5-3, i: w. s 1... . 4.. inuni...‘ .. z .t "2.3.2... 1h AME... ‘ . ‘ 3 . Eli. Viv-.1. x. 55:3... .1 340.136: . 51 .c 1...: 35.13.5335... 21.25.. ... 26 . c; .13 v. 4):: .x . a..- i V ‘. . flaky: . , ; .. I..I ._ fiéixfifii . ; “3...; ,. ._ 1...; 01' 13"}: .33. a . .x; Doc} This is to certify that the dissertation entitled HIGGSLESS ELECTROWEAK SYMMETRY BREAKING FROM THEORY SPACE presented by ROSHAN FOADI has been accepted towards fulfillment of the requirements for the Ph.D. degree in Physics M7€jfil¥9€ Major Professor’s Signature 22, Z 005 Date MSU is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University -.-.—u—.-.a—u-o-i-—--—.-----v--u-o-u--o--u--n-o--n-u-u-n-c-n— PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/DateDuepGS-sz HIGGSLESS ELECTROWEAK SYMMETRY BREAKING FROM' THEORY SPACE By Roshan Foadi 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 2006 ABSTRACT HIGGSLESS ELECTROWEAK SYMMETRY BREAKING FROM THEORY SPACE By Roshan Foadi We propose a Higgsless model of electroweak symmetry breaking, with inspiration from the physics of one compactified extra-dimension. The gauge sector consists of an SU(2) Yang-Mills theory on an extra-dimensional interval, with boundary con- ditions breaking SU(2) to U(1) at one end, and large brane kinetic terms on both boundaries. Exchanges of Kaluza—Klein modes are shown to postpone the unitarity violation of longitudinal gauge boson scattering amplitudes to energy scales higher than the customary limits of Dicus-Mathur or Lee-Quigg-Thacker. Fermions are first implemented into the model as brane-localized fields, and then as bulk fields with large brane kinetic terms on both boundaries. Only in the latter case can unitarity and precision electroweak constraints coexist, as long as the amounts of leakage into the bulk of the Standard Model gauge bosons and the Standard Model left-handed fermions are properly related. In order to achieve a realistic top mass without violat- ing unitarity of gauge boson scattering amplitudes, the gauge-sector compactification radius, R9, and the fermion-sector compactification radius, R f, are made indepen- dent by breaking the five-dimensional Lorentz invariance. Unitarity and experimental constraints are shown to impose, respectively, upper and lower bounds on l/Rg and 1/12,. I dedicate this thesis to my parents, Amalia and Hushyar, and to my brothers and sisters. iii ACKNOWLEDGMENTS I would like to thank my thesis advisor, Prof. Carl Schmidt, whose support and advice have helped me to succesfully complete my research projects during my years at Michigan State University. I am especially grateful to him for teaching me how to focus on the important aspects of the research in theoretical physics, a field in which it is often easy to waste time on unessential technicalities. I am also grateful to Prof. R. Sekhar Chivukula. Since his arrival to Michigan State, in the Fall of 2003, he has been for me an endless source of knowledge and advice. I am grateful to the Physics and Astronomy Department, and to the professors of the High Energy Physics Theory Group for their financial support. I also want to thank the Chairperson for the Graduate Programs, Prof. Bhanu Mahanti, for his advice, and the members of my thesis guidance committee, Prof. Kirsten Tollefson, Prof. Vladimir Zelevinsky, and Prof. Carlo Piermarocchi, for useful guidance. I am grateful to Prof. Scott Pratt for his participation to my thesis defense. I would like to thank Prof. Wayne Repko, Prof. Wu—Ki Tung, and Prof. C.-P. Yuan for their advice, and Prof. Kaustubh Agashe, from Syracuse University, for use- ful discussions. I am further grateful to our current and former post-docs: Alexander S. Belyaev, Shrihari Gopalakrishna, Yu Jia, Daisuke Nomura, and Kazuhiro Tobe for illuminating discussions; to my graduate students colleagues: Aous Abdo, Jorge Benitez, Qing-Hong Cao, Chuan-Ren Chen, Stefano DiChiara, Hyeonngan Kim and Baradhwaj Panayancheri Coleppa, for the happy time in the office. I also want to thank Thomas Rockwell and the members of the departmental Computer Center for their technical support. I am sincerely grateful to Debbie Simmons and Brenda Wenzlick for their help and kindness. iv Contents LIST OF FIGURES ............................. vii 1 Introduction 1 2 Unitarity in the Standard Model 14 2.1 Partial Wave Expansion ......................... 14 2.2 Longitudinal Gauge Boson Scattering in the Standard Model ..... 17 2.3 Unitarity and the Equivalence Theorem ................. 22 3 Unitarity without the Higgs Boson 26 3.1 Compactified Extra-Dimension ...................... 27 3.2 Symmetry Breaking by Boundary Conditions .............. 30 3.3 Unitarity in Extra-Dimension ...... . ................ 3 7 3.4 Unitarity and the KK Equivalence Theorem . . ............. 41 3.5 Deconstructed Models ...... '. I .................. 44 4 Higgsless Electroweak Symmetry Breaking 51 ' 4.1 Higgsless Models on a Warped Background ............... 52 4.2 Brane Kinetic Terms ........................... 56 4.3 A Minimal Higgsless Model ....................... 59 4.3.1 The SU(2)0xSU(2)1xU(1) Model ................ 59 4.3.2 The SU(2)xSU(2)N xU(1) Model ................ 66 4.3.3 The N —> oo Limit ........................ 70 5 Coupling to Matter Fields and Experimental Constraints 78 5.1 Model I .................................. 79 5.2 Experimental Constraints on Model I .................. 81 5.2.1 Direct Constraints on Heavy Boson Production ........ 81 5.2.2 Indirect Constraints on the Low Energy Fermion Lagrangians 81 5.2.3 Indirect Constraints on the Low Energy Gauge Lagrangian . . 84 5.3 Model II .................................. 86 5.3.1 Fermion Masses and Wave Functions .............. 88 5.3.2 Generation Mixings ........................ 5.4 Experimental Constraints on Model 11 ................. 5.4.1 Direct Constraints on Heavy Boson Production ........ 5.4.2 Indirect Constraints on the Low Energy Fermion Lagrangians 5.4.3 Indirect Constraints on the Low Energy Gauge Lagrangian . The Top Sector 6.1 The Top Mass in Theory Space ..................... 6.2 Unitarity of Fermion Scattering Amplitudes .............. 6.3 Bounds on the Model Parameters .................... Conclusions Solutions for the SU(2)0xSU(2)1xU(1) Model Solutions for the SU(2)xSU(2)N xU(1) Model Coupling Constants in Model II C.1 Gauge Boson Couplings ......................... C.2 Fermion Couplings ............................ vi 95 96 97 97 99 100 100 103 106 110 113 115 120 120 123 List of Figures 1.1 1.2 1.3 2.1 2.2 2.3 (a) Structure of a linear moose diagram. The circles represent gauge groups Gj, and their couplings gj. A line connecting two circles repre- sents a non-linear sigma model field, Ej, and its VEV fj. Each 2 field transforms bilinearly under the adjacent gauge groups. (b) Moose dia- gram of a deconstructed model in a flat background: All gauge groups, couplings, and VEVs are identical, because of the translational in- variance of the underlying five-dimensional theory. The only possible exception is for the first and the last site, depending on the boundary conditions .................................. 