I I I I I l I I I I I I I I I I I 80 20 jIjIlIIIIIrII11III'IjIII b-—.u lllllJlJlLllllllllllllLJ O 50 100 150 200 250 Figure 8.12: Mb; distributions in W + b5+ jet (solid) and W + b5 (dash) events after applying the cuts listed in (8.11). 81 800 I I I I l I I I I I I I I j I I I I l 800 400 200 lllllLllllJlllllll .5 Figure 8.13: The rapidity distribution of the spectator quark q, after cuts in Equa- tion (8.12), for the signal q’b -+ qt(—) bW+(—) (+11)), and of the spectator quark 3 for the major background 9%] -> hbW+(—> (+11) (dots), for m, = 180 GeV (solid) and 140 GeV (dash), at the Di-TeV. 82 .- I I I I I I I I l I I I I I Ij I I I I I I I I I I I I .4 2000 '— '1 — l- l I ,4 I : . —. ' “z I 'I q 1500 F E I I — .. !- | _. " i I ‘ 1000 - I — l- I .. F q 500 *— —‘ : ___-._ : O P 1 1 1 [#1'..1H.l.j‘f'”1"1"1 :1--_ - 0 25 50 75 100 125 150 P: [GeV] Figure 8.14: PT distribution of the b quark, after requiring P} > 15 GeV along with all the other cuts in (8.12), for the signal q’b —) qt(—> bW+(-+ 8+ 11)), and the major background ([6 —) bbW+(—-> (+11), at the Di-TeV. 83 500). I I I I [fl I I I l I I I I l j I I I d 400— n- - y- q“J": ‘I I E 'Id-r'd : : 300'— |1 P- __— . .I I . . ,5 |l| I - t .. 200— 1007- H. t 0.4 l l l I l l l L L l l 1 l l l l l 1 d -10 —05 0.0 05 10 c030,.I Figure 8.15: cos 9&1 distribution for the signal q’b —> qt(—+ bW+(—> (+11)) and back- ground q’q —> bbW+(—> (+11) at the Di-TeV. 84 _1 I I I I I I I I I I I I1 I I I I I T I I r] IIIII I q 5 2000 —— F. -— .n ’ ,4: - > r I ' ' D " I - U P I I -4 no 1500 _. : . — h '- | d o _ I . a _ . . W . ‘6 ' ' . | . "5’ 1000 — : . _ :3 ’ , I ‘ I: r I I - '5 Z I : 2 3 500 — _' , A W - | l q . | 1‘ a i- I. _ d o P 1 1 1 l 14 1 1 '1 J 1 1 “151-1— 1 1 1 1 + 0 50 100 150 200 250 300 111‘ [GeV] Figure 8.16: The m, distribution after the cuts Equation (8.12) and Equation (8.4) for m: = 180 GeV (solid) and 140 GeV (dash) at the Di-TeV, including both the signal and background events with W* —> 6* or 11*. 85 II I I I l I I I I I I I I I l’ I I I I I I I I I l I I I I L- - g: 2000 I'- — 3 L q > 0 ~ '1 u . . no 1500 — _ — h " .. a - I h D d 0 '- 'l g 1000 — — I: P 1 o! " .I ‘3 * T > _ q a 500 _ . I Z 0 l l L l O 300 Figure 8.17: Same as Figure 8.16 but with detector resolution effects as described in Equation (8.9). 86 50 I I I I I I I I T7 fr I fiIW I I Ifi I I fI r l T I l l q : 1': : 40— I L _ b I cl E ' ' I ‘ 30— _. I I I p l- _. .- l - 20— I_ —. r- l - i 5' ‘ |_ .. - J 10— .fl - '1 I- '—- q p ...... ""--.~... -t“-_ O I 1 1 L11 111 1 1 J LLJ_1 1L11'”"‘11"1“1~1-J_-.1 O 25 50 75 100 125 150 p: [GeV] Figure 8.18: PT distribution of the b quark, after cuts i_n (8.13), for the signal q’b —) qt(-> bW+(—) (+11)), and the major background q’q —> bbW+(—> flu), at the LHC. 87 20 I IfI l I I I I l I I I I1 I I I I7 15 I I l I I I I I I 10— (J- 1— b I I I: Pr' ‘ ' 'cI ,- l- E' Ll“1 r d : 1.: - 5— — h d I- d I- . ........................ -I F- 0 ”1 1 1 1 144 1 14 1 1 J i 1 1min; -4 -2 0 2 4 Figure 8.19: The rapidity distribution of the spectator quark q, after cuts in Equa- tion (8.13), for the signal 9’ b -—> qt(-) bW+(—> (+11)), and of the spectator quark 5 for the major background q’ q —) bbW+(—> [+V) (dots), for m, = 180 GeV (solid) and 140 GeV (dash), at the LHC. 88 Figure 8.20: cos 0;, distribution for the signal q’b —> qt(—) bW+(—-> £+u)) and back- ground q’q -) bbW+(-) (*1!) at the LHC. 89 20000 IIIIIIIIITIIIITIIIIIIIIIIIIII 5 L I .o > 15000— — o . - o _ 1 a b d I... a b 1 L 10000— — O )— |I .. on f' 5 'I ” I, .. fl 4.: " 'I - ‘3 5000— 'I — ° I > . I . III II 'I I- ‘L q 0 50 100 150 200 250 300 Figure 8.21: The m, distribution after the cuts Equation (8.13) and Equation (8.4) for m, = 180 GeV (solid) and 140 GeV (dash) at the LHC including both the signal and background events with Wi -+ ei or p*. 