WESiS MIC CHIGANS IIUMMWHHINNIH!!!I!!!HIHHH'UUIWIW 55 3195 W This is to certify that the dissertation entitled "Probing the Symmetry-Breaking Mechanism through the Electroweak Interact ions of the Top Quar " presented by EhabMalkawi has been accepted towards fulfillment”. of the requirements for Ph. D. degree in PhVSiCS %M/%W/m a )0!‘ prostor Date May 21, 1996 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State UnIversIty PLACE IN RETURN BOX to tomovothb ohookout Itom your rooord. TO AVOID FINES return on or bdon date duo. DATE DUE DATE DUE DATE DUE MSU Io An Affirmative Action/Equal Opponunlty Inotltwon mm: PROBING THE SYMMETRY-BREAKING MECHANISM THROUGH THE ELECTROWEAK INTERACTIONS OF THE TOP QUARK By Ehab Omar Malkawi 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 1996 ABSTRACT PROBING THE SYMMETRY-BREAKING MECHANISM THROUGH THE ELECTROWEAK INTERACTIONS OF THE TOP QUARK By Ehab Malkawi Since the top quark mass is of the order of the symmetry-breaking scale, the top quark is likely to provide useful hints about the symmetry-breaking mechanism responsible for generating the gauge boson masses and at least connected with the fermion mass generation mechanism. I propose to probe the electroweak symmetry- breaking sector by measuring the effective couplings of the top quark to the gauge bosons. Different scenarios of electroweak symmetry-breaking will imply different correlations among these couplings. Using precision LEP and SLC data, I constrain the nonuniversal couplings of the top quark to the gauge bosons using the electroweak chiral Lagrangian framework. Constraining these couplings will provide an estimate for possible deviation in the gauge universality advocated in the SM. At the order of m? In A2, in which A ~ 47w is the cutoff scale of the effective theory, new physics in the left-handed neutral current is already constrained by LEP data. In models with an approximate custodial symmetry, a positive new physics contribution in the left-handed charged current is preferred. The right-handed neutral current can be constrained by studying the direct detection of the top quark at the Tevatron and the LHC. At the LC, the neutral current can be better measured. It is also interesting to note that due to the nonstandard couplings of the top quark to the gauge bosons, the upper bound on the top quark mass, from radiative corrections, can be raised from the SM bound m. < 200 GeV to as large as 300 GeV. That is to say, if there is new physics associated with the top quark, it is possible to say that (from radiative corrections) the top quark is heavier than what the SM predicts. Also, I present a theoretical frame work to extract the pure m. corrections to the low energy data in the chiral Lagrangian framework. The result is useful and inter- esting for him reasons: First, it simplifies the whole process of calculating radiative corrections. Second, this approach is shown to clearly identify observables which are sensitive to the symmetry-breaking sector of the electroweak theory. Finally, I present a self-contained model which demonstrates how the nonstandard top quark couplings to the gauge bosons can be generated. The model has a very rich structure and significant implications at low and high energy scales. Using the low energy data I discuss the possible constraints on the model. On the other hand, high energy colliders will provide further tests and demonstrate possible new physics especially interesting FCNC processes. To my parents, my lovely wife, and our expected baby. iv ACKNOWLEDGEMENTS I would like to express my deepest gratitude to the people who have contributed to me and to this work: To my thesis advisor C.-P. Yuan for teaching me high energy physics (up to date), for his calmness, and his support; To the members of my Thesis Committee: C. Brock, W. Repko, S. D. Mahanti and A. Brown, for their support; To my colleagues in the High Energy Theory group: Doug Carlson, Fransisco Larios, Tim Tait, Mike Wiest, Csaba Balasz, Liang—Hung Lai and Xiaoning Wang. Also to Kate Frame and Simona Murgia from the High Energy Experimental group; To my lovely wife, Amal, for her patience, her support, and for the long nights she was waiting and standing by; To my parents who wished to be present during my defense but could not, thanks for everything. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES 1 2 The Standard Model and Precision Tests 1.1 The Electroweak Standard Model .................... 1.2 Radiative Corrections ........................... 1.2.1 QED Corrections ......................... 1.3 Renormalization .............................. 1.4 The f Parameters ............................. 1.5 The SM Heavy m, and m” Contributions to the Low-Energy Data 1.5.1 Heavy Top Quark Contributions ................. 1.5.2 Heavy Higgs Boson Contributions ................ 1.6 Status of the SM ............................. The Chiral Lagrangian 2.1 Physics Beyond the SM .......................... 2.2 Model Independent Analysis ....................... vi xi 10 .13 28 35 35 38 39 48 48 50 2.3 Introduction to the Chiral Lagrangian ................. . 53 2.4 Heavy Top Quark Contribution to the 5 Parameters in the Chiral La- grangian .................................. 61 Global Analysis of the Top Quark Couplings to the Electroweak Gauge Bosons 65 3.1 Motivations and Perspectives ...................... 65 3.2 The Top Quark Couplings to Gauge Bosons in the Chiral Lagrangian Framework ................................. 72 3.3 Low Energy Constraints ......................... 77 3.3.1 General Case ........................... 78 3.3.2 Special Case ............................ 92 3.3.3 At the SLC ........................ , . . . . 107 3.4 Heavy Higgs Boson Limit in the SM ................... 114 3.5 Direct Measurement of the Top Quark Couplings ........... 116 3.5.1 At the Tevatron and the LHC .................. 117 3.5.2 At the LC ............................ 121 3.6 Discussion and Conclusions ....................... 122 Heavy Top Quark Effects and the Scalar Sector 125 4.1 Introduction . . . . ............................ 125 4.2 Large m, effects in the SM ........................ 127 4.3 Large m, Effects In the Chiral Lagrangian ............... 131 4.3.1 Effective Lagrangian ....................... 136 vii 4.3.2 Renormalization ................. . ......... 140 4.3.3 Low Energy Observables ..................... 142 4.4 One Loop Corrections in the SM .................... 144 4.5 One Loop Corrections with Nonstandard Top Quark Couplings . . . . 145 4.6 Conclusions ................................ 148 A Model of Strong Flavor Dynamics for the Top Quark 150 5.1 Introduction ................................ 150 5.2 The Model ............................ _ ..... 151 5.2.1 The Bosonic Sector ........................ 152 5.2.2 The Fermion Sector ........................ 156 5.3 Low Energy Constraints ......................... 162 5.4 High Energy Experiments ....................... 172 Discussions and Conclusions 180 Renormalization Schemes 184 The S, T, U parameters 194 Heavy Mass Expansion 199 Non-Linear Realization 203 D.1 Linear Realization ............................ 203 D2 Non-linear Realization .......................... 204 viii LIST OF REFERENCES 213 ix 1.1 1.2 3.1 5.1 5.2 LIST OF TABLES The input parameters used to calculate the SM predictions in Table 1.2. 46 Experimental and predicted values of electroweak observables for the SM. The SM predictions are calculated using the input values in Table 1.1. Columns a and b are for a, = 0.125 and 0.115, respectively. . . . 47 The confined range of the couplings, KL and KR for various top quark masses and for my = 65 GeV ....................... 100 Experimental and predicted values of electroweak observables for the SM and the proposed model (with different choices of parameters) for a, = 0.125 with m, = 175 GeV and my = 300 GeV ........... 178 Experimental and predicted values of electroweak observables for the SM and the proposed model (with different choices of parameters) for a, = 0.115 with m, = 175 GeV and my = 300 GeV ........... 179 1.1 1.2 1.3 1.4 1.5 1.6 2.1 3.1 3.2 3.3 3.4 3.5 LIST OF FIGURES a: The Feynman diagrams contributing to the running a up to one- loop level. b: The Feynman diagrams contributing to the running a summed to all orders ............................ The whole set of one-loop corrections needed to renormalize the SM parameters 0, Op, and M2. ....................... The one—loop corrections to the coupling 7—e‘-e+ ............ a: The vacuum polarization function of the Z boson, up to one loop. b: The one-loop corrections to the p decay. All corrections, except the W self-energy, are collectively denoted by JG“; ............. The top quark contribution to the epsilon parameters, at the one-loop level ..................................... The Higgs boson contribution to the epsilon parameters, at the one- loop level. ................................. The relevant Feynman diagrams in calculating the top quark contribu- tion to the epsilon parameters using the chiral Lagrangian approach. . The relevant Feynman diagrams, for the nonstandard top quark cou- plings case and in the ’t Hooft—Feynman gauge, which contribute to the order 0(mt21n A2). .......................... A two—dimensional projection in the plane of K50 and xfiC, for m, = 160 GeV (solid contour) and 180 GeV (dashed contour). The Higgs boson mass is fixed, m” = 65 GeV. ................... A two—dimensional projection in the plane of K}: C and NEC, for m, = 160 GeV (solid contour) and 180 GeV (dashed contour). The Higgs boson mass is fixed, m” = 65 GeV. ................... A two—dimensional projection in the plane of male and NEC, for m, = 160 GeV (solid contour) and 180 GeV (dashed contour). The Higgs boson mass is fixed, my = 65 GeV. A two—dimensional projection in the plane of sic and K150, for m; = 160 GeV (solid contour) and 180 GeV (dashed contour), and for the heavy Higgs boson mass my = 1000 GeV. ............... xi 14 16 19 20 '37 4O 64 82 87 88 89 3.6 The allowed region of KL and KR, for m, = 170 GeV, my = 65 GeV. (Note that m, = #20 = 2K€C and KR = KEG.) ............. 3.7 The allowed range of (KL — 253) as a function of the mass of the top quark and for my = 65 GeV. (Note that KL = KQ’C = 2n?) and K}; = KEG.) ................................ 3.8 The allowed range of the coupling REC = ref C / 2 = KL / 2 as a function of the mass of the top quark and for m” = 65 GeV ........... 3.9 The allowed range of the coupling KEG = rag C / 2 = KL / 2 as a function of the mass of the top quark and for m H = 300 GeV. ......... 3.10 The allowed range of the coupling KEG = a}? C / 2 = KL / 2 as a function of the mass of the top quark and for my = 1000 GeV .......... 3.11 The allowed region of 51:6 and team, for models without a SM Higgs boson and for m; = 170 GeV. ...................... 3.12 A comparison between our model and the model in Ref. [95 . The allowed regions in both models are shown on the plane of KLC and nfic, for m, = 170 GeV and my = 300 GeV ............... 3.13 The allowed region of KL and ran, using the SLC measurement Aug, for m, = 170 GeV and m” = 300 GeV. ................. 3.14 The allowed region of KL and KR, using LEP data and the SLC mea- surement of Aug, for m. = 170 GeV and my = 300 GeV. ....... 3.15 The overlapping of the two measurements A, and A L R as a function of the top quark mass. Negative values of It indicates overlapping, while positive values indicates no overlapping. ................ 3.16 The Feynman diagrams needed to calculate the effective couplings of the top quark to the W and Z gauge bosons ............... 3.17 The allowed Ingcl and KEG are bounded within the two dashed (solid) lines for a 20% (50%) error on the measurement of the single—top pro- duction rate, for a 175 GeV top quark. ................. 4.1 The Feynman diagrams which contribute to p and T to the order 0(mt21n A2) ................................. 5.1 The Feynman diagrams for the process 7' -> ppp. ........... 5.2 The lower bound on the heavy Z’ mass as a function of sin2 (I) for 3a, sin2 ,6 = 0 (solid) and sin2 fl = 1 (dashed) and a, = 0.125 ....... 5.3 The lower bound on the heavy Z' mass as a function of sin2 (I) for 3a, sin2 ,6 = 0 (solid) and sin2 3 = 1 (dashed) and a, = 0.115 ........ 5.4 The event number of pir": produced at the LC, with a c.m. energy of 500 GeV, and for two choices of parameters. .............. xii 100 101 102 103 104 106 110 112 113 115 120 146 166 169 170 176 Chapter 1 The Standard Model and Precision Tests The Standard Model is a SU(3)C x SU (2) L x U (1)y gauge invariant quantum field theory of the strong and electroweak interactions. The Standard Model has been very successful in describing the fermions’ (quarks and leptons) interactions via the force mediators (gauge bosons). The gauge symmetry S U (3)0 is associated with the strong interaction (QCD) of the quarks via the corresponding gauge bosons known as the gluons. The remaining symmetry 3 U (2) L x U (1)y known as the electroweak symmetry is composed of the weak S U (2) L and the hypercharge U (1)y symmetries. The electroweak symmetry governs the unified weak and electromagnetic interac- tions, collectively, know as the electroweak interactions. To allow for the generation of the weak gauge boson masses the gauge symmetry S U (2) L x U (l)y must be bro- ken into the electromagnetic symmetry group U (1)em. This breakdown is triggered through the spontaneous symmetry breaking. The intimate connection between the electroweak interactions and the symmetry breaking mechanism constitutes the major issue I will explore throughout this body of work. Therefore, throughout this study I will be mainly concentrating on the electroweak interactions. In the next section I will outline the basic features of the electroweak Standard Model. For a more detailed 2 discussion the reader can refer to existing literature [1, 2, 3, 4, 5]. 1.1 The Electroweak Standard Model The electroweak Standard Model (SM) is based on the gauge group SU (2) L x U (1)y with the following basic structures: 0 The fermions are assembled in three families or generations with left-handed doublets and right-handed singlets under the S U (2);, group: 9 9 R9 Ra Ra e L d L y La 3 L9 ”Ra Ra R) V, t (1. )La (b)La T39 bRa tR- (1.1) The hypercharge quantum number Y] of a fermion f is fixed by the Gell-Mann- Nishijima relation Y Q; = T31 + 31’ (1.2) where Q I is the electric charge in units of e and T3; is the weak isospin quantum number. For example, the left-handed electron 8;, has Qe = -—1 and T3e = —1/2. A left—handed fermion doublet will be denoted by In, whereas a right-handed fermion singlet denoted by \IIR. The fermionic Lagrangian is given by cm = ZEI7”D,,\II£ + ZIP—giyuppxrf, , (1.3) f f where f = 1,2,3 is a family index. Also, Ta 13er = (a, - 2'9 2 Y w; — ig’EBp) \IIL, (1.4) 0er = (a, — ig’QBp) tn, (1.5) where a = 1,2,3 is an isospin index. The gauge boson field W; with the gauge coupling 9 and the gauge boson field B” with the gauge coupling 9’ are associated with the gauge groups SU (2) L and U (1),», respectively. The Pauli matrices 'r“’s are normalized according to the relation: Trace('r“'rb) = 26"". - (1.6) There are 4 gauge bosons transmitting the electroweak force: A (photon), W+, W", Z. (1.7) The self-interactions of the gauge bosons are given by 1 I! 1 a a V LGK = jaws" — ZWWW # , (1.8) where 3,“, = 6,3,, — 3.3,, (1.9) W3, = 6,.W3 — aw; + geabcwgwg, (1.10) and with the field definitions W11FIW2 w=t = ————, 1.11 fl ( ) Z=cosOW3—sin03, A=sin0W3+cosdB, (1.12) where 0 is the weak mixing angle and cos2 0 = 1 — sin2 0 . 4 o A complex scalar doublet field with hypercharge Y = —1 implements the spon- taneous symmetry—breaking = ((v+H+i¢o) /\/§) .4)- (1.13) where the fields ¢o and 45* are called the would-be Goldstone bosons and the neutral field H is called the Higgs boson. The scalar doublet field has a non-vanishing vacuum expectation value (v.e.v) < >, where <>= (”a/5). (1.14) The components (13*, ¢° are unphysical and can be gauged away in a specific gauge known as the unitary gauge. The scalar Lagrangian is given by A 122’ 2 cu. = (D..1*(D”> - 5 («NW — 7) . (1.15) where . r“ . 1 D,, = (a, -- 193%: + 1933,.) <1. (1.16) Because of the v.e.v, spontaneous symmetry-breaking is triggered and the Wi and Z bosons are rendered massive: 2 2 MW_2'1 MZ=_——\/9+5’” 2 , 2 (1.17) The fermion masses are generated through the fermion interactions to the scalar doublet (Yukawa interactions), e.g., for the third generation m___¢\/2 fimb ( £Yukawa = —-—(tL bL) (DIR + IL EL) (—IT2(I)') b}; + Jim, (. _ v 11;, TL) (—i'rg"') T}; + hermitian conjugate. (1.18) 5 Due to the generation of quark masses for both the up and down type quarks, mixing between different quark families is possible. The mixing is described by a unitary matrix VCKM which consists of three mixing angles and one phase. No mixing in the lepton sector can be generated within the SM because neutrinos are massless. The full electroweak gauge invariant SM Lagrangian can be written as ['SM = £FK + CGK + £¢K + LYukawa- (1.19) By examining Eq. (1.19) one concludes that the SM has a certain number of free parameters: 9, g’ , v, A, in addition to the fermion masses m; and the quark mixing matrix 1. Those free parameters have to be determined from experimental data. The parameter /\ can be traded for the Higgs boson mass m H = vx/X. The Higgs boson mass is experimentally constrained to be above 65 GeV [6]. Also, theoretical arguments suggest an upper bound on mg of about 1 TeV [7, 8]. All fermion masses [9] except the top quark mass [10, 11] are below the M 2 scale [12]. Throughout this work, I will refer to fermions with masses below the M 2 scale by light fermions. Also, by low energy experiments I mean experiments operating at the M 2 scale or below. 1.2 Radiative Corrections At tree level the low energy electroweak observables can be entirely written in terms of the three free parameters 9, g’, and v in addition to the light fermion masses and the quark mixing matrix VCKM. The top quark mass m, and the Higgs boson mass my only enter through radiative corrections. Thus, given the light fermion masses and the quark mixing matrix, three additional measurable quantities are needed to fix the free parameters 9, g’, and v. The three observables chosen must be precisely 1Also, from the QCD sector, there is the strong gauge coupling 9,. 6 measured and theoretically calculable in a clean way. After this procedure of defining the physical input, other observables can be predicted and compared with the cor- responding experimental data. Different choices of the basic measured observables correspond to different renormalization schemes. In general, loop calculations are divergent and therefore, one needs to regularize the theory. Different regularization schemes are used in the literature, e. 9., using a momentum-cutoff, dimensional reg- ularization, etc. The tree-level (bare) parameters in the Lagrangian have no direct physical meaning and can be replaced by the renormalized parameters, e. g., the bare gauge coupling go can be written in terms of the renormalized coupling 9 9=Qo+59, (1.20) where 69 is called the coupling counterterm. The renormalized parameters are finite by definition and can be fixed by a set of measurable quantities. By this redefinition of the parameters the one-loop amplitudes are rendered finite. However, only the divergent part of the counterterm is fixed by the requirement of divergence cancela- tion. The finite part is somewhat arbitrary and thus calculated quantities depend on the renormalization scheme. To any perturbative order, the difference in a calculated quantity using two different renormalization schemes is of a higher order, e.g., for a one-loop calculation the difference is of the order of a two-loop correction. All renormalization schemes are equivalent if calculations are performed to all orders. Since only a perturbative fixed order calculation is usually possible, one should be consistent in defining the renormalization scheme and using the calculated numbers in that particular scheme. Different renormalization schemes are implemented in the literature. In appendix A, I discuss in some detail different renormalization schemes. Here, I briefly mention some of the most commonly used schemes: 0 The Z-pole scheme, where the input observables are chosen to be: 7 — The electromagnetic coupling a = e2/47r measured from electron-proton (e-p) scattering in the limit of zero momentum transfer q2 —+ 0 (Thomson limit) [9] (1'1 = 137.0359895(61). (1.21) - The Fermi coupling constant G p measured from the muon lifetime “r” and the theoretical formula 1 @7715 mg a 25 2 2a m,, '1; "" 192773 (1—8m2) (1+; (RT—fl. )) (1+5-7Tln me), (122) p where QED corrections to the four-fermion interaction includes the one- loop correction and the leading correction in 02. The value of G p is [9]. Gp = 1.166389(22) x 10“5 GeV—2. (1.23) - The Z mass [12] M2 = 91.1885 :1: 0.0022 GeV . (1.24) In this scheme the weak mixing angle sin2 0 is defined to all orders by the relation 1 41m ”2 '20:.1— 205-1-1—-——— . 1.25 sm cos 2( [ fiGngj ) ( I By definition sin2 0 has no dependence on the top quark mass m, and the Higgs boson mass m”. o The on-shell scheme where a, M z, and MW are the input observables. In this scheme the weak mixing angle sin2 0 is defined to all orders as M2 - 2 w sm 0 =1— -——. (1.26) Mg Unfortunately, the W mass is not determined as precisely as M z [12, 13] Mw = 80.26 :1: 0.16 GeV. (1.27) 8 Therefore, sin2 0 is usually extracted from other data. In this case sin2 0 has a strong dependence on the top quark mass m,. o The MS scheme where only the divergent pieces in the loop calculations are absorbed by the counterterms. In this scheme, sin2 0 is defined as 8-3 = 1 — _, , (1.28) where MW and M; are the W and Z masses defined in the MS scheme. In this scheme the weak mixing angle 33 has some sensitivity on the top quark mass. For more information on renormalization schemes, the reader can refer to ap- pendix A. The main goal of this work is to browse through the top quark effects to low energy data. Therefore, it is advantageous to isolate the top quark effects to low energy data completely from the input parameters. For this reason, I will work with the Z-pole scheme since sin2 0, by definition, does not depend on the top quark mass. The basic observables a, G p, and M 2 can be written in terms of the parameters 9, g’, and v as follow 929,2 GP: 1 _\/92+9'2v = , _, M ______, 1.29 a W «82 Z 2 ‘ ’ These equations can be solved in favour of g, g’, and v as a function of a, G p, and Mg as follow - 1/2 1): 1 , = 47m, 9,: x/47ra, (1.30) J20)? s1n0 6080 where 1 47m ”2 - 2 2 _ sm0=1-c030=— 1—1——— 1.31 2 ( fiGFMfii ) ( ) 9 All other derived parameters in the Lagrangian can be written in terms of the input parameters. For the W mass we have M3}, = cos2 0 Mg. (1.32) The photon coupling to the fermions f—7 is given by Qn/mn- (1.33) The W* coupling to the fermions f- f’ is given by V47ra mfill " 75)- (1'34) The Z coupling to the fermions f-7 is given by ‘ 1/2 («.20ng) 7u(9Vf " 9/1175) , (1.35) where 9Vf = T3; - 2Q] Sinzg, 9A; = T3f~ (1.36) For the electron QC 2 —1, T3,. = —1/2. If we do not care at all about radiative corrections, it is already possible to predict the low energy observables. However, the existed low energy data are precise enough to force us into considering the radiative corrections. In the rest of this chapter I will confine the discussion to the physics at the Z -mass scale, i.e., the physics at LEP and SLC (e"’e" colliders at the Z pole). At one-loop level, the amplitude for the process e‘e"’ —+ f7 near the Z resonance can be factorized and written as A(e‘e+ —1 f7) = A, + Az, (1.37) 10 where A, and A z are the photon— and Z-exchange amplitudes, respectively. In Eq. (1.37) I have ignored contributions from box diagrams which have a negligi- ble effect at the Z pole [14]. Near the Z peak the total cross section as a function of the center of mass (c.m.) energy («3) can be written as 0,7(3) = az’ffis) + awfls) + a7z‘f7(s) . (1.38) Here, 0207(8) is the Z contribution to the cross section defined as _ 0 sI‘2Z WT“) ’ ”f7 (s — M3,)? + era/M3 ’ (1.39) where 0?? is the peak cross section which is connected to the Z partial decay widths for Z -) e‘e+ and Z —-> f7, 127rI‘CI‘, 0'0- : —- . u Mgrié (1.40) Eq. (1.39) is taken as the definition of the Z mass and the total decay width [‘2 [15, 16]. Notice that Eq. (1.39) is different from the Breit-Wigner shape by the s- dependence of the width [14]. The terms a7’f7(s) and a7z,f7(s) in Eq. (1.38) stand for the pure photon and the Z -photon interference contributions to the cross section. Near the Z peak, the largest contribution to the cross section 0,7(3) comes from the Z contribution. For the LEP physics it is customary to extract the pure QED effects and to isolate them from the Z contribution. Below I discuss the QED correc- tions. 1.2.1 QED Corrections The pure photon and the Z-photon interference contributions to the cross section 07.f7(8) and 07z,17(3)1 are theoretically calculated and subtracted from 0,7(8). The QED corrections can be summarized in the following items 11 o The initial-state photon radiation is the largest source of correction to the Breit- Wigner Z-line shape. This correction causes a reduction of about 25% in the peak cross section [15]. It can be counted for through a ”structure function” G(a:, s) that can be deconvoluted from the measured cross section 0'1“?“(3) 0}“79”(s) = jde(a:,s)a,7((1—:r)s), (1.41) where a: is the fraction of the initial momentum of the electron or the positron carried by the photon. The function G(:1:,s) is theoretically predicted to a good accuracy through renormalization group methods [17]. o The largest QED effect to the physical observables at the Z —pole comes from the change (running) in a due to the change in the energy scale from (12 = 0, where a is measured, to q2 = M; where physical observables are evaluated. This effect is related to the photon vacuum polarization function which can be written as Illuqz) = —ig,,,,q2F7"(q2) + qpqy terms. (1.42) Note that 1133(0) = 0 due to QED gauge invariance. In defining the running coupling a at the Z-pole I only consider the light fermions contribution to the photon vacuum polarization function. Hence, from the photon exchange at one loop (see Figure 1.1a) we have 00 00 a a 32- ,2 - 329F770?) = q—gu — 177702)], , (1.43) where no is the bare coupling as discussed in the next section. A redefinition (renormalization) of an is performed to render it finite, a, —> a[1+ 1177(0)] . (1.44) 12 Therefore, we get a(q2) = 0(1— F77(q2) + F77(0)) E a(1+ Aa(q2)) , (1.45) where Aa(q2) is a finite correction to the electromagnetic coupling a. The improved summed expression for a is (see Figure 1.1b) a In Aa(q2) I do not include the gauge bosons or the top quark contributions which are conventionally included in the remainder of the corrections. I write the light leptons and quarks contributions to Aa(q2) as Aa¢(q2) and Aaq(q2), respectively. At the Z -mass scale, the leptonic contribution to a is determined accurately, Aa((M§;) = -0.03142 [18], whereas for the quark contribution Aug, 3 perturbative calculation is not possible. The determination of A0,, is done using a dispersion integral over the measured cross section of e+e‘ -> hadrons [18]. This gives Aaqulg) = —0.0286(9) and therefore one obtains [18] a-1(M§) = 128.89 :1: 0.09, (1.47) where the largest uncertainty is coming from the hadron contribution. The effect of the running a is significant in the radiative correction analysis near the Z pole. With the coupling cr(q2 = 0) being replaced by the running coupling (1(q2 = Mg) in the tree-level Z contribution to the cross section, one obtains what is called the improved Born approximation for physics at the Z-pole. Final state-photon radiations are accounted for by including a factor (1 + 3aQ}/47r) multiplying the Z partial widths. Similarly, for the QCD correc- tions we include the factor RQCD in the hadronic partial widths, where [19] 2 3 RQCD = 1+ (9—5?) —1.4(-a—‘(-:—4§—)) — 12.4%?) , (1.48) 13 and a, is the QCD coupling, a, = gf/477. o The remaining QED corrections due to proper vertex correction and fermion self-energies are negligibly small at' the Z-pole and vanish for a real photon (s —) 0) [14]. The typical size of these corrections to the form factors is about 10‘3 relative to the tree-level [14]. By taking care of the photonic corrections theoretically and experimentally, one can concentrate on the Z contribution to the Z -pole physics. Even with the inclusion of the running coupling a(M§) in the Z -pole observables it has been shown that the weak radiative corrections are becoming more apparent and significant in low energy data [20]. In this case one needs to go through the whole renormalization procedure. 1.3 Renormalization In considering the renormalization procedure at one loop one needs to consider the whole set of one-loop level contributions, which can be summarized in the following corrections: 0 The vacuum polarization functions of the gauge bosons (see Figure 1.2a) written in the following form “39(92) = —z'g,,,, (742(0) + q2F‘j(q2)) + qpqu terms, (1.49) where i, j = W, Z, 7(photon). Alternatively instead of using Z and 7 one can use i, j = 3, 0 for W3 and B, respectively. The relation between the two cases is as follow A33 = cos2 0 A” + 2 sin 9 cos 07472 + sin2 0 A?1 , (1.50) 14 Figure 1.1: a: The Feynman diagrams contributing to the running a up to one-loop level. b: The Feynman diagrams contributing to the running a summed to all orders. 15 A30 = — c080 sin BAZZ + (cos2 0 — sin2 0 )A72 + 0056 sin (9747'1 , (1.51) A00 = sin2 0 A22 - 2sin 0 cos 0.472 + cos2 0 A77 , (1.52) and similarly for F '7 . In Eq. (1.49), there is a small imaginary part which I will ignore since it does not contribute at the one-loop level. 0 The contribution to the vector and the axial form factors at q2 = M g in the Z -+ f7 vertex from prOper vertex diagrams and fermion self energies only (see Figure 1.2b) 1/2 (750mg) 1.09” — 641,15). (1.53) o All the one-loop corrections except the W vacuum polarization to the ,u-decay amplitude at zero external momentum (see Figure 1.2c). All these corrections will be denoted by (SCH/.3 [16] 6—?7-‘1245EMI- 791/.) (W‘(1 — 19v.) . (1.54) The input bare parameters can be written in terms of the renormalized ones and the above one-loop corrections [16]. (1) For the electromagnetic coupling I write the bare parameter do in terms of the renormalized coupling a and the counterterm do as a=ao+6a. (1.55) The one-loop corrections to the effective coupling 7-e“-e+ at zero momentum transfer (12 = 0 are summarized in Figure 1.3. The bare coupling 7-e‘-e+ can be written, using Eq. (1.55), as 1 6 6‘7” (1 — 53') , (1.56) C! 16 Figure 1.2: The whole set of one-loop corrections needed to renormalize the SM parameters a, Gp, and Mg. 17 where 60/01 = 266 / e. The one-loop proper vertex and self-energies calculation can be written as 67p (59% - 7559/93) - (1-57) The photon wave function can be written as (1 + 1 /26Z.,) where (SZ7 = —F77(0). Adding all the one-loop corrections, one finds 1 (la v,z A7z(0) 1 87” (I — 2:7- — 2sin0c030 Mg + 69% + 2627) ac AIZ(0) 6 e — , . 4.67““ ( 9A 2sin0cosl9 Mg ) (1 58) where ve = -1/2 + 2sin20 and ac = —1/2 are the vector and axial-vector couplings of the Z boson to e" e+. In pure QED no 7-2 mixing exists and using the QED Ward identity one finds dgVe = 59.47: = 0 [21]. Therefore, the renormalized coupling reduces to 160 1 e(1— 5"; + 5527) . (1.59) By requiring that, in the limit q2 = 0, the renormalized coupling reduces to (e) we conclude that —— = ($27 = —F77(0), (1.60) In the SM case we have to include the 7-Z mixing which is mainly due to the W- boson loop. In the SM case one finds that the electromagnetic Ward-Takahashi identity guarantees [21] 1 1472(0) 4sin0cosl9 Mg 18 Also, one finds l—4sin20A’1Z )__ sin20 A720) 4sin9cos0 Mg _ sin9cos0M§ ' 69v. + (1.62) Therefore the coupling reduces to 160 l sin20 1472(0) e (1 - 2‘0— + 2627 — sin9cosl9 Mg ) ' (1'63) From which one finds . .72 ég:_F7./(O)_2sm0A (0) a c080 Mg (1.64) For the Z—mass renormalization I consider the one-loop corrections to the Z boson propagator (see Figure 1.4a). One finds 1 1 A22 0 21722 2 92 " M20 92 " M20 92 " M20 The full or dressed propagator is 1 —+ 1 1 _ «12 - M40 42 - Mg. 1+ “lghfigjz‘q’l 1 (1 66) 92 - M20 + A“(0) + 921722012) ' ° The physical mass M Z is identified with the position of the pole q2 — Mg, + AZZ(0) + 1121:2201?) = 0 at q2 = Mg. (1.67) Hence defining the mass counterterm as M; = 143,, + 6M3, (1.68) one finds 6M2 Azz 0 Z - — ( )— FZZ(M§). (1.69) M%_ M% 19 Y e e e 7 19%9 e Figure 1.3: The one-loop corrections to the coupling 'y-e’—e+. 20 Figure 1.4: a: The vacuum polarization function of the Z boson, up to one loop. b: The one-loop corrections to the p decay. All corrections, except the W self-energy, are collectively denoted by 60143. 21 (3) For the Fermi coupling G p, I compute the one-loop correction to the p decay (see Figure 1.4b). I denote all the proper vertex, box and self energies corrections by 6GV,B. Writing the counterterm as GP = GPO + 507*, (1.70) and adding the one-loop calculations with the counterterm to the p—decay am- plitude, one finds WW 2 WW 2 GP 1_§_G_1~_A (0)+9P2‘ (q)+6Gv,B . (1.71) GF q2 — MWO GP At q2 = 0 we have 60,. AWW(0) 601/3 1 — —— —- ——'— . . Gp( GP + Mgv + GP (172) By requiring that Eq. (1.72) reduces to CF one finds 6 AWW 0 6 GP - ( ) + -———GV'B . (1.73) “6*?" M3,, Gp All other derived quantities can be written in terms of the renormalized a, G p, M 2. and their counterterms. The renormalized weak mixing angle can be derived as follows sin20 = sin200 + 6811129, cos26 = cos2 00 + 6cos29, (1.74) where - 2 2 "0‘0 sm 90cos 00 = -———————. 1.75 fiGFoMgo ( ) Therefore, one finds 6cos20 = —6sin20 = sin29cos20 (_6a 6G; 6Mg), (1.76) c0820 —sin20 3+ Gp + Mg 22 where sin2 0 is defined to all orders as 1 4770: “2 sm0_1—c030=§(1—[1————\/§F g] ) (1.77) Equivalently, 1ra . 2 2 Sin 0 cos 0 = —— «20ng and c0826 2 1— sin2 0 . (1.78) The renormalized IV mass can be derived from the tree-level relation M3”, = cos2 OoMgo. (1.79) One finds M3,, = M3“, + 6M3} = cos2 GoMgo + 6M12v = cos2 6 Mg + 6M}; 2 2 2 M2 COS 9 (sin2 0 6—0 - sin2 6 EE— — cos2 0 6%) - (L80) sin20 — c0326 0 Op Mg By demanding that MW coincides with the physical (on-shell) mass, the counterterm 6M3], is fixed In summary, the tree-level (bare) S matrix 30 = 30(001GF01M20)1 (1-32) is replaced at one-loop level by a finite physical 3 matrix so -+ S(a,Gp, Mg) + 65(01, Gp, Mg, 111., m”), (1.83) where 65 encompasses the one-loop corrections and the induced shift in the renor- malized quantities. Notice that m, and my only appear in the radiative corrections. 23 The tree-level amplitude Z —+ f7 can be written as fiGmM§O_ 1 . 