6 (a) Deconstruction of the SU(2)LxSU(2)RxU(1) model on a flat back- ground and brane kinetic terms on the Planck brane. (b) The same model can be unfolded to a single chain of SU(2) groups, and a chain of U(1) groups. .............................. 9 Moose diagram for an SU(2)xSU(2)N xU(1) Higgsless model. The gauge couplings of the internal sites, and the 2 field VEV’s are identi- cal, as the underlying five-dimensional model is translationally invari- ant. If g, 9' << é/x/N + 1, g and g’ are approximately equal to the SM SU(2)xU(1) gauge couplings. ...................... 11 The partial wave coefficients of elastic scattering amplitudes are com- plex numbers which must lie on the unitarity circle. When a tree-level amplitude — which is real — reaches :l:1 / 2, radiative corrections become as large as the leading order contribution, forcing the theory to be strongly interacting. ........................... 16 Tree-level diagrams for the HEM—’1: —> I'VZWI: elastic scattering, in the SM. .................................. 17 J = 0 partial wave amplitude for the W3 WI: —> WEI/VI: scattering, with (blue) and without (dashed red) the contribution of the Higgs bo- son, for m H =400 GeV. Without the Higgs boson, the amplitude grows like 3, leading to unitarity violation at ~1.6 TeV. The Higgs exchanges unitarize the amplitude by exactly canceling the energy growing terms. 19 vii 2.4 2.5 2.6 2.7 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 4.4 J = 0 partial wave amplitude for the WEWI': —+ W; W; scattering in the SM, for 171}; = 0.6 TeV (green), well below the critical value (2.18), and m H = 1 TeV (red), above the critical value. In the first case unitarity is satisfied for all non-exponentially large energies. In the second case unitarity is already violated at energies above the Higgs pole. .................................... 20 Tme—level diagrams for the n+7r‘ —+ n+7r' scattering, in the SM. . . 22 Moose diagram for the GWS model with the Higgs boson integrated out. 23 Diagrams for the 7r+7r_ —’ n+7r— scattering in the SM without the Higgs boson: (a) Tree-level amplitude. (b) One-loop corrections. . . . 24 Equivalence between the SO(4)~SU(2)LXSU(2)R gauge theory on a five-dimensional interval, with 30(4) broken down to SU(2)diagonal at one end, and a single SU(2) with double interval length. ....... 34 Diagrams contributing to the elastic scattering AgAi’, —+ Ail/lg: Con- tact interaction plus s-channel, t—channel, and u-channel exchanges. . 37 Rectangular Wilson loop in the (it, 5) plane, with the fifth-dimensional interval on a lattice. ........................... 45 Moose diagram for the deconstructed SU(2) gauge theory on a five- dimensional interval, with BC’s breaking SU(2) to U(1) at one end of the interval. ................................ 48 Moose diagram of the deconstructed SU(2)L> W’ZW 1': elastic scattering, in the SU(2)0xSU(2)1xU(1) NLSM model. ................ 61 The coefficient of the (E /mW)2 term in the WEWI: ——> WEWL- scat- tering amplitude in the SU(2)0xSU(2)1xU(1) model (blue) as a func- tion of the Z' and W' mass difference, with mW/ = 500 GeV fixed. The same quantity in the SM without a Higgs boson (red) is also plotted. The vertical line indicates the position where m , — m2 ,, = mg — may. 62 Z M viii 4.5 4.6 4.7 4.8 4.9' 4.10 4.11 4.12 4.13 4.14 5.1 5.2 5.3 The quantities f1 (blue) and f2 (red) as a function of the Z’ and W’ mass difference, with mWi = 500 GeV fixed. The vertical line indicates the position Where 171%, — me’ = mzz — mfiv. ............. g’ = 9’2/47r (red), and of, = @2/47r (green), as a function of the Z’ and W’ mass difference, with mW/ = 500 GeV fixed. The vertical line indicates the position where mZZ, — m3,” = 17122 - map .................... The coupling constants (19 = 92/47r (blue), 01 The J = 0 partial wave amplitude as a function of ,/§ for the SM with- out a Higgs boson (red) and the SU(2)0xSU(2)1xU(1) model (blue) with mW/ = 500 GeV and mzz, — mf/V’ = mg — ma,“ ......... Moose diagram for the SU(2)xSU(2)NxU(1) Higgsless model. . . . . Tree-level diagrams for the WZWI: ——> W 2' WI: elastic scattering, in the SU(2)xSU(2)N xU(1) model. .................... The J = 0 partial wave amplitude as a function of J3 for the SM with- out a Higgs boson (red) and the SU(2)xSU(2)NxU(1) model (blue) for N = 1 to 100 with mWi = 500 GeV. ................ The coupling constants ag = 92/41r (blue), (19/ = g'2/47r (red), and 095 = Qg/4rr (green), as a function of 1/ R. ............... Probability density for the position of the W boson (red), W1 (green), and ”’2 (blue) in the extra-dimensional interval, for 1 / R =500 GeV, with My) replaced by a narrow Gaussian ................. Probability density for the position of the photon (dashed), the Z boson (red), Z] (green), and 22 (blue) in the extra-dimensional interval, for 65 66 68 69 73 .74 1 / R =500 GeV, with 6(y) and 6(7rR — y) replaced by narrow Gaussians. 75 Unitarity violation curve for the W3 W; —> W; W; scattering, in the (fi, 1 / R) plane. For a given value of l/R, unitarity is satisfied for values of J3 below the curve. ...................... Probability density for the position of the left-handed electron (red), and its KK resonances, 6L1 (green), and 6L2 (blue) in the extra- dimensional (or TS) interval, for 1/R 2500 GeV, and tL = A/\/3, with My) and 6(7rR - y) replaced by narrow Gaussians ......... Probability density for the position of the right-handed electron (red), and its KK resonances, 6R1 (green), and 6122 (blue) in the extra- dimensional (or TS) interval, for 1/R 2500 GeV, and tL = /\/\/3, with 6 (y) and 6(7rR — y) replaced by narrow Gaussians ......... Masses of fermions, as a function of the bulk mass, for t L = 10‘1, tR = 1, and l/R = 500 GeV. ...................... ix 76 92 93 6.1 6.2 6.3 6.4 C.1 C.2 Mass of the lightest fermion, as a function of the bulk mass, for t, L chosen to adjust the S—parameter to zero, and 1/R = 500 GeV. The curves correspond to several values of t , from 10"1 to infinity. . . . XR (a) Diagrams contributing to the tf —+ WEI/VI“: tree-level scattering amplitude in the SM. (b) Same process, in our Higgsless model. Bounds imposed by unitarity constraints on the if -—+ W; W I: scat- tering at \/3 =10 TeV (upper curve), and the WELWI: —» W; W I: scattering at \/E =10 TeV and \/5 =5 TeV (vertical lines), in the (1 / R9, 1 / R f) plane. Specifically, we have assumed the requirement of (to < 1/ 2 for both scattering processes. The tf —> W; W; scattering at \/s =5 TeV imposes no bound, since at this energy the no Higgs boson is required to unitarize the amplitude. The two curves on the bottom correspond to the minimum value of 1/Rf which allows a top mass of 175 GeV to be a solution of the mass equation for M = 0 (lower), and the minimum value of 1 / R f which gives a th right-handed coupling in agreement with the experimental constraint (upper). The vertical curve on the left corresponds to the experimental bound on le from analysis of the W W Z vertex. ...................... Right-handed th coupling, in units of e/s, for l/Rg = 550 GeV, as a function of l/Rf. The horizontal line corresponds to the experimental bound of Ref. [80]. ............................ Feynman rules for gauge interactions ................... Feynman rules for charge-current and neutral-current interactions. . . 101 104 107 122 124 Chapter 1 Introduction The SU(3)C010r> U j——11 (1:)Ej(1:)U j(.r). In a deconstructed model all gauge groups are of course identical, and the 2 fields correspond to the value of the gauge-field fifth com- ponent at discrete points. If a flat background is assumed, all couplings and VEVs are also identical. The only possible exception is represented by the first and the last site, where boundary conditions (BCs) and brane kinetic terms, in the underly- ing five-dimensional theory, can result in different gauge groups and couplings in the deconstructed model. We will come back to this point later in this introduction. The moose diagram of a deconstructed model looks therefore like the one in F ig.1.1(b). In deconstructed models the coefficient of the E2 term does not vanish precisely, but it is suppressed by 1/N2, for N —> 00. As a consequence, the delay of unitarity violation is not as large as in the full five-dimensional models [36] [41] [42]. Nonetheless deconstruction is still a powerful tool for model building for at least two reasons. First, 2In Georgi‘s model fermions are coupled in a peculiar way, which makes the interpretation as a deconstructed five-dimensional model not viable. Nonetheless such interpretation is valid for the gauge sector. Figure 1.1: (a) Structure of a linear moose diagram. The circles represent gauge groups Gj, and their couplings gj. A line connecting two circles represents a non- linear sigma model field, 23-, and its VEV fj. Each 2 field transforms bilinearly under the adjacent gauge groups. (b) Moose diagram of a deconstructed model in a flat background: All gauge groups, couplings, and VEVs are identical, because of the translational invariance of the underlying five-dimensional theory. The only possible exception is for the first and the last site, depending on the boundary conditions. it allows for common four-dimensional UV completions, like linear sigma models (as in Georgi’s model) or dynamical symmetry breaking. This turns out to be particularly useful in Technicolor-like models, since a strongly coupled physics at the TeV scale is disfavoured by electroweak precision (EWP) data [43] [44], and deconstructed models with a large symmetry structure can raise the scale of unitarity violation by a factor 10 [45] [46]. Second, deconstructed models allow for more freedom in model building than strict five—dimensional theories. This feature is especially welcome when it comes to coupling a model to matter fields. As opposed to five-dimensional theories, where the interaction of fermions with the gauge-field fifth component has the same strength as the ordinary gauge interactions, in deconstructed models the Yukawa coupling to the 2 fields can be made independent of the gauge interactions. Moreover, terms which are not local in a five-dimensional theory make perfect sense in deconstructed models. Theories represented by moose diagrams are commonly referred to as models from theory space (TS). As the number of sites goes to infinity, a theory-space chain of gauge groups becomes an extra-dimensional interval only under special assumptions. We will discuss the relation between extra-dimension and theory space in section 3.5. Having established that extra-dimension and theory space provide new interac- tions which delay unitarity violation, the next step is to inquire whether EWSB can be implemented in these frameworks. Models with a compactified extra-dimension are usually mapped onto five-dimensional intervals, where the interval ends are four- dimensional branes with rather special properties, depending on the BC’s. The fifth- dimensional interval is commonly referred to as the bulk of the extra-dimension. Csaki et al. showed that an appropriate choice of the gauge group in the bulk, supplied with suitable BCs, can give a symmetry breaking pattern which contains EWSB. There- fore, Higgsless EWSB is indeed a potentially viable alternative to the more traditional models with a Higgs boson in the GeV range [36] [42] [45] [46] [47] [48] [49] [50]. The W boson would then be interpreted as the lightest mode of a charged KK tower, while the photon and the Z boson would be interpreted as the lightest modes of a neutral KK tower; The elastic scatterings W+W" —) W+W— and WiZ —> WiZ would be unitarized by exchanges of virtual Zn and W5“ KK modes, respectively, where n = 1,2,... [36] [42]. However the cancellation of the terms which grow like E2 is not enough to ensure unitarity. In gauge theories with a Higgs boson, we argued that unitarity requires an upper bound on the Higgs mass. Similarly, in extra-dimensional models the delay of unitarity violation imposes an upper bound on the compactifica- tion scale [46] [50]. Of course unitarity is not the only constraint we must consider. First, a realis— tic model of EWSB must reproduce the mass spectrum of the observed particles, top quark included. Second, it must satisfy the constraints imposed by EWP data. Third, low-energy anomalous interactions must be within the experimental bounds. Consid- erable effort has been recently spent in the attempt to meet all these requirements. A model, in particular, has emerged as a potentially serious candidate: the SU(2)L xSU(2)RXU(1) B— L five-dimensional gauge theory, coupled to a warped anti-de Sit- ter (AdS), Randall-Sundrum (R81) model [51], where BCs break SU(2)LX SU(2)R down to SU(2ldiagonal at the infra-red, or “TeV” brane, and break SU(2)RX U(1) B— L down to U(1)y at the ultra-violet, or “Planck” brane [46] [47] [52]. The electroweak symmetry SU(2)LXU(1)y is therefore localized on the Planck brane, and is broken by the extra-dimensional bulk and the BCs on the TeV brane. The SU(2)L xSU(2)R 'T’SU(2)diagonal structure is designed to satisfy the bounds on the p parameter. The latter is defined as the ratio between the strength of the charged-current interactions and the neutral-current interactions at zero momentum. Experimental results show that p differs from unity by less than 2.5-10'3 [53]. The natural way to meet the constraints on the p parameter is to guarantee that a global isospin symmetry is still present even after EWSB, and is only broken by hyper- charge and Yukawa interactions [54]. Such global symmetry is known as custodial isospin, and is naturally embedded in the SM. In the extra-dimensional model, the SU(2)L xSU(2)R —>SU(2)diag0nal structure guarantees a custodial symmetry if the five-dimensional profile of the matter fields is appropriately chosen. For example, with matter fields localized on the x5 = 0 brane the tree-level p parameter does not differ from unity: This has been proved for a wide class of deconstructed models, with arbitrary gauge couplings and f —constants [55]. However in general the charged- current and neutral-current interactions are not necessarily mediated by the W and the Z bosons only, since exchanges of the heavy KK modes might be equally impor- tant. This is potentially problematic, because a sizable contribution of the heavy KK exchanges means that the first heavy charged and neutral gauge bosons are relatively