90 zmoofIIIlIrIIIIIIrIIIIIlIIfiIIIIII l l l 1 I I I I 15000 10000 5000 Event number per 5 GeV bin 0 50 100 150 200 250 300 Figure 8.22: Same as Figure 8.21, but with detector resolution effects as described in Equation (8.9). Chapter 9 Discussions and Conclusions We discussed the physics of top quark production and decay at hadron colliders, such as the Tevatron, the Di-TeV and the LHC. We showed how to use the invariant mass distribution of rm, to measure the mass and the width of the top quark, produced from either a single-top or a tt- pair process. It has been shown in Reference [41] that the distribution of mu is not sensitive to radiative corrections from QCD interactions. Thus it can be reliably used to test the polarization of the W—boson from t decay (hence, test the polarization of the top quark from the production mechanism) and to measure the mass of the top quark using the observed value of f1“,g (the fraction of longitudinal W’s from top decays). We also discussed how well the couplings of t-b—W vertex can be measured to probe new physics, and how well the CP properties of the top quark can be tested in electron or hadron colliders. In Reference [24] we showed that an almost perfect efficiency for “kinematic b tagging” can be achieved due to the characteristic features of W—gluon fusion events. In addition, the ability of performing b—tagging using a vertex detector increases the detection emciency of a heavy top quark produced via the W—gluon fusion process. A detailed Monte Carlo study on how to identify the characteristic features of the signal events (i.e., the transverse momentum and the rapidity distributions of the 91 92 spectator quark which emitted the virtual W) and therefore suppress the background events was performed in Chapter 8.1 For an integrated luminosity of 1 fb'l, there will be about 75 (105) single-t or single-f events reconstructed in the lepton+jet mode for m, = 180 (140) GeV at x/IS'. : 2 TeV. (The branching ratio of W -—> e, or p is included, and the b-quark tagging efficiency is assumed to be 30% for P,“ > 30 GeV with no misidentifications of a b-jet from other QCD jets.) The dominant background process is the electroweak-QCD process W + bl.) whose rate is about 80%(60%) of the signal rate in the end of the analysis. The tf events are not as important to our study. The results for J8 = 4TeV at the Di-TeV and for \/§ = 14 TeV at the LHC were also discussed. Although the W“ —) t5 rate in the SM is not as large as the W—gluon fusion rate for producing a heavy top quark, it remains a complementary process for probing new physics in the single-top quark event. The W" process is particularly useful for detecting new physics through some possible high mass resonance in the theory. In that case, its rate will be highly enhanced by the resonance effects. We however did not study such a possibility in this work because its rate depends on the details of the models considered. lThe fortran code, ONETOP, used for this study is available by request. In Appendix F we briefly describe the processes included in this program. Appendix A Helicity Amplitude Method In this appendix we would like to display the rules for doing calculations at the ampli- tude level using the Helicity Amplitude Method. The method breaks down the algebra of four-dimensional Dirac spinors and matrices into equivalent two—dimensional ones. This algebra is easy to program and more efficient than computing the Dirac algebra as it stands. All diagrams are summed and squared numerically. In what follows we introduce the Weyl representation of Dirac spinors and matri- ces. We also include several example calculations to illustrate the finer points of the method. Throughout this paper we use the Bjorken-Drell metric g‘“’ = diag(1,—1,—1,—1). (A.1) The four-momenta have the form in spherical coordinates: p“ = (E, Ifil sin0 cos 05, |fi| sin9 sin db, |15'| c030) (A.2) with E2 — m” = m2. We define the right-hand (R), left-hand (L) and longitudinal (0) polarization vectors for a spin-l field as l M 5(a) = e—fi(0,i sin¢ - cosdJ c030, —z' cosqb — sin¢ cosB,sin0) 1For a massless spin-1 field, only the right-handed and the left-handed polarizations are physical. 