2 1 W247“ (—§ '1' 28111 60 + 755) ue X7177” (T3, — 2Q, sin200 — 7573,) 11;. (1.84) A(e’e+ —> Z —+ f?) Next, I include all the one-loop corrections and the counterterms to the amplitude A. Notice that the photon—Z mixing will also contribute to the amplitude. One finds A(e‘e+ —> Z —) f7) = 2 2 22 «220111422 (1 _ 56p __ 6M: _ FZZ(Mg) __ 11192 (M2)) q - M2 GP Mz dq 9:7,. (9v. + Agv. — 75 [91c + A9181) u. x 1777» (9v; + Agw - 75 [94; + A9411) "1, (1-85) where to simplify the notation, I defined 9v; = T3] - 2Qfsin20, 9/4; = T31, (1.86) . 2 . “472(0) Z 2 Agvf=2Qfdsln 0 +6gvf—2Qfsm0cosl9 M2 +17.1 (M2) , (1.87) 2 A91; = 5911;- (1-88) This result is slightly different from the result given in Ref. [16] by the inclusion of the term A7Z(0) in Eq. (1.87). At one-loop level, fermions do not contribute to 1472(0), i.e., A}Z(0) = 0. However, there is a small contribution from the W—boson loop [14]. Determining the amplitude enables us to calculate all physical observables at LEP, namely the partial decay widths, the forward-backward asymmetries, and the polarization measurements Ac and A,. The partial-decay widths can be written as follow N 1 - — — F M — — M 6N2 C< Gp Mg ( Z) dq2 ( Z) X ([gv; + Agvf]2 '1' [9A] + AgAf]2) , (1.89) NZ -* f7) 24 where NC- is a color factor, NC = 3 for a quark final state and 1 otherwise. The leptonic forward-backward asymmetry can be written as 3lgAz + AQAzlzlgvc + Agvzl2 2 1 ([9111 + AQA112 + [9w + AQVtI2) A‘FB = (1.90) where I have assumed lepton universality in A‘FB. In the SM, lepton universality is a valid statement. However, In chapter 5 I will discuss a model, different from the SM, which may break lepton universality. In the following discussion I will continue with the assumption of lepton universality. The e and 7' polarization observables A, and A, can be written as f __ 219.4! + AQAII [W] + Agv 1] ([94! + A9402 + law + Agwl?) where f stands for e or T. (1.91) If we ignore for the moment the non-oblique corrections to the form-factors Agvf and Ag,” ((5gi = 69,4, = 0) then it is clear that only oblique corrections (corrections independent of the fermion flavor) are present. Also, one finds that all LEP observ- ables can be written in terms of three combinations of oblique corrections. The first correction is the one multiplying the partial decay width in Eq. (1.89), namely - F”(M2) — fizz-(M2)) (192) 6Gp 6M g (lg2 ”'37 - M; The second one is seen clearly from the ratio 7Z £911; = 4|Qf| (68in20 -— sin0cos€ [AMgm + Flz(M§)]) - (1-93) 9.4; Z The third one is given in terms of the W mass, which can be seen clearly from Eq. (1.80). From Eq. (1.80) one can extract the third oblique correction, M117 coszoMg =1+ 6Mg, _ sin20 (£3 _ 60p 00820 6Mg) (194) Mg, sin20 -cos20 0 Op - sin 0 Mg 25 To summarize, ignoring all non-oblique corrections, there are only three combinations of oblique corrections which affect the Z —pole observables. To include the non-oblique corrections one has to make some general assumptions to make the analysis simple and useful as a model independent tool. Before I do this I consider the measurements of I‘(Z —) p'p+), A763, and MW with the non-oblique corrections included. I combine the term A9,,” with the oblique correction to the decay width I‘(Z -) 11" 11+) and define the first parameter Ap where 2 ZZ ApE _5_c_;_,: — LA? — FZZ(Mg F —— 2 — Gr M2 ) dq2(Mz) 489A”. (1.95) Using the form-factors ratio extracted from A323 one finds W}: + AgVu =1—4'n20—2A +21—4'26A . 1.96 9.1,. + A9,,” s1 91/11 ( 8m ) 911,1 ( ) I define the second parameter Ak’ where 9Vu+AgVu __ -2 r ————=1-4sm 61+Ak . 1.97 9M+A9Au ( ) ( ) From which one concludes , 1 Ak - 2sin20 (AgV, —- (1 — 431112 (9)4941) . (1.98) Using the parameters Ap and Ak’ one can write the observables F(Z —) [171") and .4ng GFM3 , 2 _ + _ Z 2 I . I‘(Z—>p 11 )-24N§(1+Ap)([1—4sm 6(1+Ak)] +1), (1.99) 2 3 1—4sin20(1+Ak’ 747.3: [ )1 (1.100) (1+ [1 - 4sin20 (1 + Ak’)]2)2 . 26 It is worth mentioning that one needs to assume only e-p universality to extract Ak’ from the forward-backward asymmetries Aim. So far there is no need to include the 1' letpon in the universality assumption. Finally, using Eq. (1.80) and the expression for (1 — 5:712?) 5%., I introduce the third parameter Arw where Z Z (1 M3,) M3,, na(Mg) -— —E . 1.101 M2 M3. fiapMHl-Arw) ( ’ Using Eq. (1.80) one finds 6_G_p_ 60 + cos20 6Mg + sin26 — c0329 5M3], G"? a $11126 M; 311129 143,, ‘ AI‘w: (1.102) One should remember that the term 60/0 in Eq. (1.102) does not include the light fermions contribution which has been absorbed in 0(Mg) as discussed earlier. Notice that in the improved Born approximation (including only the QED corrections) Ap, Ak’, and Arw vanish by definition. In terms of the one-loop corrections we found earlier one can write explicitly the quantities Ap, Arw, and Ak’ as follow AP = 14:20) " figég)‘ " Mz— Liz: (M2)- 62:8 — 469A!» (1103) A... = {$3.33 (A139 — A335”) 353186141 1 111(0) _sin 2:; :08 20 FWW(M?V) + 2:1:00A320) + 62:3 , (1.104) cozs’jien 30;:00 A120) + 2sin6cos 6F72(M?)+ msg#21112 0 62:3 + 691,, — (1 — 4 sin? a )591, . (1.105) 27 Now I expand my scope and take a further step into the analysis. The definition of the parameters Ap, Arw, and Ak’ in terms of the physical observables I‘(Z —-) p" M”), 1452.3, and MW nominate these parameters to be used as a model independent probe. The parameters Ap, Arw, and Ak’ are equipped to describe radiative corrections due to any model which is identical in its tree—level low energy part to that of the SM. Therefore, I will proceed with this aim in mind. To include additional observables in the model independent analysis I will make some general assumptions [16, 22, 23]. I start by assuming e-p-T lepton universality. As mentioned before, lepton universality is a valid statement in the SM. Consequently, all leptonic partial decay widths and forward-backward asymmetries are uniquely determined by Ap and Ak’. Similarly for the polarization measurements Ae and A,. In the case of lepton universality one can use the average lepton measurements I‘(Z -+ (“(+) and A533 in defining Ap and Ak’. To include the quark measurements, i.e., the hadronic widths and the forward-backward asymmetries, further assumptions are needed. I assume that all relevant deviations from the SM are only contained in the vacuum polarization functions, i.e., through oblique corrections. This may be a reasonable assumption except for the b quark which I handle separately. In the b-quark case, there is a large non-oblique correction to the decay width I‘(Z —> b5) due to vertex diagrams involving the top quark. This correction can not be expressed in terms of the parameters we defined above. Therefore, I follow Refs. [16, 22, 23] by introducing an additional parameter 6,, to describe this non-oblique correction. I will discuss this parameter in the next section. To summarize this section, all precision electroweak observables at the Z pole, under some general assumptions, can be written in terms of four quantities Arw, Ap, Ak’, and £5 (to be discussed below). These quantities can be deduced easily from the experimental measurements I‘(Z -+ £1”), A‘FB (assuming lepton universality), 28 MW, and I‘(Z —-) ()5). This result provides a straight-forward method to check for the validity of the theory against the experimental data. The theory under investigation includes a more general set of models and not exclusively the SM. All models which are identical in their low-energy tree-level sectors to that of the SM and only differ by radiative corrections can be included in the model independent analysis. Examples include the Minimal Supersymmetric Standard Model (MSSM), Technicolor (TC) models, multi-Higgs doublets models, etc. 1.4 The 6 Parameters From the previous section, we found that all electroweak radiative corrections can be parameterized by a set of four parameters. This result is interesting because one does not need to worry about the whole renormalization procedure any more. To parameterize the electroweak radiative corrections, it is useful to separate different possible effects into different parameters. In other words, to disentangle new physics effects it is very useful to choose the parameters so that some are sensitive to specific types of new physics. In this work where I am interested in the top quark contributions to low energy data, the temptation is high to choose some parameters to be very sensitive to the top quark mass effect. A well-known parameterization can be implemented to our case based on the scheme used in Refs. [16, 22, 23], where the electroweak radiative corrections can be parameterized by 4 independent parameters, three of those parameters 61, 62, and 63 are proportional to the popular T, U, and S parameters [24]. The fourth one; 6;, describes the relevant non-oblique corrections to the proper vertex Z —) b5 [16, 22, 23]. A similar parameterization to 65 is discussed in Ref. [25]. In appendix B, I discuss briefly the S, U, and T parameters. Also, I discuss their relation to the epsilon 29 parameters. I write the electroweak precision observables as 4 0.- : 0,-IB (1+ 20.56)) , (1.106) j=l where, 0.13 is the corresponding prediction of the theory in the improved Born ap- proximation (i.€., with the QED and QCD corrections). The four dimensionless pa- rameters £1, 62, 63, and 6), contain the genuine electroweak radiative corrections. All dependence on m, and my comes through these parameters. The quantities agj are fixed numerical constants. In principle there are many different ways to parameterize the electroweak cor- rections. In this work, I follow the parameterization in Refs. [16, 22, 23] given in terms of the epsilon parameters. The parameters 61, 62, 63 are given in terms of the previously defined parameters Ap, Ak’, and Arw £1 = Ap, (1.107) sin2 0 _ 2 62 _ COS 9 AP + cos2 0 — sin70 Arw — 231112 0Ak’, (1.108) 63 = cos2 6Ap + (c0320 — sin2 9 )Ak'. (1.109) For the parameter c), I write the partial-decay width Z -> b5 as follows Gng a 2 2n¢§ (1+AP) ((951)2+(gfi,)2) (1+ (M2) Pb = 121r ) RQCD, (1.110) where RQCD is the QCD correction given in Eq. (1.48), R.(_)(M_z)(_w) (1....) 7f 1r 30 and a, is the QCD coupling, a, = gf/47r. The quantities 9%, and 9%, are defined as follow [16, 22] 1 9gb = ‘30 + 60), (1.112) g5), _ 1 —— 4/3(1+ Ak’)sin20 + 6,, 1.113 9X0 1 +6!» ( ) One should notice that Eqs. (1.112) and (1.113) are just auxiliary equations. So far they do not mean that relevant physics is only coming through the bL-E-Z vertex. In fact, from Eq. (1.110), the correct assumption is that relevant physics will modify the decay width I}. To put it differently, the combination of relevant physics to both the bL-FZ-Z and bR-EE-Z vertices which contributes to I], is simply parameterized by q, (cf. Eq. (1.124)). However, if one is interested in other observables, e.g., A), (see below), then an additional assumption is needed. The assumption is that the only relevant new physics is coming through the bL-EZ-Z vertex. The reason is simply that another combination of the neutral left- and right-handed b quark couplings will enter these observables. I will adopt this assumption since the observable A), is not sensitive to the top quark mass (see Eq. (1.141)) and also it does not show a deviation from the SM (see Table 1.2). In terms of the one-loop corrections we found for the parameters Ap, Arw, and Ak’ one can write the arameters £1 62 and e as follow 1 a a 3 5GV,B 61 = 61 — 85 -— - 459,“ , (1.114) Gr . 2srn9 6G 62 = 82 — 81112084 — C082985 - m 83 — ‘G—‘gB' " 69V! - 36g,“ , (1.115) c -e+cos206 cos2de +cosfle_+-cos26l —sin20 l+2sin20(s (1116) 3 - 3 4 5 6 2sin§0 9v: 2sin20 gAl- - sin0 31 Here I define the following quantities — AZZ(0) — AWW(0) 81 — Mg M3,, , (1.117) e. = FWW(M3,,) — F33(M§), (1.118) c030 30 1.119 83: sinBF (M2) ( ) 64 = F77(0) — F77(M§), (1.120) 2dFZZ e5- .. Mz— dq 2 (M2), (1.121) 1172(0) = . 1.122 86 Mg ( ) For the parameter 6),, I write the one-loop corrections to the vector- and axial- form factors (5ng and 69,45, which are due to proper vertex diagrams and b-quark self energy, as 1 1 59% = --2- (80 + 59w) , 59.10 = *5 (80 + 59M) ~ (1-123) In Eq. (1.123) I split the corrections to (5ng and 69,“, into two parts, the first part c), encompass all non-oblique corrections dueto the top quark mass. (Notice that the relevant m, corrections are only in the left-handed current.) The second part includes the vector and axial-vector corrections independent of the top quark mass ng and 9,“, which are identical, in the SM, to the d—quark vertex corrections. Calculating the I“), decay width using Eq. (1.89) and comparing with Eq. (1.110) one finds ‘6 = “rggm [(1'58‘“ ”)‘ng‘Sng + 2 4 8 2173—1115718- (l—gsinz 0)6gw—§(— -5+§sin2 (9)69”). (1.124) 32 Note that the parameters 61, 62, 63, and 6,, can be written in terms of the basic measurements F” = I‘(Z —+ p”,u+), A’;B, MW, and I}. To do this let us first rewrite the parameters Ap, Arw, and Ak’ in terms of 61, 62, and 53. Using Eqs. (1.107)— (1.109) one finds Ap = 61, (1.125) C082 0 cos2 0 — sin2 9 _ — 2 1.126 Arw sin2 0 1 sin2 0 62 + £3 ’ ( ) Ak' = 1 (63 - cos2 0 61) . (1.127) cos2 0 — sin2 0 Using Eqs. (1.99)—(1.101) and (1.110) one finds I), = I‘,,|3(1+1.2061— 0.2663) , (1.128) Alf‘B = Afiglg (I + 34.6061 — 45.0063) , (1.129) 314-??- = Mi [3 (1+14361-10062—08663) (1130) P5 = Pb]3(1+1.4161— 0.5463 + 2.9365) , (1.131) where OIB stands for the observable O in the improved Born approximation (including the QED and QCD corrections only). Other observables can be written in terms of the epsilon parameters [26] as long as they satisfy the conditions we mentioned before. 0 The charged leptonic decays will be identical to I‘” as long as lepton universality is satisfied. In this case, one can use the average leptonic decay width I“. Similarly for the leptonic forward-backward asymmetries 145,3. 33 o The polarization measurements Ac and A, are identical, assuming leptonic uni- versality, where c A e e A e (gVe + AgVe) + (gAe + AgAe) One finds A, = A. = Aela (1 + 17.3.;1 — 22.553) . (1.133) c The observable A“; is measured at SLC using polarized initial 6‘ 6+ beam and at the Z peak. Under the general assumptions I mentioned before, the observables Ac at LEP and ALR at SLC measure the same quantity, i.e., (ch + AgVe) (gAe + AgAe) A” = 20». + Ayn)? + (g... + Agra? z A" (”34) Thus, one simply has ALR = A8 = Ae]3(1+17.361— 22.563) . (1.135) s The total decay width of the Z boson 1‘2 [‘2 = FZIB (1 + 1.3561 — 0.4663 + 0.356),) . (1.136) 0 The observable R; = I‘h/I‘g, the ratio of the partial widths Z -) hadrons and Z -+ 3" 6+, R; = Rglg (1 + 0.2861 - 0.3663 + 0.5061,) . (1.137) s For the hadronic peak cross-section 0),, where _ 121rI‘J‘). O’h - m, (1.138) one finds 0'}. = 011180 + 0.0361— 0.0463 + 0.2065) . (1.139) 34 o The observable Rb = I‘b/I‘h, the ratio of the partial widths Z -+ b5 and Z —+ hadrons, R, = Rblg (1 + 0.0661 - 0.0743 + 1.795.) . (1.140) c For the polarization measurement of Ab (at SLC), the dependence on the non- oblique corrections is not exactly given by 6;, (see Eq. (1.124)). Nevertheless, with the assumption that the relevant effect enters through e), one can ignore the other contributions, i.e., 69v.) and 69,“ defined in Eq. (1.123). In this case, one finds A), = Ab|3(1+ 0.2361 — 0.2963 + 0.166),) . (1.141) s The polarization measurement of A, (at SLC) Ac = Ac[3(1+1.7161 — 2.2263) . (1.142) 0 The b—quark forward-backward asymmetry Apr A2,, = AgrBlB (1 + 17.5361— 22.8063 + 0.166),) . (1.143) a The c-quark forward-backward asymmetry Ah; Ali‘s = A}BIB(1+19-0161— 24.7263) . (1.144) 0 Finally, the measurement of RC = I‘C/I‘h, the ratio of the partial widths Z —) c8 and Z —-) hadrons, RC = Rc|3(1+ 0.11661— 0.15163 — 0.56),) . (1.145) Therefore, using the above equations and the low energy experimental data, one can fit the parameters 61, 62, £3, and 6),. A direct comparison with the theory is possible by calculating the theoretical values of these parameters and then comparing with the extracted experimental values. 35 1.5 The SM Heavy mt and mm Contributions to the Low-Energy Data The important property of the epsilon parameters is that, for all observables at the Z pole, the whole dependence on m, and m H enters only through these parameters. Therefore, to calculate the m, and my contributions to the low energy data it is enough to calculate their contributions to 61, 62, 63, and 6),. In this section, I treat the top quark mass m, and the Higgs boson mass m H as heavy mass scales and calculate their contributions at one-loop level to £1, 62, £3, and e), in the SM. Using the heavy mass expansion (discussed in appendix C) and keeping only the leading contributions of m, and m H I determine their contributions to the epsilon parameters. 1.5.1 Heavy Top Quark Contributions In this case, only the vacuum polarization functions and the quantity e), are sensi- tive to the top quark mass (see Figure 1.5). One can perform the calculations in any gauge, since the result is gauge invariant. I use dimensional regularization and define A=—n_2_4—7—ln47r, . (1.146) where n is the space-time dimension and 7 = 0.577. . . is the Euler’s constant. I keep only the leading contributions of 712,. o For the photon 1477(0) = 0, . ‘ (1.147) 92 16 sin2 0 [MINA/1%) = 167r2 9 (A — ln m?) . (1.148) 36 o For the Z boson 1422(0) = g2 3771? —161r22c0820 (A-lnmf) ’ 2 FZZ(M§) g 1 (l — fi-sini‘iél + 166-sin40) (A — ln m?) . =167r2c0820 2 3 o For the W boson 2 2 9 3m 1 AW”) = razT‘ (“A ‘ 2 + 1m?) ’ 2 g FWW(M§) = 16712 (A — lnm?) . o For the 7-Z mixing 4172(0) -- 0. 2 o 72 2 = ——-g ___4srn0 (1-3 ' 2 ) _ 2 F (M2) 167r23cosl9 2 38111 0 (A lnmt). s For the proper vertex 8,, 2 2 9 mt ch = —167r2 2M3V ' Therefore, we find AZZ(0) _ AWW(0) _ 92 3m? _ 30pm,2 = A = — - —_ 61 )0 Mg M34, 167r2 4M3V 8\/’27r2 ’ G M2 £2 = FWW(M[%/) _ F33(M§) = — 43.2.13, 11107712), _ 0030 30 2 __ GPA/112v 2 E3 - sin0 ( Z) — 12\/21r2 11mm), 0pm? 65 = -4\/27r2 . (1.149) (1.150) (1.151) (1.152) (1.153) (1.154) (1.155) (1.156) (1.157) (1.158) (1.159) 37 all Figure 1.5: The top quark contribution to the epsilon parameters, at the one-loop level. 38 1.5.2 Heavy Higgs Boson Contributions For the heavy Higgs boson case, I calculate the vacuum polarization functions to the leading order in my (see Figure 1.6). Due to the screening theorem [27], at one-loop level, the leading dependence on the heavy Higgs boson mass in low energy physical observables can be at most a logarithmic dependence. All vertex corrections are negligible because of the small light fermions masses. 0 At tree level, the Higgs boson does not couple to the photon. Thus, the Higgs boson does not contribute to A77(O) and F77(M§) up to one loop. Similarly, there is no contribution to A72(0) and F7Z(M§). o For the Z boson 2 2 22 - _9__ _flli_§ 2 2 A (0) - 16712C0820 ( 8 4len(m,,)) ’ 2 ln(m2) Fzz 2 ___ _ 9 H . (M2) 167r2cos20( 12 ) o For the W boson 2 2 g m 3 2 ln m2 FWW(M§)=—1gni’( (1211)) Therefore, one finds AZZ(0) AWW(0) 30,514? 31520 6 = A = — .— ___— : _ W 2 1 p Mg M1241 8\/§7I2 C032 0 n(mH) 7 62 = FWWWfi») - 1"33(1‘4221)= 0, (1.160) (1.161) (1.162) (1.163) (1.164) (1.165) 39 c080 €3=m 30( ———l . 1.166 24x/27r2 n(mH) ( ) M2) = The parameter q, is not sensitive to the Higgs boson mass because the Higgs boson couples to the b quark with a coupling strength proportional to the b-quark mass m), (Yukawa coupling). In fact, in the limit of ignoring the bottom quark mass m5, the Higgs boson contribution to q, vanishes. 1.6 Status of the SM In general, precision tests performed at LEP, SLC, and the Tevatron have con- firmed to a large accuracy the SM predictions. One-loop and even in some cases two-loop corrections to the SM have been implemented and checked against the low energy data [26, 28, 29]. Using the input values in Table 1.1, the SM predictions for the low energy physical observables have been calculated [30]. In Table 1.2, I tabulate the new data [12] and the corresponding SM predictions [30]. The SM pre- dictions are calculated for two values of a,(M§), 0.125 and 0.115, respectively. The top quark and Higgs boson masses used in the SM predictions are m, = 175 GeV and my = 300 GeV, respectively. The only sensitive measurements for a,(M§) are the total Z decay width I‘z, the ratio R; = I‘h/I‘g, and the hadronic peak cross-section a), (see Eq. (1.138). With all the accumulated success of the SM, there are only a few hints of possible deviations from the SM that have been reported recently. Among those measurements are: the LEP observation of a small excess in the measurement R), = I‘b/I‘h, the ratio of the partial widths Z -> b5 and Z —) hadrons, of about 3.50‘ [12]. Also at LEP there is the measurement of RC = I‘c/I‘h, ratio of the partial widths Z -+ c? and Z -—) hadrons, with a deficit of about 2.50 [12]. At the SLC, a deviation from the SM has been seen in the measurement of Am of about 2.80 [31]. At the Tevatron, an 40 Figure 1.6: The Higgs boson contribution to the epsilon parameters, at the one-loop level. 41 excess of large E, jets has been reported [32]. Finally, there is the issue of the QCD coupling a,(M§) [33]. From the LEP electroweak fit at the Z pole, the extracted value for the QCD coupling is a,(M§) = 0.125 :1: 0.004 [12]. The LEP and SLC combined fit gives a,(M§) = 0.123 :1: 0.004 [12]. These fits are not consistent with the evolved value a,(M§) ~ 0.11 coming from the measured a, at Iq2| << ME in deep inelastic scattering (DIS) [9], or with the value of 01, ~ 0.115 from lattice QCD [9]. As it is argued by Shifman [34], the tendency of lower values of a,(M§); determined form low energy observables, as compared to the higher values of a,(M§); measured at LEP and SLC, presents a serious discrepancy that could be a signal for new physics. The new LEP and SLC data show some interesting features which did not manifest themselves in the old reported LEP and SLC data (before the summer of 1995) [35]. In fact, by the time I started this work the new data was not available. The old data reported an excess of about 2.00 in R5. No significant deviation in R6 was observed then (within 1.00). All other measurements like Re = Fh/Pg, A573, A,, A7, I‘h, Ab, Ac, and ALR where consistent with the SM. As I discussed in the previous sections, based on a few general assumptions, all low energy data at LEP, SLC, and the Tevatron measurement of MW can be expressed in terms of 4-independent parameters 61, 62, 63, and 6b. The assumptions made are that new physics appear in the vacuum polarization functions and/or the Z -1 b5 vertex. The epsilon parameters contain the genuine electroweak corrections including the top quark and the Higgs boson contributions. Unfortunately, in defining the 6 parameters no relevant new physics in the observable PC is assumed. Given the above assumptions in defining the 6 parameters, the reported anomaly in Rc = I‘c/I‘), has a very small dependence on 6;, [see Eq. (1.145)] which cannot explain the large deficit in RC. Therefore, to use the 6 parameters one has to ignore the anomaly in Re. In fact, as argued by Shifman [33], the size of the anomaly in He cannot be given by any 42 perturbative physics. Using the leptonic data, i.e., I“ and the asymmetries combined with the W mass, the fit of 61 and 63 shows interesting features [36]: the good agreement with the SM, the evidence for weak corrections, and the preference for a light Higgs boson. Lighter Higgs boson implies lighter top quark mass. Including the hadronic data (except RC) does not alter the correlation between 61 and 63. However, the fitted 61, is departed from the SM by about 20. Therefore, the conclusion is that, by ignoring the anomaly in R." the SM predictions for £1, 62, and 63, are in good agreement with the fitted experimental values. However, the SM prediction for e), is not consistent with the low energy data at the 20 level. The electroweak data can be used to predict quantities like the top quark mass, the W mass, the Higgs boson mass, and a,(M§). The new data predicts the top quark mass to be [12] m, = 170 :1: 10 :1: 19 (1.167) where the central value and the first error refer to my = 300 GeV. The second error corresponds to the variation of the central value when varying m” in the interval 60 GeV 3 m H _<_ 1000 GeV. This is consistent with the top quark mass reported from the observation of the top quark at the Tevatron by GDP [10] and DO [11]. Up to one loop, the Higgs boson contribution to the low energy observables can be at most a logarithmic contribution [27]. This makes the determination of my more difficult than m¢. Also, because the effects of m, and my are correlated, the deter- mination of m” is more difficult without a precise measurement of m,. Electroweak precision tests show a preference for a light Higgs boson [6, 36]. However, as argued in Ref. [37], LEP precision data, upon excluding the observables R), and RC, does not imply a strong bound on the Higgs bosons mass, i.8., my can still be as large as 1 43 TeV. If one takes the new discrepancies seriously, i.e., is not mere statistics, then one needs to understand the sources of these anomalies. Before the announcement of the new data, the observed anomaly in Rb in the old data, has triggered the hope to detect new physics. In fact, it was observed [34, 38] that if there was new physics in R5 then the LEP fit of a,(M§) would go down to a,(M§) ~ 0.11 in better agreement with the DIS and QCD values of a,(M§). With the inclusion of the new data, the need for new physics is indeed more apparent. In fact, as discussed below, many attempts have been made to explain the observed anomalies. Even though my work is not completely oriented toward explaining these anomalies, part of it is. In the next chapter I will discuss my general motivations for launching into this work. Below, I will mention briefly mine and other’s efforts to understand these deviations. If we assume that the reported anomalies are not mere statistics, then we can advocate specific types of new physics which can tackle these experimental anomalies. The problem is that the anomalies reported recently as a whole represent a confusing picture. The argument given [36] is as follows: The measurement of Pb at the Z pole is precise with uncertainty of about i3 MeV. However, the access in [‘5 by 11 MeV and the deficit in PC by 32 MeV amounts to a deficit in the sum I}, + I}, which enters the hadronic width I‘h, by 21 MeV. This deficit is far too large compared with the accuracy of the measurement of Ph. Even with the inclusion of the ambiguity of a,, coming from the M 2 scale or the DIS physics, 6a, = :l:0.007 which corresponds to :l:4 MeV shift in Pb, the total shift in F], does not amount to that of I), + PC. Therefore, a shift in the partial decay widths of the light quarks is needed. This shift must be tuned up to account for the accuracy of Ph. The problem then is how to produce the shifts seen in Rb and Re while not affecting other precise data, especially that of the leptonic sector. This is not a trivial task and I think building a natural model to cure 44 all these anomalies altogether is extremely difficult (ad hoc). One possibility to explain the anomaly in Rb is through the top quark. As I discussed in section 1.4, the decay width [‘5 has a strong dependence on the top quark mass. Therefore, assuming the top quark couplings to the gauge bosons [39, 40, 41] are different from the SM may be a plausible explanation to the anomaly in R5. In chapter 3, I will discuss indetail a general treatment for studying the anomalous top quark couplings from the low energy data using a model independent analysis. Unfortunately, such a scenario can not accommodate the anomaly in B, because, in the general model discussed in chapter 3, there is no mechanism to affect the charm couplings through the top quark. In chapter 5, I discuss a special model in which the third generation of fermions undergoes a different SU (2) interaction from the first two generation of fermions. The SH (2) symmetry associated with the third generation exhibits a strong flavor dynamics which leads to a modification in the Z -boson couplings to the fermions (as compared to the SM). Nevertheless, due to the accuracy of LEP data the constraints on the free parameters of the model are so severe that this model can account for the deviation in Rb from the SM at the 30 level. Even though RC is shifted in the needed direction, the predicted value is still outside the 20 range of the data. Therefore, one cannot explain the anomaly in RC entirely based on this proposed model. By the time this part of my work was going through a final revision, a similar model was proposed in Ref. [42]. Also, I have become aware of another similar model discussed before [43]. Other efforts to explain these anomalies have been done by many theorists. The reported anomalies can not be understood within the fully MSSM [44, 45]. According to Ref. [44], the MSSM with additional high energy scale effects may be a suitable candidate to explain the observed anomalies in the data. A fit on a,(M§) = 0.116 45 with large Rb and large A“; can be achieved. However, such a model predicts light superpartners below 100 GeV, a result which may rule out such a model if light superpartners are not found at LEP2 and FNAL. In Ref. [45] a SUSY model with four generations is proposed to explain Rb while ignoring the problem of Re. In Ref. [46], a composite model of a forth heavy fermion family has been assumed. The model can explain the anomaly in Rb but not the anomaly in Re. A common feature among these models is that they all predict some light particles with masses around the MW scale. Such models may be ruled out if nothing is observed at LEP-II and at the Tevatron. As it is argued in Refs. [43, 47], non-commuting extended technicolor (ETC) is a possible candidate to give rise to a large Rb which is consistent with the data. Other efforts are oriented towards the inclusion of an extra Z’ gauge boson [48, 49, 50]. Some of these models seem to explain the Rb anomaly but not Re. In Ref. [49] an extra Z’ is coupled to the third family through an additional U (1) gauge symmetry. This model may explain the anomaly in Rb and in a,(M§). Other authors [50] claim to explain the R." anomaly, a,(M§), and RC on the expense of building ad hoc models. The anomaly in the SLC measurement of ALR is difficult to realize because LEP measures a similar quantity A, which is in a very excellent agreement with the SM. In fact, in the SM and in many other extensions the two observables are identical. It is true that the two measurements refer to different observables, nevertheless, _it is hard to imagine new physics which would affect one and not the other. In Ref. [51], the authors discuss the possibility of new physics which would reconcile the measurements of A L R and Ac. They conclude that the only possible way to reconcile the two measurements is through an additional Z’ boson coupled almost exclusively to quarks and with a mass almost degenerate with the Z boson mass. They also conclude that the Z' must have an almost vanishing coupling to the leptons. 46 Table 1.1: The input parameters used to calculate the SM predictions in Table 1.2. Input Value a,(M§) a: 0.125 b: 0.115 CAME 128.89(9) Mz 91.1885(22) GeV Gp 1.166389(22) x 10"5 GeV?! sinzd 0.2312(3) m, 175 GeV my 300 GeV Summarizing this section [30, 36], o The precision electroweak experiments at LEP and SLC test the SM predictions at a few times 10’s. A need for electroweak corrections is demonstrated. 0 All data agree well with the predictions of the SM except for R5 which shows an access of- about 3.50, Re which shows a deficit of about 2.50, and A L R with a deviation of about 2.80. Combining the two data Rb and Rc alone rule out the SM at 99.99% confidence level (C.L.) for mt > 170 GeV [30]. (i.e., a lower mt is preferred.) 0 There are many different scenarios which can explain the access in R5. So far, it is not possible to understand the anomaly in Be, it may even be beyond the perturbative region [33]. It is also extremely difficult to understand the anomaly in ALR- 47 Table 1.2: Experimental and predicted values of electroweak observables for the SM. The SM predictions are calculated using the input values in Table 1.1. Columns a and b are for a, = 0.125 and 0.115, respectively. I Observables Experimental data [ SM 1 _ a b LEE gv. —0.0368 :1: 0.0017 —0.0367 -0.0367 9A. -0.50115 :5 0.00052 —0.5012 -0.5012 sum/9vc 1.01 a: 0.14 1.00 1.00 gAp/gA, 1.0000 :1: 0.0018 1.0000 1.0000 gvf/gv, 1.008 :1: 0.071 1.000 1.000 gA,/g,., 1.0007 :1: 0.0020 1.0000 1.0000 I‘z 2.4963 :1: 0.0032 2.4978 2.4922 R. 20.797 :1; 0.058 20.784 20.716 12,, 20.796 4: 0.043 20.784 20.716 R, 20.813 :1: 0.061 20.831 20.716 0?, 41.488 :1: 0.078 41.437 41.490 A, 0.1390 :1: 0.0089 0.1439 0.1439 A, 0.1418 :1: 0.0075 0.1439 0.1439 A53 0.0157 :1: 0.0028 0.0157 0.0157 A53 0.0163 3: 0.0016 0.0157 0.0157 A,” 0.0206 :1: 0.0023 0.0157 0.0157 R, 0.2219 3: 0.0017 0.2157 0.2157 3: 0.1543 :t 0.0074 0.1721 0.1721 QLQ Am 0.1551 :t 0.0040 0.1439 0.1439 R, 0.2171 :1: 0.0054 0.2157 0.2157 A, 0.841 :t 0.053 0.934 0.934 A, 0.606 t 0.090 0.666 0.666 1919111911 MW 80.26 a: 0.16 80.32 80.32 Chapter 2 The Chiral Lagrangian 2.1 Physics Beyond the SM As I discussed in chapter 1, the SM has proven to be very successful in accom- modating all precision measurements available so far; with the exception of a few observables. Among those observables are R1, and R, at LEP with deviations of about 3.50 and 2.50, respectively [12]. Also at the SLC there is the A“; measure- ment with a deviation of about 2.80 [31]. At the Tevatron there is the observation of an excess of large E, jets [32]. Despite the success of the SM as a whole, there are two main points regarding the SM one should bear in mind. The first point is that some parts of the SM remain untested, examples include the top quark interactions with the electroweak gauge bosons which is the main theme of this work. It is only recently that the first direct observation of the top quark has been made [10, 11]. With a top quark mass much larger than the Z mass, physics of the top quark is still premature. In chapter 3, I will discuss in detail the possibility of anomalous top quark couplings to the gauge bosons. A second example is the study of the gauge boson self-interactions [52]. The current available energy at LEP which is around the Z-mass scale, does not permit a direct test on the gauge boson self-interactions. It is only through radiative corrections that such couplings can be examined. There has 48 49 been several attempts to study the SU (2) L x U (1)y gauge structure using the low energy data at LEP/SLC and other low energy data [25, 53, 78]. With the upgrade collider LEP-II with an energy of about twice the Z mass, a direct test will be avail— able in the future [54]. A third example is the Higgs sector, which is responsible for the symmetry-breaking mechanism. The Higgs boson is still a hypothetical particle. Direct search at LEP only led to a lower constraint on the Higgs boson mass to be larger than 65.2 GeV [6]. From radiative corrections, the Higgs boson mass still can be as large as 1 TeV [37]. The second point about the status of the SM is that despite the celebrated success of the SM, there is little faith that the SM is the final theory. The reasons behind this are fundamental and basic. Some of these reasons are 0 No unification in the gauge couplings g, g’, and 9, can be attained within the SM framework. Furthermore, gravity is a totally ignored subject in the SM. 0 The SM contains many arbitrary parameters with no apparent connections, e.g., fermion masses and quark mixings. o The SM provides no prediction for the fermion masses, the quark mixings, the Higgs boson mass, and why the number of generations is 3. o The SM provides no satisfactory explanation for the symmetry-breaking mech- anism which takes place and gives rise to the observed mass spectrum of the gauge bosons and fermions (see chapter 3). The Higgs boson is still a hypothet- ical particle. In fact the sole existence of the Higgs boson by itself is a debated issue [55]. o The SM does not explain the large and unnatural difference between light and heavy quark masses. The question of why the top quark is so much heavier 50 than the other quarks, or to put it differently why all the quarks, except the top quark, have small masses relative to the symmetry-breaking scale 12 = 246 GeV. 0 Other issues are not contemplated within the SM, examples include neutrino masses, the observed CP violation in the Kaon system, strong CP violation, etc. If one holds the belief that the SM is not a fundamental theory, then many ques- tions arise. How come the SM is so successful? Is the success of the SM an accident, or is there a link between the ”fundamental” theory and the SM? Where is the new physics? How to reach for it? . . . These are some of the questions that one hopes to answer in the on going efforts to understand nature. 