light, and could be below the direct—search experimental bounds. The AdS warping factor solves this problem, because it pushes the lightest KK ecitations toward the TeV range, while keeping the W and Z masses light [47]. This in turn means that the charged-current and neutral-current interactions, at zero momentum, are almost entirely mediated by the SM gauge bosons. (b) Figure 1.2: (a) Deconstruction of the SU(2)LXSU(2)RXU( 1) model on a flat back- ground and brane kinetic terms on the Planck brane. (b) The same model can be unfolded to a single chain of SU(2) groups, and a chain of U(1) groups. Warping seems therefore to bc a necessary ingredient for a five-dimensional Hig- gsless model to meet the experimental contraints on the direct search of heavy gauge bosons. However, integrating out a large slice of AdS5 in the proximity of the Planck brane leaves an effective field theory with a nearly flat background, and a kinetic term localized on one of the boundaries, which “mimics” the warping of the extra- dimcnsion [56] [57] [58]. In this way we can still work with a flat extra-dimension, which means that we can work with sines and cosines, rather than Bessel functions, thereby simplifying the mathematical content of the model. This approach is only valid as long as the energy is much smaller than the curvature of the AdS5 profile, which lies in the Planck scale, and begins to break down as the energy increases. The corresponding deconstructed model is shown in Fig. 1.2(a). The brane kinetic terms force the first SU(2) and U(1) gauge groups to have different couplings. The same model has been unfolded in Fig. 1.2(b). Notice that the unfolded moose diagram shows explicitly the equivalence between an SU(2)LXSU(2)R gauge symmetry in the bulk, broken to SU(2ldiagonal on one brane, and a single SU(2). We will demonstrate this in section 3.2. Of course there are more constraints to be considered, other than the p parameter. If the contribution of the new physics on the low energy observables is approximately oblique — that is, it does not change the form of the interactions, but only modifies their relative strength - then it can be parametrized in terms of the Peskin-Takeuchi S, T, and U variables, where S = T = U = 0 corresponds to the SM [43] [44] [59]. T is directly related to the p parameter, and for models with a custodial symmetry is zero at tree level. U is usually expected to differ from zero by only a percent of T, and is therefore negligible. The constraints on the S parameter are usually more difiicult to meet, since a small S often requires the new physics scale to be higher, which is harmful for unitarity. In Higgsless models from extra-dimension the value [of S depends on the fermion profiles. It is well known that when the matter fields are localized on the Planck brane, the S parameter is positive, while matter fields localized on the TeV brane give a negative S value [60]. It is then clear that in order for S to vanish, the matter fields must be delocalized. This has been proved for a wide class of Higgsless models [46] [55] [61] [62] [63] [64]. We will return to these results in chapter 5. The model of Fig. 1.2(b) can be simplified by reducing the number of U(1) groups to one, as in Fig. 1.3. By eliminating the chain of U(1) groups, we necessarily in- troduce interactions which would be non-local in a deconstructed five-dimensional theory [42] [45]. In fact, in order to maintain gauge invariance, all fermions in the bulk must couple to the U(1) group with left-handed hypercharge YL. However from a purely TS point of view this is perfectly legitimate. This is a first example of what was mentioned before, namely that TS offers more freedom, in model building, than the rigid structure of a real extra—dimension. The SU(2) x SU(2)N >< U(1) model of Fig. 1.3 is arguably a minimal Higgsless model 10 SU(2) SU(2) SU(2) SU(2) U(1) Figure 1.3: Moose diagram for an SU(2) xSU(2)N xU(1) Higgsless model. The gauge couplings of the internal sites, and the 2 field VEV’s are identical, as the underlying five-dimensional model is translationally invariant. If g, 9’ << 57/ m, g and g’ are approximately equal to the SM SU(2)xU(1) gauge couplings. of EWSB, and is the model we will consider in this dissertation. The SU(2) and U(1) gauge groups on the two edges of the moose chain act approximately as the SM SU(2)L XU(I)y, and the SU(2)N gauge group represents approximately the new- physics contribution. This approximation is valid as long as the the effective SU(2)N gauge coupling, g/m, is large, if compared to the coupling of the first SU(2) group, 9, and the coupling of the U(1) group, 9’. Then 9 and 9' have approximately their SM values. The fermion sector follows the same pattern of the gauge sector. If matter fields are delocalized, as required by the experimental contraints on the S parameter, then the SM fermions are mainly, but not exclusively, charged under the SU(2) and U(1) gauge groups on the chain edges, with SM quantum numbers. The new, heavy fermions are mainly coupled to the SU(2)N group, with approximately vector-like couplings [42] [45]. As N -——+ 00, this theory becomes a model from continuum TS: The discrete site index of Fig. 1.3 becomes a continuous variable, but the non-local couplings of matter fields disfavour the extra-dimensional interpretation. Despite this, the continuum TS limit is more convenient for calculational purposes: Recurrence relations and eigen- value equations become differential equations and transcendental equations, which are simpler to solve. Therefore, in this dissertation we will carry out most of the calculations in the continuum limit. The main source of tension, in Higgsless models of this kind, is between the uni- 11 tarity bounds, and the constraints from EWP data and the top-quark sector. In par» ticular, a heavy top mass is problematic, if the compactification scale is to be within the unitarity bounds. In order to solve this problem, two compactification scales are introduced: one for the gauge sector, and one, higher, for the fermion sector. This makes the extra-dimensional interpretation even more problematic: It corresponds to a microscopic breaking of the five-dimensional Lorentz invariance, in addition to the macroscopic breaking due to compactification. On the other hand, from a discrete TS standpoint this just corresponds to Yukawa couplings being independent of gauge couplings [20] [45]. - This dissertation is organized as it follows. In chapter 2 the unitarity of longitudi- nal gauge boson scattering amplitudes is analyzed in the SM, as well as the role of the Higgs boson and the corresponding bounds on its mass. In chapter 3 the role of extra- dimension as a tool to delay unitarity violation is discussed in details. Deconstructed models, and TS are then shown to be viable alternatives. The goal of this chapter is to show how to build theories without Higgs bosons, which are weakly coupled and unitary up to high energy scales. In chapter 4 these concepts are applied to models of EWSB. We first consider the extra-dimensional SU(2)LxSU(2)RxU(1) model. Then we introduce the model which is the object of our study. The gauge