93 94 -54, 6(2) = E—(OJ sin 05 + cos (0 cos 0, —i cosqb + sin ¢ cos 0, —sin 0) (A.3) \/§ 8" — 1 " E ° 0 E ' 0 ' E 0 (0) — Rflpl, Sln cosab, sm smcb, cos ). The above equations satisfy the identities sz) = —5(‘£), 6’50) = 5(3), ppefh) = 0 and efflef‘.) = -6,.,.,, for h, h’ = R, LorO. In four component form we define the following. In the Weyl basis Dirac spinors have the form w = (3:) (AA) where for fermions _ “2:” = win/2 A 5 2Ibzt — ”(Az—l _ w ( ) :t - $X-1/2 and anti-fermions 0:1) v = :l:w _ wi : { va=-1) - 45X 1/2 (A.6) n — :Fwin/2 With w: = “E :l: '5'. The xA/g’s are eigenvectors of the helicity operator h=I5-5,15=fi/|13| (A-7) with eigenvalue A where A = +1 is for “spin-up” and A = —1 is for “spin-down”. _ cosO/2 _ —e“¢’sin0/2 X1” _ (e‘fsinG/Z) ’ X—1/2 _ ( cosG/2 ) ° (A8) 95 Later it proves useful to represent xA/g’s using bra-ket notation where |15+) E X1/2, '15—) E X-1/2- (A.9) Gamma matrices in the Weyl basis have the form 0 1 - 0 —0- 1 0 0_ J_ J 5_ _ 7 _(1 0)’ 7 ‘(o,- o ) 7 ‘75‘(0 —1)' (A10) where a,- are the Pauli 2 x 2 spin matrices 0 1 0 -z' 1 0 01—(1 0), 0'2—(2. 0), 03—(0 _1). (A.11) The Chirality projection operators are defined by 1 5 Notice that P+ (P.) projects out the “right-handed” (“left-handed”) component of the Weyl spinor effectively reducing the algebra from one involving four component spinors and matrices to one involving two component spinors and matrices. P—w = (3 ?)(.‘Zi)=(£_) (010M? (1,) ((1, 3) = (fl 0) (A13) 213P+ In the Weyl basis )1 has the form #Epp7p=(po—O&'.fi Po+0&'-I3°)E(:- %+)Epp(% ’75:) (A.14) where 7; = (1,:pa). . (A.15) Products of these 7i’s have the following useful property when Lorentz indices are contracted: (filij(7u+)u = (7’30 (711-)kl = 2l5ij6u - 6.1%] (A.16) 96 d ——> U ——> wl _'_. b -1" Figure A.1: Diagram for the (2 —> 2) process u b —> dt. and (vi)n(7n_)u = (7‘1)ij(7n+)u = 26.1%, (A.1?) where the Roman indices are not vector indices in the usual sense, but are labels identifying bras and kets. For instance, for arbitrary kets Ii), | j ), lie) and II) we have (7¢)ij(’7u—)u = (il‘rileleu—ll) = 2(i|1)(k|j) = 26i16kj- (Ax-18) Equations (A.16) and (A.17) are simply the two-dimensional version of the well known Fiertz identities. A.1 Helicity Amplitudes for ub —> dt To illustrate the use of helicity amplitudes we calculate the matrix element for the (2 -> 2) process ub —> dt which contributes to the total rate for W—gluon fusion. Figure A.1 shows the Feynman diagram for this process with the t decay included. In this example we use the generalized t-b—W coupling 1‘” = (1 + In“) 7“P. + “607%., (A.19) where KLCC and Race parameterize deviations from the Standard Model in which IcLCC = 0 and K300 = 0. We calculate the matrix element in the ’t Hooft-Feynman gauge ignoring for simplicity the factors due to vertices and propagators. We obtain M: [17(t tI‘)“(bu ()][u(d )_'y,,Pu( 21)] (A20) 97 where u, b, d and t are the momenta of the external legs and we retain only the mass of the top quark. We use the algebraic properties of the projection operators Pi = PbPiP; = 0 and Pi7" = "WP; to project out the Chirality states. In this example the amplitude contains both the left-handed and right-handed currents. For pure vector or axial vector currents one must first insert 1 = R, + P. or 75 = R, — P. respectively. Therefore M takes the form M = (1+ KLCCHui (t)viu-(b)l[u1(d)7n+u—(u)] + nRCC[uL(t)7fi “+(b)ll“t- (d)7,,+u- (1‘)] (A21) According to Equations (A.5) and (A9) we see that we = I/2—Eila—> u_(d) = (273.1%) M» = (355%) Mb) = x/éEII'H) w) = { (E. - It"! I£+> ‘/E¢+|{||{—) § + A N V l { V Et + If] lid") (A22) «El-lfllf_) Therefore, M(+) = (1 + KLCCNEI - It'l (”Ml 3-)(3-|7u+| a -> + KRCCV E: + It"| (5+I75I 3+)< Ci-l’hH-l '1 ‘) M(-) = (1 + KLCCNE: + IFI (f-lviil 5->(Cl-l7e+l 1‘1 -) + KRONE. —Ii'1 (A23) 98 where M(:t) denotes the amplitude with t helicity A, = i=1 and we ignore for now the common factor of \/2Eu \/2Eb\/2Ed. One is now tempted to move on ahead and contract the Lorentz indices as in Equation (A.16). However, contracting 