2.2 Model Independent Analysis Based on the belief that the SM is not the whole story, the search for physics beyond the SM is a continuous effort. One can investigate the possible existence for physics beyond the SM through a systematic model by model study. An example may be a grand unified theory (GUT) valid up to some high energy scale. Evolving that theory down to the electroweak scale permits a direct comparison between the prediction of the theory and the precision low energy data [56], or at least one may be able to set some constraints on the free parameter space of the model under study. Such an approach provides a consistent analysis for low energy data because the full theory under investigation is at hand. Other examples of this approach include Supersymmetry (SUSY) models [57] and TC along with its revised versions [47, 58, 59]. (For a review see Refs. [25, 60].) However, such approach is cumbersome and time consuming because of the need to go over the whole model and pin point 51 all of its characteristics at low energy. A useful alternative method in searching for new physics is the model independent analysis. There are two common approaches in searching for new physics using a model independent analysis. The first approach is by characterizing all energy effects at low energy using a few parameters. As I discussed in chapter 1, such an approach is possible, e.g., using the epsilon parameters [16, 22, 23] or the S, T, U quantities [24] (see appendix B). Then by simply calculating theses quantities in any model and comparing with the corresponding values extracted from the low energy data one can judge whether a specific model is compatible with the low energy data or not. However, one should admit that using these parameters does not allow for a general treatment of all possible models. this is true because in defining these parameters certain assumptions are implemented. For example, in these parameterizations one assumes that at the tree level the low energy part of any model should reduce to that of the SM. Difference is only allowed in radiative corrections. Other assumptions include lepton universality in defining the epsilon parameters. In the S, T, U parameters one assumes that the only relevant physics is coming through the vacuum polarization, i.e., oblique corrections, and so on. The second approach to the model independent analysis is the effective Lagrangian approach [61, 62, 63]. In this case, under general assumptions, one can effectively describe new physics effects to low energy data with no regard to the actual mechanism at the high energy scale. The use of effective Lagrangians began with the introduction of the non-linear a-model [64] in the early 60’s as an effective theory for the strong interactions. The theoretical basis of the effective Lagrangian method was formulated by the late 60’s [65, 66]. In general, realistic models of new physics at high energy scales generate a host of effective operators at the low energy scale. The effective theory is valid below some 52 high energy scale A above which the effective theory breaks down. Dependingion the underlying physics one can resort to two different approaches in constructing the efl'ective Lagrangian. The first approach is if the underlying physics is decoupling [67]. In this case, the effective operators are usually suppressed by some power of a high mass scale A. Therefore, the effective Lagrangian can be expanded in powers of 1 / A or equivalently in terms of increasingly suppressed higher dimensional operators [68]. (For this effective Lagrangian, the S U (2) L x U (1)y gauge symmetry is linearly realized by inserting more Higgs doublet fields in the effective operators.) The second approach is if the underlying physics is non-decoupling. In this case, the contributions due to physics above A need not be suppressed by powers of 1/ A. Therefore, in the non- decoupling case, an expansion in powers of momenta is performed and the effective Lagrangian is known as the chiral Lagrangian [68, 69]. (For the chiral Lagrangian, the SU(2)L x U (1)y gauge symmetry is non-linearly realized, see appendix D.) The effective Lagrangian can be taken to respect the gauge invariance of the SM. The disadvantage of the effective Lagrangian is that it involves a large body of free parameters, making the whole approach a tedious one for a general treatment. To simplify the whole approach usually one resorts to a limited class of effective operators which are sensitive to the case under study. Some recent efforts have been implemented to account for a broader class of effective operators than has previously been considered [70]. The chiral Lagrangian constitutes a powerful approach in describing the phe- nomenon of spontaneous symmetry breaking [71]. It provides a systematic way to effectively incorporate new physics without adhering to a limited scenario of the symmetry-breaking mechanism as long as the electroweak symmetry is spontaneously (as opposed to dynamically) broken. For example, one does not have to be confined to the assumed SM Higgs mechanism and therefore, the Higgs boson is not an essential 53 part of the basic structure of the chiral Lagrangian. In the next section I discuss in some detail the structure of the electroweak chiral Lagrangian. For more details, the reader can refer to the literature [62, 72, 73, 74, 75, 76, 77, 78]. In the effective Lagrangian approach one can understand the reason why the SM is so successful, it is because the SM is viewed only as a low energy effective theory. The success of the SM simply indicates the irrelevant effect of higher dimensional operators. Another thing to mention is that the use of the effective Lagrangian does not necessarily stem from our ignorance of the full dynamics. In fact, as pointed out by H. Georgi in Ref. [76], the effective field theory framework is not only simpler and more transparent, but it actually provides a deeper insight into the relevant physics at the distance scale that is relevant to the current experimental data. 2.3 Introduction to the Chiral Lagrangian The chiral Lagrangian approach has been used in understanding the low energy strong interactions because it can systematically describe the phenomenon of spon- taneous symmetry-breaking [71]. In fact, the chiral Lagrangian found its greatest development in the context of strong interactions [79], e.g., 7777 scattering. Recently, the chiral Lagrangian technique has been widely used in studying the electroweak sector [62, 72, 74, 75, 76, 77, 78], to which this work has been directed. The chiral Lagrangian can be constructed based solely on symmetry principles with no other assumptions regarding any explicit dynamics. Thus, it is the most general effective Lagrangian that can accommodate any truly fundamental theory possessing that symmetry at low energy. Since one is interested in the low energy behavior of such a theory, an expansion in powers of the external momentum is 54 performed in the chiral Lagrangian [69]. In general, one starts from a Lie group G which breaks down spontaneously into a subgroup H, hence a Goldstone boson for every broken generator is to be introduced [80]. I consider the case where G = S U (2) L x U (1)y and H = U (1),,m. There are three Goldstone bosons generated by this breakdown, 45“, a = 1, 2,3 which are eventually eaten by Wi and Z and become the longitudinal degrees of freedom of these gauge bosons. The Goldstone bosons transform non—linearly under G but linearly under the subgroup H. A convenient way to handle this is to introduce the matrix field 2 = exp (i¢:Ta) , (2.1) where 'r“, a = 1,2,3 are the Pauli matrices normalized as: Trace(7'°'rb) = 2605. Be- cause of U (1),3m invariance v1 = 1);» = 0, but is not necessarily equal to 113. The matrix field 2 transforms under G as a 3 E —+ 2’ = exp (£02741) 2exp(—iy:2—) , (2.2) 1,2 where a ’3 and y are the group parameters of G. In the SM, being a special case of the chiral Lagrangian, v = 246 GeV is the vacuum expectation value of the Higgs boson field. Also 11;; = v arises from the approximate custodial SU(2) symmetry in the SM. It is this symmetry that is re- sponsible for the SM tree-level relation va p = = 1 , (2.3) 5.. a. where c3 = 1 — 33 and 33 is the weak mixing angle defined in the on-shell scheme (see appendix A). Low energy data already constrains p to be 1 within about 0.1% 55 accuracy [60]. Therefore, in this work, I will assume the full theory guarantees that v1 = v2 = 113 = 12. Out of the Goldstone bosons and the gauge boson fields one can construct the bosonic gauge invariant terms in the chiral Lagrangian l a V0 1 V 1 £3 = —ZWWW" -— ZB,,,,B“+ 2521‘: (DpElD’E) , (2.4) where the covariant derivative 7'3 0,2: 73,-2 gw;—-2:2+ig',,.2B-T§— (2.5) Under G, the covariant derivative transforms as 07a 3 2 ) DpE exp(—iy:2—). (2.6) 1),,2 —+ (D,2)' = exp (4“ In the unitary gauge 2 = 1, one can easily see how the gauge bosons acquire their masses. In the unitary gauge, the mass term in the Lagrangian reduces to _ 12 1 12 a" ' 2 c... _ Zv"r11-(1),,219#2)—1:1-v'rr([— 9W 2 +g'Bp2] — l112(2w0WW—2 ’W3B"+ ’23 B”) (27 - 8 g ,. 99 ,. 9 1. . -) Using the field definitions in Eqs. (1.11) and (1.12) one recovers the gauge boson masses given in Eq. (1.17) 22 92 ”2 + p— 9 v [1 £m= 4 —W’‘ W + 8coszflz”z . (2.8) The W and Z masses are 2 2 2 2 M3,,=-g—3-, Mg: 9" . (2.9) 56 The difference between the non-linear and the linear realizations is that in a general gauge, the non-linear realization produces other complicated terms in powers of the Goldstone bosons. (See appendix D for details.) In general, one finds c M2W+W“’ M322“ 6 +6” - 1 3a” 3 m: w # +—'2_ p +p¢ ([2 +§p¢ ¢+..., (2.10) where the fields 63* are defined as i_¢‘=1=i¢2 ¢ .. ([2 . (2.11) Fermions can be included in the context of the chiral Lagrangian by assuming that they transform under G= SU(2)L x U (1);» as [74] f -+ f’ = e‘”Q’f, (2.12) where Q I is the electromagnetic charge of fermion f. Out of the fermion fields f1, f2, with the condition Q f, — Q f, = 1, and the Goldstone bosons matrix field 2 the usual linearly realized fields \II (see section 1.1) can be constructed. For example, the left-handed fermions [SU(2)L doublet] are constructed as ‘11,, = EFL = 2 (Q). (2.13) One can easily show that \IIL transforms under G linearly as follows a 3 3 \IIL -) \IIL’ = exp (2,027“) 2 exp (—iy-T2—) exp (iy [:g— + g]) FL aa Ta 2 ) exp(iy%)\11, (2.14) = exp (i where in Eq. (2.14) I wrote the fermion charge Q f in terms of the Gell-Mann—Nishijima relation defined in Eq. (1.2) 3 Y 7' 57 Therefore, under the group G=SU(2)L x U(1)y WI. —> ‘I”L = 9 ‘I’L .’ (2.16) where g = exp(ia fungi?) e G. (2.17) Linearly realized right—handed fermions \I! R [S U (2) L singlet] simply coincide with FR: i.e., ‘I’R = F]; = (;;) . (2.18) R Out of the defined fields with their specified transformations it is straightforward to construct a Lagrangian which is invariant under SU(2)), x U (1),». In constructing the low energy chiral Lagrangian, one can use either the fermionic basis \IIL and \PR or the basis f1 and f2. The two basis lead to an identical physical S matrix in virtue of Coleman’s theorem [66, 81]. In constructing the chiral Lagrangian, I will follow Ref. [74] and define the com- posite fields as 2° = —-;-Tr('r“ElD,,2). (2.19) 14 Under the gauge transformation element 9 E G and using Eqs. (2.2) and (2.6), one finds that the composite fields transform as: a I a i ' T3 0 - T3 1 2,, —) 2,, = —-2-Tr exp(-zy-§-)T exp(zy-2—)23 DPS . (2.20) From which one concludes that under a general gauge transformation I3 23 —> 2 y = 22, (2.21) 58 and :l: _ ii :1: 2: -1 2', - e 112,, , (2.22) where fj=‘/_(E], =1: 72f) (2.23) The field 22 behaves as a neutral matter field while the two fields Eff behave as charged matter fields with Q = :t1. By a matter field I mean a field which does not transform as a gauge boson field under the symmetry group H. In a general gauge, 22 can be expanded as 1 1 g ' _ _ 2?: = 72: a_p¢3 2cosOZ"— 35(“745 _ WP 45+) —; (¢+a,,¢- — ¢-a,,¢+) + . .. (2.24) The composite field E; can be expanded as 2+-i +1W+£2+ oz '64 3W+ ,, — v2 ”(1) —Zg ”_v(¢ [cos p+sm.,,]—¢) p) i +;(¢+Bp¢3-¢3511¢+)+m 12.25) The component 2; is just the Hermitian conjugate of 2;. In the unitary gauge 2 = 1 one finds that the composite fields reduce to the physical gauge bosons, i.e., 1 9 3 _— 2" 2c050 " ’ (2'26) and 2:: -—;gWi. (2.27) Using the non-linear realization technique for G=S U (2) L x U (1)y and H=U(l)em one finds that the conserved generator Q (electric charge) is associated with the B” 59 gauge boson rather than the photon A” [77]. Therefore, in constructing the chiral Lagrangian, the covariant derivative of a fermion I will include the Bu gauge boson field rather than the Au boson field (see below). The main observation is that the bosonic fields 22, 2; combined with the fermionic fields f1, f2, .. . , only feel the electromagnetic transformation U (1)em even though the whole gauge symmetry is SU(2)L x U (l)y. This is a very important observa- tion because it enables us to write an invariant Lagrangian under the gauge group SU(2)L x U (1)y simply by requiring the Lagrangian (constructed from 2;, B”, and FAR) to be invariant under U (1),,m. For example , consider the top and bottom quarks. From Eqs. (2.13) and (2.18) one has t F: (b) =FL+FR, (2.28) with f1 = t and f2 = b. The SM Lagrangian involving the t and b quarks can be deduced from —. . , Y 73 — £0 = Fry” 8p—zg 3+3- Bp F—FMF — E7”T°FLEZ + £3 , (2.29) where the hypercharge number is Y = 1/ 3 and M is a diagonal mass matrix M = (’3‘ 72b) . (2.30) £0 is invariant under G and the electric charges of t and b quarks are given by the relation Y/ 2 + T3, where T3 = 13/2 is the weak isospin quantum number (T,3 : 1/2 and T: = -1 / 2). The terms in Eq. (2.29) are not the only possible gauge invariant dimension four operators. In fact, the chiral Lagrangian permits the inclusion of other terms while keeping the gauge invariance maintained. In chapter 3, I discuss how to parameterize the anomalous top quark couplings using the chiral Lagrangian framework. 60 Here is a final note regarding the physical Higgs boson. It is known that the gauge bosons acquire their masses through the spontaneous symmetry-breaking mechanism. The bosonic Lagrangian £3 in Eq. (2.4) only involves the gauge bosons and the unphysical Goldstone bosons. The Higgs boson is not a part of the Lagrangian £3. This indicates that the chiral Lagrangian can account for the mass generation of the gauge bosons without the actual details of the symmetry-breaking mechanism. Furthermore, the fermion mass term is also allowed in the chiral Lagrangian, _mfaflfi a (2.31) because it is invariant under G, where the fermion field f,- transforms as in Eq. (2.12). From this it is clear that the Higgs boson is not a necessary element in construct- ing the low energy effective Lagrangian. This indicates that the SM Higgs mechanism is just one example of the possible spontaneous symmetry-breaking scenarios which might take place in nature and still be described within the chiral Lagrangian frame- work. However, a Higgs boson field can be inserted in the chiral Lagrangian as an additional field (S U (2) L x U (1),» singlet) with arbitrary couplings to the rest of the fields. To retrieve the SM Higgs boson at tree level, one can simply substitute the fermion mass m, by 9,1) and v by v + H in fig [see Eqs. (2.4) and (2.29)], where g, = m {/11 is the Yukawa coupling for fermion f and H is the Higgs boson field. Hence, one gets the Higgs Lagrangian L” = gapHa“H — émgHQ — V(H) + $511 Tr (0,21022) + i-H”'1‘:(D,,2*D#2) , (2.32) where V(H) describes the Higgs boson self—interaction. In the SM, V(H) is given by 2 V(H) = "2% (4vH3 + H4) . (2.33) 61 Each term in Eq. (2.32) is separately gauge invariant because H is electromagnetically neutral. The coefficients of the last two terms in Eq. (2.32) can be arbitrary for a chiral Lagrangian with a scalar field other than the SM Higgs boson. 2.4 Heavy Top Quark Contribution to the 6 Pa- rameters in the Chiral Lagrangian In chapter 1, I calculated the top quark leading contributions to the epsilon pa- rameters in the SM. In this section, I repeat these calculations using the chiral La- grangian framework. There are some differences in the Feynman diagrams between the SM and the chiral Lagrangian case. The dimension-four gauge boson couplings to the fermions are the same as in the SM. However, the Goldstone bosons couplings to the fermions are different. What is important to our case are the vertices ¢3-t- f, and (bi-t3. In Figure 2.1, I show the relevant Feynman diagrams to the epsilon parameters in the chiral lagrangian. For the case of the 61, parameter one needs to include an additional Feynman diagram (see Figure 2.1f) which appears because of the non—linear realization of the chiral Lagrangian. The calculations are performed in the ’t Hooft-Feynman gauge. Calculations have been cross-checked in the Landau and unitary gauges, they all agree as they should. Using the heavy mass expansion discussed in appendix C, one finds o For the photon A77 (0) = 0 , (2.34) 92 16 sin2 0 167r2 9 F77(M§) = (A — ln m?) . (2.35) o For the Z boson 2 2 22 _ 9 3m, A (0) - —167r? 2 cos2 0 (A — In 171?) , (2.36) 62 2 1 1 4 . 16 . FZZ(M§) = %?m (5 — §s1n20 + -9—s1n4 0) (A - 1n m?) . (2.37) o For the W boson 2 3 2 l AWW(0) = 1%,??? (-A — 2 + In 711?) , (2.38) ww 2 92 2 F (M2) = 173;; (A — In 772,) . (2.39) o For the 7-Z mixing 2472(0) = 0 , (2.40) 2 o 72 2 =_-2_£_S_lfl (1-2 ' 2 ) _ 2 F (MZ) 1671’23C080 2 38m 0 (A lnmt). (2.41) o For the proper vertex 6;, one has to include all the relevant Feynman diagrams (see Figure 2.1c,d,e,f). One finds 0pm? _ , 2.42 4\/21r2 ( ) 61,: Therefore, we find €1=AP=W__M&,_-=Tfifr3m-=W’ (2.43) e: = FWW - mm) = €3ng mm?» (2.44) e. = :3 301 3.) = —192‘{/1§—ff,-1n(m?>. (2.451 and 5., = —f5§’:§2. (2.46) 63 The result we obtained here is identical to the result in section 1.5, as expected. By virtue of Coleman’s theorem [66, 81] the linear and non-linear realizations lead to the same physical observables. The equivalence between the two approaches is verified at one loop for the heavy top quark contribution to the low energy data. 64 (a) W W (b) b b + /—’——' Z ¢// ’/ lit (C) ’\/V\/\/\/\z<\\ \ ¢. \\—<— b (e) (f) Figure 2.1: The relevant Feynman diagrams in calculating the top quark contribution to the epsilon parameters using the chiral Lagrangian approach. Chapter 3 Global Analysis of the Top Quark Couplings to the Electroweak Gauge Bosons 3.1 Motivations and Perspectives As I discussed in chapter 2, there is little faith that the SM is the final theory. The reasons behind this are fundamental and basic (see chapter 2). One of the most important reasons is the assumed SM symmetry-breaking scenario. The SM assumes the symmetry breaking is a result of the dynamics of a complex scalar doublet . The scalar doublet acquires a v.e.v along the direction of the Higgs boson field. The simple SM scenario suffers a great difficulty if the SM is viewed as a fundamental theory. If the SM is valid up to all energies then the scalar sector is trivial (non-interacting) [82]. This ’triviality is problematic because of the need for a self-interacting scalar sector to generate the W and Z masses through the symmetry-breaking mechanism. To solve the triviality aspect of the SM scalar sector one has to assume that the scalar potential V‘(¢) is only valid below some energy cutoff scale A. Therefore, the SM can only be viewed as a low energy description of an effective theory which is only valid up to the energy cutoff scale A. Another problem in the SM scalar sector is that the loop corrections to the Higgs boson mass are quadratically divergent and 65 66 counterterms must be adjusted order by order in the perturbative expansion to cancel the quadratic divergencies. If the SM is embedded in a larger fundamental theory with heavy mass scales then the divergencies in the loop corrections to my depend on the square of the heavy mass scales. In order to keep my light, as required by unitarity conditions discussed below, there must be a deliberate cancelation of all the quadratic divergencies by the appropriate counterterms and to all orders. This cancellation is viewed by many as an unnatural aspect in the scalar sector which is known as the fine-tuning problem. The most recent experimental bound on the Higgs boson mass comes from the direct search at LEP [6] my > 65.2 GeV at 95% CL. (3.1) As concluded in Refs. [6, 36], electroweak precision tests show a preference for a light Higgs boson although the allowed range of m H is still wide enough due to the soft non- decoupling Higgs boson contribution to low energy data at one—loop level (logarithmic contribution [27]). On the other hand, as indicated in Ref. [37], LEP precision data, upon excluding the observables Rb and RC, does not imply a strong bound on the Higgs bosons mass, i.e., my can still be as large as 1 TeV. The requirement of the consistency of the SM scalar sector implies some theoretical bounds on the Higgs mass. Unitarity of the perturbative partial wave amplitudes [8] implies an upper bound on the Higgs mass [83, 84] my < 860 GeV. (3.2) The triviality aspect of the scalar sector and the requirement of new physics to appear at a high energy scale A (of the order of 1 TeV) implies a perturbative upper bound 67 on the Higgs boson mass [83] my < 1 TeV. (3.3) The higher the energy scale A is the lower the Higgs boson mass will be, e.g., for A ~ A”, and for m, ~ 175 GeV one finds [85]: m” < 200 GeV, , (3.4) where ApL is the Planck scale. Careful estimates of the triviality using the lattice [86] places a somewhat tighter limit my < 640 GeV. (3.5) Theoretical lower bounds on the Higgs mass come from the vacuum stability [87]. The lower limit is a function of the top quark mass and of the cutoff scale A. The dependence on the top quark mass comes because the Higgs boson coupling to the top quark is proportional to m, (Yukawa coupling). Recent analysis of the vacuum stability [85, 88] concludes that if the cutoff scale A is as large as the Planck scale Apt, then for m, ~ 175 GeV, the SM Higgs boson mass must be larger than about 120 GeV. Many attempts to offer alternative scenarios for the symmetry-breaking mecha- nism are discussed in literature. A general trend among all alternatives is that new physics appear at or below the TeV scale. Examples include MSSM [89], technicolor models [58, 89] and possibly extended technicolor sectors to account for the fermion masses [47, 59]. Other examples include top-mode condensate models [90] and a strongly interacting Higgs sector [91]. In this chapter, I study how to use the top quark to probe the origin of the spontaneous symmetry-breaking and the generation of fermion masses. I start the 68 discussion with a brief description of the top quark. The top quark is the T3 = +1 / 2 member of the weak isospin doublet containing the bottom quark with an electric charge Q = +2/ 3. The existence of the top quark has been confirmed recently by the CDF and D0 experiments [10, 11] at the Tevatron proton-antiproton collider with a center of mass (c.m.) energy of 1.8 TeV. Before this direct observation of the top quark, there were strong experimental and theoretical arguments suggesting that the top quark must exist [92]; e. g., the measurement of the weak isospin quantum number of the left—handed b quark T3 = —1/2 suggests that the b quark should have an isospin partner, namely the top quark. By 1994, from a negative result of direct search at the Tevatron, assuming SM top quark, DO concluded that mthas to be larger than 131 GeV [93]. In the same year, data were presented by the CDF group at FNAL to support the existence of a heavy top quark with mass m, ~ 174 i 20 GeV [94]. In 1995, both CDF and D0 announced the discovery of the top quark. From the recent observation of top quark events at GDP and DO a fit of the mass distribution leads to the CDF top quark mass [10] m, = 176 :f: 9 GeV , . (3.6) and the DO top quark mass [11] m, = 170 a; 18 GeV, (3.7) where the second error is the estimated systematic uncertainty. Furthermore, a recent study [12] based on the analysis of all available LEP data concludes that the SM top quark mass is m, = 178 :1: 8 :33 GeV. (3.8) The central value and the first error quoted refer to my = 300 GeV. The second 69 errors correspond to the variation of the central value when varying m H in the interval 60 GeV _<_ mg g 1000 GeV. Despite the recent success in the observation of the top quark and its mass mea- surement, there are no compelling reasons to believe that the top quark couplings to the weak gauge bosons should be of the SM nature. The GDP and DO measurements of the top quark mass are from a fit on the mass distribution, i.e., from kinematics, no conclusions can be drawn regarding interactions or dynamics. Therefore, using the top quark radiative corrections to the low energy data is the only available approach, so far, to study the top quark interactions. Because the top quark is heavy relative to the other observed fundamental particles, one expects that any underlying theory at high energy scale A >> m, will easily reveal itself at low energy through the effective interactions of the top quark to the gauge bosons. Also, because the top quark mass is of the order of the Fermi scale 1) = (x/2G'p)_l/2 = 246 GeV, which characterizes the electroweak symmetry-breaking scale, the top quark may be a useful tool to probe the symmetry-breaking sector. It is this connection between the top quark weak- interactions and the symmetry-breaking mechanism that I would like to investigate throughout the discussion of this chapter. Since the fermion mass generation can be closely related to the electroweak symme- try-breaking, one expects some residual effects of this breaking to appear in accor- dance with the generated mass [47, 59, 95]. This means that new effects should be more apparent in the top quark sector than any other light sector of the theory. Therefore, it is important to study the top quark system as a direct tool to probe new physics effects. In the SM, which is a renormalizable theory, the couplings of the top quark to gauge bosons are fixed by the linear realization of the gauge symmetry S'U (2);, x U (1)y. However, the top quark mass is a free parameter in the theory (SM) and 70 need to be measured experimentally. If the top quark is not a SM quark, then the couplings of the top quark to gauge bosons can be different from the SM. Also, the effective theory describing the top quark interactions at low energy can be non— renormalizable. Therefore, to conclude upon the properties of the top quark from the radiative corrections is less vital and predictive. Nevertheless, precision data at this stage are our best hope to look for any possible deviation in the top quark sector from the SM, until a direct measurement of the top quark interactions can be made. To study the couplings of the top quark to the gauge bosons, I will first use the precision data at LEP/SLC to constrain these couplings in a model independent approach, then I will examine how to improve our knowledge about the top quark couplings at the current and future colliders. In addition, I will discuss how to use this knowledge to probe the symmetry-breaking mechanism. It is generally believed that new physics is likely to come in via processes involving longitudinal gauge bosons (equivalent to Goldstone bosons) and/or heavy fermions such as the top quark. One commonly discussed method to probe the electroweak symmetry sector is to study the interactions among the longitudinal gauge bosons in the TeV region. Tremendous work has been done in the literature [96]. 'However, this is not the subject of this work. As I argued above, the top quark plays an important role in the search for new physics. Because of its heavy mass, new physics will feel its presence easily and eventually may show up in its couplings to the gauge bosons. If the top quark is a participant in a dynamical symmetry-breaking mechanism, e.g., through the it condensate (top-mode Standard Model) [90] which is suggested by the fact that its mass is of the order of the Fermi scale v, then the top quark is one of the best candidates for the search for new physics. An attempt to study the nonuniversal interactions of the top quark has been carried out in Ref. [95] by Peccei et a1. However, in that study only the vertex 71 t-t-Z was considered based on the assumption that this is the only vertex which gains a significant modification due to a speculated dependence of the coupling strength on the fermion mass: m,- S 0 (fl), where ngj parameterizes some new dimensional- four interactions among gauge bosons and fermions i and j. However, this is not the only possible pattern of interactions, e.g., in some extended technicolor models [59] one finds that the nonuniversal residual interactions associated with the vertices bL-E-Z , tL-fZ-Z , and tL-E-W to be of the same order. In Section 3.4, I discuss the case of the SM with a heavy Higgs boson (my > m,) in which we find the size of the nonuniversal effective interactions t L-fZ—Z and t L-E-W to be of the same order but with a negligible bL-b—i—Z effect. Here is the outline of my approach to the analysis. First, I use the chiral La- grangian approach [66, 69, 71, 72] to construct the most general SU(2)], x U (1)y invariant effective Lagrangian including up to dimension—four operators for the top and bottom quarks. Then I deduce the SM (with and without a scalar Higgs bo— son) from this Lagrangian, and only consider new physics effects which modify the top quark couplings to gauge bosons and possibly the vertex bL-E-Z . With this in hand, I perform a comprehensive analysis using precision data from LEP/SLC. I include the contributions from the vertex t—b—W in addition to the vertex t-t—Z , and discuss the special case of having a comparable size in b—b—Z as in t-t-Z . Second, I build an effective model with an approximate custodial symmetry (p z 1) connecting the t-t-Z and t-b—W couplings. This reduces the number of parameters in the effective Lagrangian and strengthens its structure and predictability. After examining what we have learned from the LEP and SLC data, I study how to improve our knowledge on these couplings at the Tevatron, the LHC (Large Hadron Collider) and the LC (Linear Collider) [97]. (I use LC to represent a generic e'e+ supercollider.) The rest of this chapter is organized as follows. In Section 3.2, I parameterize-the 72 anomalous top quark couplings using the chiral Lagrangian framework. In Section 3.3, I present the complete analysis of the top quark interactions with gauge bosons using LEP data for various scenarios of symmetry-breaking mechanism. Also, I discuss how the SLC measurement of A“; can contribute to the study of the top quark couplings to the gauge bosons. In Section 3.4, I discuss the heavy Higgs limit (my > m,) in the SM model as an example of our proposed effective model at the top quark mass scale. In Section 3.5, I discuss how the Tevatron, LHC, and LC can contribute to the measurement of these couplings. Some discussion and conclusions are given in Section 3.6. 3.2 The Top Quark Couplings to Gauge Bosons in the Chiral Lagrangian Framework I will concentrate my discussion on the b and t quarks. Precision. tests at LEP and SLC have shown that the interactions among the other light fermions and the gauge bosons agree very well with the SM, with the exception of the recent observed deviations in A“; at SLC and in Rb and R6 at LEP. As discussed in chapter 1, if the anomaly in Re persists in the future then it is unlikely for us to be able to understand such a large deviation within any perturbative model. Therefore, I will ignore this measurement in my discussion. (I will come back later to say a few words about Re.) To simplify the discussion on the proposed new top quark interactions, I will ignore all possible mixings of the top and bottom quarks with the other light fermions. In case there exists a fourth generation with heavy fermions, there can be a substantial impact on the Cabibbo—Kobayashi—Maskawa (CKM) matrix element V“. To be discussed below, this effect is effectively included in the nonstandard couplings of t—b—W of the effective Lagrangian at low energy scale. 73 Taking advantage of the chiral Lagrangian approach (see chapter 2), nonstandard interaction terms, invariant under SU(2)], x U (1)y, are allowed in addition to the standard terms in Eq. (2.29). These terms are £1 = —K’ZCt—I:‘)’ptl,22 — Kng-E‘thnzz — fixgcfi'y’bbfij — fiflgClEZ-Vyth; " finficfi'y"b32: — fiN%ClE7"tRE; a (3.9) where nfc, KIA/C are two arbitrary real parameters, REC, REC are two arbitrary com- plex parameters, and the superscript NC and CC denote neutral and charged cur- rents, respectively. The composite fields 22, a = 1, 2, 3 are discussed in chapter 2. In the unitary gauge, the Lagrangian above reduces to £1 = Isa—3.7;? (#130741 — 75) + nficwl + 75)) t Z. 9 - CC .CC + + 1,7240% 7"(1—75)+'~R 7”(1+'75))pr 9 - CC? .60? — + 72““ 211—75)“, 7"(1+75))tW,.