sector is an SU(2) Yang-Mills theory on an extra-dimensional interval, with BCs breaking SU(2) down to U(1) on one end. In order to obtain the right masses for the SM gauge bosons, large kinetic terms are added on both branes, without affecting the delay of unitar- ity violation. Unitarity upper bounds on the compactification scale are calculated numerically in terms of the cutoff scale. Deconstruction is then shown to preserve the important features of the extra-dimensional model, in a familiar four-dimensional context. In chapter 5 matter fields are coupled to the model in two different ways: First, with brane-localized fermions, and second, with slightly delocalized fermions. In both cases the extra-dimensional interpretation is not viable, because non-local interactions must be introduced. Therefore, the N —> oo limit of the deconstructed model should be interpreted as a continuum TS. The constraints from'the EWP data are analyzed for both ways of coupling fermions, including lower bounds on the 12 mass of the W1 and 21 bosons. While the tree-level T parameter is naturally sup- pressed by custodial isospin in both cases, the bounds on the S parameter can only be satisfied by the delocalized model, as long as the fermion leakage into the bulk is appropriately tuned. Also, multiple generations and fermion mixings are shown to be naturally implemented in the delocalized model. In chapter 6 a heavy top mass is shown to be unattainable, when the unitarity bounds of gauge boson scattering amplitudes are imposed. However, from a continuum TS standpoint, the compactifi- cation scales for the gauge sector and the fermion sector are shown to be independent quantities. This allows to accommodate the top mass without violating the gauge- sector unitarity bounds. Unitarity of if —> W2" 71:, and experimental constraints on the right-handed th coupling are translated into upper and lower bounds, respec- tively, for the fermion-sector compactification scale. Finally, in chapter 7 we offer our conclusions. 13 Chapter 2 Unitarity in the Standard Model Any respectable quantum theory must return probabilities between zero and one. In scattering theory this requirement is guaranteed by unitarity of the S matrix: The wavefunctions of scattered particles differ from the wavefunctions of incident particles by unitary transformations. This forces the partial wave amplitudes to lie on a radius- one circle in the complex plane. In this chapter we derive the precise formulation of this constraint, and find its implications for the scattering of longitudinal vector bosons in the SM. 2.1 Partial Wave Expansion Let us consider a two-particle elastic scattering process in the spin-0 channel. In the center-of-mass (COM) frame, this process is equivalent to a one-particle scattering off a spinless fixed target, and can be described in the context of ordinary one-particle quantum mechanics. With all couplings turned off, the total wavefunction is just a plane wave propa- gating, say, along the z axis. An expansion in spherical waves gives 00 w, = em (2.] +1) [(—1)Je—ip’" — elm] PJ(C086) , (2.1) 2pr J=0 where p is the magnitude of the particle momentum in the COM frame, 6 is the scattering angle, and r is the radial distance from the COM. This expansion is a 14 superposition of incoming and outgoing spherical waves, 6‘27”" and em" , respectively. The scattering center can only affect the outgoing wave. For an elastic scattering, with no absorption, unitarity of the S matrix implies that the corresponding partial wave coefficients are multiplied by phase factors, which we denote by €273.70». Therefore, the total wavefunction is ‘ w . . - - I/total— _ figs (2 I +1 )[(— (—1)Je—W — e2wJ(p)esz] PJ(cosa) . (2.2) The scattered wave represents the difference between the outgoing waves in UI’total and 1h, ' eipr oo €2i6J(p)_ 7(scattered : 2 (2J + 1l—PJ(COS 6) pr J20 2i eil’T where F (p, 6) is the scattering amplitude, 00 )62idl(p) _1 The scattered outgoing flux in a solid angle (19, through a sphere of radius r, is 'L’Ol‘vbscatteredl27'2dQ = ”OlF(P16)l2dQ , where vo is the outgoing particles speed. This expression is by definition equal to the product of the scattering cross-section and the incident flux, vii/5111i" = vi: v0[F(p, 6)|2dQ = vida . Since the collision is elastic, vi = vo, and d0) 2 -— = W2?, 9)| - (2.5) (d0 COM, elastic In quantum field theory, at high energy, with Lorentz-invariant normalization of the quantum states, equation (2.5) reads (1 w ,0 2 (l) =——-—" (5’ ”2, (26) ‘19 COM, elastic 64” '41) 15 Im(aJ)‘l (glee. ‘_ 1/2 Re(a:) Figure 2.1: The partial wave coefficients of elastic scattering amplitudes are complex numbers which must lie on the unitarity circle. When a tree-level amplitude — which is real — reaches :l:1 / 2, radiative corrections become as large as the leading order contribution, forcing the theory to be strongly interacting. where M (p, 6) is the scattering amplitude [65]. Equating the right-hand side of (2.5) and (2.6), and using (2.4), leads to the partial wave expansion of M (p, 6), with the appropriate normalization factor, 00 1W(p, 6’) = 167r 2 (2J +1)aJ(p)PJ(c086) , (2.7) J=0 where 62i6J(p) _ 1 aJ09) = T - (2-8) In the complex plane, aJ(p) lies on a radius-1/2 circle centered on i / 2, the unitarity circle, as shown in Fig 2.1. Since tree-level scattering amplitudes are real, a J(p) can only lie on the circle if loop corrections are included. These become more and more important as the tree-level amplitude increases in magnitude. When the latter reaches :l:1/2, radiative corrections become as important as the leading-order contribution. Therefore, at tree level, either , (2.9) NIH la-J(P)| S 16 M, 7+ L z + + 7+2 7+Z W; +E+§ + 1H H W t, Figure 2.2: Tree-level diagrams for the W; W I: —’ WZ' WI: elastic scattering, in the SM. W; or the theory must be strongly interacting. 2.2 Longitudinal Gauge Boson Scattering in the Standard Model We now apply last section results to the scattering amplitude of longitudinal W bosons in the SM. As before, we consider the scattering in the COM frame, with initial momenta along the z-axis. The four momenta, for the scattering particles and the scattered particles, are, respectively, (E,0,0,:tp) and (E, 0, ip sin 6,ipcos 0), where E = M. The polarization vectors are respectively (p, 0,0, iE)/mw and (p, 0, iE sin 6, :l:E cos 6) /mW. At tree level, in unitary gauge, there] are seven diagrams, as shown in Figure 2.2. A simple power counting shows that the amplitude can diverge as badly as (E / mw)4, for large values of E / mW, because each polariza- tion vector grows like E / mw, and longitudinal gauge boson propagators do not fall off with energy. 