7,35 with the same Chirality introduces extra terms into the matrix element. It would be more useful with processes containing many branchings from decay, such as in Supersymmetry, if there were a way to utilize Equation (A.17) instead. To this end, we digress a moment. Consider some current of the form ¢}il7m¥ ' ' ' 7un¥lwii (A24) where n is odd. We note in passing, recalling the properties of the projection op— erators, consecutive matrices of the same sign would give zero current. Since Equa- tion (A24) is just a number, it is identical to taking its transpose as ((11%: ' ' ' 7;¥]¢;i ° (A25) We now utilize the following algebraic properties of the Pauli matrices and in partic- ular 02: 0202 = 1, a; = —02, 027;;02 = 7,“: (A26) and define £5; E 502$; - (A.27) By inserting pairs of 02 between each pair of objects in Equation (A.25) we therefore obtain ~t ~ ¢}¢l7u1¥”'7ud=l¢ii = 11):": [mat ' "’miltbfn. (A.28) It is easy to show that (iagz/ziif = II: . In addition, for an even number n of gamma matrices we have 11"}il7ui? ' ' ' 7». ill/’1'? = "171;th ' ' ' 7millz’zt ' (A'29) 99 An important result has occurred, which allows us to take advantage of Equa- tion (A.17) avoiding the number of terms that would otherwise occur. 2 For a fermion or anti-fermion with momentum 5 and helicity A, (bi is proportional to either I15+) or Ifi-). It is easy to show that (15+) = —Ie—), '13:) = +|p+), (n+1 = -(13—l, (ET-I = +(zs+l. (A30) Finally, recalling Equation (A.23) the term (oi—Inn a —> = (Mn—I ci+) (A31) giving, via Equation (A.17), M(+) = 2(1+ nLCCh/E, — (Fl (t'+| ci+)(a+|IS—) +2KRCCW(£+lfi-l(€i-|5+) M(—) = 2(1+ KLCCh/E, + (5| (i—l d+)(&+| (3—) + 2KRCCW(f—I n—)(&—| (3+). (A32) We now have the matrix element in the form we require for our Monte Carlo package ON ETOP3, remembering to include coupling constants, propagators, color factors and WWI/2?}. 2502 acts as a kind of charge conjugation operator on the Chirality states of the Weyl spinors. 3A FORTRAN code. 100 A.2 Helicity Amplitudes for ub —+ dt in the CMS. To illustrate our claim that in the SM (ale. KLCC = 0 and KRCC = 0) only the left-handed top quark is produced from the ab —) dt process in the d-t center of mass frame (CMS), we evaluate the matrix element in terms of CMS variables. Define the four-momenta: u” = (\/§/2,0,0,—\/§/2) b“ = (Vi/2,0,oA/E/2) d” = (t, —t sin 0,0, —t cos 0) t” = (E,, t sin 0, O,t cos 0) (A33) where E, = (s' + m?)/2\/§, t = (.6 — m?)/2\/§ and we have chosen 45 = 0 to be the scattering plane. Using these, we obtain from Equations (A.5), (A8) and (A9) E i. l 101 cosG/2) sin0/2 ( If+> = (cone/2) ( sin0/2 - -sin0/2 |t—) _ c059/2) (A34) Therefore, (a+|(3-) = —1 (d—|b+) = cosO/2 ({+|d+) = 0 (t+|u—) = cosO/2 (t—|d+) = —1 (f—Ifi—) = —sin0/2. (A35) Including the common factor J2Efl/2Ebszd = \/\/§(§ — m?) (A36) and E,+|t"| = «E 2 .. m E — t = -L i l I \/§ we see that M(+) = 2(nRCC)(/§(§—m3)cos20/2, (A.37) M(—) = 2(1 + nLCC),/§(§ - m3) -— 2(ICRCC)\/m?(§ - m?) sin 0/2 nose/2. Notice in the SM, the top quark is 100% left-hand polarized in the CMS. 102 Figure A.2: Diagrams for 119 —> dt(—) bW+(—> 8+ V())b. Having outlined the general procedure for calculating amplitudes using the helicity amplitude method, we list the matrix elements contributing to single top production and the major background Wbb in the Standard Model. We include the decay of t —> bW+ and W+ -) (+115 in the final form. A.3 Helicity Amplitudes for u g —-> dt(—+ bW+(—> 6+ 113)) 0 In this and the following sections, we give the diagrams for the process listed, indi- cating the momentum flow and particle momentum labels: w’s are for W+ bosons, b’s for b or b quarks, e for 8+, n for V, and u, d, t and g are for u,d,t quarks and gluon, respectively. In Figure A.2, we show the diagrams for u g —> dt(—> b W+(—) 3+ 113)) b. M;(hg, A5,) will represent the ith diagram (2' = 1,2 from left to right in Figure A.2) where by represents the two transverse gluon polarizations and Ab, represents the two helicity states of b. The matrix element in the helicity amplitude formalism for this process is Mi(hg.