- (3.10) A few remarks are in order regarding the Lagrangian L; in Eqs. (3.9) and (3.10). 1 In principle, £1 can include nonstandard neutral currents E7 bL and bz'y b3. 1‘ :4 For the left-handed neutral current 527pr I discuss two cases: (a) The effective left-handed vertices t LIE—Z , t L-b-Z-W , and bL¥b_[,'-Z are com- parable in size as in some extended technicolor models [47, 59]. In this case, the top quark contribution to low energy observables through radia- tive corrections is of a higher order, i.e., the top quark contribution will be suppressed by 1/167r2 relative to the b-quark contribution. In this case, as I will discuss in the next section, the constraints derived from low energy data on the nonstandard couplings are so stringent (of the order of a few (2) (3) 74 percent) that it would be a challenge to directly probe the nonstandard top quark couplings at the Tevatron, the LHC, and the LC. (b) The effective left—handed vertex bL-bI-Z is small as compared to the t-t-Z and t-b-W vertices. I will devote most of this work to the casewhere the vertex bL-b—i-Z is not modified by the dynamics of the symmetry breaking. This assumption leads to interesting conclusions to be seen in the next section. In this case one needs to consider the contributions of the top quark to low energy data through loop effects. A specific model with such properties is given in Section 3.4. I will assume that bR-bE-Z is not modified by the dynamics of the electroweak symmetry-breaking. This is the case in the extended technicolor models dis- cussed in Ref. [47, 59]. The model discussed in Section 3.4 is another example. The right-handed charged current contribution “$0 in Eqs. (3.9) and (3.10) is expected to be suppressed by the bottom quark mass. This can be understood in the following way. If b is massless (mb = 0), then the left- and right—handed b fields can be associated with different global U (1) quantum numbers. (U (1) is a chiral group, not the hypercharge group.) Since the underlying theory has an exact S U (2) L x U (1),» symmetry at high energy, the charged currents are purely left—handed before the symmetry is broken. After the symmetry is spontaneously broken and for a massless b the U (1) symmetry associated with b]; remains exact (chiral invariant) so it is not possible to generate right—handed charged currents. Thus K20 is usually suppressed by the bottom quark mass although it could be enhanced in some models with a larger group G, i.e., in models containing additional right—handed gauge bosons. I observed that in the limit of ignoring the bottom quark mass, KEG does not 75 contribute to the low energy data through loop insertion at the order m? In A2, therefore one cannot constrain KEG from the LEP data. However, at the Tevar tron and the LHC age can be measured by studying the direct detection of the top quark and its decays. This will be discussed in Section 3.5 It is worth mentioning that the photon does not participate in the new nonuni- versal interactions as described by the chiral Lagrangian £1 in Eq. (3.10) because the U (1)9", symmetry remains an exact symmetry of the effective theory. Any new physics can only contribute to the universal interactions of the photon to charged fields. This effect can simply be absorbed in redefining the electromagnetic fine structure con- stant 0, hence no new t-t-A or b—b—A interaction terms will appear in the effective Lagrangian after a proper renormalization of a and the wave functions of the particle fields. In this analysis, I will discuss an effective model with and without the Higgs boson. In the case of a light Higgs boson (my < 111,) I will include the Higgs boson field in the chiral Lagrangian as a part of the light fields with no new physics being associated with it. In the heavy Higgs boson case (my > m,), one should integrate out the Higgs boson field from the tree-level Lagrangian. Thus, one is left with an effective Lagrangian which contains the heavy Higgs boson effects and the additional nonstandard couplings nfc, nfic, EEC, and KEG. The Higgs boson contribution to the low energy effective Lagrangian (for energies E < m,) is only relevant in the gauge sector. This is true because, as discussed in section 1.5, the Higgs boson couplings to light fermions are negligible. However, the top quark couplings to the gauge bosons will be affected by the heavy Higgs boson due to the large Yukawa coupling (mg/v). In fact, a heavy Higgs boson may be the source for the nonstandard couplings of the top quark nfc, ago, KEG, and KEG. Finally, I will consider the possibility of a spontaneous symmetry-breaking scenario without including a SM Higgs boson in the 76 full theory. In this case I consider the effects on the low energy data from the new physics parameterized by the nonstandard interaction terms in £1 in Eq. (3.10) and the contributions from the SM without a Higgs boson. As I discussed above, one possibility of new physics effects is the modification of the vertices b-b-Z , t-t—Z , and t-b-W in the effective Lagrangian by the same order of magnitude [47, 59]. In this case, only the vertex b-b-Z can have large contributions to low energy data while, based on the dimensional counting method, the contributions from the other two vertices t-t-Z and t-b—W are suppressed by 1/167r2 due to their insertion in loops. In this case, one can use F], (the partial decay width of the Z boson to bb) to constrain the b-b—Z coupling. Denote the nonstandard b-b-Z vertex as g < WNW 1- ’75) , (33-11) which is purely left-handed. In some extended technicolor models, discussed in Ref. [47, 59], this nonstandard effect arises from the same source as the mass generation of the top quark, therefore K. depends on the top quark mass. As I discussed in chapter 1, the nonuniversal contribution to I}, is parameterized by a measurable parameter denoted as q, [16, 22, 23]. Using all LEP data, a fit on 6;, yields the value [38] e, (103) = 0.0 a: 3.9. (3.12) The SM contribution to q, is calculated in Ref. [26], e.g., for a 170 GeV top quark eff“ (103) = —6.15 . (3.13) The contribution from K, to 6,, is 6;, = -K.. V (3.14) 77 Within a 95% confidence level (C.L.), one finds that —14.2 g n (103) g 1.9. (3.15) As an example, the simple commuting extended technicolor model presented in Ref. [59] predicts that 1 2 m, z - -— 3.16 K 26 47m ’ ( l where E is of order 1. Also in that model the top quark couplingsnfc, Ego, and KEG, as defined in Eqs. (3.9) and (3.10), are of the same order as re. For a 170 GeV top quark mass, this model predicts K (103) x 27.5 {2. (3.17) Hence, such a model is likely to be excluded by using the low energy data constraints [see Eq. (3.15)]. 3.3 Low Energy Constraints In this section, I will devote the discussion to models in which the nonstandard b—b—Z coupling can be ignored relative to the t-t-Z and t-b-W couplings. In this case, the nonstandard couplings contribute to low energy observables at the quantum level, i.e., through loop insertion. I will first discuss the general case where no relations between the nonstandard couplings are assumed. Later, I will impose a relation NC between nL and KEG which are defined in Eqs. (3.9) and (3.10) using an effective model with an approximate custodial symmetry. 78 3.3.1 General Case In general, the chiral Lagrangian has a complicated structure and many arbitrary coefficients which weaken its predictive power. Still, with a few further assumptions, based on the status of present low energy data, the chiral Lagrangian can provide a useful approach to confine the coefficients parameterizing new physics effects. In this subsection, I provide a general treatment for the case under study with minimal imposed assumptions in the chiral Lagrangian. I only impose the assumption that the vertex b—b—Z is not modified by the dynamics. In the chiral Lagrangian £1, as defined in Eqs. (3.9) and (3.10), there are six independent parameters (K’s) which need to be constrained using precision data. Throughout this paper, I will only consider the insertion of K’s once .in one—loop diagrams by assuming that these nonstandard couplings are small; KNC’ CC < 1. At the one—loop level the imaginary parts of the couplings do not contribute to those LEP observables of interest. Thus, hereafter I drop the imaginary pieces from the effective couplings, which reduces the number of relevant parameters to four. Since the bottom quark mass is small as compared to the top quark mass, the non-standard coupling NRC does not contribute to low energy observables up to the order m? In A2 in the mb —) 0 limit. With these observations we conclude that only the three parameters n50, 741,30 and NEC (to bear in mind, this is really Re(IcEC)) can be constrained at LEP. The nonstandard coupling Kfic can be studied using the CLEO measurement of b -) 37. An effective model with only one nonstandard coupling ECG was studied in Ref. [98], a constraint on the right-handed charged current coupling KCC was set using the CLEO measurement of b -) $7. The authors concluded that nfic is well constrained to within a few percent from its SM value (116%C = O). This provides a complementary information to our constraint on KNC ,nfic, and 1620 79 As I discussed in chapter 1, the new physics contribution to low energy observables, under a few general assumptions, can be parameterized by 4-independent parameters 61, 62, 63, and 61,. In our case, the general assumptions are satisfied, namely all the contributions of 162’ 0, nfic, KEG, and nfic to low energy observables are contained in the oblique corrections, i.e., the vacuum polarization functions of the gauge bosons, and the non-oblique corrections to the vertex b-b-Z . Therefore, it is enough to calculate the new physics contribution to the 6 parameters in order to isolate all effects to low energy observables. As discussed in chapter 1, the 6 parameters are derived from four basic measured observables, Pg (the partial width of Z to a charged lepton pair), AfipB (the forward—backward asymmetry at the Z peak for the charged lepton Z), Mw/Mz, and [‘5 (the partial width of Z to a bb pair). The expressions of these observables in terms of 6’s are given in chapter 1. Since the top quark will only contribute to the vacuum polarization functions and the the vertex b-b-Z , I am only interested in the following terms 61 = 61 — e5 , , (3.18) 62 = 62 — cos2 0 65, (3.19) 63 = 63 - cos2 0 e5 , (3.20) 65 = e), , (3.21) where 61, 82, 63, 65, and 8;, are defined in chapter 1 as follow _ AZZ(0) _ AWW(0) e, _. Mg M31» , (3.22) e. = FWW — F33(M§), (3.23) e - E9‘"—91<““’(1\42) 3 24 3 ‘ sin0 Z ’ l ' l 80 szz 65 = M3752— (Mg). (3.25) The quantity eb is defined through the proper vertex corrections _ 1 _. v, (z —+ bb) = —2£Ceby,,-—233. (3.26) As we found in section 1.5, the parameters 61 and £5 depend quadratically on 171,. Therefore, 61 and q, are sensitive to any new physics coming through the top quark. On the contrary, 62 and 63 have at most a logarithmic dependence on m,. Hence, in our effective model, the significant constraints on the parameters n50, xfic, and KEG are only coming from £1 and 64,. Non—renormalizability of the effective Lagrangian presents a major issue of how to find a scheme to handle both the divergent and the finite pieces in loop calculations [99, 100]. Such a problem arises because one does not know the underlying theory; hence, no matching can be performed to extract the correct scheme to be used in the effective Lagrangian [61]. One approach is to associate the divergent piece in loop calculations with a physical cutoff A, the upper scale at which the effective Lagrangian is valid [74]. In the chiral Lagrangian approach this cutoff A is taken to be 41w ~ 3 TeV [61]]. For the finite piece no completely satisfactory approach is available [99]. I assume that the underlying full theory is renormalizable. In this case, the cutoff scale A serves as the infrared cutoff of the operators in the effective Lagrangian. Due to the renormalizability of the full theory, from renormalization group analysis, one concludes that the same cutoff A should also serve as the ultraviolet cutoff of the effective Lagrangian in calculating Wilson coefficients. Hence, in the dimensional regularization scheme, 1/6 is replaced by ln(A2 / [12), where 6 = (4 - n)/ 2 and n is the 1The scale 41w ~ 3TeV is only meant to indicate the typical cutoff scale. It is equally probable to have, say, A = 1 TeV. 81 space-time dimension. Furthermore, the renormalization scale )1 is set to be m,, the heaviest mass scale in the effective Lagrangian of interest. To perform calculations using the chiral Lagrangian, one should arrange the con- tributions in powers of 1/47rv and then include all diagrams up to the desired power. In a general RE gauge (2 rfi 1), the couplings of the Goldstone bosons to the fermions should also be included in Feynman diagram calculations. These couplings can be easily found by expanding the terms in L, as given in Eq. (3.9). The relevant Feyn- man diagrams are shown in Figure 3.1. Calculations are done for a general R5 gauge. (I have also checked the calculations in the ’t Hooft—Feynman gauge, the Landau gauge, and the unitary gauge. All agree as expected.) I calculate the contribution to 61 and 6,, due to the new interaction terms in the chiral Lagrangian (see Eqs. (3.9) and (3.10)) using the dimensional regularization scheme and taking the bottom mass to be zero. At the end of the calculation, as I discussed above, I replace the divergent piece 1/6 by ln(A2/m?) for e = (4 — n)/2 where n is the space-time dimension. Since I am mainly interested in new physics associated with the top quark couplings to gauge bosons, I will restrict myself to the leading contribution enhanced by the top quark mass, 116., of the oder of m? In A2. The result of the calculation is as follows 0 The vacuum polarization function of the Z boson is (see Figure 3.1a) was: 1 - .2 («W/40>- 1 (3.27) 6 1422(0) o The vacuum polarization function of the W’ boson is (see Figure 3.1b) M2 31712 1 WW W .00 c Figure 3.1c yields the result 1 e ' 2 29 m .60 4c030 47r2tv2 (-2h’4 )7” (1 " '75) (3.29) 82 Z Z (a) t D W W (b) 1 b t z : |¢ t l (d) b b // t 2 ¢/// ’ A b b WW 1 t \ L I t \ ¢\\ \\...—// (e) \ 4 4’ b Figure 3.1: The relevant Feynman diagrams, for the nonstandard top quark couplings case and in the ’t Hooft—Feynman gauge, which contribute to the order 0(mt21n A2). 83 A similar diagram with the other b quark leg attached to the Goldstone boson yields the same result. 0 Figure 3.1d yields the result 1'9 m? 4 cos 0 43%2 1 l (—2 cos20C KL 0+ Znfic — 1450) 7,, (1 — 75) -€— (3.30) c Figure 3.1e yields the result is m? 1 1 _ 2 _ cc _ _ 4cos€41r2v2 ( COS 0 + 2) KL 7" (1 75) e (3.31) o The b-quark self energy (Figure 3.1f) yields the result 1 3m2 167,2; 2 “up” (.g ) (1— 75) 2 (332) Therefore, one finds Gp A2 6f} = m3mt2(—KIZC + KIA/C + REC) In 171—? , (3.33) 5e— m? -—1 NC+~NC 1 1‘: (334 b— 232—“ 2m 4K}; A'L n m? , . ) where 661 denotes the new physics contribution to 61 and similarly for 561,. Notice that 62 and 63 do not contribute at this order. It 13 interesting to note that KL CC does not contribute to 6,, up to this order which can be understood from Eq. (3.10). If REC = -1 then there is no net t-b-W coupling in the chiral Lagrangian after including both the standard and nonstandard contributions. Hence, no dependence on the top quark mass can be generated, i.e., the nonstandard REC contribution to 6;, must cancel the SM contribution when KEG = —-1, independently of the couplings of the neutral current. From this observation and because the SM contribution to 61, is finite, I conclude that REC cannot contribute to 6;, at the order of interest. In Ref. [101] a 84 similar calculation for 6;, was performed and the author claimed to get a different result from the one above. However, the author included only the vertex corrections to calculate the physical quantity 65, which is not complete because the wave function corrections to the b quark must be included. In Eqs. (3.33) and (3.34) I set the renormalization scale 11 to be equal to m,, which is the natural scale to be used in this study because the top quark is considered to be the heaviest mass scale in the effective Lagrangian. I have assumed that all other heavy fields have been integrated out to modify the effective couplings of the top quark to gauge bosons at the scale m, in the chiral Lagrangian. Here, I ignore the effect of the running couplings from the top quark mass scale down to the Z boson mass scale which is a reasonable approximation for this study. To constrain these nonstandard couplings one needs to have both the experimental values and the SM predictions of (’8. The experimental data is given in Table 1.2. Pkom these low energy data, a fit for 61 and Q, yields the values [36] app-(103) = 3.8:i:1.5, egxp-(103) = 0.0239. (3.35) The SM contribution to 6’3 have been calculated in Ref. [26]. I include these contributions in the analysis in accordance with the assumed Higgs boson mass. If the low energy theory contains a SM Higgs boson, i.e., there is a light Higgs boson (my < m,), then the calculated values of the e’s include both the SM contribution calculated in Ref. [26] and the new physics contribution derived from the effective couplings of the top quark to gauge bosons. In the heavy Higgs boson case (my > 111.), one should integrate out the Higgs boson field from the tree—level Lagrangian. Thus, one is left with an effective Lagrangian which contains the heavy Higgs boson effects and the additional nonstandard couplings Kilt/C, n’gc, KEG, and 163720. Up to 85 one loop, the Higgs boson contribution to the low energy effective Lagrangian (for Mw < E << mg) is only relevant in the gauge sector. This is true because, as discussed in section 1.5, the Higgs boson couplings to light fermions are negligible. The effective Lagrangian after integrating out the heavy Higgs boson field can be written as L _ 1W+ w-pu 1 1 w3 W34“, IB 8”” _ l . 0 a B W3!” eff — -—§ 1w —4( —e2) #0 —4 m, 2s1n cos 83 w ma-Mgzpz" - Mg cos? 0 (1 + el)W,,+W"‘ , (3.36) where 2422(0) AWW(0) 3GpM3V sin20 = — = — 1 2 , 3.37 61 Mg Mp2], 8\/27r2c0826 n(m,,) ( ) e. = FWWME.) — F3306) = o, (3.38) c086 GpM2 = M2 = W1 2 . 3. 9 83 sin6F3O( 2) “fl”? “(mH) ( 3) However, the top quark couplings to the gauge bosons will be affected by the heavy Higgs boson due to the large Yukawa coupling (mt/v). In fact, a heavy Higgs boson may be the source for the nonstandard couplings of the top quark 16f C, 162m, KEG, and ngc. In section 3.4, I calculate the heavy Higgs boson effect to the nonstandard top quark couplings. Finally, I consider the case of a spontaneous symmetry scenario without a Higgs boson. In this case, I subtract the Higgs boson contribution from the SM values of thee parameters given in Ref. [26]. In this scenario, the nonstandard top quark couplings to gauge bosons are viewed as not due to an assumed heavy Higgs boson but possibly to some residual effects of a new symmetry-breaking mechanism. First, I consider the light Higgs boson case (171” < 171,). Choosing m. = 160 and 180 GeV, respectively, and taking my = 65 GeV, I span the parameter space defined by —1 g ICIZC g 1, —1 g 16],)“: g 1, and —1 g REC g 1. Within 95% CL. 86 and including both the SM and the new physics contributions, the allowed regiOn of these three parameters is found to form a thin slice in the specified volume. The two—dimensional projections of this slice are shown in Figures 3.2, 3.3, and 3.4. These nonstandard couplings (16’s) do exhibit some interesting features. (1) As a function of the top quark mass, the allowed volume for the top quark couplings to the gauge bosons shrinks as the top quark becomes more massive. (2) New physics prefers positive 1650, see Figures 3.2 and 3.3. For nfic = 0, 1413’ C is constrained within —-0.05 to 0.17 (0.0 to 0.15) for a 160 (180) GeV top quark. (3) New physics prefers REC z —K’,}’C. This is clearly shown in Figure 3.4 which is the projection of the allowed volume in the plane containing team and KEG. The preference for a positive 162/C is triggered by 61,. For m, = 170 GeV and my 2 65 GeV, the experimental value of 6,, is efixp‘ = 0.0 :1: 3.9 , (3.40) which is larger than the SM value cf” = —6.15. (3.41) Therefore, the new physics contribution to 6;, favors positive values in order to be compatible with the experimental measurement. Hence, in view of Eq. (3.34), 162“: prefers positive values. Also, since the main contribution to q, is coming from It? C (for K’s< 1), one finds that It? C is well constrained relative to nfic and KEG. The preference for leg“: z —n(i0 can be understood from the nonstandard contribution to 61 (see Eq. (3.33)). Since the experimental fit on 61 with its small uncertainty is compatible with the SM and because ngc is well constrained by 65, there is not much freedom in the sum 141;,” + KEG, i.e., Kfic + REC ~ 0. 87 1.0 _— 0.5 Icgc 0.0 _ —0.5 —1.0 — I l I I I I I I I I I I I I I I I J "‘ —1.0 -0.5 0.0 0.5 1.0 NC ’CL Figure 3.2: A two—dimensional projection in the plane of 162' C and Riga, for m, = 160 GeV (solid contour) and 180 GeV (dashed contour). The Higgs boson mass is fixed, my = 65 GeV. 88 I I I I I I I I I I I I I I I I I I _- l I l l l 4 1.0 — —‘ 0.5—— r- -1 o - -1 OH — _— 2 O O _ _ ,— .— -0.5 — '— — -1 )— _ -1.0 — '— V l | l J | ‘ I I L I I I I I I I I I I I L I I I -1.0 -0.5 0.0 0.5 1.0 4:0 Figure 3.3: A two-dimensional projection in the plane of it? C and NEC, for m, = 160 GeV (solid contour) and 180 GeV (dashed contour). The Higgs boson mass is fixed, my = 65 GeV. 89 PIIIrIIIIIIIIIIIIIIIIIl—I- 1.0— 0.5— 3,; 00:— —O.5— —1.0— — IIILIIIIIIIIIIIIIIIIJIII‘ -1.0 -O.5 0.0 0.5 1.0 NC ’51: Figure 3.4: A two—dimensional projection in the plane of K1,)“: and NEC, for m, = 160 GeV (solid contour) and 180 GeV (dashed contour). The Higgs boson mass is fixed, my = 65 GeV. 90 Now, I comment on the heavy Higgs boson (my > m.) and the no—Higgs boson cases. A different Higgs boson mass does not have a large influence on the allowed parameter space. As discussed in section 1.5, Q, is not sensitive to the Higgs boson mass at one-loop level, while 61 has at most, for a heavy Higgs boson, a logarithmic dependence on m 3. Since K210 is mostly constrained from Q, one does not expect any noticeable effect on the allowed range of 161:0 as a function of the Higgs boson mass. Also, since the fit on 61 is compatible with a wide range of Higgs boson masses, the relation nfi’c + REC ~ 0 is maintained. In Figure 3.5 I show the parameter space of 162’ C and nfic for the Higgs boson mass m” = 1000 GeV and for two values of the top quark mass, m, = 160 GeV(solid contour) and 180 GeV (dashed contour). One finds that for age = 0, 11ch is constrained within —0.03 to 0.2 (0.0 to 0.2) for a 160 (180) GeV top quark. Next, I consider the possibility of a new symmetry-breaking scenario without a fundamental scalar such as a SM Higgs boson. In this case, I simply subtract the Higgs boson contribution from the SM results obtained in Ref. [26]. In this case one expects to find no noticeable difference from the light Higgs boson case shown in Figures 3.2, 3.3, and 3.4. This is true because a light Higgs bosons has a smaller contribution to 61 than a heavy Higgs boson [21]. Therefore, the case of the no- Higgs boson scenario has a similar effect as the light Higgs boson case. In the next subsection, I will discuss the heavy Higgs boson and the no—Higgs boson cases in more detail. In Ref. [95] a similar analysis has been carried out by Peccei et al. However, in their analysis they did not include the charged current contribution REC and assumed only the vertex t-t-Z gives large nonstandard effects. The allowed region they found simply corresponds, in my analysis, to the region defined by the intersection of the allowed volume and the plane Kfc = 0. This gives a small area confined in the 91 _ I I I I I I I I I I I I I I I I I I I I I I - 1.0 _— —‘ 0.5 — '— 2;,“ 0.0 r- _. I 1 -0.5 — —~ -1.0 r— -‘ I L I I I l l I I I l I I I I I l I I I I I I- -1.0 —0.5 0.0 0.5 1.0 NC ”t Figure 3.5: A two—dimensional projection in the plane of nil/C and nfiC, for m, -—f 160 GeV (solid contour) and 180 GeV (dashed contour), and for the heavy Higgs boson mass my = 1000 GeV. 92 NC vicinity of the line KL = nfiC. This can be understood from the expression of 61 derived in Eq. (3.33). After setting KEG = 0, one finds ( NC _ NC) 3 42 61 or KR KL . ( . ) In this case one notes that the length of the allowed area is merely determined by the contribution from 6),. I will elaborate on a more quantitative comparison in the second part of this section. 3.3.2 Special Case The allowed region in the parameter space shown in Figures 3.2, 3.3, and 3.4 contains all possible new physics (to the order m? In A2 ) which can modify the cou- plings of the top quark to gauge bosons as described by 162’ C, nfic, and REC. In this subsection, I would like to examine a special class of models in which an approximate custodial symmetry is assumed as suggested by low energy data. The SM has an additional (accidental) symmetry called the custodial symmetry which is responsible for the tree-level relation 3V = _ = 1 . p Mg 3% ’ (3 43) where (:3 = 1 - 33 and 33 is the weak mixing angle defined in the on-shell scheme (see appendix A). This symmetry is slightly broken at the quantum level by the S U (2) doublet fermion mass splitting and the hypercharge coupling 9’ [102]. Writing p = l + 6p, 6p would vanish to all orders if this symmetry is exact. Low energy data indicate that tip is very close to zero. In fact, low energy data constrains p to be 1 within about 0.1% accuracy [60]. Therefore, I will assume that the underlying theory has a global custodial symmetry. In other words, I require the global group SU (2)1, associated with the custodial symmetry to be a subgroup of the full group 93 characterizing the full theory. I will assume that the custodial symmetry is broken by the same factors which break it in the SM, i.e., by the fermion mass splitting and the hypercharge coupling g’ . In the chiral Lagrangian this assumption of a custodial symmetry sets 1);; = v, and forces the couplings of the top quark to gauge bosons W: to be equal after turning off the hypercharge and assuming mb = m,. If the dynamics of the symmetry breaking is such that the masses of the two S U (2) partners t and b remain degenerate then one expects new physics to contribute to the couplings of t-t-Z and t-b—W by the same amount. However, in reality, m1, << mt; thus, the custodial symmetry has to be broken. I will discuss how this symmetry is broken shortly. Since I am mainly interested in the leading contribution enhanced by the top quark mass at the order m? In A2, turning the hypercharge coupling on and off will not affect the final result up to this order. I construct the two Hermitian operators J L and J R, which transform under G as J" = 420,2? —> gLJ£g[ , (3.44) J“ = £210,123 —> gRJgg}, , (3.45) where g, = exp(ia“121) E SU(2)), and 93 = exp(iy%). In fact, using either JL or JR will lead to the same result. Hence, from now on I will only consider JR. The SM Lagrangian can be derived from £0 = $21761)ng + $21761)wa — (972M913 + h.c.) 1 0 Va 1 I! v2 —ZW,,,W# — ZBWB" + Z-'1‘r(J;;J,,,,) , (3.46) where M is a diagonal mass matrix. I have chosen the left—handed fermion fields to 94 be the ones defined in Eq. (2.13): t b L The fermion field \IJL transforms linearly under G=S U (2) L x U (1),» , 1.6., ‘1’], -§ \I’IL = g‘IIL , (3.48) where g 6 SU (2);, x U (1),». The right—handed fermion fields t R and bR coincide with the original right—handed fields (see Eq. (2.18)). Also, , 'r“ a , Y D: = 0,, — zygwp — zg'-2—B,, , (3.49) . Y 7'3 D"? = 8,, - 29' (E + -2—) BF . (3.50) Note that in the nonlinear realized effective theories using either set of fields (\IILJ; or Fug) to construct a chiral Lagrangian will lead to the same S matrix [66]. The Lagrangian [.0 in Eq. (3.46) is not the most general Lagrangian one can construct based solely on the symmetry of G/H, for G=SU (2),, x U (1)y and H = U (1),,m. Taking advantage of the chiral Lagrangian approach one can derive additional interaction terms which deviate from the SM. This is so because in this formalism the SU(2)L x U (l)y symmetry is nonlinearly realized and only the U (1)em is linearly realized. Because the SM is so successful one can think of the SM (without the top quark) as being the leading term in the expansion of the effective Lagrangian. Any possible deviation associated with the light fields can only come through higher dimensional operators in the Lagrangian. However, this assumption is neither necessary nor prefer- able when dealing with the top quark because no precise data are available to lead to 95 such a conclusion. I will include nonstandard dimension—four operators for the cou- plings of the top quark to gauge bosons. In fact this is all I will deal with and will not consider operators with dimension higher than four. Note that higher dimensional operators are naturally suppressed by powers of 1 / A. One can rewrite J R as Ta J53 = 113°; . (3.51) with J‘” = Tr (ran) = iTr (7021912) . (3.52) The full operator J R posses an explicit custodial symmetry when g’ = 0 as can easily be checked by expanding it in powers of the Goldstone boson fields. Consider first the left—handed sector. One can add additional interaction terms to the Lagrangian £0 1:, = xlfiypmgztm + amnnmgztm + ng‘wjypmgmfim, (3.53) where m is an arbitrary real parameter and K2 is an arbitrary complex parameter. Note that Cl still is not the most general Lagrangian one can write for the left—handed sector, as compared to Eq. (3.9). In fact, it is our insistence on using the fermion doublet form and the full operator J R that lead us to this form. For shorthand, £1 can be further rewritten as c, = $27,2KLJ52WL + \II—L'ypEJfiIr'jzle , (3.54) where K L is a complex diagonal matrix with three real parameters. These new terms can be generated either through some electroweak symmetry- breaking scenario or through some other new heavy physics effects. If 171,, = m, 96 and g’ = 0, then we require the effective Lagrangian to respect fully the custodial symmetry to all orders. In this limit, 192 = O in Eq. (3.53) and K L = 1611, where 1 is the unit matrix and 161 is real. Since m), << 111,, one can think of K2 as generated through the evolution from m), = m, to m), = 0. In the matrix notation this implies K L is not proportional to the unit matrix and can be parameterized by ‘ 0 KL = (60,, xi.) , (3.55) with 14', = g + .42, (3.56) and K2 = g — K2 . (3.57) In the unitary gauge one gets the terms g — g 1 — +2c0802Re(KtL)tL7ptLZp + “£062 + If: )tL'y"bLW,f 9 — _ 9 — +Eoc‘; + nibbnmwp — 2cos 02Re(n';,)b,,)"b,z,,. (3.58) As discussed in the previous section, I will assume that new physics effects will not modify the bL-bZ-Z vertex. This can be achieved by choosing 161 = 2Re(x2) such that R6062) vanishes in Eq. (3.57). Later, in section 3.3, I will consider a specific model satisfying this assumption. Since the imaginary parts of the couplings do not contribute, at one-loop level, to LEP physics of interest, I simply drop them hereafter. With this assumption one is left with one real parameter K‘L which will be denoted from now on by KL / 2. The left—handed top quark couplings to the gauge bosons are 1,, — t, — 2: {Emma — )5), (3.59) 97 9 “L t —b —W: —— 1— . 3.60 L L 2fi27p( 75) ( ) Notice the connection between the neutral and the charged current, as compared to Eq. (3.10): “IZC = 2K€C = KL . (3.61) This conclusion holds for any underlying theory with an approximate custodial sym- metry such that the vertex bL—E-Z is not modified as discussed above. For the right-handed sector, the situation is different because the right—handed fermion fields are S U (2) singlet, hence the induced interactions do not see the full op- erator J R but its components individually. Therefore, one cannot impose the previous connection between the neutral and charged current couplings. The additional allowed interaction terms in the right—handed sector are given by _ 9 1 NC— 3 9 CC— + £2 — 2—CKR tR'y"tRJRu + ERR tR’7beJnfl +%Kgclblz’7ptn~]np — éfilfi 19127pran , (3-62) NC NC . . . where 16}; and K3; are two arb1trary real parameters and REC 13 an arb1trary complex parameter. Note that in £2 we have one more additional coefficient than we have in L1 (in Eq. (3.53)), this is due to our previous assumption of using the full operator JR in constructing the left—handed interactions. I assume that the bR-b-E-Z vertex just as the bL-E-Z vertex is not modified, then the coefficient 15'}ch vanishes. Because «(if does not contribute to LEP physics in the limit of m), = 0 and at the order m? In A2 we are left with one real parameter KENC which will be denoted hereafter as KR. The right—handed top quark coupling to Z boson is t); - tn - Z: %K37p(1+ 75) . (3.63) 98 (As compared to Eq. (3.10), KR— — KIA/C.) In the rest of this section, I consider the models described by £1 and £2 with only two relevant parameters KL and ’63. Performing the calculations as I discussed in the previous subsection one finds K A2 661: 232w ———2F3m, (KR — %)ln(;?-) , (3.64) m2 1 A2 56:1, 3‘75?” 2m ¢(—-I€R +ICL) 1110-77-1?) . (3.65) These results simply correspond to those in Eqs. (3.33) and (3.34) after substituting _ __ NC _ . 162/C - 2KEC — KL and KR — an. The constraints on KL and KR for models with a light Higgs boson, a heavy Higgs boson, and without a Higgs boson are presented here in order. Let us first consider a light Higgs boson with m H = 65 GeV. I include the SM values for 61 and 6), given in Ref. [26], the experimental fit on 61 and 6), given in Eq. (3.35), and the nonstandard contribution given in Eqs. (3.64) and (3.65). I span the plane defined by KL and RH for a top quark mass of 170 GeV. Figure 3.6 shows the allowed range for those parameters within 95% CL. As a general feature, one observes that the allowed range is a narrow area aligned close to the line 16;, = 2163 where for m, = 170 GeV the maximum range for KL is between —0.03 and 0.23. In Table 3.1, I give the allowed range of the couplings KL and K}; for different top quark masses. As the top quark mass increases this range shrinks and moves downward and to the right away form the origin (KLJCR) = (0,0). This behavior can be understood if we notice that the width of the allowed area is controlled by 61 because of the small error in the value of efxp'. Whereas, ezxp. controls the length of the allowed region. As the top quark mass increases, the value of 6?“ increases. In order to be compatible with the experimental data, the nonstandard contribution to £1 prefers negative values, 3.8., 2163 — KL 3 0. 