17 The contact interaction, plus the photon and Z' exchanges give the amplitude 2 Mgauge = i4— [p2E2(—2+6cos6)-E4 311126] 771W 1 F 2 2c0826 + T e— + g___2_vr (—4p2(p2 — 3E2)2) 0036 mW _ s s -— mZ ] 1 ' 2 2 26 ‘ 2 + 4 €—+gC—OS-QE- [—4E2 (p2+(E2—2p2)cos6) mW _ t t — mZ _ 2 —2p2(1 + cos 6) (2E2 — p2 — E2 cos 6) ] , (2.10) where s = 4E2, and t = —2p2(1 — cos 6). In the limit of large E /mW, this expression becomes Mgauge 2 4cos2 6W(1 — cos 6) + O ((le/E)2) . (2.11) 21+cos6( E )2+gQ3-2cos26w(1—cos6)2+cos26 mw The term proportional to (E / mW)4 vanishes because of gauge invariance, which guarantees the special relation e2 = 92 sin2 6W between different coupling constants. Nonetheless the term proportional to (E /mW)2 does not cancel. Therefore, if only interactions from the gauge sector were to be included, the J = 0 partial wave would violate unitarity at \/§ : Sfimw 8’“ 6W ~ 1.7 TeV, (2.12) e and the J = 1 partial wave would violate unitarity at J; 2: 2.9 TeV. However there are two more diagrams in F ig.2.2 which have not been considered yet, the Higgs exchanges. Their contribution to the scattering amplitude is [\[Higgs _ __ 92 (3 " 2mfi’)2 (f + 261%; COS 9):? 2 13 — 4 2 2 + t 2 ' ( ‘ ) mW S ‘— mH _ mH For E2,m%{ >> me this becomes 2 MHiggs 2 _g21+:059( E) mW 2 2 g m s t — —- 2H 2 + 2 . (2.14) 4 mW s — mH t. — mH A comparison of (2.14) and (2.11) shows that the coefficients of the (E/mw)2 term exactly cancel. The J = 0 partial wave amplitude for the IVER/"I: —> WITH]: 18 0.5 . l / l , [ / L 3 [ / > l / 0-4 ’ l l , ' , ‘ 'l / .3 f / i l » l 1 ’ _ 0.3 5, 1 /, g 1 l , __ 5 1l / . ,1 1 / 002 ll \‘ / / l l, / [i \, , ’ b 1' \ I O] r v: ’ / /\<‘\ \x‘-_ 0 0.25 0.5 0.75 l 1.25 1.5 \/§(TeV) Figure 2.3: J = 0 partial wave amplitude for the W E W; -—+ WI-f WI: scattering, with (blue) and without (dashed red) the contribution of the Higgs boson, for m H =400 GeV. Without the Higgs boson, the amplitude grows like 3, leading to unitarity viola- tion at ~1.6 TeV. The Higgs exchanges unitarize the amplitude by exactly canceling the energy growing terms. scattering is then shown in Fig 2.3, with and without the Higgs boson contribution, for m H =400 GeV. (A small m7=1 GeV photon mass has been included, in order to regulate the singularity in the t-channel. This is inconsequential in the high energy region in which we are interested.) It is evident that the Higgs boson exchanges are essential to maintain unitarity in the TeV range. Notice that near the Higgs pole the amplitude is tamed by finite-width effects, which are sufficient to keep a0 below 1 / 2. The cancellation of the quadratically divergent term is not enough to prevent uni- tarity violation at sufficiently high energies, since the term proportional to (mW / E )0 can be of order one. The contributions to the J = 0 partial wave of the con- tact interaction, plus the photon and Z exchanges, give an 0 ((mW/E)0) term of order (92/3211') log(E/mw), and is therefore safe for non-exponentially large ener- gies. (The logarithmic growth comes from the integration near cos6 = 1, where 19 l \ 0.8» 0.4 * laol 0.2 * 0 0.5 l 1.5 2 2.5 3 fi/mu Figure 2.4: J = 0 partial wave amplitude for the MEI/ll: ——> WEI'VL— scattering in the SM, for my = 0.6 TeV (green), well below the critical value (2.18), and my = 1 TeV (red), above the critical value. In the first case unitarity is satisfied for all non-exponentially large energies. In the second case unitarity is already violated at energies above the Higgs pole. mQZ/t becomes large, and high energy regions become important.) On the other hand, the contribution of the Higgs exchanges give an 0 ((mW / E )0) term of order (92/321r)(m%1/m%[;), and can be large, depending on the Higgs boson mass. There- fore, for s,m%1 >> mahrngz, the WEWE —> WEL W E scattering amplitude is to a good approximation given by the second term of (2.14), M(l/l'+W‘ W+li’“) 92 mf’ [ S + t ] (215) "LL—*’L"L1‘—2 _2 _.2’ - 4 mW s mH t "1” and the J = 0 partial wave amplitude is 2 2 2 2 ' 7— 7 y— m m m 8 (10(14wa —4 ”fl/1L): Tiff—2H [2+ ”.2 — H log (1+ —.,—)] . 7f mW S — mH S mH (2.16) In the limit of large energy, this amplitude approaches a constant value: 2 2 m iii/Til?“ —» W+W— 2: ——9— H . 2.17 aOl L L L L l 3271, mf/l ( l 20 Then unitarity demands 4 f I 1' < fir”: 3mg“ ~ 0.9 TeV. (2.18) mH If the Higgs mass is well below this critical value, the J = 0 amplitude satisfies the unitarity bound at all (non-exponentially large) energies. If the Higgs mass attains or exceeds the critical value, unitarity is already violated at energies above the Higgs pole, as shown in Fig.2.4. It is possible to refine the bound (2.18) by considering the neutral four-channel system Will/VII? (1/\/2)ZLZL, (1/\/2)HH, and HZL, rather than just WZWE [10]. Then (2.17) is replaced by a 4x4 matrix: / 1 1 l 78 79 0) 2 2 l 3 1 0 a0=—i—1"7H- 75 Z Z (2.19) 8 1 This scattering matrix has a surprising simple eigenchannel structure. The largest eigenvalue corresponds to the elastic scattering of the channel 2W3 W; + Z L + Z L + H H , and leads to the most stringent unitarity bound on the Higgs mass: m H < Sfimg‘ésm 6W ~ 0.6 TeV . (2.20) At this point, two questions naturally arise: (i) Why do terms growing like E2 exactly cancel out from longitudinal vector boson scattering amplitudes ? (ii) What makes the four neutral-channel system WITWI: , (1 / x/2) Z LZ L, (1 / V2)H H , and H Z L, so special and so simple, compared to other two-body neutral channels ? The answer to these questions can be found in the Goldstone boson equivalence theorem, which relates amplitudes for absorption and emission of longitudinal vector bosons, to am- plitudes for absorption and emission of the corresponding eaten Goldstone bosons. This will be the subject of the next section. 21 \ / , \ \ / \ / \ / \ \ / \ 1 \ 1 \ 1 Y _ \ \ / ‘ — x + >— —~— —< ‘l' i H I \ / H \ I x I \ 1 \ k l \ l \ l \ I \ 71‘- 11 — x \ / / \ \ / . \ .. \ / ‘, / + ,- / + (g 7 + Z , 7 + Z . A I K Figure 2.5: Tree-level diagrams for the n+7r" —+ n+7r‘ scattering, in the SM. 2.3 Unitarity and the Equivalence Theorem i are the Goldstone Consider the 7r+7r_ —+ n+7r— scattering in the SM, where 7r bosons eaten by the W5c boson. At tree level there are seven diagrams, which are shown in Fig. 2.5. For 3, mg, >> ma), m2Z, the interactions from the Higgs sector are dominant, because the corresponding couplings are enhanced by a factor ruff/map Therefore, the amplitude is approximately given by the first three diagrams. In a general R5 gauge, and ignoring the Goldstone boson masses, we obtain 2 2 m s t M(7r+7r-—>rr+7r-)2—%1— 2H 2 + 2 ., mW s—mH l—mH (2.21) in agreement with (2.15). This is an expected result. In fact, the Goldstone boson equivalence theorem tells us that the amplitudes for absorption or emission of a longi- tudinal gauge boson approach, at high energies, the same amplitudes with the gauge boson replaced by its eaten Goldstone boson [9] [11] [13] [14] [15] [16] [17] [18] [19]. We are therefore in a position to answer the two questions posed at the end of sec. 2.2. (i) The cancellation of the terms growing like (E /mW)2 occurs because at high energy longitudinal gauge boson scattering amplitudes become identical to Goldstone boson scattering amplitudes. These cannot steadily grow with energy, as a simple power counting shows. In fact, in a general R5 gauge, gauge boson propagators 22 SU(2) U(1) Figure 2.6: Moose diagram for the GW S model with the Higgs boson integrated out. fall off with energy, and — working in a linear representation of the gauge group — the coupling with the Goldstone bosons involves only one power of momentum. Then gauge boson exchange amplitudes approach a constant value, of order 92. In the