=I=) =4(62—Ifi+)<é+l It Ia?+>X i\/Ebi=F|b-il(fi+li—Wl)+|51i)‘Fmb\/Ebiilb-i|(u+lWig )—|5I=!=> (42 “77%) 103 mm?) = i4I/Eb1 2: I61 I <62 -|fi+)(fi+|61:t)>< (é+l #— ¢(hg)+ /. Ié+> +m?<é+l ¢(hg)— Ici+> (T2 _ m?) (A.38) where we have indicated the four different helicity states involved in this process. We keep the mass of the b parton to avoid the case where the b propagator goes on shell. For simplicity we have omitted a common factor of gay)“ \/(2Eu)(2Ed)(2Eb2)(2Ee)(2En) (A39) 73 (t2 — mow? — Max:123 — Ma» and color matrices.4 We note that the polarization vectors for spin-1 gauge bosons may be expressed in terms of spin-% bras and kets. We define I+> = ((1,) H = (‘1’), (A.40) then for the transverse polarizations f’ wémxg—I if" ifilg—><9+I (A41) and for massive spin-1 gauge bosons m A?) = (L915) <+I+I-><—I)+2§I§+>(+l + l-)<—|) + 2% a-xa—I. (A42) m A.4 Helicity Amplitudes for ub —-> dt(—-> bW+(—> 5+ 113)) 4The color factor for the amplitude squared is 3 x 4 x 1 / 3 x 1/8 for this process. 104 Figure AA: Diagram for uda bt(—) bW+(-> 8+ 14)). In Figure A.3, we show the diagram for ub —+ dt(-—> bW+(—-> 3+ 112)). The matrix element in the helicity amplitude formalism for this process is M=4(21+Ibi—)(é+l V—IJ+><52-Ifi+) (A43) where we ignore the b parton mass. We have omitted a common factor of (fl) JI2E.)(2E.)<2EI. )(2Eb2)(2Ee)(2En) (M4) fl (t2 - m?)(wi - MEI/)(wg - Mt) and color matrices.5 A.5 Helicity Amplitudes for ac? —-> W“ —> bt(—-> b W+(—> 6+ 113)) 5The color factor for the amplitude squared is 3 x 3 x 1/3 x 1/3 for this process. 105 II ——. * .jI ’/ \" quI e A / —.—l -a-' M “‘4‘ .\ Figure A.5: Diagrams for 213-) bbW+(—) 3+ 113). x. \" ’/ \’ Y '/ In Figure A.4, we show the diagram for at? -) W‘ -> bt(—) bW+(—-) 3“ u()). Aside from a possible phase, the matrix element in the helicity amplitude formalism for this process (W‘ production) is identical to that of Figure A.3. This is because one process is the cross diagram of the other and therefore only the momentum assignments are different. For clarity, it is M =4(w22> - Mat) and color matrices.6 A.6 Helicity Amplitudes for u (I —> bb W+(—-> 5+ ug) In Figure A.5 we show the diagrams for 216? —+ bbW+(-> 3+ Vt), the major W + 2 jets background to W-gluon fusion including b-tagging. The matrix element in the helicity amplitude formalism is M(+,-) = 4<&-Ifi+><61+Ia—><é+l I1. Ib‘2+)/q2 +4(3-|52+)(é+|fi-)(51 +| /- | fi+)/r2 6The color factor for the amplitude squared is 3 x 3 x 1/3 x 1/3 for this process. 106 M(—.+) = 4<&—Ifz+>(62+lfi—><é+l II.II>'1+>/q2 +4(&—|61+)(é+|22-)(52+| /. |1i+)/r2 (A.47) where we have indicated the helicity states of b and b as .M(/\;,1 , A5,). We have once again left out the factor of 9w 2«(213.»(2152:)(2EI.)(2Is"».)(2E.)<2E..) 9% E (92w -—M3v) (“8) and color matrices. 7 7The color factor for the amplitude squared is 2 x 1 / 3 x 1 / 3 for this process. Appendix B Event Rate of the (2 -—> 3) Process ug -—> dtb Monte Carlo integration is an indispensable tool in phenomenology. However, when performing a calculation one often encounters singularities which make it impossible to obtain meaningful results from a Monte Carlo program. In the case of delta functions one is forced to integrate by hand. Other singularities may occur when propagators go on mass—shell. These types of divergences may be regularized by applying suitable cuts on the external particles in the process. However, when one is interested in obtaining a total rate, part of the calculation must be performed by hand if there is any singularity present. To obtain the total rate for the (2 —) 3) process ug —) dtb via Monte Carlo Figure B.1: Diagrams for ug —> dtb. 107 108 integration, the idea is to integrate out the W+g -> tb sub-cross section in Figure (8.1) by hand. The alternative is to use Monte Carlo for the full 3-body phase space, but the singularity due to the small mass of the b quark in the right diagram would require too much computer time even for