99 ”a Figure 3.6: Theallowed region of KL and KR, for m, = 170 GeV, my 2 65 GeV. (Note that KL = 161:0 = 2KEC and K]; = 16920.) The observation that the allowed region shifts to the right, toward positive KL, as the top quark mass increases can be understood from the behavior of 61,. As the top quark mass increases, the value of 6E“ decreases forcing the nonstandard contribution to be more positive, 1.6., moving toward positive KL. The deviation from the relation KL = 2N}; for various top quark masses is given in Figure 3.7 by calculating KL — 2163 as a function of m,. Note that the SM has the solution KL = K}; = 0, i.6., the SM solution lies on the horizontal line shown in Figure 3.7. This solution ceases to exist for m, _>_ 200 GeV. The special relation KL = 2N}; is a consequence of the assumption of an approximate custodial symmetry which I imposed in connecting the left—handed neutral and charged currents. As discussed above, the SM contribution to 6b [26] is lower than the experimental 100 Table 3.1: The confined range of the couplings, KL and K}; for various top quark masses and for my = 65 GeV. m, (GeV ) KL RR 120 -0.16 —— 0.31 —0.08 —— 0.20 130 —0.12 — 0.28 —0.07 —— 0.17 140 —0.09 — 0.26 —0.05 -— 0.15 150 —0.07 — 0.25 -0.05 — 0.14 160 —0.04 — 0.24 —0.03 -—-— 0.13 170 —0.03 — 0.23 -—0.03 — 0.12 180 -0.01 —— 0.22 —0.02 — 0.11 190 0.00 — 0.21 —0.02 — 0.10 200 0.01 — 0.20 —0.02 —— 0.09 210 0.02 —— 0.20 —0.02 — 0.09 220 0.03 — 0.19 —0.01 — 0.08 230 0.04 — 0.19 -0.01 —— 0.07 1.0 h I I I I I I I I I I I I I I I I I I I I I 0.5 — _— ). q 6 i 1 “3 0.0 _ 5' _ 1 r . _ p I I I I I I I I I I I I I I I I I I I I I 1 100 150 200 250 300 Int GeV Figure 3.7: The allowed range of (KL — 2K3) as a function of the mass of the top quark and for my = 65 GeV. (Note that KL = 16f C = 2:650 and NR = KEG.) 1.0.. I I I I I I ‘l’ I I I I I I l I I I I I7 I q 0.5— _ 8: 0,0 HHHILHII F’ u —o.5—-— —— _lopkl I I I I I J I I I I I I 4 JJ I I I I 1 ' 100 150 200 250 300 m.GeV Figure 3.8: The allowed range of the coupling KEG = ref C / 2 = KL / 2 as a function of the mass of the top quark and for m H = 65 GeV. central value. This is reflected in the behavior of KL which prefers being positive to compensate this difference as can be seen from Eq. (3.65). This means that in models of electroweak symmetry-breaking with an approximate custodial symmetry, a positive KL is preferred. In Figure 3.8, I show the allowed values for KEG = ref C / 2 =- nL/2 as a function of m,. With new physics effects (1% 75 0) m, can be as large as 300 GeV, although in the SM (KL = 0), as seen from Figure 3.8, m, is bounded below 200 GeV. Now, I would like to discuss the effect of the Higgs boson mass on the allowed range of these parameters. It is easy to anticipate the effect; since 6;, is not sensitive to the Higgs boson contribution up to one loop, the allowed range is only affected by the Higgs boson contribution to 61 which slightly affects its location relative to Figure 3.9: The allowed range of the coupling REC = 162’ C / 2 = KL / 2 as a function of the mass of the top quark and for my = 300 GeV. the line KL = 2kg. One expects that as the Higgs boson mass increases the allowed area moves upward. The reason simply lies in the fact that the standard Higgs boson contribution to 61, up to one loop, becomes more negative for heavier Higgs boson (see section 1.5). Hence, 21c}; prefers to be larger than KL to compensate this effect. However, this modification is not significant because 61 depends on the heavy Higgs boson mass only logarithmically. In Figures 3.9 and 3.10 I show the allowed range, within 95% C.L., for the parameters KL and KR for m, = 170 GeV and for two choices of the heavy Higgs boson mass my = 300 GeV and my = 1000 GeV, respectively. Figures 3.9 and 3.10 are consistent with what one expects. Now I consider the possibility of a new symmetry-breaking scenario without a fundamental scalar such as a SM Higgs boson. In this case, I simply subtract the 103 ”a Figure 3.10: The allowed range of the coupling REC = K72, C / 2 = KL / 2 as a function of the mass of the top quark and for m” = 1000 GeV. Figure 3.11: The allowed region of K}: C and Islam, for models without 3. SM Higgs boson and for m, = 170 GeV. Higgs boson contribution from the SM results obtained in Ref. [26]. Figure 3.11 shows the allowed area in the KL and KR plane for a 170 GeV top quark in such models. In this case one expects to find no noticeable difference from the light Higgs boson case shown in Figure 3.6. This is true because a light Higgs bosons has a negligible contribution to 61 as compared to a. heavy Higgs boson [21]. Therefore, the case of the no-Higgs boson scenario has a similar effect as the light Higgs case. What we have learned is that to infer a bound on the Higgs boson mass from the measurement of the effective couplings of the top quark to gauge bosons, one needs a very precise measurement of the parameters KL and KR. However, from the correlations between the effective couplings (K’s) of the top quark to the gauge bosons, one can infer if the symmetry-breaking sector is due to a Higgs boson or not, i.e., we 105 may be able to probe the symmetry-breaking mechanism in the top quark system. To illustrate this point, I would like to compare my results with those in Ref. [95]. Figure 3.12 shows the most general allowed region for the couplings n.1,? C and nfic, N i.e., without imposing any relation between KL0 and KEG. This region is for a top quark mass of 170 GeV and covers the parameter space —1.0 S n2“: , 51);": S 1.0. One finds —o.15 3 Kg“? $0.35, —1.0 < 4031.0. I also show on Figure 3.12 the allowed regions for our model (nfc = 1 / 21f 0) and the model in Ref. [95] (nfc = 0). The two regions overlap in the vicinity of the origin (0, 0) which corresponds to the SM case. Note that for m, S 200 GeV the allowed region of Ic’s in all models of symmetry-breaking should overlap near the origin because the SM is consistent with low energy data at the 95% CL. For K50 2 0.1, these two regions diverge and become separable. One notices that the allowed range predicted in Ref. [95] lies along the line nfc = KQ’C whereas in our case the slope is different 51:0 = ZnQ’C. This difference comes in because of the assumed dependence of KEG on the other two couplings 161:6 and team. In our case KEG = 4‘} C / 2, and in Ref. [95] REC = 0. If we imagine that any prescribed dependence between the couplings corresponds to a symmetry-breaking scenario, then, given the present status of low energy data, it is possible to distinguish between different scenarios if my C, nfi’c and KEG are larger than 10%. Better future measurements of 6’s can further discriminate between dif- ferent symmetry-breaking scenarios. Next, I will discuss how the SLC can contribute to the study of the nonstandard couplings. 106 r I 1 I I I 1 I r I I I I 1 r - -1 0'5 _ . : : : : -— z“ - . . _ 2 0.0 . . . “0‘5 — . : : : : — I- ..... q 1 1 l 1 1 1 1 l 1 l 1 1 I 1 1 Figure 3.12: A comparison between our model and the model in Ref. [95]. The allowed regions in both models are shown on the plane of Kfc and leg/C, for m, = 170 GeV and my = 300 GeV. 107 3.3.3 -At the SLC The measurement of the left—right cross section asymmetry A L R in Z production with a longitudinally polarized electron beam at the SLC provides a further test of the SM and is sensitive to new physics. As I discussed in section 1.6, the reported measurement of A”; [31] shows a deviation of about 2.80 from the SM prediction. By the SM prediction, I refer to the values in Table 1.2 with the reference masses m. = 175 GeV and my = 300 GeV. In the previous discussion on the allowed space of the nonstandard couplings nfc, n’l‘z’c, and Kfic, I only concentrated on LEP data. It is interesting to investigate if our effective model can offer some explanation for the observed anomaly in A L R. In section 1.4 we found that AL}; = (ALRHB (1 +17.361 — 22.563) , (3.66) where (ALR)|B is the improved Born value for ALR. In fact, as discussed below, the effect of the SLC measurement of A L R on possible new physics in the top quark couplings depends on the way one incorporates AL}; with LEP data. There are two methods by which one can incorporate the SLC measurement of A L R with the other existing low energy data at LEP. The first method is to combine and average ALR with all LEP data. In this case, the anomaly in ALR is washed away due to the large number of LEP measurements consistent with the SM. One finds that including the SLC measurement A L R with all LEP data yields a new fit on the epsilon parameters with a slight decrease in the central value of £1 [38] efo' = 3.5 :1: 1.5 , (3.67) while keeping the fit on q, the same. As discussed in the previous section, the non- standard coupling 51:0 is mostly constrained by 61,. Therefore, no significant change 108 in the allowed range of 161:0 is expected. The effect of averaging the SLC and LEP data can be easily seen in the special model I discussed previously (KEG = nfC/Z). In this case, the length of the allowed area is not affected since it is controlled by 65. Since the uncertainty in 6?“ remains almost the same after including the A“; measurement, the width of the allowed area is also hardly modified. The only effect will be to shift the allowed area slightly downward (toward 2m; < KL). This con— clusion is simply due to the preference for a more negative new physics contribution to accommodate the smaller value of 61"“. It is interesting to note that the effect of including ALB with the other LEP data is similar to the effect of a light Higgs boson. The more interesting approach in dealing with the SLC measurement of A“; is to ask whether our new effective model, checked against LEP data, can give some insight into the status of the SLC measurement of Aug. In our effective model with nonstandard top quark couplings, the theoretical prediction for the observables A8 at LEP and A“; at SLC are identical. Therefore, it is not possible to explain the anomaly in .453, at the 10 level, without affecting the value of A, which is in a very good agreement with the SM. However, at the 20 level, one may be able to find a solution (notice that the SM is not a solution) which is compatible with both LEP and SLC measurements. From Eq. (3.66) and using our effective model contribution to AM, one concludes that ALR = (ALRHSM (1 + 17.3561) , (3.68) where 661 is the nonstandard contribution to 61, and A2 __2 t G _ 661 = REF-:Bmfl—Kfc + n12,” + REC) In (3.69) Since the reported measurement of Am is larger than the SM prediction as seen in 109 Table 1.2, the nonstandard contribution 561 prefers positive values, i.e., Kfic + KE0 2 fife. Therefore, the SLC measurement A L R indicates a preference for that particular region of the parameter space. It is much easier to appreciate the A“; effect in dealing with the special model discussed in the previous section. In that special model with the approximate cus- todial symmetry, i.e., K’ZC = 2n?) = KL, the SLC A“; measurement will have a significant effect on the allowed region in the KL and KR plane. In this case, one has 661 = i3nz?(—nb + 2K3) In A: . (3.70) 4J27r2 m? In Figure 3.13, I plot the allowed region for the parameters KL and K]; using the SLC measurement of A“; and 61, extracted from the LEP data, for m, = 170 GeV and my = 300 GeV at 95% CL. From Figure 3.13 it is clear that the SLC measurement of ALR indicates a preference for positive nonstandard contributions to 661, i.e., RR 2 21st,. Also, one notices that the SM is excluded by the A L R measurement at 95% CL. It is interesting to search for a solution which is compatible with the LEP and SLC data. In Figure 3.14, I plot the allowed region, the very narrow band, for m, = 170 GeV and m H = 300 GeV consistent with LEP data, at 95% C .L., and also compatible with the SLC measurement A L R at the same level of accuracy. One can understand the result of Figure 3.14 as follows. First, as discussed before, there is no effect on the length of the allowed area since it is controlled by 61,. Second, the measurement A“; prefers positive values for 661, i.e., the region where 21m > KL. In other words it prefers the region above the line 2163 = KL. This is the reason why we find the narrow band above the line 2K}; 2 KL. Obviously, in this case, the SM is excluded by the data for n’s=0. To understand the effect of the top quark mass m, on the result let us concentrate 110 NC R Figure 3.13: The allowed region of KL and ’63, using the SLC measurement ALR, for m. = 170 GeV and m” = 300 GeV. 111 on the two measurements: A. at LEP and A L R at SLC. The two observables A, and A“; have the same dependence on the quantity 661 [see Eq. (3.68)]. For a heavier top quark, the allowed band from the LEP fit shifts downward, this is because the SM prediction of Ae increases. Similarly, the band consistent with ALB also shifts downward, since the SM prediction of A L R is identical to Ac. Therefore, the two bands shift in the same direction. Furthermore, for a fixed KL, the difference between the exp. central values of the experimental measurements, A = A L R — Afixl", is proportional to the quantity A at maxi“? — Kg“) = mEAR, (3.71) where Kim is the central value, for a fixed KL, of the allowed band extracted from the SLC measurement, similarly, for xiii”. Therefore, the difference A 3 decreases as a function of the top quark mass. Nevertheless, the widths of the allowed bands decreases also as a function of the top quark mass. Thus, even though the two allowed bands from LEP and SLC move closer for a heavier top quark, their widths decrease rapidly such that the overlap in LEP and SLC data decreases as the top quark mass increases. In Figure 3.15 I plot the width of the overlapped region h, due to the measurements A6 and A L R, as a function of the top quark mass. Negative values of h indicates an overlap in the measurements while positive values indicates no overlap. One can see that the overlap |h| decreases as a function of the top quark mass. Nevertheless, a consistent solution for both measurements still exist for a wide range of m,. The overlap does not depend on the Higgs boson mass m” because the difference A R is independent of my From Figure 3.14 it is clear that there will be no effect on the length of the allowed region which in our approximation is solely determined by 61,. Hence, a more accurate measurement of 61,, i.e., I‘(Z —) b5), is needed to further confine the nonuniversal interactions of the top quark to gauge 112 NC R _o.2 I I I A L I I I I I I l I I I I I I I I I -o.2 0.0 0.2 0.4 0.6 Figure 3.14: The allowed region of KL and KR, using LEP data and the SLC mea- surement of Am, for m, = 170 GeV and my = 300 GeV. 113 0.010 '- I I I I 1 I I I I I I I I I I I I I I I I I I l I I I I .4 Z 1 0.005 "— '—_ 1- 4 0.000 :- _. C I .c: -0.005 — — t I -0.010 — — _ 1 -0.015 :- — 3 l r q _o.020 IIIIIIIlIIIlIIIIIIlIllLllllLl 100 125 150 175 200 225 250 m, GeV Figure 3.15: The overlapping of the two measurements Ac and A“; as a function of the top quark mass. Negative values of h indicates overlapping, while positive values indicates no overlapping. 114 bosons to probe new physics. 3.4 Heavy Higgs Boson Limit in the SM The goal of this study is to probe new physics effects, particularly the effects due to the symmetry-breaking sector, in the top quark system by examining the couplings of the top quark to the gauge bosons. To illustrate how a specific symmetry-breaking mechanism might affect these couplings, in this section I consider the Standard Model with a heavy Higgs boson (171" > 171,) as the full theory, and derive the effective NC NC CC couplings KL , KR , KL , and Kfic at the top quark mass scale in the effective Lagrangian after integrating out the heavy Higgs boson field. Given the full theory (SM in this case), one can perform a matching between the underlying theory and the effective Lagrangian. In this case, the heavy Higgs boson mass acts as a regulator (cutoff) of the effective theory [103]. Figure 3.16 shows the Feynman diagrams needed to calculate the effective couplings of the top quark to the W and Z gauge bosons. While setting mb = 0, and only keeping the leading terms of the order m? ln m?“ I find the following effective couplings , 9 GP ‘1 2 1 2 "If! t— t — Z . 4—CQ\/§7r2 (-8—m,7,,(1 -' '75) + gmflpu + ’75)) In (7n? , (3.72) g G —1 m2 t- b — W I Wig—«é; (E) mf'yufl — ‘75)111 (777%) . (3.73) From this result one concludes n§C=2ngC- GP (—l)m?ln - — -m—% (3.74) 2x/27r2 8 m? ’ G 1 1712 NC _ F 2 H 115 (a) ‘ b s, s . s () Figure 3.16: The Feynman diagrams needed to calculate the effective couplings of the top quark to the W and Z gauge bosons. KEG = 0 . (3-76) Note that the relation between the left—handed currents (n50 = 2KEC) agree with our prediction because of the approximate custodial symmetry in the full theory (SM) and the fact that vertex b-b-Z is not modified. The right-handed currents KEG and nfic are not correlated, and nfic vanishes for a massless b. Also, note that an additional relation in the effective Lagrangian between the left— and right—handed effective couplings of the top quark to Z boson emerges, i.e., n’g’c =..- «’50. (3.77) This means only the axial vector current of t-t-Z acquires a nonuniversal contribution while its vector current is not modified. 116 As discussed in Section 3.2, due to the Ward identities associated with the photon field there can be no nonuniversal contribution to either the b-b—A or t-t-A vertex after renormalizing the fine structure constant a. This can be explicitly checked in this model. Furthermore, up to the order of m? In m?“ the vertex b—b—Z is not modified which agrees with the assumption I made in Section 3.2 that there exists a dynamics of electroweak symmetry-breaking such that neither b WEE-Z nor bL-b_L-Z in the effective Lagrangian is modified at the scale of m,. From this example one learns that the effective couplings of the top quark to gauge bosons arising from a heavy Higgs boson are correlated in a specific way: namely, 1.50 = 216g: = —x’,¥c. (3.78) can be arbitrary, and are not necessarily 1/2 and 1 / 4, respectively). In other words, if the couplings of a heavy top quark to the gauge bosons are measured and exhibit large deviations from these relations, then it is likely that the electroweak symmetry- breaking is not due to the standard Higgs mechanism which contains a fundamental heavy scalar boson. This illustrates how the symmetry-breaking sector can be probed by measuring the effective couplings of the top quark to gauge bosons. Next, I study how the Tevatron, the LHC, and the LC can contribute to the measurements of the nonstandard couplings. 3.5 Direct Measurement of the Top Quark Cou- plings In Section 3.2, I concluded that the precision LEP data can constrain the cou- plings nfc, Kfi’c and KEG, but not ngc (the right—handed charged current). The nonstandard coupling 165,330 can be studied using the b —> 37 measurement [98]. Also, 117 I discussed how the SLC measurement of A“; can contribute to the study of the non- standard couplings nfic, team and nEC. The conclusion that the SM is not compatible with the combined LEP and SLC data may be an indirect evidence for the anoma- lous couplings of the top quark. In this section, I examine briefly how to improve our knowledge on these couplings at the other current and future colliders. 3.5.1 At the Tevatron and the LHC In this section, I study how to constrain the nonstandard couplings of the. top quark to the gauge bosons from direct detection of the top quark at hadron colliders. At the Tevatron and the LHC, heavy top quarks are predominantly produced from the QCD process 99,116 —> ti and the W-gluon fusion process qg(Wg) -> tbfb. In the former process, one can probe x90 and ngc from the decay of the top quark to a bottom quark and a W boson. In the latter process, these nonstandard couplings can be measured by simply counting the production rates of signal events with a single t or 5. More details can be found in Ref. [104]. To probe KEG and rcfic 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. For a massless b, the W boson from top quark decay can only be either longitudinally or left—handed polarized for a left—handed charged current (REC = 0). For a right—handed charged current (KEG = —1) the W boson can only be either longitudinally or right—handed polarized. (Note that the handedness of the W boson is reversed for a massless b from t- decays.) In all cases the fraction of longitudinal W from top quark decay is enhanced by mf/2M3V as compared to the fraction of transversely polarized W. Therefore, for almore massive top quark, it is more difficult to untangle the Kgcand REC contributions. The W polarization measurement can be done by measuring the 118 invariant mass (mu) of the bottom quark and the charged lepton from the decay of top quark [105]. We note that this method does not require knowing the longitudinal momentum (with two—fold ambiguity) of the neutrino from W decay to reconstruct the rest frame of the W boson in the rest frame of the top quark. Consider the (upgraded) Tevatron as a pp collider at \/§ = 2 or 3.5 TeV, with an integrated luminosity of lor 10 fb‘l. Unless specified otherwise, we will give event numbers for a 175 GeV top quark and an integrated luminosity of 1 fb‘l. The cross section of the QCD process gg, ch -) tt— is about 7 (29) pb at a J5 = 2 (3.5) TeV collider. In order to measure REC and Ego we have to study the decay kinematics of the reconstructed t and / or t-. For simplicity, let us consider the 3* + Z 3 jet decay mode, whose branching ratio is Br = 2% = %, for 8+ = 8+ orp+. We assume an experimental detection efficiency, which includes both the kinematic acceptance and the efficiency of b—tagging, of 15% for the U- event. We further assume that there is no ambiguity in picking up the right b (b) to combine with the charged lepton 8+ (0') to reconstruct t (f). In total, there are 7pb x 103 pb'1 x 28—7 X 0.15 = 300 reconstructed tf events to be used in measuring EEG and KEG at fl = 2TeV. The same calculation at ([9- = 3.5 TeV yields 1300 reconstructed tf events. Given the number of reconstructed top quark events, one can in principle fit the ma distribution EC and REC. We note that the polarization of the W boson can also to measure K, be studied from the distribution of cos 9;, where 9; is the polar angle of f in the rest frame of theW boson whose z—axis is the W bosons moving direction in the rest frame of the top quark [105]. For a massless b, cos 6; is related to mg. by 2 2m“ * C089! 9...’ m '— 1 . (3.79) However, in reality, the momenta of the bottom quark and the charged lepton will be smeared by the detector effects and the most serious problem in this analysis. is 119 the identification of the right b to reconstruct t. There are two 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 lepton B with large transverse momentum. Another is to identify the nonisolated lepton from b decay (with a branching ratio Br(b —> p+X) ~ 10%). Both of these methods will further reduce the reconstructed signal rate by an order of magnitude. How will these affect our conclusion on the determination of the non— universal couplings KEC and REC? This cannot be answered in the absence of detailed Monte Carlo studies. Here I propose to probe the couplings KEG and ago by measuring the production rate of the single—top quark events. A single—top quark event can be produced from either the W—gluon fusion process qg (W+g) —+ th, or the Drell-Yan-type process qtj -) W" -) tb. Including both the single—t and single-t- events, for a 2 (3.5) TeV collider, the W—gluon fusion rate is 2 (16) pb; the Drell-Yan type rate is 0.6 (1.5) pb. The Drell-Yan-type event is easily separated from the W-gluon fusion event, therefore it will not be considered hereafter [106]. For the decay mode of t —) bW?‘ —> b€+u, with 3+ = 8+ or 11", the branching ratio of interest is Br = g. The kinematic acceptance of this event at \/§ = 2TeV is found to be 0.55 [106]. If the efficiency of b—tagging is 30%, there will be 2pb x 103 pb'l x g x 0.55 x 0.3 = 75 single-top quark events reconstructed. At J? = 3.5 TeV the kinematic acceptance of this event is 0.50 which, from the above calculation yields about 530 reconstructed events. Based on statistical error alone, this corresponds to a 12% and 4% 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, we examine two scenarios: 20% 0.50 I T T I I I I I I I I I I I I I W I T I It I : PP. VS=2 TeV ~ m,=175 GeV 0.25 I.- 507 : ----- 20% L f 0.00 ~~~~~~~ _— ~ -‘~ ~ ‘ ‘ 8,; —o.25 ‘ILIIIIIIIIIIIIIIIII —0.50 — d b \‘ C x .. \ d -0.75 :- \ _L _ \ .1 _ \ .7 -1.00 I ' 1 1 1 l 1 1 1 1 l m 1 1 1 l 1 1 1 1 I 1 1 1 1‘ I 0.0 0.2 0.4- 0.6 0.8 1,0 Figure 3.17: The allowed Ingcl and ngc are bounded within the two dashed (solid) lines for a 20% (50%) error on the measurement of the single—top production rate, for a 175 GeV top quark. and 50% error on the measurement of the single—top cross section, which depends on both KEG and K5330. (Here we assume the experimental data agrees with the SM prediction within 20% (50%).) We found that for a 175 GeV top quark KEG and REC are well constrained inside the region bounded by two (approximate) ellipses, as shown in Figure 3.17. These results are not sensitive to the energies of the colliders considered here. The top quark produced from the W -gluon fusion process is almost 100% left— handed (right—handed) polarized for a left—handed (right—handed) t-b-W vertex, therefore the charged lepton 6" from t decay has a harder momentum in a right— handed t-b-W coupling than in a left—handed coupling. (Note that the couplings 121 of light-fermions to W boson have been well tested from the low energy data to be left—handed as described in the SM.) This difference becomes smaller when the top quark is more massive because the W boson from the top quark decay tends to be more longitudinally polarized. A right—handed charged current is absent in a linearly S U (2) L invariant gauge the- ory with a massless bottom quark. In this case KEG = 0, then KEG can be constrained to within about —0.08 < KEG < 0.03 (-0.20 < KEG < 0.08) with a 20% (50%) error on the measurement of the single—top quark production rate at the Tevatron. This means that if we interpret (1 + ICEC) as the CKM matrix element 14),, then Va, can be bounded as V“, > 0.9 (or 0.8) for a 20% (or 50%) error on the measurement of the single—top production rate. Recall that if there are more than three generations, within 90% C.L., V“, can be anywhere between 0 and 0.9995 from low energy data [9]. This measurement can therefore provide useful information on possible additional fermion generations. Measuring the DreIl-Yan-type single-top production rate can further improve the measurement of 14),. We expect the LHC can provide similar or better bounds on these nonstandard couplings when detailed analyses are available. 3.5.2 At the LC The best place to probe nfc and rcfic associated with the t—t-Z coupling is at the LC through e'e+ —> A, Z —+ tt_. A detailed Monte Carlo study on the measurement of these couplings at the LC including detector effects and initial state radiation can be found in Ref. [107]. The bounds were obtained by studying the angular distribution and the polarization of the top quark produced in e‘e+ collisions. Assuming a 50 fb'1 luminosity at ([3. = 500 GeV, we concluded that within a 90% confidence level, 122 it should be possible to measure K’IYC to within about 8%, while “1,1110 can be known to within about 18%. A lTeV machine can do better than a 500 GeV machine in determining £20 and rage because the relative sizes of the t 3(f) R and t [,(f) L produc- tion rates become small and the polarization of the tf pair is purer. Namely, it is more likely to produce either a t1, (3) R or a t 3(f) L pair. A purer polarization of the tt- pair makes 1:50 and ”go better determined. (The purity of the tt- polarization can be further improved by polarizing the electron beam.) Furthermore, the top quark is boosted more in a lTeV machine thereby allowing a better determination of its polar angle in (the tf system because it is easier to find the right b associated with the lepton to reconstruct the top quark moving direction. Finally, we remark that at the LC REC and KEG can be studied either from the decay of the top quark pair or from the single—top quark production process, W- photon fusion process e’e+(W7) —+ tX, or 8")(W7) —> t—X, which is similar to the W—gluon fusion process in hadron collisions. 3.6 Discussion and Conclusions In this chapter I have applied the electroweak chiral Lagrangian to probe new physics beyond the SM through studying the couplings of the top quark to gauge bosons. First, I examined the precision LEP data to extract the information on these couplings. Second, I discussed how the SLC measurement A L R can contribute to the constraints on the nonstandard couplings nfc, nfiC, and REC. Third, I discussed how to improve our knowledge about the top quark nonstandard couplings at current and future colliders such as at the Tevatron, the LHC, and the LC. Because of the non—renormalizability of the electroweak chiral Lagrangian one can only estimate the size of these nonstandard couplings by studying the contributions 123 to LEP observables at the order of mfln A2, where A = 47712 ~ 3 TeV is the cutoff scale of the effective Lagrangian. Already I found interesting constraints on these couplings. Assuming b-b-Z vertex is not modified, I found that nfc is already constrained to be -o.05 < n2"? < 0.17 (0.0 < nfc < 0.15) by LEP data at the 95% CL. for a 160 (180) GeV top quark. Although nfic and KEG are allowed to be in the full range of :tl, the precision LEP data do impose some correlations among nfc, nfic, and KEG. (7.53," does not contribute to the LEP observables of interest in the limit of mm = 0.) In my calculations, these nonstandard couplings are only inserted once in loop diagrams using dimensional regularization. Inspired by the experimental fact p z 1, reflecting the existence of an approximate custodial symmetry, I proposed an effective model to relate mic and KEG. I found that the nonuniversal interactions of the top quark to gauge bosons parameterized by nfc, 162,6, and KEG are well constrained by LEP data, within 95% CL. The results are summarized in Table 3.1 (see also Figures 3.6—3.10). Also, the two parameters NC KL =nL NC and NR = n R are strongly correlated. In my model, KL ~ 2K3. I note that the relations among rc’s can be used to test different models of elec- troweak symmetry-breaking. For instance, a heavy SM Higgs boson (my > m,) will modify the couplings t-t-Z and t-b—W of a heavy top quark at the scale m, such that NC___ NC 9 KL NC _ CC , and nfic = 0. Another example is the effective model discussed in Ref. [95] where, 1.920 = KEG = 0. In this model the low energy precision data impose the relation nfc ~ nfic. Also, the simple commuting extended techni- color model presented in Ref. [59] predicts that the nonstandard top quark couplings are of the same order as the nonstandard bottom quark couplings. It is also interesting to note that the upper bound on the top quark mass can 124 be raised from the SM bound m, < 200 GeV to as large as 300 GeV if new physics occurs. That is to say, if there is new physics associated with the top quark, it is possible that the top quark is heavier than what the SM predicts. However, for a SM top quark, m, should be less than 200 GeV, as shown in Figures 3.7 and 3.8. Also, I discussed how the present SLC measurement of AL); can contribute to the constraints imposed on the nonstandard couplings nfc, nfi’c, and NEC at LEP. I found that if one uses the LEP constraints to predict the new physics contribution to the SLC measurement Aug, then for the special model, 52" = KQ’C/2, it is possible to reconcile the LEP and SLC data at 95% CL. for a wide range of the top quark mass. This is shown in Figure 3.15. Undoubtedly, direct detection of the top quark at the Tevatron, the LHC, and the LC is crucial to measuring the couplings of t-b—W and t-t-Z . At hadron colliders, go and 1:920 can be measured by studying the polarization of the W boson from top K quark decay in tt— events. They can also be measured simply from the production rate of the single top quark event. The LC is the best machine to measure 5’30 and xfic which can be measured from studying the angular distribution and the polarization of the top quark produced in e‘e+ collision. Details about these bounds were given in Section 3.5. Chapter 4 Heavy Top Quark Effects and the Scalar Sector 4.1 Introduction In chapter 3, I calculated the one-loop level quadratic contribution of the top quark mass m,, i.e., m? In A2, to the parameters 61 and £5. The calculation is based on the Lagrangian £0 + [.1 (see Eqs. (2.29) and (3.9)). In general, in performing the one loop level calculation, one needs to consider a gauge invariant set of Feynman diagrams in which massive gauge bosons can appear as external and / or internal lines. However, at one-loop level, I found that to extract the mfln A2 dependence of the low energy observables (equivalently, 61 and 6),) one only needs to include the massive gauge bosons as external fields. Figure 3.1 shows the relevant Feynman diagrams needed to extract the m? dependence in a general R5 gauge. Only the Goldstone bosons and the top quark appear as internal (propagating) fields. The gauge bosons behave as classical (non-propagating) fields. This result is expected since extracting corrections in power of m. is equivalent to a perturbative expansion in the Yukawa coupling 9: = mt/v which has nothing to do with the gauge structure [108]. This is true because the m? corrections are present even for vanishing gauge couplings ie 9 and g’ —) 0. The m? corrections are a consequence of the symmetry-breaking mechanism 125 126 which controls the Yukawa interactions connecting the scalar fields and the fermion sector. This observation is important because one can generalize it to include'the pure dependence on the top quark mass to all orders and not merely to the one-loop level. Furthermore, with the gauge couplings switched off in the Lagrangian, one can derive a set of Ward identities which relates the physical quantities, as discussed in the next section, to appropriate renormalization constants of the reduced Lagrangian (with gauge couplings switched off). This observation was made for the SM case in Ref. [108], where explicit calcula- tion of the two loop m‘,‘ corrections to the low energy observables was performed for arbitrary values of the Higgs mass m H. The calculation was performed by considering the Lagrangian of the SM in the limit of vanishing gauge coupling constants. The gauge bosons play the role of external sources and the relevant propagating fields are the top quark, the massless bottom quark, the Higgs boson field, and the charged and neutral Goldstone bosons (25*, 453. This reduced Lagrangian is called the Gaugeless Limit of the SM [108]. In this chapter, I develop a similar formalism to calculate, at one loop level, the contribution to 61 and 65 that grows like 111? in the chiral Lagrangian framework. The formalism holds for all contributions which do not vanish when setting the gauge couplings g and g’ to be zero. Therefore, generalizing the result in Ref. [108] to incorporate a larger set of effective models. Similarly, I find that to extract the m? dependence in the chiral Lagrangian framework, one needs to concentrate only on the Goldstone bosons, the top quark, and the massless bottom quark. The calculation of m? dependence in the new formalism gives an identical result to the one I found in chapter 3. Similar to the'procedure in chapter 3, I consider an effective field theory describing the nonstandard top couplings to the gauge bosons. I show how to conveniently 127 relate various radiative corrections important for testing the standard model (SM) in a rather elegant and clear way. More importantly, this approach is shown to clearly identify observables which are sensitive to the symmetry-breaking sector of the electroweak theories. In section 2, I briefly review the Gaugeless limit of the SM [108]. In section 3, I present new formalism, in the chiral Lagrangian framework, to study the large top quark mass contribution (in powers of m,) to low energy physics. I show that all large m, effects enter through two quantities p and 7' [108], which are, in this limit, equivalent to the quantities 61 and 65. Section 4 contains some of my conclusions. 