Higgs sector, the coupling Higgs-Goldstone does not depend on momentum, thus the Higgs exchanges fall off with energy, for J3 > m H- Therefore, as the energy increases, only the contact interaction — which is constant, and of order 92m2 Mia, — becomes relevant. (ii) The four channels W L,(1/\/2)ZLZL, (1/\/2)HH, and HZL have a simple eigenchannel structure becauseL they correspond, via the equivalence theorem, to the Higgs-sector neutral channels. n+1r , (1/\/-2_)7r0 0, (1/\/2)H H , and H No, where no is the Goldstone boson eaten by the Z. The simplicity of the scattering matrix (2.19) is then a consequence of the underlying global symmetries of the Higgs sector. Of course the equivalence theorem by itself is not enough to guarantee the can- cellation of the terms growing like (E /mw)2: The key ingredient here is the Higgs boson. This has been shown explicitly in sec. 2.2, and can be seen also using the equivalence theorem. If we take the GWS model, and integrate out the Higgs boson, what is left is an SU(2)L> n+7)” scattering in the SM without the Higgs boson: (a) Tree-level amplitude. (b) One-loop corrections. The ira’s are the usual SM Goldstone bosons, and the 00’s are the Pauli matrices. Expanding the exponential, the first non—zero interacting term is a quartic interaction: 5(4) _ 1 [((W+)2(a,,—)2 + (nT)2(87r+)2 — 27r+7r_(67r+)(37T-)) - 6112 — 2 ((87r+)((‘)7r_)(7r0)2 + 7r+7r7(87r0)2 — (877+)7r_(87r0)7r0) ]. (2.24) + The tree-level n+7r— —2 7r 7r‘ scattering is only given by a contact interaction, Fig.2.7(a). The corresponding amplitude is + - + — _ U M(rr 7r ——>7r 7r ) — —v—2 2 1+cos6 E in agreement with the leading term of (2.11): The equivalence theorem still works. But the term growing like (E / mW)2 does not cancel. The difference, with the Higgs sector of the SM, is in the couplings, which involve derivatives of the Goldstone fields, and thus the external momenta. This happens because a model with only Goldstone bosons and no physical scalars is necessarily in a non linear realization of the symmetry group. Then the Lagrangian can only be built out of derivative terms, because terms without derivatives vanish. Therefore, in an SU(2) x U(1) gauge theory the Higgs boson is a necessary ingredient for unitarity. This example sheds some light on the connection between unitarity violation and non-renormalizability. As the tree-level amplitude grows like E2, the one-loop cor- rections, F ig.2.7(b), grow like E4, by simple power counting. Then, at the unitarity 24 violation scale the loop corrections become as important as the leading order con- tribution, and the perturbative expansion breaks down. Moreover, as the external momenta grow, higher order terms must be added to (2.22), and new terms in the expansion of the 2 field must be considered, because they cease to be negligible. Thus, as the energy increases, the non renormalizable operators become important. Above the unitarity violation scale a new theory must take over. Such theory must reproduce the low-energy physics of the SU(3)c010rxSU(2)L x U(1) model, and restore unitarity, or delay unitarity violation to higher energy scales. 25 Chapter 3 Unitarity Without the Higgs Boson The non-linearly realized SU(2)LxU(1)y gauge theory has been shown to violate unitarity of longitudinal gauge boson scattering amplitudes at few TeV’s. In order to restore unitarity, or delay unitarity violation to higher energy scales, new particles must come into play, and mediate interactions which cancel the bad high energy behavior. In the SM, the Higgs boson suffices to restore unitarity at (almost) all energies. Recently, models with one compactified extra-dimension have been shown to violate unitarity at energy scales higher than the customary limit of Dicus-Mathur or Lee-Quigg-Thacker. The violation delay is mediated by a tower of massive gauge bosons, rather than a scalar particle. In this chapter the physics of one compactified extra-dimension is introduced and discussed. From a four-dimensional standpoint, the extra-dimensional compactifica- tion breaks a countable infinity of gauge symmetries, with a corresponding generation of towers of Goldstone bosons and massive gauge bosons. The residual gauge symme- try can be further broken by an appropriate choice of BCs. Gauge symmetry breaking via compactification and BCs is shown to be soft (spontaneous), rather than hard (ex- plicit), for all BCs consistent with the variational principle. Deconstructed models, where the extra-dimension is put on a lattice, and models from theory space are introduced as viable alternatives. 26 3.1 Compactified Extra-Dimension The structure of our universe may be larger than the ordinary space—time four dimen- sions. There might be a compactified spatial extra-dimension, substantially larger than the Planck scale, but small enough to elude detection in the past generation of hadron and linear colliders [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]. Being compact, the fifth-dimension is usually taken to be an interval. For instance, a circular fifth-dimension of radius R can be mapped onto a [0,27rR] interval with periodic BCs. As a second example, if a Z2 symmetry is imposed on the circle, the fifth-dimension can be conveniently represented on a [0, 7rR] interval, with BCs which follow from the field transformations under Z2. In order to maintain generality, it is therefore convenient to specify a gauge theory on a [0, 7rR] interval, where R is an arbitrary length, and derive the most general class of allowed BCsl. For an arbitrary gauge group G, the action of a Yang-Mills theory on a flat background is all 1- 1 / Sgauge = /d4x/0 dr"[— —F1fINFaMN 4952) — 1 (6 .4““+§85Aa5)2 (3.1) 2.035 ,1 i where FffIN = avidlv — (UV/Ill! + f “India/lit! - (3-2) Here a is the gauge index, f “be the structure constants, and 95 the five-dimensional gauge coupling, with mass dimension -1 / 2. Notice that with this normalization, the gauge fields have mass dimension 1, as in 4D. The five-dimensional coordinates are labeled by M,N 6 (41,5), with p E (0,1,2,3), and the metric tensor — assuming a flat background — is G M N = diag(1,-1,-1,-1,-1). The gauge-fixing term we have chosen explicitly violates the five-dimensional gauge invariance. However the latter is already broken by compactification, which forces the AZ and Ag fields to behave differently. It is therefore convenient to get rid of the B5Afit9t‘A‘5' mixing term from F3517 “"5, and the gauge-fixing term of (3.1) is especially designed for this purpose. 1This and the next. two sections follow closely the content of Ref. [36]. 27 The variation of the action (3.1) leads to the equations of motion (EOM) 1 BMFM'I" _ fachbAIVAfi/I + Eat/80A: _ 8118514551 : O , 8"F,‘,‘5 — fabCF§5Aw + 6565A“ — £83. 