the powerful technique applied in VEGAS, a fortran code for calculating multiple dimensional integrations [64]. An added benefit of performing the sub-cross section integration in the way shown in this appendix is an understanding of the validity of the effective—W approximation[33]. The differential cross section for this process is Pl] 0'3" a — 23 (27r)3(2E,,) ”b where - d3t (131) Uaub = [(2%)3(2E¢) (2W)3(2Eb) WP (21046402 + k — n — t — b). (3.2) Since a”), is a Lorentz invariant, we choose to evaluate it and the matrix element in the tb center of mass (CMS) frame. We express the 4-vector components of t, b, e, n and k explicitly as t = (Ehp sin0 cos (15,17 sin0 sin ¢,p cos 0) b = (Eb, —p sin9 cos 45, —p sin0 sin 4), —p cos 0) e = E,(1,sin0,,0,cos0e) n = En(1,sin0n,0,coso,,) k = Ek(1,0,0, 1). (13.3) In the tb rest frame 5 E (t + b)2, so §+m?-m§ E; = A NE ~ 2 2 s—m,+mb 2v: 109 p = él—fiw- (m.+m.,)2IIs“- (m. —mb)21. (3.4) It is easy to see in the tb CMS, that E, 1?, ii all lie in the same plane. Momentum conservation (e+k = n+t+b) and {45: flimply é’+ic.= ii. It only takes two vectors to describe a plane and any vector which is a linear combination of those two vectors lies in that plane. Therefore, we choose the vectors 6, E, ii to define the 23-2 plane with the momentum of the gluon (1:) along the z—axis. For a 3-body final state, there are only 5 (= 3 x 3 —4) independent variables. (The minus 4 is for energy-momentum conservation.) We shall choose the 5 independent variables to be 6, <13, Eu, 0,, and 45,, where ¢n can be trivially integrated as qufin = 21r. This is a consequence of the arbitrariness of the choice of scattering planes. Hence, we should be able to express all vectors in terms of the 4 variables 0, <15, En and 0“. What we have done so far is express all 4-vectors in the tb CMS. It is now a fairly trivial exersize to evaluate cub} We found _ _1_ .13. “‘—2 0“, _ 4(27T)2 J; ldcosfld¢|M| . (13.5) Having done so, we now have a Lorentz invariant expression for Usub = Usub(31Ena on) To compare our result with the effective—W approximation, we express En and 0,, in terms of ck CMS quantities with the aid of some projection operators defined below. To extract out the information of e, k and n in the tb CMS frame we use the fact that 5+ 12— ii = 0 and (e + k — n)2 = 5‘. We then define the projection operators P E and P“ as follows: (8 + k -— n) - V \/§ 1This integration is done analytically to avoid bad convergence in the numerical integration method. All the singular terms are of the nature of ln(m§). Because the analytic form for this result is long, it will not be explicitly given here. PEV E (B.6) 110 —k - V + EkEV El: PHV E (13.7) where V = 8,1: or k. P E and P" project out the energy of V and component of V parallel to k (i.e., the z-axis) respectively. We express the 4-vector components of e, n and k explicitly as (1,0,0,1) IE ”IS = (1,0,0,-1) ~ M =E (1, —sin 0;, 0, —cosO;,) :3 (BS) where s = (e + k)2. From now on we will distinguish ek CMS components from tb CMS components by a prime. We assume I n E =E;(1—x)=-\2—/g(l-x) and use the following results :3 w2 (e+k—n)2=xs (e — n)2 = -§(1— 1:)(1— c030,"). Using the projection operators it is easy to show El: ell "ll §I 1 (2:3 — W2) 2 l 2.5:: 1 2V5? El: (3 + W2) 3(1—27) E c030 - -——1--(W2 + 3(1 3)) + E e C — 2Ek n Encosfln = if: + E. (3.10) 111 and therefore Usub = asub(sax1 W2)- Also, the Lorentz invariant phase space integral in terms of ek CMS components becomes d3n' 1 1 [_(21r)3(2E;) -_— (4775/0 d2: _‘(l_r)dW2. (3.11) Finally, the following cross section 1 1 o = —— 2 su W2 ' a 2s(47r)2./odx/;,(1—z)dw a b(s,:1:, ) (B 12) may be folded in with parton distributions and safely evaluated using a Monte Carlo program. We note that W172 is the virtuality of the W—boson line emitted from the u—quark line (with momentum e). (1 —:c) is the fraction of the incoming u—quark energy carried away by the outgoing d—quark line in the ek CMS. If desired, one can approximate the above equation as the result