4.2 Large mt effects in the SM In this section, I briefly review the analysis performed in Ref. [108] which is a study of the large m, contributions to the low energy observables in the SM. In this case, one is interested in corrections in powers of the Yukawa coupling 9, = mt/v, while, corrections in powers of the gauge coupling 9 are ignored. In other words, one is considering the perturbative expansion in the Yukawa coupling 9, = mt/v rather than the gauge coupling 9. To fully extract the pure m, corrections in a general R: gauge, the massive gauge bosons do not appear in loops. Thus, gauge bosons can be treated as classical (non-propagating) sources. Consequently, there is no need to break the gauge invariance of the SM Lagrangian in order to perform the loop calculations. The exact gauge invariance of the Lagrangian (in the limit of ignoring corrections in power of the gauge coupling g) leads to a set of ward identities valid to all orders. These Ward identities relate the n-point functions of the gauge bosons to those of the scalar Goldstone bosons. Thus, by connecting the n-point functions of the gauge bosons to those of the Goldstone bosons, one can relate the physical 128 observables to the n-point functions of the Goldstone bosons. The Ward identities can be easily derived in the path integral formalism using the generating function technique. As I discussed in chapter 1, all radiative corrections to low energy observables, under a few general assumptions, can be written in terms of the gauge boson vacuum polarization functions and the proper vertex correction of Z-b-b. Thus, using the derived Ward identities, one can simply relate all physical radiative corrections to a set of corrections involving the Goldstone bosons and the fermion sector. The derived Ward identities relate the vacuum polarization functions of the gauge bosons to those of the Goldstone bosons [108] as follow 2 2 q”q”fl,w(q) = 430:261101), 92v? (fquum) = THHQ), (4-1) where IIW(q) is the vacuum polarization of the Z boson, Hfu(q) is the vacuum po- larization of the Wi boson, U(q) and 11*(q) are the self energy of the Goldstone bosons (1)3 and (bi, respectively. Similarly, the Z -b-b proper vertex correction 1",, can be related to the proper vertex ¢3-b-b correction I‘ as follow (p’”—p“)1‘,.(p’,p)=i 9” I‘+ 2c059 _ , P sin20 P sin20 _ co"s,,,(s,.~‘(—,£- 3 >—(—,i*-— 3 >590») (4.2) where PAR = (1 q: 75) / 2 and 851(1)) is the self energy of the massless b quark with momentum p. S;1(p) is parameterized as Sm = izbverPL + WMPR. (4.3) At tree level, Z" = 1. However, higher order corrections contribute to Z” whereas the right handed self-energy of the massless b quark does not get modified. This is 129 true since the right—handed quark, b3, is singlet under 5' U (2),, symmetry. Therefore, the right-handed quark, b3, does not couple to the Goldstone bosons in the limit of vanishing b quark mass. The 2-point functions of the W, Z gauge bosons, and the Goldstone bosons can be expanded in powers of q2 since this is the only relevant low energy scale in the loop calculations after turning off the gauge couplings. In the limit q2 -) 0 the 2-point functions are parameterized as 22 H.401) z figflz - 1)g..... (4.4) 2 2 113.01) z 3%(Zi -1)g,... (4.5) II(<1) z (ZS - 1)q2. (4.6) II*(q) z (Z? - 1M”. (4.7) where at tree level Z = Z i = 223 = Z;t = 1. The energy scale q2 = 0 is the only relevant low energy scale in the loop calculations. The internal fields are the massless Goldstone bosons, the heavy top quark, and the Higgs boson. Therefore, there are only three mass scales in the calculations, a low energy scale q2 = 0 and two high energy scales m, and m H. Using the identities in Eq. (4.1), one obtains Z = 2.3, (4.8) 2* = 2;. (4.9) Since, as discussed in chapter 1, oblique corrections to physical observables can be written in terms of the quantities Z and Z*, it is therefore possible to relate the 130 quantities Z3, 2% to direct physical observables, or equivalently to the parameters 6p, Ale, and Arw. The quantities 223 and Z; are related to the quantities Azz(0) and AWW (0) defined in Eq. (1.49) as follows 22 2 2 9 v 2;, AWW(0) = iii—22*. (4.10) 4 cos2 6 AZZ(0) = I will discuss this connection in the next section after dealing with the chiral La- grangian case. The prOper Z-b-b vertex, can be parameterized as [108] i 2 . 2 . Fu(p',P) = —2cgsa ((1 — §SIn20)ZnuPL — Esmzfi'prR) , (4.11) for p’ z p, i.e., q = p’ — p z 0. Similarly, for the proper 053-b-b vertex m __ r = 2%,,” “PL. (4.12) 221 Therefore, using the Ward identity in Eq. (4.2) one finds 2 - 2 3 2 - 2 b (1 — 58m (9)21: Zl +(1— §sm 0)Z2. (4.13) To get the physical Z-b-b vertex one needs to renormalize the left-handed b quark. Therefore, one finds the physical Z -b-b vertex to be __ .9 -2-2 a: _2.2 V”— 2cosG [(1 331n0+Zg)7pP 331n0PR. (4.14) The conclusion is that to calculate the pure m, corrections to low energy data, one simply has to calculate the quantities Z3, Z}, Zi‘, and Zé’ which are calculated from a set of Feynman diagrams involving only fermions (top and bottom quarks) and scalar bosons ( Goldstone bosons and the Higgs boson) but not gauge bosons. Thus, in extracting the large m, corrections to low energy observables in the SM, one starts with the tree-level Lagrangian, involving the Higgs doublet (I), the third 131 generation left-handed quark doublet 11),, = (tL,bL) and the right-handed top quark field t R /\ 112 2 c = (3,4)*(a”<1>)—-2-((<1>*<1>)2—-2-) + Win/"c1411, + $1??th + ({L 0L) (I’ t3 + [1.0. (4.15) flmt 1) From which one calculates the needed quantities 23, Z%, Z f, and Z3. 4.3 Large mt Effects In the Chiral Lagrangian In this section, I am interested in the chiral Lagrangian formulated electroweak theories in which the gauge symmetry SU(2)L X U (1)y is nonlinearly realized. A mentioned in chapter 3, the chiral Lagrangian can be constructed solely based upon the broken symmetry of the theory, and it is not necessary to specify the detailed dynamics of the actual breaking mechanism. Hence, it is the most general effec— tive Lagrangian that can accommodate any underlying theory with that pattern of symmetry-breaking at the low energy scale. In this section, as a matter of convenience, I define the composite fields in a slightly different way from the ones I defined in chapter 3. I define w; = —.'Tr(r°2*D,,2) (4-16) and Bu = g’B” , (4.17) where I define the quantity 7.0 0,2 = (a, —ig 2 W3) >3 . (4.18) 132 The quantity DpZ as defined above is not a covariant derivative. Its transformation under SU(2)L x U (1),» can be checked using the W and 2 field transformations (see appendix D). In my notation W; and B" are the gauge bosons associated with the SU(2)L and U (1)y groups, respectively. Also, 9 and g’ are the corresponding gauge couplings. The composite fields transform under 3 U (2),, x U(1)y as ”fiawvfi=Mfi—,g, (4w) W3 -> W’ff = ei‘vwf, (4.20) maq;afimw am where Wi = W. (4.22) “ fl I also introduce the composite fields 2" and .24" as 2.. = W3 + 3,. , (4.23) 32.44,1 = 32W: — C23,, , (4.24) where 32 5 sin2 0, and c2 = 1 — 32. In the unitary gauge (2 = 1) W: = —gW: a (4'25) 9 Z” = —;Z“ , (4.26) 8 A” = -§A,, , (4.27) 133 where I have used the relations 6 = gs = g’ c, W3 = cZu+sAm and B“ = —sZ,,+cA,,. The transformations of Z” and A” under S U (2) L x U (1)y are aaq=4, am A, —+ A; = A, — $61.3, . (4.29) Hence, under SU (2);, x U (1)y the fields Wj‘ and Z“ transform as vector fields, but '41: transforms as a gauge boson field which plays the role of the photon field A”. Using the fields defined as above, one may construct the SU(2)L x U(1)Y gauge invariant interaction terms in the chiral Lagrangian 1 1 , ____ _ _ 0 am! __ 11V .63 4g,w,,,w 4g, 13,..3 02 + _,. 02 ,1 + —4-w,, w + 3'2”}: + . .. , (4.30) where wgu = auw: —- aw; + eabcwfjw; , (4.31) 13,... = 0,3,, — 0.3,. , (4.32) and where . . . denotes other possible four- or higher— dimensional operators [78, 72]. It is easy to show that1 WZJ“ = —g2*W5,r°2 (4.33) and W3,W°P" = gzwguwal‘” . (4.34) lUse W37“ = —21'21D,,E , and [7“,71’] = 2ieabcrc. 134 This simply reflects the fact that the kinetic term is not related to the Goldstone bosons sector, i.e., it does not originate from the symmetry-breaking sector. The mass terms in Eq. (4.30) can be expanded as 2 2 ”7ij4‘ + ”E232” = ay¢+aflqr + gap¢3afla3 2 2 2 2 +g—Z—Wgwp’ + %Z,,Z" + . .. (4.35) At the tree level, the mass of Wat boson is MW = gv/2 and the mass of Z boson is M2 = gv/2c. The above identity implies that the radiative corrections to the mass of the gauge bosons can be related to the wave function renormalization of the Goldstone bosons, cf. Eq. (4.51), and therefore sensitive to the symmetry-breaking sector. Fermions can be included in this context by assuming that each flavor transforms under SU(2)], x U(1),» as [74] f -+ f’ = e‘”Q’f. (4.36) where Q I is the electric charge of f. My goal is to study the large Yukawa corrections to the low energy data from the chiral Lagrangian formulated electroweak theories. I will separate the radiative corrections as an expansion in both the Yukawa coupling 9, and the weak coupling g. (g, = mt/v, where m, is the mass of the top quark.) With this separation one can then consider the case where corrections of the order 9 are ignored compared to those of g,. This case is similar to the analysis in Ref. [108] where the gauge bosons were considered as classical fields so that the full gauge invariance of the SM Lagrangian was maintained, and a set of Ward identities was derived to relate the Green’s functions of the Goldstone bosons and the gauge bosons. Hence, large g, 135 corrections can be easily obtained from calculating Feynman diagrams involving only fermions (top and bottom quarks) and scalar bosons (e.g., Goldstone bosons and possibly the Higgs boson) but not gauge bosons. The same conclusion can be drawn using the chiral Lagrangian approach in a far more elegant and clear way, as shown below in this section. Why is the chiral Lagrangian formulation useful in finding large g. corrections beyond the tree-level? In general to perform a loop calculation, one needs to fix a gauge and therefore explicitly destroys the gauge invariance [SU(2)L x U(1)Y] of the Lagrangian. However, to find the large g, corrections one does not need to include gauge bosons in loops [108]. Thus, there is no need to fix a gauge and the full gauge invariance of the eflective Lagrangian is maintained. Because the chiral Lagrangian possesses the SU(2)L x U(1)Y invariance (nonlinearly) and the U(1)cm invariance (lin- early) at any given order of the perturbative expansions, and all the loop corrections can be reorganized using the composite fields Wf, Z,“ and A“ in a gauge invari- ant form, therefore, it is the most convenient and elegant way to find g. corrections beyond the tree-level. This is obvious because the leading radiative corrections (in powers of m.) are products of the spontaneous symmetry breaking (SSB) and there- fore independent of the weak gauge coupling g. One notes that in the expansion of the field 2 2,, = 56,453 — £2, + (4.37) there is always a factor g associated with a weak gauge boson field. Hence, loop corrections independent of the gauge coupling g can be obtained by simply considering the scalar and the fermionic sectors in the theory. In the following discussion, I will show how this is done. 136 4.3.1 Effective Lagrangian To obtain the large contributions of the top quark mass (in powers of m.) to low energy data, one needs only to concentrate on the top-bottom fermionic sector (f1 = t and f2 = b) in addition to the bosonic sector. The most general gauge invariant chiral Lagrangian can be written as . p .283 .- n .33 £0 = II’)’ 0,,+z—-3—.A,, t+zb7 0p—13A,‘ b 1 I 23‘2 _ —2s2 ._ — (— - -—° + 620) tn"tLZ.. — ( 3 0 + “20) tRVHRZu —1 33 .— 14 33.— p — —2- + ? bL’)’ ()sz - 3037 bRZp -% (1 + KEG)fin/"@142;L - -—1— (I + K201) fir/"QW; V2 1 cc— 1 ccf— - -——rc t ”b W+ — —n b "t W ‘5 R 727 R ,1 \/2 R R7 R p —m¢Zt + . . . , (4.38) where 32m, K120, REC, and REC parameterize possible deviations from the SM predic- tions, and . . . indicates possible Higgs boson interactions and other higher dimensional operators. Here I have assumed that new physics from the SSB modify the interac- tions of the t0p quark to the electroweak gauge bosons. On the other hand, the bare b-b-Z couplings are not modified in the limit of ignoring the mass of the bottom quark. The subscript 0 denotes bare quantities and all the fields in the Lagrangian £0, Eq. (4.38), are bare fields. Needless to say, the composite fields are only used to organize the radiative cor- rections in the chiral Lagrangian. To actually calculate loop corrections one should expand these operators in terms of the Goldstone boson and the gauge boson fields. The gauge invariant result of loop calculations can be written in an effective La- 137 grangian with a form similar to Eq. (4.38). Denoting the fermionic part of this effective Lagrangian as Lie”, then _ 2_ 1 2 2 _— £eff = iZbe'ypa"bL + 21339bL’7‘1bLA” + '2' (Z: — 22%!) bL’YflbLZ” , — 3(2)— 33— +!Zfbn7uapb3 + Z3§bn7prAu — Z4§bmrubRZ" + . . . , (4.39) in which the coefficient functions 21, 22, Z3, Z4, Zf, Zf, and 25‘ contain all the loop corrections, and all the fields in Le” are bare fields. Since the gauge invariance is maintained one can write Cd, in a from similar to Eq. (2.29), i.e., in terms of the B gauge boson field rather than the composite field A. Explicitly, Eel, can written as Leff = iZ:FZ’Yp6PbL — 21%E7prB” + $25157pr3” +izffiyuaubfl — 23%337pb38" + . . ., (4.40) where 8,. == 33(2) - A“) , (4.41) derived from Eqs. (4.23) and (4.24). Note that as shown in Eqs. (4.17) and (4.21) the field 8,, is not composite and transforms exactly like By. Comparing Eq. (4.39) with (4.40), one conclude that the coefficient functions 21, Z2, 23, and Z4 must be related and Z2 = Z} , (4.42) Z4 = 23 . (4.43) All the radiative corrections to the vertex b-b—qb3 in powers of m, are summarized by the coefficient function Z? because, from Eq. (4.37), 1 _ 1_ EsznflbLzfl = Zfabnprawy’ + (4.44) 138 Since the effective Lagrangian Le” possesses an explicit U (1)em symmetry and under G the field A” transforms as a gauge boson field and 2,, as a neutral vector boson field, therefore, based upon the Ward identities in QED one concludes that in Eq. (4.39) 21 = 2,? , (4.45) and Za = Z? . (4.46) Hence, the effective Lagrangian £6” can be rewritten as 2 2 £4, = 2'sz”" (a, —- 2733—04.) 1),, + £2,511.37" (a, — £3394”) b3 1 L L283 — p 1283— p +-2- Zv- b— bL'l’prZ —Z,, 3514pr123 +~~ (4-47) This effective Lagrangian summarizes all the loop corrections in powers of m, in the coefficient functions 2f, 25“, and Z5. Recall that up to now all the fields in L,” are bare fields. To compare with the low energy data I prefer to express £6” in terms of the renormalized fields. In Eq. (4.47), the kinetic terms of the bL and b3 fields can be properly normalized after redefining (renormalizing) the fields bL and b); by (Zffilbl, and (Zf)%b3, respectively. In terms of the renormalized fields bL and b3, .6,” can be rewritten as 2 2 L,” = Eh" (8,, — £32.44”) bL + 52'7" (0,, -— 2339/1”) b); +2 L 2 2 1 (Z: _ Q) Why _ tummy.” + . (4.48) z, 3 3 Before considering the physical observables at low energy let us first examine the bosonic sector. Similar to our previous discussions, loop corrections to the bosonic 139 sector can be organized using the effective Lagrangian 1 1 B = —-—W" W‘“ — —B ”3"" 4.1% + —# xvg p +Z 7,-qu +Z —8—Z,,Z +... . (4.49) Note that in the above equation I have explicitly used the subscript 0 to indicate bare quantities. The bosonic Lagrangian in Eq. (4.30) and the identity in Eq. (4.34) imply that the Yang—Mills terms (the first two terms in LB) are not directly related to the SSH sector. Hence, any radiative corrections to the field W3” must know about the weak coupling 9, i.e., suppressed by g in our point of view. This also holds for operators, of dimension four or higher, which include W; in the chiral Lagrangian where all these gauge invariant terms are suppressed by the weak coupling 9 [72, 78]. (The same conclusion applies to B,,.,.) Therefore we conclude that the fields Wf, Z,“ and A” in £6” and £53” do not get wave function corrections (renormalization) in the limit of ignoring corrections of the order 9, namely the renormalized fields and the bare fields are identical in this limit. Expanding the mass terms in Eq. (4.49) we find 2 2 z¢§°wgw-“ + zxis‘lzyzr‘ = z¢ap¢+a#¢- + gzxapqfiaw + 2 2 2 2 2¢Mij-” + Zxfl‘lzpz‘ + . .. (4.50) 4 age, It is clear that Z 45 denotes the self energy correction of the charged Goldstone boson (13*, and ZX denotes the self energy correction of the neutral Goldstone boson (:53. Since W3: and 2,, do not get wave function correction in powers of m,, therefore the gauge boson masses are 9202 M3. — z¢—°4°=24M3v., 92v? M; = Zxfi =ZXM§0. (4.51) 4C3 140 In summary, all the loop corrections in powers of m, to low energy data can be organized in the sum of L,” [in Eq. (4.48)] and L?” [in Eq. (4.49)]. Comparing them to the bare Lagrangian £0 in Eq. (4.38), we find that in the limit of taking 9 —) 0 the chiral Lagrangian £0 behaves as a renormalizable theory although in general a chiral Lagrangian is nonrenormalizable. In other words, no higher dimensional operators (counterterms) are needed to renormalize the theory in this limit. The same feature was also found in another application of a chiral Lagrangian with 1/N expansion [109]. 4.3.2 Renormalization Now we are ready to consider the large m, corrections to low energy data. I choose the renormalization scheme to be the a, G p, and M z scheme (the Z-pole scheme). With 47m .93 = 8(2) 0 (4'52) and s c = , 4.53 or, 1/2 1 47m s2=-1—(1- ° ) . 4.54 Define the counterterms as a = ao+6a, Gr = GF0+5GF. M3, M3,, + (SM; , (4.55) 141 and 32 = sg+632=sg—6c2, (:2 = c3+5c2, (456) then s2c2 + (c2 — .92) 602 = 7m ( (SQ 6GP 6M%) (4 57) —— 1——+—+ fiGFMg a GI“ Mg As shown in the above equation, even after the counterterms 6a, 60p, and 6M3 are fixed by data [e.g., the electron (g-2), muon lifetime, and the mass of the Z boson], I still have the freedom to choose 602 by using a different definition of the renormalized quantity 3202. In this case, I will choose the definition of the renormalized 32 such that there will be no large top quark mass dependence (in powers of m,) in the counterterm 602. I will show later that for this purpose the renormalized 32 satisfies 2 = _ , 4.58 s c x/I‘Z'GpMép ( ) where p is defined from the partial width of Z into lepton pairs, cf. Eq. (4.75). With this choice of 32 and the definition of the renormalized weak coupling 92 = 17‘?“ , (4.59) 3 one can easily show that the counterterm 6g2 (= g2 — 93) does not contain large m, dependence. (Obviously, 50 will not have contributions purely in powers of m,.) Namely, in this renormalization scheme, a, g, and 52 do not get renormalized after ignoring all the contributions of the order g. Hence, all the bare couplings go, g6, and 83 in the effective Lagrangians £8” and [If], do not get corrected when considering the contributions which do not vanish in the limit of g —> 0. The only non-vanishing 2If one defines .n’zc’2 = na/fiGng, then 82 = s’z(1 + An’) with Ax' = —c""6p/(c’2 - 8’2), and the counterterm of 3’2 will contain contributions in powers of m;. 142 counterterm needs to be considered in Eq. (4.49) is 6'02 (= v2 — '00). From Eq. (4. 51) and MW = gv/2, one finds 24.: = 62 , (4.60) because neither 9 nor W’: (or Wi) gets renormalized. Thus, 1 1 G).0 = —\/—_—_ — 2“ «v? = Z¢GF . (4.61) Consequently, 93 BGFOM§0_ 83—ng Z" (:3 = T — T Z_" , (4.62) and the effective Z-b-b coupling is _ 90 u _Z_v_L_?_S_§ p_.2:_8_§p — 2—‘607 [(Z: 3 L R — (:ng z» 25 4s2 25 fm 2x [(277)—2575 ’ (“3’ where PAR = (11F 75)/2. 4.3.3 Low Energy Observables A discussed in chapter 1, all the radiative corrections to low energy data can be categorized in a model independent way into four parameters: 61, 62, 63, and q, [16, 22, 23] or equivalently, the S, T, U, . . . [24] (see appendix B). The parameters 61, 62, 63, and 65 can be derived from four basic measured observables, such as I‘,, (the partial decay width of Z into a 11 pair), 14%,; (the forward-backward asymmetry at the Z peak for the p lepton), Mw/Mz (the ratio of W* and Z masses), and l"), (the partial decay width of Z into a bb pair). The expressions of these observables in terms of (’8 can be found in chapter 1. 143 In this section, I only give the relevant terms in e ’s that might contain the leading effects in powers of m, from new physics. Denote the vacuum polarization for the W1, W2, W3, and B gauge bosons as 1164,.(6) = —z'g..., [A‘J‘(0) + qQF‘Hqfi] + 6qu terms, (4.64) where i, j = W, Z, '7, respectively. Then, 61 = e1 — e5 , (4.65) 62 = 82 - cos2 0 285 , (4.66) 63 = 83 - cos? 0 285 , (4.67) 65 = e), , (4.68) where _ AZZ(0) — AWW(0) 81 — Mg W 9 (4.69) 62 = FWW(M3v) - F33(M§), (4-70) 63 = 22S—0F3°(M2) (4 71) sing Z ’ ' szz 66 = Mg-EqTM/Ig), (4-72) and e), is defined through the vertex corrections to Z —+ bb - _ 9 1 - 75 v, (z —+ bb) _ 72m, 2 . (4.73) Both 61 and 6), gain corrections in powers of m,, and are sensitive to new physics coming through the top quark. On the contrary, 62 and 63 do not play any significant 144 role in our analysis because their dependence on the top mass is only logarithmic. Hence, £1 = 5p + corrections of the order g , 6;, = T + corrections of the order g, 62 = corrections of the order 9 , £3 = corrections of the order 9 , (4.74) where 6p = p — 1. The parameters p and 1' are defined by _ — Gng 2 2 Pp = I‘(Z"*IJ # )=Pm(9pv+gm1) ’ _ - 6'ng 2 2 Pb = “Z "’ bb) = Pm (95v + 95.4) » (4-75) where 1 l gI‘V 2 —§ (1—482), gpA=_§ , 1 4 1 gbv = -'2' (1—§S2+T), 9b,; =—-2-(1+7') . (4.76) Hence, comparing to Eq. (4.63) we conclude Zd’ 6p = "Z—x' - 1 , zL 4.4 One Loop Corrections in the SM The SM, being a linearly realized SU(2)L x U(1)Y gauge theory, can be formu- lated as a chiral Lagrangian after nonlinearly transforming the fields (see chapter 2). Applying the previous formalism, I calculate the one-loop corrections of order m? to p and 1' for the SM by taking REC = MAC = KEG = rcfic = O in Eq. (4.38). These 145 loop corrections can be summarized by the coefficient functions Z X , Z ¢, Zlf‘, and Zf which are calculated from the Feynman diagrams shown in Figure 4.1(a), 1(b), 1(c), and the sum of 1(d) and 1(e), respectively. I find zx = 1+1—g-:%§(A—lnmf), Zd’ =1+—§-—n;:2(A+-—ln2m,), 2: = 3:12:20), Z: = ngg:2(— A+1nm,-g) . (4.78) One notes that Figure 4.1(e) arises from the nonlinear realization of the gauge symme- try in the chiral Lagrangian approach. Substituting the above results into Eq. (4.77), one obtains 3Gpm? 6p , 8\/§7r2 2 = _Nw , (4.79) which are the established results (see section 1.5). 4.5 One Loop Corrections with Nonstandard Top Quark Couplings In chapter 3, I calculated the one-loop corrections (of order m? In A2) to p and 7 due to the nonstandard couplings of the top quark to the electroweak gauge bosons. The set of Feynman diagrams we considered contained external massive gauge bosons lines. In this section, I show how to reproduce those results by considering a set of Feynman diagrams which contains only the pure Goldstone bosons, the top quark, and the bottom quark lines, as described in Section 4.3. 146 X 3 3 . ¢ ¢ Z ° _____________ . (a) I 4” ¢- ¢ . Z . ———————————— (b) b ¢+ L b l/ \\ b Z b I: A l: ‘ I: (C) (d) (6) Figure 4.1: The Feynman diagrams which contribute to p and 7' to the order 0(mfln A2). 147 Non-renormalizability of the effective Lagrangian presents a major problem on how to find a scheme to handle both the divergent and the finite pieces in loop calculations [99]. Such a problem arises because the underlying theory is not yet known, so it is not possible to apply the exact matching conditions to find the correct scheme to be used in the effective Lagrangian [61]. One approach is to associate the divergent piece in loop calculations with a physical cutoff A, the upper scale at which the effective Lagrangian is valid [74]. In the chiral Lagrangian approach this cutoff A is taken to be 47w ~ 3TeV [61].3 For the finite piece no completely satisfactory approach is available [99]. To perform loop calculations using the chiral Lagrangian, one should arrange the corrections in powers of 1/41rv and include all the Feynman diagrams up to the desired order. Figure 4.1 contains all the Feynman diagrams needed for our study. I calculate the leading contribution to p and 7' due to the new interaction terms in the chiral Lagrangian using the dimensional regularization scheme and taking the bottom quark mass to be zero. At the end of the calculation, I replace the divergent piece 1/6 by ln(A2/m,2) for e = (4 — n) / 2, where n is the space-time dimension. Effectively, I have assumed that the underlying full theory is renormalizable. The cutoff scale A serves as the infrared cutoff of the operators in the effective Lagrangian. Due to the renormalizability of the full theory, from renormalization group analysis, I conclude that the same cutoff A should also serve as the ultraviolet cutoff of the effective Lagrangian in calculating Wilson coefficients. Hence, in the dimensional regularization scheme, 1/6 is replaced by ln(A2/p2). Furthermore, the renormalization scale p is set to be mt, the heaviest mass scale in the effective Lagrangian of interest. Since I am mainly interested in new physics associated with the top quark couplings to gauge bosons, I will restrict myself to the leading contribution enhanced by the 3This scale, 41w ~ 3 TeV, is only meant to indicate the typical cutoff scale. It is equally probable to have, say, A = 1 TeV. 148 top quark mass, i.e., of the order of (m? In A2). Inserting these nonstandard couplings in loop diagrams and keeping only the linear terms in it’s, one finds zx = 1+ 12:2; (2nfic—2K§C)In:—:,, Zd’ = 1+igr—Zlingc £1; , Z]; = 1 — 162:2)? (652C — 4KEC + n20) 1n g;- . (4.80) Thus the nonstandard contributions to p and 1' are - 3Gpm? CC NC NC A2 which agree with my previous results obtained in chapter 3. 4.6 Conclusions In chapter 3, I performed a one-loop level calculation of the leading quadratic mt corrections by considering a set of Feynman diagrams, derived form the nonlinear chiral Lagrangian, whose external lines were the massive gauge boson lines. The leading corrections (in power of 172,) to the low energy observables were found not to vanish in the limit of vanishing g (the weak coupling) because they originate from strong couplings to the SSH sector, e.g., through large Yukawa coupling 9,. Therefore, the result in chapter 3 should in principle be reproduced by considering an effective Lagrangian which involves only the scalar (the unphysical Goldstone bosons and 149 probably the Higgs boson) and the top—bottom fermionic sectors. This was shown in Section 4.3. I discussed how to relate the two corresponding sets of Green’s functions for the low energy observables of interest. I showed that by considering a completely different set of Green’s functions (without involving any external gauge boson line) from that discussed in chapter 3, I obtained exactly the same results. My result for T is different from that given in Ref. [101] where the wave function correction to the bottom quark was not included. Chapter 5 A Model of Strong Flavor Dynamics for the Top Quark 5.1 Introduction In chapter 3, I discussed a phenomenological model in which new physics appears in the top quark interaction with the gauge bosons. In that phenomenological model I did not specify an explicit dynamics which triggers the top quark nonstandard couplings. In general, these couplings could be due to different dynamical models, e. 9., extended technicolor models, models with extra gauge bosons, etc. In this chap- ter, I construct a specific model which triggers the top quark nonstandard couplings. It also leads toiother interesting physics at low energy. Therefore, one has to study all of the aspects and effects of the model at low energy. The construction of this model is based on the theoretical observation of the hier- archy of the fermion mass spectrum. The relatively large mass of the third generation of fermions may suggest a dynamical behavior for the third generation different from that of the first two generations. In this model, the third generation undergoes a dif- ferent flavor dynamics from the usual weak interaction proposed in the SM. I assume this flavor dynamics to be associated with a new S U (2) gauged symmetry. Therefore, a new spectrum of gauge bosons emerges in this model. No modifications to QCD 150 151 interactions are considered here; this case has been discussed elsewhere [110]. 5.2 The Model The model is based on the flavor symmetry G: SU(2), x S U (2);, x U (1),» . Where the third generation of matter (top quark, t, bottom quark, b, tau lepton, 7', and its neutrino, 11,.) experience a strong flavor interaction, instead of the weak interaction advocated by the SM. On the contrary the first and second generations only feel the weak interaction supposedly equivalent to the SM case. The strong flavor dynamics is attributed to the S U (2);, symmetry under which the left-handed fermions of the third generation transform in the fundamental representation (doublets), while they remain to be singlets under the S U (2), symmetry. On the other hand, the left-handed fermions of the first and second generation transform as doublets under the SU(2), group and singlets under the S U (2),, group. The U (1)y group is the SM hypercharge group. The right-handed fermions only transform under the U (1),» group as assigned by the SM. Finally the QCD interactions and the color symmetry SU(3)C are the same as in the SM. The symmetry breaking of the Lie group G into the electromagnetic group U (1)....” is a two stage mechanism, first SU(2), x SU(2)), x U(1)y breaks down into SU(2)L x U (1)y at some large mass scale. The second stage is where S U (2) L x U(1)y breaks down into U (1)em at a scale of the order of the SM electroweak symmetry-breaking scale. The spontaneous symmetry-breaking of the group SU (2); x S U (2);, x U (1)y is accomplished by introducing two scalar matrix fields 2 = a + Mar“ 1 and with the transformations 2 ~ (2,2)0 , ~ (2,1)1 , (5.1) l 1"”3 are the Pauli matrices and Tr(‘r°1'°) = 260b- 152 i.e., the 2 field transforms as a doublet under both SU (2), and SU(2)), and as a singlet under U (1)y. On the other hand the Q field transforms as a doublet under SU(2)), as a singlet under SU(2),” and has a hypercharge quantum number Y = 1. Thus, the scalar doublet Q is equivalent to the SM Higgs doublet. However, as to be shown later, the Yukawa sector is different. As a realization of the symmetry I define the field transformations as 2 '4 .9129; a ‘1’ —’ 9193/4) , (5-2) where 91 6 SU(2)), 92 E SU(2);,, and 9}» E U(1)y. In this section I discuss fully the structure of this model. 5.2.1 The Bosonic Sector Under the gauged S U (2), x S U (2) h x U (1)y , I introduce the covariant derivatives of the scalar fields, D”2 = am + ig,W,"E — ighzwg , (5.3) DWI) = aw +ig1W,”Q + gg'Bpo , (5.4) where the gauge boson fields W} :— IV,“T° / 2 and W), E Wf'r“ / 2 corresponds to the gauged groups S U (2), and S U (2);,, respectively. The gauge coupling 9;, is assumed to be larger than g; even though I will restrict myself to the region where the perturbative calculation still holds. With these definitions, the gauge invariant bosonic Lagrangian is .63 = gppqfipflo + imppztpflz) + V(Q, 2) 1 a a 1 a a 1 _ZW’ ”W: ” - ZWhpWh” — 113qu , (5'5) 153 where V(Q,E) is the scalar potential. I assume that the first stage of symmetry breaking is accomplished through the 2 field, i.e. by acquiring a vacuum expectation value u, (2) = (g 2) . (5.6) Hence the symmetry SU (2); x SU(2)), x U (1)y is broken into the diagonal group SU(2)L x U (1)y , and the symmetry-breaking scale is set by the vacuum expectation value u. The next step is to break the SU(2)], x U (1),» symmetry into the U (1)em symmetry through the scalar Q field, i.e. by acquiring a vacuum expectation value v. <> = (0) . (5.7) where v, as we will see later, is of the order of the SM symmetry-breaking scale. Because of this pattern of symmetry breaking, the gauge couplings are related to the U (1)em gauge coupling 6 by the relation 1 —=—2+'—2'+—. (5'8) Here I define e e , e =-.—— gh=.—-.— g = srn0cos¢ ’ smflsmgb ’ cosfl’ 91 (5.9) where 0 is the usual weak mixing angle and (b is a new parameter in this model. The scalar fields, except Re(¢°) from the Q doublet and a from the 2 matrix field, become the longitudinal components of the physical gauge bosons. The surviving Re(¢°) field behaves similar to the SM Higgs boson except that it does not have the usual Yukawa couplings to the third generation. To get the gauge boson mass eigenstates, I first concentrate on the charged gauge bosons. As a first step I rotate the gauge fields by the angle <25. W?” = cos ¢Wfp + sin 43ny , W?” = — sin ¢Wfp + cos ¢W§kp , (5.10) 154 where I’Vf” = (VI/,1“ ¥ iW,2p)/\/§, and similarly for Whiu. The mass matrix reduces to 1 -tan¢ M‘2‘,=Mg (-—tan¢ m+tan2¢) a (5.11) where 2 2 2 2: 8'” :9. 512 Mo -m x- (- > Next I consider the neutral sector W12, W112» and B“. Define W13p = cos 0(cos (WI/,3” + sin ¢W,?”) — sin 08,, , (5.13) A, = sin 0(cos ¢W,3p + sin 451fo) + C0893”, (5.14) and W23” = — sin ¢W,3p + cos (pH/,1?” . (5.15) The gauge field A” is massless, corresponding to the physical photon field, while the remaining fields have the mass matrix M3 1 2—cosOtan45 2 __ :ccos 0 2 2 z — c0320 (—cos€tan¢ m+cos Otan <15) . (5.16) To get the mass eigenstates and the physical masses of the gauge bosons, I further diagonalize the mass matrices M3,, and Mg. In this model, I am concentrating on the case where g). > g1, (equivalently tanqfi < 1) but with g}: _<_ 47r (which implies sin2¢ Z 92/(471’) ~ 1/30) so that the perturbation theory is valid. Similarly, for g), < 9;, we require sin2¢ S 0.96. Furthermore, I focus on the region where a: >> 1, 155 though another region of interest could be :1: ~ 1 (11 ~ v), but in this case the one-loop level contributions due to the heavy gauge bosons should also be included because they are of the same order as the SM one-loop contributions. In the limit a: >> 1, I expand terms only up to the leading order in 1/1'. Thus, in dealing with this limit one can ignore all higher order corrections, since they are suppressed by higher powers of 1 / 1:. To the order 1 / 1:, the eigenstates of the light gauge bosons are sin3 45 cos d) W: = W?” + W3”, (5.17) sin3 (1) cos (p Z”=ZI"+ xcosfi Z2” - (5.18) While for the heavy gauge bosons one finds - 3 WE: = —WWF’, + W?” , (5.19) sin3 ¢cos¢ 2:1: —W 1’, + Z2” . (5.20) To the same order, the gauge boson masses are ' 4 M3,. = 1402(1— 3‘“ 15), (5.21) M2 sin4 (b 2 o _ M2 _ COS, 0 (1 x ). (5.22) While for the heavy gauge bosons one finds - 2 M2 I = 2 :1: sm qb W i M0 (sin2 czbcos2 ¢ + cos2 45 ’ (5'23) - 2 M2, = 2 :1: sm ¢ . 2 M0 (sin2 (bcos2 ¢ + cos2 45 (5'24) 156 It is interesting to notice that the heavy gauge bosons are degenerate up to this order, i.e., M’Z = erzt. This is due to the fact that the heavy gauge bosons do not mix with the hypercharge gauge boson field, B“. 5.2.2 The Fermion Sector Now I focus my attention on the fermionic sector. The quarks transform un- der the usual color SU(3)C gauge group as in the SM. As discussed before, only the third generation interacts with the SU(2)), gauge bosons. The first and sec- ond generations only interact with the SU(2), gauge bosons. Explicitly, under the SU (2); X S U (2);, x U(1))» symmetry, the transformation of the first and second gen- eration is as follows Left-handed quarks: (2,1)1/3 , Left-handed leptons: (2,1)-1 For the third generation, we have Left-handed quarks: (1,2)1/3 , Left-handed leptons: (1,2)-1 For all the right-handed fermions, we have Right-handed quarks and leptons: (1,1)Q , where Q is the electric charge of the right-handed fermions. Because of this assign- ment, the model is anomaly free since cancelation of anomalies are satisfied family by family. I denote left-handed doublets of the first and second generation by 29'”), and right-handed singlets by ‘11)?) . On the other hand, I denote the left-handed dou- blets of the third generation by Q2 and right-handed singlets by ‘11:}? The fermionic Lagrangian is L, = III—Z(l’2)i7"D,,\II(,3'2)+TP;(1’2)2'7”D,,\IIS§’2)+ \p—Li‘iyflppq’i + fi3i7”D,,\II%, (5.25) 157 with 0,293) = (a, — 79,12:er — 7,7333”) 2‘33), (5.26) 072%” = (6.. “56285231”, (5.27) 0,511), = (6,. + £9.32:ng + ig'g-Bp) 1113,, (5.28) Du‘P'le = (all + ig’QBp) Win (529) where Y is the hypercharge generator of the U (1)y group, and the relation Q = T,3 + T,‘,’ + 1;- (5.30) is satisfied, where T311) is the third isospin component of S U (2);“). In terms of the mass eigenstates of the gauge bosons W5", Z”, WLi, and Z; the interaction Lagrangian is sin2 ¢ Lint __ 8 W112)?” [II-Th +7711: + :1: 2 i - 2 (1.2) :t f _sin0 (T,I cos ¢—T, srn 45)] \IIL Wp-l- sin2 (1) 17 5:02)?" [71,3 + T)? — Q sin2 0 + (cos2 <15T,,i — sin2 ¢T,*)] Wilma, sin 9 cos 0 e —3 pp_sin¢ i cosqS i:__sin3cos i i 3 ,i +sin0‘pL7 _ cos¢TI +sin¢Th :rcos20 (Th +T’) WLWH L—s p '_sin¢ 3 cos¢ 3_sin3¢cos¢ 3 3_ ,2 3 , +sin0wl’ 7 L cosqb ' sing”) h xcos20 (Th +T’ ern 9) ‘I’LZH ° 2 . 3 —.' ”,- _ erm 0—-.' p,- _sm ¢cos¢ , +8Qf7 f A” sin0c030 ”7 f“ Z" xcosO Z” ' (531) Now I consider fermion mass generation and mixing. The first and second gener- ations acquire their masses through the Yukawa interactions to the Q doublet field, 158 with < Q >= 1). The Yukawa Lagrangian is _l e e ['Yukawa = ‘I’L <1) [911311 + 912”}! + gis’rlz] __2 c e c + ‘I’L ‘1’ [921812 + 922HR + 923712] + h-C- (532) For the third generation one can not generate the fermion masses through the usual Yukawa terms (dimension four operators), as it is not allowed by gauge invariance. It is only through higher dimension operators that one can generate these fermion masses. This characteristic of this model may be significant in understanding the fermion mass generation problem especially in understanding the observed mass hi— erarchy. Thus, although the masses of the first and second generations are generated through the Yukawa interactions as in the SM. The mass spectrum of third generation must be generated by a different mechanism. This conclusion may be attributed, in this model, to the strong flavor dynamics which may be evident at adequate high energy. At high energy where the interactions are strong enough, the masses of the third fermion generation are assumed to be generated possibly within some dynam- ical framework. I do not offer an explicit scenario for such a picture, however, an extended technicolor scenario may offer such a solution. The conclusion is that the strong flavor dynamics may be an essential player in the mass generation mechanism and an understanding of the strong dynamics at the high energy scale is required to solve the mass generation issue. Also, the strong dynamics may be responsible for the large masses of the third generation as compared to the first and second generations. In this discussion, I limit my self to the region where the strong flavor interaction is being under a perturbative control. Effectively, one can generate the third generation masses using dimension five operators, e.g., for the 1' lepton one can generate its mass through the following mass terms 1 AIfif’z‘lcp [931812 + 932MB + 933%] + h-C-a (5-33) 159 where Q: = (”:)L, and A characterizes some large mass scale associated with the strong flavor interaction. It is reasonable to assume that A ~ it >> 12, and thus 933 is of order 1. An early version of the model Ref. [111] with an additional scalar doublet couples only to the third generation through the usual Yukawa interactions is another scenario for generating the third family masses. With the fermion mass matrices being generated, one can obtain their physical masses by diagonalizing the mass matrices using bilinear unitary transformations. For example, for the lepton sector, the lepton mass matrix M, can be read out from the mass Lagrangian written above in Eqs. (5.32) and (5.33). I introduce the rotational unitary matrices L, and R, with the transformations, e}, —> Lifei, e), —+ 72:14,. (5.34) Hence, the physical mass matrix is given by M3‘%- = LLMCRC. (5.35) Because the third family interacts differently form the first and second generation, I expect in general Flavor Changing Neutral Currents (FCNC) may occur at tree level. For example, in terms of the leptonic weak eigenstates the left-handed neutral and charged currents are given by ___—e — — — . sin4 sin2 8L 23in0cosl9 ( 8L “L TL )7” [-1+281n26+ a: d) — x ¢G] “L Zn, (536) TL and e __ _ __ sin4¢ sin2¢ V‘L fisin0(e L L )7” [1— a: + :1: G] ”FL Wpi’ (5-37) 160 where 000 0:000. (5.38) 001 In terms of the fermion mass eigenstates it follows that the left-handed neutral current interactions are . 4 2 8L e (51:71?fi)7”[-1+2sin20+Sl:¢-SinLLIGLeI‘](L)Zu- 2 sin 0 cos 0 TL (5.39) For the left-handed charged current interactions, one has e (tr—r) P1—Si"4¢+5in2¢L;GL 56L W’+hc (540) fisin0 LLL L 7 a: "L “ H ' 3.? ”TL For the neutrino sector, the neutral currents are 2 sin9cos€ - 4 2 Va]. 6 (1722173375)7”[1-81:¢+ sin xLLLIGL](V ”L)Z,,. (5.41) VTL Similarly, for the quark sector I introduce the unitary matrices L” and L4, in terms of the mass eigenstates one finds the following interaction terms: .4 2 UL . ()1_ WW )2“ 2sin9cos€ :r t L (5.42) . d ‘9 (___) F (1-5“‘4¢)L*L+ 8‘“ ¢LLGL L W’+hc fisinfl L L L 7 I " L+ d p H, 30a...) 161 and . . d e — _ — 2 . sm4¢ sm2¢ L 28in0c080( L 3L LL )7#[_1+ ~3-s1n20+ T — a: LLGLL] ( :L Zu- L The right-handed fermion couplings to the neutral gauge bosons Z and Z’ are, respectively, e sin 0 cos 0 (—Q sin2 0) , (5.45) and sin3 ¢cos¢> . (5.46) e , 2 — 9 sin0 (Q srn xc0820 The fermion couplings to the photon are the usual electromagnetic couplings. As shown above, if g), > 9, then the heavy gauge bosons would couple strongly to the third generation and weakly to the first two generations, and vice versa. For the charged-current interactions in the quark sector, one observes that in the case of ignoring the new physics effect, the quark mixing is described by the unitary matrix V = LI, Ld which is identified as the usual Cabibbo—kobayashi-Maskawa (CKM) mixing matrix. When new physics are turned on, the mixing acquires an additional contribution proportional to sin2 (p/x. Since the ofl-diagonal elements in the CKM matrix are small, one can approximate the quark mixing as 04 .2 .4 02 LL(1_smq§+sm ¢G)Ld 2 LlLd_sm¢+sm (LG 2: a: :1: a: =v_ ° 4 ° 2 8111 $111 <15 + d> 13 17 G. (5.47) Interesting new features emerge in this model, e.g., lepton mixing is an exciting possibility. In addition to that, neutrinos can mix through their weak interactions, 162 making it an exciting feature that may be connected to the solar neutrino problem. Quark mixing also has interesting features, e.g., FCNC are a possibility in the model which could be investigated. However, the quark sector has more free parameters (to describe the mixing) than the lepton sector which makes the analysis more tedious and less predictive. Hence, I will not consider FCN C in the quark sector. For the lepton sector, there are already significant constraints on lepton universality and lepton number violation from the low energy data that one has to check against from the start. First, one must examine the already existed data on lepton number violation to see if such mixings are allowed. As an example is the almost vanishing branching ratio of the process F),- _,c—e+e- which forbids any mixing between the first and second generation. Other lepton number violation processes, especially those involving the third family, are not as severe as I‘p—_,e—.,+¢—. In the next section, I will discuss in more details the constraints imposed by the low energy data. 5.3 Low Energy Constraints As discussed in chapter 1, the input parameters of the SM 0, Gp, and Mg are defined through three experimental measurements, e.g., through the e-p scattering, the p decay, and the Z peak at LEP. Similarly, one needs to fix the input parameters in the proposed model. The input parameters can be chosen as a, Gp, M 2, sin 03, and 1:. I fix the first three parameters in a similar way to the SM case. The last two parameters will be treated as free parameters to be constrained through the low energy data. It is advantageous to express the input parameters in this model in terms of their corresponding SM ones. In the case of the electromagnetic coupling both values coincide, i.e., a = as”. In the case of weak coupling constant I use the p decay to define G p. I calculate the p-decay width as predicted in this model and 163 including the W and W’ contributions. I find that G p = G?“ (equivalently v = vs“) as long as one demands no mixings between the first and second lepton families (see below). Finally, I define M 2 using the Z peak at LEP, i.e., M2 = M2”. Now, one can write other bare parameters in the proposed model in terms of the input parameters. For example consider Eq. (5.22) ' 4 Mz ——3'-’—— (1 - 8’“ d’) . (5.48) = 2 sin 0 cos 6 2:1: In the SM, M z can be written as 6’0 Z = 2sin05M cos 05M ’ (5°49) where sin 03M is the SM weak mixing angle. Since M z is defined in both the SM and the proposed model through the Z peak at LEP, one concludes that +L _ 29:2 ___ , 6v , (5,50) 2 snn 0 cos 0 2a: 2 sm 03M cos 93M One finds that sin4 4) 2sin9cos€ = 25in 63M cos 03M (1 — 2x ) . (5.51) Hence, solving for sin2 0 = 1 — cos2 0 in terms of sin2 03M, one finds that 20 - 4 . sin20 = sin2 95M 1 — COS sm d) . (5.52) cos 26 SM .7: Therefore, observables can be predicted in the proposed model using the the input quantities a, G )5, M2, and sin2 03M. (The values of these quantities are given in chapter 1.) In addition to those quantities one has two additional free parameters sin2 55 and 0:. As previously discussed, lepton mixing is an interesting consequence of this model. However, the almost null measurement of p —§ e‘e+e‘ forbids any mixing between 164 the first and second lepton families. Still mixing may be allowed to exist between the third family and either the first or the second one. It is natural to assume that the mixing strength between leptons may be directly related to the lepton masses. If so, one would expect the mixing between the second and the third families to be more significant than the first and the third families. In the following discussion, I will assume that in the case of leptons, mixing will be only between the second and the third families. Because the mixing matrix is of the form LlGLe, where the matrix G is given in Eq. (5.38). One finds that the mixing matrix for the second and third lepton families can be expressed by one free parameter. The 2 x 2 mixing matrix can be written as ( sin2 ,6 cos B sin fl) , (5.53) cos 3 sin 6 cos2 6 where sinfl is a free parameter of the model for describing the mixing between the second and third lepton families. To test this model by the low energy constraints on lepton mixing and FCNC processes, it is necessary to understand the form of the four-fermion current-current interactions at zero momentum transfer. The four-fermion charged-current weak in- teractions are [112] 2 . . 2 . 5501 + 11:)? + 0333’ (554) and the neutral current four-fermion interactions are 2 .3 .3 o 2 . 2 2 .3 . 2 . 2 o 2 55(3) +]h — sm 03m) + $50,, — srn cpsm 9.7m) , (5.55) where j”. are the left-handed charged currents corresponding to the first two genera- tions and the third generation, respectively. Similarly, jg), refers to the left-handed T3 currents, while jem represents the full electromagnetic current of the three families. I 165 conclude that if there is no mixing in the lepton families then all leptonic decays are identical to the SM, e.g., the T lifetime can not furnish any new information about this model. (This also explains why G p = 0%” from the u-decay if there is no mixing between the first and second lepton families.) However, it is more general to allow mixing in the leptonic families, so I will investigate this possibility more carefully. Be- cause of the almost vanishing branching ratio of the decay I‘(,,-_,,—¢+,-), BR 5 10‘12 [9], I will only allow mixing of T and p and their neutrinos. This mixing will mod- ify the lifetime of the T lepton, which depends on one free parameter, sin2 H. The constraints on sin2 3 come from: the ALEPH measurement (in terms of the effective couplings ratio gr/gp, cf. Table 5.1) of the branching fraction for T decay into p and the determination of the T lifetime [113], the lepton number violation decay of T —) mm, with a branching ratio BR < 4.3 x 10’6 (at 90% CL.) [9], and the FCNC search at LEP with BR(Z -> piTT) < 1.7 x 10'5 (at 95% CL.) [114]. Figure 5.1 shows the Feynman diagrams for the process T —) pup. At zero momentum transfer both diagrams are of equal importance. One finds that - 2 2 2 I‘(,.-_,,,-,,-,,+) = I‘(,__,,,—,,rpu)sm 5:208 fl (sin flz - 45in2 03in2 (b) . (5.56) All other fermionic processes at zero momentum transfer, such as the p decay, K -T{. mixing, and B-3 mixing, are identical to the SM predictions. In this model, the low energy predictions depend on the values of l/x, sin2 d), and sinzfl in addition to the measured values of a(Mz), G p, and Mg. Using the most recent LEP measurements [12] (the total width of the Z boson, Re, R”, R.” the vector gv and axial-vector gA couplings of e, the ratios gv(p, T) / gv(e), gA(p, T) /gA(e), the lepton forward-backward asymmetries, the T and e polarization asymmetry, the hadronic pole cross section 02, and the ALEPH measurement of g.,/gp [113]) combined with the FCNC measurements of T‘ —) p'p‘u” and Z —+ [.l—T+, I determine the 166 (b) Figure 5.1: The Feynman diagrams for the process T —> ##4#- 167 allowed values of sin2 ()5, sin2 ,3, and M 2,. I do not include the controversial observables Rb and Rc as a part of the fit. Instead, I treat them as a prediction and discuss later whether the proposed model is able to explain the anomaly in these measurements. The experimental values of the electroweak observables [12] and their SM prediction [30] are given in Table 5.1. I calculate the changes in the relevant physical observables relative to their SM values to leading order in l/x, i.e., 0 = 0SM (1 + 60) , (5.57) where 0SM is the SM value for the observable 0 including the one-loop SM correction, and 60 represents the new physics effect to leading order in 1 /:1:. I list the calculated observables as follows, [‘2 = 1‘3“ (1+ % {—0.896sin4 as + 0.588 sin2 6]) , (5.58) RC = REM (1 + i [0.0794 sin4 (15 + 0.549 sin2 96]) , (5.59) Ru = Rf,” (1+ i- [0.0794 sin“ 6 + 0.549 sin2 ¢ — 2.139sin2 651112 42]) , (5.60) R, = R§M (1 + :1; [0.0794 sin4 (b + 0.549 sin2 65 — 2.139 cos2 flsin2 43]) , (5.61) 3.3 =( 7.3)“ (1 + g [10.44 sin4 ¢]) , (5.62) 1152.3 = (1163)“ (1 + :- [10.44.6114 d> + 1214511266612 ¢]) , (5.63) F's = (21:58)SM (1 + % [10.44 sin“ (25 + 12.14 costlflsin2 66]) , (5.64) Ac = AEM (1 + 213 [5.22sin4 ¢]) , (5.65) 168 A, = A?“ (1 + :1; [5.22 sin4 ¢ +12.14cos2 6sin2¢]) , (5.66) 02 = (absM (1 + :3 [-0.013in4 d) — 0.628 sin2 (12]) , (5.67) MW = M3,,“ (1 + g [1 + 0.215 sin“ ¢]) , (5.68) SM ' 1 9.“: = (9;) (1+ - [0.50sin26cos26]) , (5.69) 9;: 9p 3 _ __ _ + sin2flcoszfl ,2 '26 _ 2 2 BR(T —-) p )u u ) = 0.045——x2—-—(s1n fl — 43m sm ()5) , (5.70) sin2 ¢sin )6 cos )6 2 I‘(z_,y-T+) = 0.167 G6V( a: ) . (5.71) In Figure 5.2 I show the fit result, at the 30 level, of the Z’ mass as a function of sin2 d), for a, = 0.125 and for three values of the mixing parameter sin2 6 = 0 (dashed line), 0.5 (dot-dashed line) and 1 (solid line). In the case of sin2 6 = 0, I find a lower bound on M 2: approximately 1.1 TeV. For sin2 6 = 1, M z: is approximately 1.4 TeV. For sin2 ,3 = 0.5, M z: is required to be larger for smaller sin2¢ (< 0.1) due to the strong constraint from the lepton number violating process T —) mm (see Eq. (5.70)). As shown in Figure 5.2, as sin2¢ increases the lower bound on M 2: increases, and increase in M 2' is slow for sin2 (b < 0.5 and fast in the other case. This indicates that, a relatively light Z’ prefers strong interactions with the third family fermions. If I consider a 20 fit, then the lower bound on M z' is about 1.4 TeV for sin2 6 = 0 and 1.8 TeV for sin2 6 = 1. In Figure 5.3 I show the fit result, at the 30 level, For a, = 0.115 I find that My 2 1.3 TeV for sin2fi = 0 and My 2 2.2 TeV for sin2fl = 1. In Tables 5.1 and 5.2 I calculate the low energy predictions in the model understudy for different choices of sin2 (b, :13, sin2 )6 = 1, and for two values of as, 0.125 and 0.115, respectively. 169 6000VITT‘ITTT—UIIUIIIITFIUIIITU , - I- , '- 5000 — uln'p-o I _. E — - «mp-0.5 I Z - — --1n'p-1 I w 4000 “1' (99") 3000 2000 1000 L I l l l l l l 1 J 1 1 L l l l l l 1 I l l l l 0.0 0.2 0.4 0.6 0.8 1.0 315‘.» Figure 5.2: The lower bound on the heavy Z' mass as a function of sin2¢ for 3a, sin2 6 = 0 (solid) and sin2 6 = 1 (dashed) and a, = 0.125 170 6000IIIIIIIFIITIIIIIIIIIUFIT 5000 — Iln'fi-O I’ — : -— - «mp-0.5 / : .. — -s1n'fl=1 / - .. 4000 “2‘ (GeV) 3000 2000 1000 — '- llllllLlllJlelillllllll 0.0 0.2 0.4 0.6 0.8 1.0 sinz¢ Figure 5.3: The lower bound on the heavy Z’ mass as a function of sin2¢ for 3a, sin2 )6 = 0 (solid) and sin2 [3 = 1 (dashed) and a, = 0.115. 171 As discussed in chapter 1, the LEP measured quantities R5 = I‘b/I‘h and Re = I‘c/I‘). are not consistent with the SM prediction. One possibility to explain the anomaly in these quantities is to consider new physics which can affect the b and c quarks’ couplings to the Z boson. The question now is whether this model is able to give any insight regarding these measurements. The observed value R?“ = 0.2219:l:0.0017 [12] is higher than the SM value R2“ = 0.2157 [30] by about 3.50. On the other hand, R?“ = 0.1543 j: 0.0074 is smaller than the SM value R?“ = 0.1721 by about 2.50. With the allowed region of the parameter space being determined, I investigate which part of the allowed space is able to explain the anomaly in R5. Because the measured value of R5 is different from the SM value by more than 30, I expect to be able to constrain the smallest and largest Z’ mass by requiring that the new physics effect shifts the theoretical value of R5 to be within the 30 range of the measured value. In this model, Rb is given by 1 R5 = REM (1 + ; [—0.01495in4 (b + 1.739 sin2 05]) . (5.72) Ignoring the negligible sin4 (0 term, one finds Rb - REM _ sin? ¢ Rf“ — (1.739 a: . (5.73) Since Mg, 2: Mam, the Z’ mass can be constrained to be 462 < Mz: 00305 < 1481 GeV. (5.74) Thus, if I assume the anomaly in R), is mainly due to this type of new physics,'then there is an upper bound on M 2' which depends on the gauge coupling (equivalently sin (b). For example, for sin2¢ = 0.04, the upper bound (which is independent of sin2 [3) on M 2: obtained from R), is ~ 1.5 TeV. 172 For Re I find that the new modification to the SM model shifts R, in the correct direction, i.e. it decreases the theoretical value as desired. I find Rc = REM (1 + 31:- [0.038 sin4 45 — 0.549 sin2 05]) . (5.75) However, the amount of shift is very small to account for its anomaly, e.g., with the lower bound on the heavy mass coming from 1.1 TeV, I find that the theoretical value of RC is still outside the 20 range of the measured value. From these results I conclude that this model can account for the deviation in R), from the SM at the 30 level. Even though RC is shifted in the needed direction, the predicted value is still outside the 20 range of the data. Therefore, one cannot explain the anomaly in Rc entirely based on the proposed model. Also, in this model the prediction of the observable Am is identical to that of the observable A,. Thus, this model cannot explain the discrepancy between the the SLC measurement ALB = 0.1551 :1: 0.0040 and the LEP measurement A. [30]. 5.4 High Energy Experiments LEP was operating at the Z-pole with large production rates, it is therefore un- likely to better test this model at other high energy colliders at the scale of M z. I have checked that the allowed parameters in Figures 5.2 and 5.3 do not upset the measurements of W:t and Z properties at the Tevatron by CDF and D0 groups [115]. To study the possible effects due to the heavy W’ and Z’ bosons, I will con- centrate on physics at energy scales larger than M z. In this study, the interference effects from A (photon), Z and Z’ in neutral channels and the interference of W and W’ in charged channels are all included. To simplify the discussion, I will consider two sets of parameters for (:6, sin? <1), sin2 6) : (7,0.04,0) and (20.6,0.14,0.5) which cor- respond to (M z:,I‘z:) equal to (1083,291) GeV and (1050,76) GeV, respectively. The 173 conclusions, however, will not significantly depend on the details of the parameters chosen from Figures 5.2 and 5.3. At the Tevatron, it is possible to reach the high energy region where the W’ or 2’ effects can be important. CDF has reported the result of searching for new gauge bosons by measuring the number of excess di-lepton events with large transverse mass [116] or invariant mass [117]. I find that those results do not further constrain the parameters shown in Figures 5.2 and 5.3. For the Tevatron with Main Injector (a pp collider at J5 = 2 TeV with a 2fb"l luminosity), the excess in the e"e+ or e+ue rates from this model is generally not big enough to be easily observed. Since the third family leptons can strongly couple to the new gauge bosons, the rate of T lepton production can in principle be quite different from that of e or u. Furthermore, if sinfl is not zero, the production rates of pp —+ W, W’ —> (V; or pp -) 7, Z, Z’ —+ (2- will be different for E = e and )1. However, even with the maximal mixing between T and p (i.e., sin 6 = 1) this difference at the Tevatron can only exceed a 30 effect for a 10 fb'1 of integrated luminosity. At the LHC (a pp collider with J3 = 10 TeV and a luminosity of 100 fb-l), this excess cannot be mistaken. Furthermore, at the LHC, the excess in the production rates of the (V; and the (“‘8‘ events can also be individually tested. Thus, it is much easier to either find such new effects or constrain parameters of the model at the LHC than at the Tevatron. I note that this conclusion holds for either a small or large sin2 (1). Although with a large sin2 4), the new physics effects to light family fermions will be large, because of the large W’ and Z’ masses, the net effect of the new physics to the production of di-lepton pairs does not significantly depend on sin2 ¢. Another signature of the model is an excess in the top quark production, however, this excess cannot be observed at the Tevatron because of large background from the QCD processes qq, gg —) tf. At the LHC, the excess in the tt_ pair productions can 174 easily be seen in the invariant mass distributions. The extra gauge bosons can produce an excess of di-jet events in the large invariant mass region, but the parameter space remaining after imposing low energy constraints does not allow a big enough effect to explain the results reported by CDF [32]. Another possible interesting signature is the production of 11in pairs, which is unconstrained by current LEP data. At the Tevatron for (:1:,sin2 (13,sin2 6) equal to (20.6,0.14,0.5), the most favorable scenario for observing this signal, I find a total of about 20 events for 2 fb"1 of integrated luminosity, assuming no cuts are imposed. It is interesting to notice that this implies that the upgraded Tevatron can provide a better constraint on this FCNC type of event than LEP can. At the LHC, the cross section is 170 fb for this choice of parameters. At high energy electron colliders, the detection of the above new signatures be- comes much easier as long as there are enough of them produced in the collisions. In this model, neither LEP140 or LEP-II can see them, so I will concentrate on the future high energy Linear Collider (LC) [118]. Consider the proposed e+e" LC at center of mass (CM) energy \/§ 2 500 GeV with an integrated luminosity of 50 fb'l. For m. = 175 GeV the SM production rate Oiyc__ui) is 558 fb. Thus, a large number of t-f pairs is expected at the LC. Considering the set of parameters (0:,sin2 ,sin2 fl) = (7.0,0.04,0.0), I find that 0(e+,-_,,;) = 709 fb, i.e. there is about 27% increase in the total production rate compared to the SM. At the LC it is expected to measure the t-f cross section, for f’ + jets decay modes, to within a few percent. With the assumption that the expected measurement is within 3 standard deviation from the SM, one can constrain the parameters to those which produce M z; 2 2.3 TeV. I note that the same constraints hold for different choices of sin2 4‘) and :1: but with almost the same ratio sin2 ¢/:r, especially for small sin 43, since in the cross section the two parameters enter as a ratio. Because only the left-handed couplings of the top quark 175 are significantly modified in this model, measuring the angular distribution of t in the t-f CM frame can further improve these bounds if no new signal is found. Although the e+e‘ LC is suitable to probe the model under study, I notice that the pflu‘ collider is also interesting because of the possible mixing between p and T leptons. For small mixing the e+e" and the pflr" colliders lead to similar production rates as expected. For large sinfl the total production rate of 0(”+”-_,,;) becomes smaller than the SM rate which shows the opposite effect to the production of the 6+6" —) tf events predicted by this model. For the same reason, if sin2 ,6 = 1.0, then it is easy to observe the difference in the production rates of 6" 8+ and 11‘ [1+ (or T+ T") pairs at the LC. Furthermore, at the LC, if the FCN C event e‘e+ -) piT“ occurs, it can be unmistakably identified. For a 500 GeV LC with a 50 fb’1 luminosity, I expect an order of 300 such events to be observed for (:6, sin‘2 (0, sin2 6) equal to (20.6,0.14,0.5). Figure 5.4 shows the FCNC event numbers at the LC for a few choiCes of parameters, assuming no cuts are imposed. In summary, I find that due to the strong constraints to this model implied from low energy data (including Z—pole data) it is not easy to find events with new signa- tures predicted for Tevatron or LEP-II. However, at the LHC and the LC, it becomes easy to detect deviations from the SM in the productions of the third family or second family (in case of large mixing between T and p lepton) fermions. I have also checked the possible excess in the W+W‘ or the WitZ productions at future high energy colliders. It turns out that the branching ratios for Z ’ or W’ to the pure gauge boson modes are always small. One finds ezMzt sin6 cos2 45 Mg. ’ W+W" z . NZ —> ) 1927rsin29 :62 Mil, (5.76) Therefore, the gauge boson pair productions are not good channels for testing this model. 176 100 T I I I l I I I I l I I I I l I I I I I I I I 1 - 1 : - —-1n'¢-o.1. H;--1700 Gov : )- ' -1 $ 80 - sin ¢-o.4. nag-1700 GeV — .. -1 i‘ h 60 — —- g _ _ z : : ’6’ . . 6 ,0 ._ _ - / , _ \ \ . " / \ .- - / \ . / \ 20 '— / \ '— *' / \ . - / \ . - / \ . / \ O l L l l l l l l l I I l L l l J J L l I I l l L 0.0 0.2 0.4 0.6 0.8 1.0 sin‘p Figure 5.4: The event number of )1in produced at the LC, with a c.m. energy of 500 GeV, and for two choices of parameters. 177 In the process of preparing for this paper, I noticed that another similar work appears in Ref. [42]. My conclusions on the allowed parameters of the model and the predictions on the event yields for electron or hadron colliders are different from theirs. Also, I became aware of a work done in Ref. [43], in which a similar model was proposed and studied using the low energy data. 178 Table 5.1: Experimental and predicted values of electroweak observables for the SM and the proposed model (with different choices of parameters) for a, = 0.125 with m. = 175 GeV and my = 300 GeV. a: sin? 6 = 0, sin2 6 = 0.04, M; = 1.1 TeV, 1", = 288 GeV. b: sin2fl = 1, sin2 45 = 0.04, M’z = 1.4 TeV, F’z = 370 GeV. c: sin? 6 = 0, sin2 6 = 0.80, M'Z = 3.0 TeV, 1",, = 287 GeV. (1: sin2 6 = 1, sin2 6 = 0.80, M5, = 3.3 TeV, r' = 316 GeV. I Observables Expe'rimental data SM The model a b c d gv(e) —0.0368 :l: 0.0017 -0.0367 -0.0367 -0.0367 ~0.0372 -0.0371 gA(e) —0.501 15 :l: 0.00052 -0.5012 -0.5012 -0.5012 -0.5005 -0.5006 gv(p)/gy(e) 1.01 :l: 0.14 1.00 1.00 1.05 1.00 1.04 gA(p)/g,4(e) 1.0000 :L- 0.0018 1.0000 1.0000 1.0034 1.0000 1.0030 gV(T)/gv(e) 1.008 :l: 0.071 1.000 1.073 1.000 1.047 1.000 gA(T)/gA(e) 1.0007 :1: 0.0020 1.0000 1.0055 1.0000 1.0036 1.0000 [‘2 2.4963 2!: 0.0032 2.4978 2.5054 2.5025 2.4967 2.4969 R, 20.797 :1: 0.058 20.784 20.848 20.823 20.830 20.822 Ru 20.796 :h 0.043 20.784 20.848 20.671 20.830 20.690 . R, 20.813 :l: 0.061 20.831 20.648 20.870 20.717 20.869 02 41.488 i 0.078 41.437 41.293 41.348 41.343 41.359 Ae 0.139 :l: 0.0089 0.1439 0.1441 0.1440 0.1461 0.1457 A, 0.1418 :l: 0.0075 0.1439 0.1537 0.1440 0.1523 0.1457 A53 0.0157 :l: 0.0028 0.0157 0.0157 0.0157 0.0162 0.0161 A53 0.0163 :1: 0.0016 0.0157 0.0157 0.0164 0.0162 0.0167 A53 0.0206 :1: 0.0023 0.0157 0.0168 0.0157 0.0169 0.0161 g,/g,, 0.9943 :l: 0.0065 1.0000 1.0000 1.0000 1.0000 1.0000 R5 0.2219 :l: 0.0017 0.2157 0.2178 0.2170 0.2170 0.2168 R6 0.1543 :l: 0.0074 0.1721 0.1716 0.1718 0.1718 0.1718 MW 80.26 :h 0.16 80.32 80.32 80.32 80.37 80.36 A“; 0.1551 :1: 0.0040 0.1439 0.1441 0.1440 0.1461 0.1457 179 Table 5.2: Experimental and predicted values of electroweak observables for the SM and the proposed model (with different choices of parameters) for a, = 0.115 with 'm.