of the effective—W approximation. However, we shall not pursue it further here. Appendix C Helicity Amplitudes of t —> W+b and f —> W-b In Equations (6.1) and (6.2) we have listed the most general form factors for the decay processes t —> W+ + b and t- —) W’ + b. Here we use those equations to calculate the helicity amplitudes for an on—shell W—boson. (We take the limit of 172,, -—> 0 in the following for m, > m5.) For the decay process t —> W+b, the top quark is taken to decay in its rest frame where the top quark momentum is p, = (m,, 0, 0, 0). Spherical coordinates are used to describe the outgoing particles; 0 is taken from the positive Z —axis and 45 is taken from the positive X —axis in the X — Y plane. The bottom quark and the W—boson are taken on their mass shells with the four—momenta for the bottom quark (pb) and the W—boson (pw) taken as 19,, (E5, -—Eb sin9cos 45, —Eb sinflsin ¢, —Eb cos 0), pw (Ew,Eb sin0cos ¢,E¢, sinflsin<1>,ElJ cos0), (C.1) where we have neglected the bottom quark mass, and m? — M3,, 2m, . 112 Eb: 113 The angles 0 and (I) refer to the direction of the W-boson. Denote the helicity amplitudes as (h,,z\w,hb) with Aw = —,+,0 being a left- handed, right-handed, and longitudinal W—boson. After suppressing the common factor “'9 / E 2Ebmt, ((3.3) there are 8 nonvanishing helicity amplitudes in the rest frame of the top quark for mb=0z : . 0 (—0—) = (fiff+ff)sm§, 1 9 - (___, -.- «1(11+,7;11)ms,ew, (+0-) = (£11+f:)coseei¢, MW 2 (+ - —) = —J§ (if + fig) sin gem, (— 0 +) = - (33%;? + 3) 3e41, (— + +) = -\/§ (ff + Elna-ff) sin gee—2‘43, (+ 0 +) = (fiffi + f2”) sin 3, (+ + +) = —J2— (ff + 553) cos $6"? (0.4) To obtain the averaged amplitude squared, a spin factor % should be included. We note that there is no right-handed W—boson produced with a massless left-handed b from a top quark decay. Similarly, from helicity conservation, it is not possible to have a left-handed W-boson produced with a massless right-handed b from t decay. 114 For an unpolarized top quark decay, after summing over the helicities of the bot- tom quark, the amplitudes squared for various W polarizations are, apart from a common factor (g2Ebmt)1 2 mm = ->12 = ff” + ”—73 , MW 9 |M(»\w = +)I2 = |ff + —12 |M(/\w=0)|2 = 53% Wf1+f.(+ +-|-—— 3+3 (0.5) The fraction (fume) of longitudinally polarized W-boson produced in the rest frame of the top quark is defined as the ratio of the number of longitudinally polarized W—bosons produced with respect to the total number of W—bosons produced in top quark decays: hm I‘(Aw = 0) + row = —) + I‘(,\W = +) M(Aw =0) 2 WWW = 0W + |M(/\w = —) 2 + |M()\W = +)|2, (cs) where we use I‘(/\w) to refer to the decay rate for a top quark to decay into a W—boson with polarization Aw. Using a parallel definition for the process { —) W‘b, we obtain the helicity am— plitudes (hz, Aw,h5), similar to the ones listed in Equation (CA) for the t —) W+ b process, provided we replace flL by ff", f1” by f1", f2" by f2”, and f2” by f2“. (Here the superscript * means complex conjugate.) 115 The helicity amplitudes of the process W+ -) e+ue are well known. After sup— pressing the common factor (ng), they are (Aw = _) = _e-id>: (29133.) , 2 sin0“ A = o = ——£, ( W ) fl - . 1+ 0" where 0; and cf); refer to 62+ in the rest frame of W+. The helicity amplitudes (Aw) for the decay process W’ —-> 6’17e can be obtained from Equation (Q?) by replacing 0; by 7r - 0; and 45; by 7r + ¢;. In this case, 6; and 45; refer to e‘ in the rest frame of W'. Appendix D The Total Rate for W—gluon Fusion As discussed in Section 2, the total rate for the W—gluon fusion process is obtained by Total = (2 '7’ 2) + (2 —> 3) — (splittingpiece) and the rates of (2 "’ 2) = [€151 €152 qu/A(€1,l1)fb/3(€2,#)5(Q'b —> qt) + [da «152 311(53):) macaw q qt) + [ta «ta Map) 3,361.11) quq —> qt) (us) where, for instance, fb/A(£1, u) denotes the parton distribution function (PDF) of the b quark inside hadron A, carrying the fraction {1 of the hadron momentum, and p is the energy scale at which the PDF is evaluated. The constituent cross section (7 is given by the difierential cross section for p1(m1) +p2(m2) —> p’IIm’l) +pa(m’2) + - - - + p’.