‘ = 175 GeV and my = 300 GeV. a: sin? 6 = 0, sin2 6 = 0.04, M3, = 1.3 TeV, 1", = 343 GeV. b: sin2 6 = 1, sin? 6 = 0.04, M’z = 2.1 TeV, I"Z = 562 GeV. (2'. sin2 6 = 0, sin? 63 = 0.80, M’z = 3.0 TeV, I" = 287 GeV. (1: sin2 B = 1, sin2 (6 = 0.80, M’Z = 4.5 TeV, I" = 430 GeV. [ Observables Experimental data SM The model K a b c d gv (8) —0.0368 :l: 0.0017 -0.0367 -0.0367 -0.0367 -0.0372 -0.0369 gA(e) -0.50115 2t 0.00052 -0.5009 -0.5012 -0.5012 -0.5005 -0.5009 gv(p)/gv(e) 1.01 :l: 0.14 1.00 1.00 1.02 1.00 1.02 gA(p)/g,,(e) 1.0000 :1: 0.0018 1.0000 1.0000 1.0015 1.0000 1.0016 gV(T) /gv(e) 1.008 :1: 0.071 1.000 1.053 1.000 1.047 1.000 gA(T) [9,,(6) 1.0007 :1: 0.0020 1.0000 1.0040 1.0000 1.0036 1.0000 I‘z 2.4963 :1: 0.0032 2.4922 2.4977 2.4943 2.4911 2.4917 Rc 20.797 :1: 0.058 20.716 20.761 20.733 20.762 20.736 Ru 20.796 :t 0.043 20.716 20.761 20.666 20.762 20.666 R, 20.813 :1: 0.061 20.762 20.631 20.779 20.649 20.782 0]), 41.488 :t 0.078 41.490 41.387 41.450 41.395 41.448 A, 0.139 :1: 0.0089 0.1439 0.1440 0.1440 0.1461 0.1449 A, 0.1418 :1: 0.0075 0.1449 0.1510 0.1440 0.1523 0.1449 A53 0.0157 :1: 0.0028 0.0157 0.0157 0.0157 0.0162 0.0159 A53 0.0163 :1: 0.0016 0.0157 0.0157 0.0160 0.0162 0.0162 A53 0.0206 :l: 0.0023 0.0157 0.0165 0.0157 0.0169 0.0159 g,/g,, 0.9943 :l: 0.0065 1.0000 1.0000 1.0000 1.0000 1.0000 R), 0.2219 i 0.0017 0.2157 0.2172 0.2163 0.2170 0.2163 R, 0.1543 in 0.0074 0.1721 0.1717 0.1720 0.1718 0.1720 MW 80.26 :l: 0.16 80.32 80.32 80.32 80.37 80.34 Chapter 6 Discussions and Conclusions Since the top quark is heavy, the top quark can be a window for new physics, either from top quark decays to new objects, or from large radiative corrections. Because of the heavy top quark mass, new physics will feel its presence easily and eventually may show up in the effective top quark couplings to the gauge bosons. Furthermore, since the top quark mass is of the order of the symmetry-breaking scale, the top quark is likely to provide useful hints about the symmetry-breaking mechanism responsible for generating the gauge boson masses and at least connected with the fermion mass generation mechanism. The main goal of this work is to browse through the low energy precision data from LEP and SLC searching for possible new physics effects dominantly in conjunc- tion with the top quark couplings to the gauge bosons. Constraining the nonstandard couplings of the top quark provides an estimate for possible deviation in the gauge universality advocated in the SM. Furthermore, if the deviation in the gauge univer- sality for the top quark case is due to the symmetry-breaking mechanism, then the measurement of and the correlation among the nonstandard couplings can be a direct probe to the symmetry—breaking mechanism. In chapter 3, I have applied the electroweak chiral Lagrangian to probe the non- 180 181 standard couplings of the top quark to the gauge bosons using precision LEP data. Assuming b-b-Z vertex is not modified, I found that nfc is already constrained to be -0.05 < Kfc < 0.17 (0.0 < K2“: < 0.15) at the 95% C.L. for a 160 (180) GeV top quark. Although K’A’C and KEG are allowed to be in the full range of :l:1, precision LEP data do impose some correlations among 76,70, ngc, and KEG. (ng does not contribute to the LEP observables of interest in the limit of m), = 0.) Inspired by the experimental fact p z 1, reflecting the existence of an approximate custodial symmetry, I proposed an effective model to relate 61,30 and 16550. I found that the nonuniversal interactions of the top quark to the gauge bosons are well constrained by LEP data, within 95% C.L. The constraints are summarized in Table 3.1 (see also NC Figures 3.6-3.10). Also, the two parameters KL = KNC and KR = K are strongly L R correlated where KL ~ 2K3. I note that the relations among K’s can be used to test different models of elec- troweak symmetry-breaking. For instance, a heavy SM Higgs boson (my > m,) will modify the couplings t-t-Z and t—b—W of a heavy top quark at the scale m, such that 16]? C = 2nfC, mg C = -n’,‘,’c, and KEG = 0. Another example is the effective model discussed in Ref. [95] where, REC = REC = 0. In this model the low energy precision data impose the relation KQ’C ~ K’A’C. Also, the simple commuting extended techni- color model presented in Ref. [59] predicts that the nonstandard top quark couplings are of the same order as the nonstandard bottom quark couplings. It is also interesting to note that the upper bound on the top quark mass can be raised from the SM bound m, < 200 GeV to as large as 300 GeV if new physics occurs. That is to say, if there is new physics associated with the top quark, it is possible that the top quark is heavier than what the SM predicts. Also, in chapter 3, I discussed how the present SLC measurement of Am can 182 contribute to the constraints imposed on the nonstandard couplings KQ’C, 5’50, and KEG at LEP. I found that if one uses the LEP constraints to predict the new physics contribution to the SLC measurement of Aug, then for the special model, 1|ch = 7650/2, it is possible to reconcile the LEP and SLC data at 95% C.L. Undoubtedly, direct detection of the top quark at the Tevatron, the LHC, and the LC is crucial to measuring the couplings of t-b—W and t-t-Z . At hadron colliders, KEG and xfic can be measured by studying the polarization of the W boson from top quark decay in tf events. They can also be measured simply from the production rate of the single top quark event. The LC is the best machine to measure rc’g’c and nfic which can be measured from studying the angular distribution and the polarization of the top quark produced in e’e+ collision. Details about these bounds were given in section 3.5. In chapter 4, I present a theoretical frame work to extract the pure m, corrections to low energy data in the chiral Lagrangian approach. I reproduced the results in chapter 3 by considering an effective Lagrangian which involves only the scalar sector (the unphysical Goldstone bosons and probably the Higgs boson), and the top and bottom quarks. I discussed how to relate the two different approaches presented in chapters 3 and 4. I showed that by considering a completely different set of Green’s functions (without involving any external gauge boson line) from that discussed in chapter 3, I recovered exactly the same result. The new frame work is useful and inter- esting because first, it simplifies the whole process of calculating radiative corrections, as it is much easier to work with sealers than with vector bosons. Also, this approach is shown to clearly identify observables which are sensitive to the symmetry-breaking sector of the electroweak theory. This is clear since only contributions independent of the gauge structure survives. In chapter 5, I present a self-contained model which demonstrates how the non- 183 standard top quark couplings to gauge bosons can be generated. The model has a very rich structure and significant implications at low and high energy scales. Using the low energy data I discuss the possible constraints on the model. On the other hand, the high energy colliders will provide further tests and demonstrate possible new physics especially interesting FCNC processes. Appendix A Renormalization Schemes In chapter 1, I discuss in details the Z-pole renormalization scheme and very briefly mention few other schemes. In this appendix, I discuss to some extent different renormalization schemes and possible relations among these schemes. As discussed in chapter 1, to fix the low energy part of the SM one needs to specify three input quan- tities, the light fermion masses, and quark mixing. Different choices correspond to different renormalization schemes. The most common used renormalization schemes include, the Z -pole scheme, the on-shell scheme, and the MS scheme. At tree level the weak mixing angle can be written in different equivalent ways, 1 47m 1’2 M2 9’2 sin200=— 1—[1———9——] =1— W0: 0 . A.1 2 ( fiGFOMgo M20 902 + 96 ( ) Once radiative corrections are included all these definitions are are not satisfied si- multaneously. One has to pick one of these definition for any specific renormalization scheme. 0 The Z-pole scheme. In this scheme, the input observables are chosen to be: ' — The electromagnetic coupling a = 82/477 measured from electron-proton 184 185 (e-p) scattering in the limit of zero momentum transfer q2 —) 0 (Thomson limit) [9] 0‘1 = 137.0359895(61). (A.2) — The Fermi coupling constant 6'; measured from the muon lifetime 7'” [9] 06 = 1.166389(22) x 10-5 GeV'2. (A.3) - The Z mass [12] M2 = 91.1885 :t 0.0022 GeV. (A4) The Z-pole scheme is simple and precise because the input parameters are measured very well. This scheme is suitable in studying physical observables at the Z pole. As discussed in chapter 1, in analysing the Z -pole physics the pure QED corrections are treated separately from the weak corrections. In particular, the light fermions contributions to the photon vacuum polarization function are absorbed in defining the running coupling a(M§) at the M 2 scale. In chapter 1, we found 6(Mg) = 1_ A:(M§) . (A.5) where, Aa(M§) is defined as Aa(M§) = —F77(M§) + 1877(0). (A.6) Currently, there is a lively debate on what value of a(M§) to use. I quote the value [18] a-1(M§) = 128.89 a; 0.09, (A7) F; 186 In this scheme, the main theoretical error propagating into the definitions of the SM parameters is coming from the error in determining a(M§). In this scheme, the weak mixing angle is defined to all orders by the relation . 1 47ra(M2) 1’2 2 _ 2 _ Z The weak mixing angle sin? 6 is well determined theoretically [60], sin2 6 = 0.2312 :1: 0.0003. (A.9) The main theoretical error in sin26 is coming from the theoretical error in determining the running coupling a(M§). In fact by only considering the error in a(M§), there is an induced error in (isin2 6 6 sin2 0 Tsin 0 ~0.1%, (A-IO) which amounts to most of the error in sin2 0. The Z -pole scheme is defined such that all top quark and Higgs boson contribu- tions are removed from the parameters a(M§) and sin2 0. The top quark and Higgs boson contributions enter when considering other predicted observables (see chapter 1), e.g., the W mass, the partial decay widths, etc. The on-shell scheme. In this scheme, the input observables are taken to be a, M z, and Mw. In the on—shell scheme, the weak mixing angle has a simple definition. To distinguish different schemes I will denote sin20 in the on-shell scheme by 33 where, 83 is defined to all orders by the relation £438 2 -1— . M6 30- (A.11) 187 Unfortunately, the W mass is not determined precisely as M z [13] MW = 80.26 :1: 0.16 GeV. (A.12) The large error in the measurement of MW induces a large error in calculating 33 using the defining relation [29], 2 2 (£321) 836%!) 81.6%, (11.13) exp exp so 83 Ma, where I ignored the small error in M 2. Therefore, one does not rely to the mea- surement of MW to extract 33. The usual approach is to use the measurement of the p decay and the theoretical formula 1 6367712 mg a 25 2 2a m“ 7.." 192n3(1—8m2 (1+;(T’"))(1+§171”E)’ (“4’ p where QED corrections to the four-fermion interaction includes one loop cor- rection and the leading correction in 02. Performing the l-loop radiative cor- rections to the p decay (see Figure 1.4b) one finds that Gp/fi coincides with the expression [14] g: _ 77070 1 + AWW(0) fl — 2311210114150 M3,, where I used the bare parameters in the above equation and the term (ver- + (vertex, box)[ , (A.15) tex,box) denote corrections other than the W-self energy, i.e., due to vertex, box, and fermion Self energy diagrams. In fact these corrections are in corre- spondence with what I called 6G“; in chapter 1, 60KB _ W00 fl - 23729011450 The bare quantities can be written in terms of the renormalized quantities as (vertex, box) . (A.16) follow em = 01 — (la , (A.17) 188 M3,, = M; — mg, (A.18) M3,, = ME, —— (SMEV. (A.19) Therefore, one finds M2 6M2 6M2 82 =1- W°=62—cz(—£——‘l), (A20) °° M2. 9 9 M2 M6 where to all orders we have M2 Thus, one finds 95 _ 1m [113 _c_g mg ma, ‘5 _ 233M3V a 33 Mg M3,, 6 2 WW + MW + J: (0) + (vertex, box)] Mw 77a where, Ar is a finite combination of one-loop diagrams and counterterms. It is clear that Ar depends on the top quark mass m, and the Higgs boson mass my. By including and summation of higher order corrections [14], one can write the above relation as 2 2 2 2 _ (1 _M_w) MW _ 7‘" (A23) 3 c — — — — . 9 0 Mg 114% «20,6430 - Ar) One should recognize that the quantity AT is different in two aspects from the quantity Arw I defined in chapter 1. First, in calculating Ar one should use the on-shell quantities 83 and 03 rather than sin2 0 and cos2 0 defined in the Z-pole scheme. Second, in Ar, the electromagnetic a is still defined at q2 = 0 rather than at q2 = Mg. Thus, Ar contains QED corrections from the light fermions, 189 Aa(M§) as discussed in section 1.2. In fact, for the case of AT one can absorb the quantity Aa(M§) in defining a(M§) as we did for the Z-pole scheme. One can split the different contributions to AT at one loop and write AT as [14] 2 6% A 2 Ar = Aa(MZ) - 3A6 + (A1")mmainder , ( . 4) where by Ap I mean the leading contributions of the top quark mass (quadratic in mg). Therefore, this Ap coincide with the quantity Ap I defined in chapter 1, as long as one only concentrates on the quadratic terms in m,. The term (Ar)wminder includes the remaining contributions, e. g., the logarithmic depen- dence in m. and m”, gauge bosons contributions, etc. The typical sizes of Aa(M§), Ap, and (A1“)reminder are ~ 0.06, 0.03-0.05, and 0.01, respectively [14]. The measurement of 83 is usually extracted using the measured ratio MeV/Mg from low energy experiments like the neutrino-electron scattering. The present world average on 83 from experimental data is [14] M2 s: = 1 - 34—”; = 0.2253 :1: 0.0047. (A.25) Despite the simple definition of 33 in the on-shell renormalization scheme, the weak mixing angle 3% has a strong dependence on the top quark mass m,. Hence, using this scheme in any analysis requires a precise knowledge of the top quark mass. The MS scheme. The ITS scheme, also known as the modified minimal subtraction scheme, is a well-known scheme in QCD physics. In this scheme, one simply, in doing the radiative corrections, only requires that the counterterms contain the di- vergent pieces needed to cancel divergencies arising from loop calculations. In 190 other words, one only absorbs the loop divergencies in the counterterms. In dimensional regularization one simply cancels the quantity A=—nE4—7—ln47r, (A26) from the loop calculations, where n is the space-time dimension and 7 = 0.577 . . . is the Euler’s constant. Therefore, masses and couplings calculated in the MS scheme have a dependence on the renormalization scale 6. Consequently masses in the NTS scheme (running masses) have no direct physical meaning. To distinguish the MS quantities I will use a bar over the quantities. The bare quantities can be written as a0 = 6 — 66 , (A27) M3, = m — 5m, (A28) MEN = 1'77“, — 6%, (A29) The coupling ‘a' is defined using the e-p scattering at q2 = 0. From the result in chapter lwe know that the quantity 0 is given by A 2sin0 A72 (0) = _ 77 _ a do [1 F (0) c030 Mg (A.30) Writing an in terms of the MS renormalized quantity we find ._ 2.510 ALzs(0) a = a [1— FA147§(0) '— 3W , (A31) where F% is defined as F7“1 with simply subtracting the divergent piece A defined in Eq. (A.26). Similarly for the quantity ALAS" Also, 83 is sin20, defined below, in the —M_S scheme and 2% = 1 — 83. Therefore, one finds (A.32) 191 Choosing the renormalization scale to be p = M 2, one can evaluate the running coupling Zr" at the M 2 scale. Notice that the top quark mass enter the quantity A6. However, the dependence is only logarithmic. Numerically, one finds, for m, = 180 :l: 12 GeV, [14] (6)" = 128.08 a: 0.02 i 0.09, (A33) where the first error corresponds to m, = 180 GeV and the second error to the interval :l:12 GeV around the central value of the top quark mass. The renormalized masses in the VS— scheme do not correspond to the physical masses. In chapter 1, we found that summing loop corrections leads to the result ——1—— —+ l (A 34) (12 - M30 (1"- - M§o + AZZ(0) + (PFZZW) ' ' Writing the bare mass in terms of the renormalized FITS- mass M3, = W — 5X43, (A.35) one finds l 1 _, (A.36) q2 — M§o q2 — W + A%(0) + q2F—3—E ((12) ’ where the counterterm 6M3- has been chosen to cancel the divergence in Azz (O)+ MgFZZ(M§). The on-shell condition relate the on-shell mass to the M8 mass as follow, q2 - W + Ag—ZS-(O) + q2F§§(q2) = 0, at q2 = M3. (A.37) Therefore, one has Mg - A7? + A-ngm) + MgFg—gmg) = 0. (A38) 192 Thus, for the W and Z mass we have the relations between the on-shell and the H8 schemes M3. = $73 - A-f.—Z(o )— MzFZ—§(M§> . (A39) M3,, = V3; — Algae) — Map—gyms,» . (A40) The weak mixing angle can be defined in different ways, I choose the definition One can relate the weak mixing angle in the on—shell scheme to the one defined in the MS scheme as follow M2 W0( ) AQW) —2 _ w ww 22 s, _ 1—— Mg 1+-%%-+F,TS (M3,)— 33% — figmg) = 83 + ngfis’ , (A.42) where I defined AZ__Z_(0) AWW(0) X-fig E —L—’X4— ——§2—"1‘W+ F-g§2(Mz)— Fg—YS‘WMEV ). (A.43) z W Also, we have 63 =1—sg =c3(1—X-M—S). (A44) The on-shell gauge boson masses can be related using the IVE quantity 8% 2_ 2% _1_ X—l—M—SMZ= _ cgpMZ, (A.45) where I defined 1 1— fig. 25 = (A46) 193 The weak mixing angle 83 can also be related to the Z-pole scheme by calcu- lating the p decay in terms of the IVE quantities. We found earlier AWW M124» £5 —. ___—flag 1 fl — 2330M12vo + (vertex, box)] . (A.47) Writing the bare quantities in terms of the 171—S- renormalized quantities on finds Gr NZ!- AEWW(0) ] where 6(V, B)‘M_s' is all the one-loop corrections to the p decay except the W self-energy calculated in the ITS- scheme. Using Eq. (A45), one finds G 775 I 7% = W [1 — X-fig - F;—}S—‘V(M3,,) + 5(v, mm] . (A49) Summing higher order corrections yields [14] GP 7rEi 1 75 - 24,2, 83 mg 1 _ A? 4 (“0) where 1 7,- : _, (A51) 1 — XE A? = Jaws-Hwy) + 5(v, mm. (A52) For more details on the M_S scheme, the reader can refer to the discussion in Refs. [14, 29]. Appendix B The S, T, U parameters In this appendix, I present briefly a well-known parameterization for the low energy data different from the epsilon parameterization. For more details the reader can refer to Refs. [24, 25]. The parameterization I will discuss in this appendix is similar in spirit to the epsilon parameterization. First, the assumption made are [25] o The electroweak gauge group of the effective low energy theory at the weak scale is the standard 5 U (2);, x U (1)y . Therefore, the only relevant electroweak gauge bosons are the photon, W*, and Z. 0 The only relevant new physics to consider are the oblique corrections, i.e., the corrections to the gauge boson vacuum polarization functions, with the excep- tion of the non-oblique correction to Z ~b—5 vertex. 0 The earliest use of this parameterization [24], was based on a third assump- tion, namely, the new physics scale is large compared to the W and Z masses. Therefore, one can expand new physics contributions to the gauge boson vac- uum polarization functions around (12 = 0. Recent efforts have been toward retaining more higher order terms in the expansion of the vacuum polarization 194 195 functions [25]. I start the discussion concentrating on the oblique corrections. The parameters S , T, and U are chosen to parameterize the oblique new physics corrections. The convention used is to subtract the new physics effects from the SM contributions and then define the S, T, and U parameters in term of the remnant new physics effect. To do this, one must know the top quark and the Higgs boson masses. However, since our knowledge of theses masses is not precisely established, one simply picks reference masses for the top and the Higgs boson masses. Then one calculates the SM prediction using these reference masses and define the new physics contributions to low energy observables as the difference between the experimental data and the SM predictions (using reference 711, and m H). The third assumption mentioned above allows one to expand the vacuum polariza- tion functions around q2 = 0 in powers of q2/A2, where A is the scale of new physics. Using my notation for the vacuum polarization functions in Eq. (1.49) and the Z -pole renormalization scheme, the S, T, and U parameters are defined as follow 05 = 4sin0cosOF30(O) , (B.l) aU = 4sin20 [1833(0) - FWW(0)] , (3.3) where 1830(0) = — sinocosaFZZm) + (c0320 - sin20)F7Z(O) + 1777(0), (3.4) and F33(0) = cos2 0 Fzz(0) + 2 sin 9 cos 0F7Z(0) + sin? 6 1777(0) . (B.5) 196 Using the latest data [12], a fit for the S, T, and U parameters yields [25] s = -0.33:1:0.19, T = —0.17:i:0.21, U = —0.34:{:0.51. (3.6) The above fit is extracted for m, = 180 GeV, my = 300 GeV, 01‘1 (Mg) = 128.9, and a, = 0.123 [25]. Recent efforts have been implemented to relax the third assumption by allowing the new physics scale to be comparable with the Z —mass scale [25, 70]. In this case, one retains more higher order terms in the expansion of the vacuum polarization functions. It was shown in Ref. [70] that it is sufficient to introduce three mere parameters and increase the total number to six. The definition of S and U, but not T, is slightly modified as follows [70] 05 = 4sin0 cos9F3°(0) + 4 sin2 0 cos.2 9 [FZZ(M§) - 1722(0)] , (3.7) AZZ(0) AWW(0) aT — Mg -— MEV , (B.8) aU = 441112 0 [1833(0) — FWW(0)] + 4sin2 0 FFWW(M3V) — FWW(O)] + 49,1112 a 'FZZ(Mg) - F” (0)] , (3.9) The S, T, and U parameters are in one to one correspondence with the oblique corrections e3, 31, and 62 defined in chapter 1 [see Eqs. (l.117)-(1.119)]. The quanti- ties 63, 81, and 82 constitute the leading contributions to the parameters 63, 61, and 62. Therefore, concentrating on the possible large new physics contributions one can 197 relate the two set of parameters as follows - 2 S = 4srn 0563 , a T = 1661, a . 2 ' U = —481:05€2, (8.10) where, 661 is the new physics (beyond the SM) contribution to 61. Similarly, for 652 and 663. Parameterizing the non-oblique correction to the Z—b—E vertex has been imple- mented recently [25, 119]. In Ref. [119] the non-oblique corrections from new physics to the Z -b—5 vertex are expressed in terms of the effective left- and right-handed couplings of the b quark to the Z boson as 1 . 9'11: (9(1),)SM + 56sm20 + 6gi, (BM) 1 . g’}, = (95:95., + 56s1n20 + 5513,, (8.12) where 5 sin2 0 expresses the shift of the effective weak mixing angle due to oblique cor- rections, and 6g}: and 59% express non-oblique vertex corrections due to new physics. The quantities (92)“ and (g§)SM are the left- and right-handed couplings of the b quark to the Z boson in the SM, where . 1 1 . (938M = —§+§Sm29, (919w = Eli-sin”. (3.13) The decay width 1"), depends on one combination of 692 and 69’}; . While the observable A383 depends on another combination. This parameterization is similar to the one I used in chapter 1 (see Eq. (1.123)). The relation between the cases is simply as follow 1 6 6 b (68+ £1de 9A4) , 198 591;! = _% (+5QVd'2-59Ad) , (3.14) where one should remember that, in the above equation, eb, 15ng, and 69,“ contain only the new physics effect defined as the difference between new physics and the SM case using reference m, and my. Using the recent data [12], excluding RC, one obtains the fit [25] 69'}, = 0.0004i0.0037, 69'}, 0.036 a: 0.017, (3.15) where m, = 180 GeV, my = 300 GeV, a'l(M§) = 128.9, and a, = 0.123 [25]. The conclusion is that, one can express new physics effects in terms of the S, T, U, and the two non-oblique parameters 69'}, and 6932. Notice that 69';2 does not involve large m, effects since the large m, effects are coming at one-loop through the left-handed W—t-b coupling. From this brief discussion one notes some differences between the epsilon param- eters and the parameterization discussed in this chapter. 0 In the epsilon parameterization, the parameters 61, £2, £3, and q, are written directly in terms of physical observables, rather than in terms of two—point functions which is the case for S, T, and U. o The parameters 61, 62, £3, and q, incorporate the SM contribution in addition to possible new physics. On the contrary, the other parameterization only contains the new physics effect. Appendix C Heavy Mass Expansion In this appendix I discuss how to use the heavy mass expansion and tabulate some relevant integrals if there is a heavy mass in the loop calculation [120]. Loop integrals will be performed using dimensional regularization. A general two-point function has the typical integral [(11% kph” . .. ((3.1) 2w)" (k2 — m2>< m and M >> p2. The heavy propagator is expanded as follow l __ 1 2k.p + p2 ((k+p)2—M2) ’ (kz—Mz) (l—W+...). (C.2) For a three-point function a similar treatment can be done. In this case one encounters the typical integral / d"k kyle”... ((3,3) (2%)“ (k2 - mi)((k + p)2 - m%)((k +1) + <1)2 - M2)’ where m1 and m2 are two light mass scales and M >> 171;, mg. The external momen- tums p and q satisfy the condition M 2 >> 102, q2. The heavy propagator is expanded 199 200 as follow (0.4) 102—M2 1 _ 1 1_2k.p+2k.q+(p+q)2+ (k+p+q)§iM2-k2-M2 . Determining the order of the expansion depends on the desired heavy mass M dependence to be retained. For example, to keep contributions up to In M 2 needs more expanded terms than keeping terms up to M 2 only. To determine this consider the general integral / d4k Mdkpku... ( 210‘ (k2 — mam: +107— mamk +1» + q)? - M2) . .. 1 (05) where dis an integer and where the dependence on the heavy mass M in the numerator can come from propagators as in the fermions’ propagators or it can come from couplings as in the Yukawa couplings to a heavy fermion (top quark). To determine the number of propagators to retain I will use the following quantities. Denote the number of propagators to be retained by the integer Np. The number of explicit momentums in the numerator by Nm. The desired power of the heavy mass M to be retained by N M. Then the number of retained propagators is fixed by counting the dimension of the integral. The total dimension of the integral is 4 + d + Nn - 2N? where the term 4 is coming from the integral measure (We. therefore the following inequality must be satisfied 4+d+Nm—2NPZNM, (C.6) After doing the expansion, one can use the identity (k2 — m2)(k2 _ M2) = M2 _ m2 ((k2 _ M2) - W) - ((3.7) Then using the integrals below becomes straight forward. 201 Here I tabulate some useful integrals in the heavy mass expansion and using dimensional regularization. First I define 2 A—-n_4—7—ln41r, (C.8) where n is the space-time dimension and 7 = 0.577 . .. is the Euler’s constant. Also I define the quantity gym, gpuAa = guugAa + gpkgua + gyagfl (C9) d”k 1 i 2 f(21r)" [:2 — A42 _ _1671’2 (—A _ 1 + 111M) 1 (C.10) (Pl: 1 i 2 j (270" (k2 - M2)2 = _167r2 ("A + 1" M ) 1 (0.11) d"k 1 i 1 (20)" (k2 — M2)3 = _161r22W’ ((3.12) fag): (k2 —1M2)4 = @3735}le (C.13) (3'3" .2 5342 = / (if; £3533. = = 0, (C14) / (:1): 1:31:42 = _16in29flufl4 ("A ' 3 +1“ MI) 1 (C-15) / (grin “£12,452 = "101.42 9""2M 2 (—A —1+lnM2) , (C.16) / ((21:3" (1,2,3212)?‘ = -f617}'2'gZ,—V (_A + 1n M2) 1 (C.17) / d"lc lanky (2a)" (k2 — M2)4 / d"k kflkukxk, _ (217)" k2 — M2 — I d"k kpkukxk, (210" (k:2 — M2)2 / d"k kpkukik, (2%)“ (k2 _ M2)3 (1% kpkukik, (20)" (k2 _ M2)4 202 = .. i _giL. 167r212M2’ i gpuAaMG (_ _ E 2) 167r2 24 A 6 +lnIll , ' 4 ’ 9"”"°M (—A —-:-+1an) . —167r2 8 _ 2 guuAaM2 1671’2 8 i M(—A+lnM2). = _167r2 24 (—A—1+lnM2). (C.18) (C.19) (C20) (C21) (C22) Appendix D Non-Linear Realization In this appendix, we are interested in realizing the symmetry described by the Lie group G. The symmetry of the theory is assumed to be spontaneously broken, i.e., we consider the breakdown of G into a subgroup H. Given the symmetry group G (global or local) and the matter fields (leptons and quarks), the problem of constructing the invariant Lagrangian (under G) reduces to the problem of realizing the symmetry, i.e., choosing representations of the Lie group G. The realization of the symmetry can be achieved in two different ways; the familiar linear realization and the less familiar non-linear realization. D.1 Linear Realization In this realization, the problem is choosing the natural representations of the fields. By natural representation I mean representations which form a self-consistent the- ory, i.e., free from of all possible anomalies. Usually the matter fields (fermions) are assigned in the fundamental representation of the group G. On the other hand, the gauge bosons (the force mediators) are assigned in the adjoint representation. To illustrate, consider the Lie group SU(2)L x U (1)y (the SM group). Left-handed fermions \IIL are assigned in doublets (the fundamental representation) under 3 U (2) L 203 204 with the transformation under G as ‘I’L ‘1 ‘VL = QLQY ‘I’L, (D-i) where 9,, = exp(i . ) e SU(2)L, (D2) 9,, = 9444333) e 0(1),. (113) and a“, a = 1,2,3 and y are real parameters of the group G. The right-handed fermions are singlets under SU(2)], The SU(2)], gauge bosons are assigned in the adjoint representation with the transformation gWST“ —-> gWS'T“ = 91, (W373 — a”) g[ . (D4) The U (1)y gauge boson B transforms as g’YBp -+ g’YBL = 9v (YB): - 31091 , ' (13.5) where g and g’ are the gauge couplings of S U (2),, and U(1)y, respectively. D.2 Non-linear Realization A different realization of the symmetry from the linear realization can be imple- mented, namely, the non-linear realization. The starting point is the spontaneous breakdown of G into H. The mathematical situation is as follows; consider a real analytic manifold M, together with a Lie group G. Under the action of G, an element a: in M transforms as x-—)g:r xEM, gEG. (D.6) 205 We assume that there is a special point of M called the origin, described by the null vector 0. The origin 0 is invariant under the action of a subgroup H of G: h0=0, heH. (DJ) The physical situation is that of a manifold of scalar fields with the origin describing the vacuum configuration. The group G is the global or local invariance group of the theory and the subgroup H is the invariance group of the vacuum. Thus, one is dealing with the spontaneous breaking of G into H [72]. An important point is that for any non-linear realization of the fields, we can redefine the fields in such away that their transformation under H is linear [66], we call this form, in which the fields transform non-linearly under G and linearly under H, the standard form. Furthermore, the standard form as discussed in Ref. [81] still leads to the same S matrix as the original realization. The group G has n generators Z“, a = 1,2,...,n and the subgroup H has p generators V", b = 1,2, . . . , p. The symmetry described by G breaks down into the symmetry described by H. Therefore, there are n —- p broken generators A", c = 1, 2,. . . , n - p. From the spontaneous breakdown of the theory as described by the breakdown of G into H, there are n - p Goldstone bosons [80]. The Goldstone bosons 05°, c = 1, 2, . . . , n - p form the coordinates of the manifold M. Each Goldstone boson 45" is associated with a broken generators A“. Any element 9 E G can be uniquely written as [66] ia' A' eiB‘V‘ 9:. , ”1,2,...,n—p, 4:1,2,...,p, (0.8) where a“ and ,6” are real numbers. We associate with every broken generator A“ a coordinate 45" with the transformation '6: '00'66'00 -aIa-bb gent/1 =ezaAetflVer¢A =ei¢ .48qu . (13.9) 206 Notice that the object eff. A. 6 G, therefore, the transformed object ge‘f’. A. E G, this is why we wrote the result of the transformation in the above form. The transformar tion of ()5 written as ¢°($) -> d>’°(d>(:r)), (D.10) constitutes a non-linear representation of G. The above equation is taken as the definition of the transformed fields 05’ in terms of the original fields 05 and the group element g. Matter fields 2p (quarks and leptons) are required to possess definite transformation properties only with respect to element h E H, i.e., \II —> \P' = D(h)\II, (D.11) where D(h) denotes a certain linear representation of H. Therefore, the non-linear realization of the whole group G is established as 4° —+ ¢“’(¢), q; —+ 00011:. (0.12) In general, even in the case of a global transformation one is forced to introduce covariant derivatives for the matter fields, because of the a: dependence in d) (see Eq. (D.10). In the case of local transformations with gauge bosons W“, a = 1, 2,. . . , n and with their transformation under G as 0;. = 9999’ — 90,19’. I (0.13) where (2,, = gawgza, (p.14) 207 one defines the operator [66] L" = c-m.” (0,, - Qp)e‘°"4' = WZA“ + szb , (D.15) where w]: transforms as a gauge boson under the subgroup H. On the other hand 7r: transforms as a matter field (not as a gauge boson). The covariant derivative of the fermion field \II is defined as 0,0; = (a, + wgvbw, ' (D.16) with the transformation D,,\II —-> D(h)D,,\I1. (D.17) To proceed I will consider a specific example, namely the electroweak group G=SU(2)L x U (1)y and H=U(1)em. There are 3 broken generators associated with 3 Goldstone bosons (0“, a = 1, 2, 3. The generators of G are the Pauli matrices 7'“ / 2, a = 1,2, 3 and the hypercharge generator Y/ 2. The U (1)em generator can be taken as 7'3 Y V—Q—3+E. (13.18) The broken generators can be taken as TO A“ = '2— . (13.19) Rather than working directly with the complicated transformation of the scalar fields (1)“, we define the object 2: = e‘LZF', (D20) 208 where the parameters 1)“ are introduced for dimensional purposes. Under a general transformation (9 6 G), we have (see Eq. (D.9)) u I 3 L1'_ ,‘uLJ‘l 2 -) g}: = e‘ v e 2 , (D21) where g = e‘°'%em¥ . (D22) We can write the above transformation as - .r' - r3 - Y e'“ 72 = E'e’“'2'e'(“'m‘2’ . (D23) Since the left-hand side of the above equation is an element in S U (2) 1,, it follows that the right-hand side of the equation must be independent of the hypercharge generator Y. Thus, we conclude that u = fl and the transformation of 2 reduces to . _,- . ,3 cm 72 = 276*"? (D24) One can write the above transformation as . “a . ,3 2’ = 8'“ TZe“’3T = nggL, (D25) where , . fa , r3 gL = em T a 9R = 6w? - (D.26) It is clear that under G=SU(2)L x U (1),» , the field 03“ has a complicated transfor- mation, while, under the subgroup H, the field <25“ transforms linearly. For h E H and h = 856(r3+Y)/2 he“. A. = e‘fflA' h . (D27) 209 Thus, 6545"» = h 6.6.21- h] = eih¢'A‘h’ . (D28) From which it is clear that, for G=S U (2) L x U (1)y , the transformation 45616 _, ¢aITa = h¢aTaht = gR¢aTagL (D29) constitutes a linear transformation. We also find that 7r; = EIDpE (D.30) transforms as a matter field under G, namely 7r: —+ gmrfig]z , ‘ (D.31) where the covariant derivative D”): is a 3 0,23 = a”): — igwg-Tz—z: + £923.15. (D.32) Under G, the covariant derivative transforms as 73 2 D”): —+ (D,,E)' = exp (2,027“) DpE exp(—z'y ) . (D.33) A gauge invariant (under G) can be easily seen in the following term ivz'rfinpzfipfiz) . (D34) This terms can be shown to give the gauge bosons their masses. In the unitary gauge, this term reduces to 1 2 t 1 2 07'“ , 73 2 21) 'I‘I'(D”2 D”2) —) Z7.) fi(l-9Wp'-2—+gBy-2-'] 1 = § 122 (92W:W”“ — 2gg'WgB" + g’2BpB") . (D.35) 210 Using the field definitions in Eqs. (1.11) and (1.12) one recovers the gauge boson masses given in Eq. (1.17) 2v2 921,2 + p— g p TWPW + 80032022 . (D.36) Thus, the W and Z masses are 22 22 2...9_”_ 2=_fl_ MW— 4 , MZ “0826. (D37) The difference between the non-linear and the linear realizations is that in a general gauge, the non-linear realization produces other complicated terms in powers of the Goldstone bosons. In general, one finds 2 1 5,, = M3,, WjW’” + 542—2 2,2" + ay¢+a"¢‘ + iap¢3aw3 + . .. , (D38) where the fields (2* are defined as 4:51 =1: 2'45“ i - ¢ _ fl 0 (D039) Fermions can be included in the context of the chiral Lagrangian by assuming that they transform under G: SU(2)L X U(1),» as [74] f -> f’ = e‘”°f. (D40) where Q is the electromagnetic charge generator. One finds that the fermionic co- variant derivative is Dpf = (0,, +w,,Q) f, (D.41) where w“ = B” . (D.42) 211 Thus surprisingly, the gauge field is the B field and not the photon field A as one would expect. Out of the fermion fields f1, f2, with the condition Q I, — Q I, = 1, and the Goldstone bosons matrix field 23 the usual linearly realized fields \11 can be constructed. For example, the left—handed fermions [S U (2) L doublet] are constructed as __ _ f1 \IIL—EFL —E . (DAB) f 2 L One can easily show that \IIL transforms under G linearly as follows a 3 3 \IIL -) \II'L = exp 0021'“) 2 exp (—iy%—) exp (iy [T— + {D FL 2 2 027'“) exp(iy2/2—) \IIL. (D.44) = exp (2' Therefore, under the group G=SU(2)L x U (1)y , ‘11,, —) \II’L = g‘IIL. (D.45) In constructing the invariant Lagrangian, one can define the composite fields as 2° = —%Tr('r“EID,,E). (D46) P Under the gauge transformation element 9 E G and using Eqs. (D25) and (D33), one finds that the composite fields transform as: a i , 7'3 a , 7’3 2: -) 2;, = —-2-Tr (exp(—2y7)7 exp(zy§)Ble2) . (D.47) From which one concludes that under a general gauge transformation 3 3 _ 3 2,, —> 23', _ 2 (D48) p and 2:: -+ 23': = eiivzf, (D49) 212 where 1 . D: = Wm}, $123,). (D.50) The field 2?, behaves as a neutral matter field while the two fields 2: behave as charged matter fields with Q = :t1. By a matter field I mean a field which does not transform as a gauge boson field under the symmetry group H. In a general gauge, 22 can be expanded as 1 1 9 2'9 __ _ 2:: = 175 ”(253-20050 ”—E-(W’fcf) _W" ¢+) —% (¢+8,,¢‘ — ¢’8,,¢+) + . .. (D51) The composite field 2: can be expanded as 1 1 2'9 2+ = 'v—26u¢+ —§gW: — p ; (60+ [cos 02,, + sin 024”] — ¢3W:) 2' +5 (¢+a,,¢3 — 636,61“) + . .. 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