(m£.). (3.4) 116 117 as shown in Figure D1 The differential cross section is 2d . ’ ... ' (Ml n(p1+p2.p1, ,Pn) (13.5) d& = 4\/(P1 ' P2)2 " mfmg with n n (131)! , I I = 4 4 _ I_ __1__ d@n(pl +P21p11 Ipn) (271') 6 (p1+p2 Zpa) i=l-Il (27F)3dE: (D'G) i=1 and W is the square of the amplitude after summing over the final state polariza- tion and colors and averaging over the initial state polarization and colors. Notice that the differential cross section for d&(bq’ —) qt) can be obtained from d&(q’ b —> qt) by interchanging the 4-momenta p1 and p2 in the scattering amplitudes. In terms of the polar angle 0" and azimuthal angle (19“ defined in the center-of-mass frame of the partons, this means replace 0" by 7r - 9" and a)“ by 7r + 45". In Equation (D.3), the effective parton density ~ _ 015(1) 112 dz .2 +(1— 2)? e fb/A(€, l1) - 27f 111 (31—3) /-2_ [ 2 ]f9/A (Ed!) (D7) in the MS scheme. The coupling constant 127r (33 - 2n;)1n(${-§)° 0501) = (D8) Here, n f is the number of quarks with mass less than the energy scale )1. The QCD (_n_,_) MS is an experimentally determined parameter. Perturbative QCD parameter A E A is presumed to be applicable for )1 >> A. For CTEQ2L PDF, A91)§ = 190 MeV. M , I I plsml Pia m1 I p’2,m2 O O O I I p'bm‘l pn’mn Figure D.1: n—body scattering _ -- r-"l7.t.xu (AT I Appendix E The Eikonal Approximation for 0(AB(ucI) ——> bbW+ + jet) Applying the Eikonal approximation, we can write the cross section of a(AB(uJ) —) bbW+ + jet) in terms of the amplitude square of the ad —I bbW+ process as follows. a(AB(uJ) —> 5bW+ + jet) = [dQ2 fdy/dq§./d<1>3-21—S ($5153) ) x (12,/A (x... Q) [2: P33, o I,” + Pg}, o my] (:23, Q) + (El) 2 fu/A($AaQ) fJ/A($B,Q) [03111 (a?) - Earl} |M(ud—> bbW+) (2 + (A H B), where (14);, is the usual 3-dimensional phase space volume as defined in Appendix B. Q, y and 91 are the invariant mass, rapidity and transverse momentum of the (W + b + b) system. IM (ud -) bbW+) |2 is the amplitude square of at? —+ bbW+ after summing over the spin and color factors in the final state and averaging over the spin and color factors in the initial state. For a given Q, y and qT l‘A = 79—53%, 13 = -%e’y, (E2) where x/S is the center-of-mass energy of the hadrons A and B. 118 119 The splitting functions are 1+22 Pig—)1 z) = CF(1_ Z) ‘5ij (E3) + 3,534.2) = $02 + (1 — z)2) (EA) and 1 _ 19% f (P‘)°f)(x,Q) — /. 619%,) me, ). (Es) (E.6) where the indices j and 1: denote the flavor of quark or antiquark, 6,”- is equal to 1 for Ic = j and zero otherwise. In QCD, C'p = g for three colors. The “+” prescription is defined by /l dz(1+ 22) f(2) = [,1 qz<1+ 22)f(z)0(z — x) - <1 + 22” (1), (E7) 3 + 1—2 1—z where 1 for z > :1: 9(2 — x) — { 0 otherwise ' (E8) The above result holds in the soft-gluon approximation. We have also assumed that the initial state QCD radiation dominates the soft-gluon radiation from 213 —> bbW+. This should be a good approximation because the b quark is massive and is less likely to radiate gluons as compared to the initial state quark or gluon. Appendix F The Computer Program ONETOP Our analysis is based on our Monte Carlo program ON ETOP, created by modifying PAPAGENO (version 3.07), written by Ian Hinchlifi'e. ONETOP contains code for parton level analyses of single top-quark production at hadron colliders as well as the major background. The top quark decays on-shell to bW+ with branching ratio Br = 1. All final state W’s decay on-shell to eu with branching ratio Br = {DID-l In addition, we implemented QCD tt- production with the top quark decaying on- shell according to the Effective Lagrangian of Equation(6.1), which includes the most general t-b-W couplings. Only the CTEQ2 leading order parton distribution is implemented[27]. ONETOP accepts matrix elements calculated using the helicity amplitude method described in Appendix A. 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