5* = 0 . (3.3) However the boundary pieces must vanish as well. This leads to the requirement [F353AW+ (8 AW — 585.4g)5Ag]gR = 0 . (3.4) Periodic BCS (with period n1?) clearly satisfy (3.4). If the fields are not periodic, (3.4) implies that the equations F3564” = 0 , (65AM — 585.4964; = 0 (3.5) hold at x5 = 0 and r5 = 7rR. For instance, if the extra-dimension is a radius-R circle, and a Z2 symmetry is imposed on the gauge fields [66] [67], Age, 175) = 4:3(12, 4:5) , Agahf") = —Ag(x, —x5) , (3.6) the independent degrees of freedom lie on only one half of the circle, say, the upper half. Then the BCS on the [0, 7rR] interval are (”Jr/la = 0, O u , r5 = 0,7rR . (3,7) Ag = 0 It is not difficult to prove that (3.7) satisfies (3.5). However the requirement (3.5) allows for a broader set of BCs, even those which have no interpretation in terms of an orbifolded circle. There are three choices of BCs which respect the four-dimensional Lorentz invariance: Aa— — 0, As— — const. (3.8) Af, = 0 , 65.45 = 0 (3.9) 55 = 0 , Ag‘ = const. . (3.10) Notice that the orbifold BCS (3.7) correspond to the choice (3.10), although the condition F :5 = 0 is satisfied in a trivial way. In addition to (3.8)—(3.10), there are 28 also more interesting choices in which the sum of different terms in (3.5) vanish, rather than each individual term. We will return to this points later in this section. Although the EOM (3.3) are uniquely determined by the principle of least action, the BCS are arbitrary, and correspond to different physical scenarios. An illuminating analogy is given by the physics of a vibrating rod. The wave equation governs the displacements inside the rod, but the BCS determine what kind of motion can actually take place. If both ends of the red are fixed, the displacement at the boundaries is zero, which is analogous to Af‘, = 0 in the extra-dimensional model. However if only one end is fixed, the behavior at the loose end is governed by a non-trivial equation, which is analogous to PS5 = 0. We are interested in the four-dimensional implications of the five-dimensional theory. It is then convenient to expand the five-dimensional gauge fields in KK modes, that is, in eigenfunctions of —p§ E 6% satisfying the chosen BCS. For simplicity, we first consider BCs which do not depend on the gauge index a. Then we can write 40( (I, T5=ZOfTI<$AIaip(T1) ' = 2 4565426), (3.11) n=0 where a: is a four-dimensional coordinate, and the expansion coefficients are obviously :r-dependent. The eigenfunctions fn(.r5) and 45,,(335) satisfy the equations —m%fn(r5) . 5 —M2¢n(r ) , (3.12) F film") 911(95) and the BCs. These requirements determine in", Mn, fn(:r5), and 0575(335) up to a normalization constant for the wavefunctions. A canonical normalization of the four-dimensional fields requires /W:d$5fn($5)fm(15):J55nm~ [OR (12:5 071(335),$¢m(5)=ggdnm. (3.13) Inserting (3.11) in the free part of the action (3.1), and using (3.12) and (3.13), leads 29 to 00 1 I Sfree = 2 [444514314 (9“"32 - (1-E)0“3” + 9“”77131) 43w n=0 1 1 +§(a,,7rg)2 — §gMfi(ng)2] . (3.14) This action describes a tower of mass-mu gauge bosons, and a tower of mass-JEM." Goldstone bosons. The Goldstone bosons can be removed from (3.14) by going to unitary gauge, 5 —> 00, except for the massless mode, if there is any. Since the natural size for the mass spacing is 1/ R, as R —+ 00 all gauge boson masses go to zero. This shows that compactification — that is, the acquisition of a finite value for R — acts as a “geometrical” Higgs mechanism, which breaks a countable infinity of gauge symmetries, with consequent generation of a tower of massive gauge bosons, and a tower of eaten Goldstone bosons. If the BCs on Ag are 65Ag = 0 on both ends, then the lowest KK mode in the expansion of Ag is a physical massless scalar field, n8, in the adjoint representation of the gauge group. As we shall see later in this chapter, such field is not essential for unitarity. Moreover, it does not behave as a Higgs boson, because it does not have the appropriate quantum numbers. Therefore, in the following we will always impose BCS which do not allow for such massless state to exist. 3.2 Symmetry Breaking by Boundary Conditions We have just seen that compactification is a symmetry—breaking mechanism. How- ever compactification cannot break the symmetries which are localized on the four- dimensional branes. Therefore, if the gauge group G is unbroken on both branes, then the four-dimensional theory will be G—invariant, and the mass of the lowest KK mode will be zero, m0 = 0. As an example, consider the gauge group SU(2), and impose the BCS , r5 = 0,7rR . (3.15) 30 N otiee that these are the orbifold BCS (3.7). Since only the derivative of A$(r, x5) is fixed by (3.15), the gauge fields can have any value on the boundaries. This means that SU(2) is unbroken on both branes, and therefore also in the four-dimensional theory. The BCS on A31 do not allow for a massless scalar, thus in unitary gauge we only have to consider Afi. The solution of (3.12) for fn(r5) is fn(r5) = An eos(mnzr5) + B", Sin(mn:1:5) . The BCS (3.15) imply 8,, = 0 and sin(mn7rR) = 0, whence ,n=aL2H.. (3m) mn = B As expected, there is a triplet of massless gauge bosons, due to the unbroken SU(2). Notice that the wavefunction of the massless fields is flat, which means that the corre- sponding particles have equal probability of being anywhere, in the extra-dimensional interval. A more interesting case is when only a subgroup of G is unbroken on the two branes. Then the four-dimensional theory will not be invariant under the full group C, but only under the subgroup which is unbroken on both branes. In other words, BCS can be used to break four-dimensional gauge symmetries. In order to better understand this concept, we consider three examples of symmetry breaking via BCS. (1) G=SU(2), and BCS breaking SU(2) down to U(1) at one end of the interval: 85.40: , 412:0, 8rA3=0, 5 o ’u r5 u 12 0 ‘1 ,r"=7rR. (3.17) These BCS can be seen as deriving from a 22 x Z2 orbifold on a radius-2R circle, ( A,,(r, «R + 1:5) 2 PA,,(.1:, 7rR — 1‘5)P_1 , Agendr+xh==—PAgxndt—x%P-1, B18) 31 where P = diag(1, -1) is the orbifold projection operator. The BCS (3.17) do not allow for a massless scalar, thus in unitary gauge Ag 5 0, and we can focus on the four-dimensional components only. There are two different KK towers, one for the charged sector, and one for the neutral sector. The KK expansions are2 r )= Z fn,(.’135)l'V,§1(:r) , 143(131155:209n(335)2np(f13) . (3.19) The eigenfunctions are as usual combinations of sines and cosines, and the BCS (3.17) lead to the mass equations. cos(mn7rR) = 0 sin(Mn7rR) = 0, (3.20) Here and in the examples below mn (Mn) is the mass of the n-th charged-boson (neutral-boson). The solutions are n—l 2 mn: R/ ,n=1,2,..., Mn=% ,n=0,1,2.... (3.21) The lowest mode of the charged KK tower is a massive particle. However the lowest mode of the neutral tower is a massless particle: It is the massless gauge boson of the unbroken U(1) symmetry. Finally, the normalized wavefunctions are 2 2 . fn(1:5) = ficoamnxo) , 7rR . 2 2 115(1-0) = 11% cos(Mn:r5), (3.22) which include the flat wavefunction of the massless neutral boson, 90(15) = const.. 2Here and in the following, the superscript “i” in A:t refers to the SU(2) linear combination (Al r iA2)/\/§. 32 (2) G=SO(4)~SU(2)LXSU(2)R, and BCS breaking SU(2)R down to U(1) at one end of the interval, and 80(4) down to SU(2)diagonal at the other end. In addition to the gauge fields A6131)! and Alli’M’ it is convenient to define A; M by A“ :tAa 311w:— --’--—-'—LM\/§ RM (3.23) Then the BCS read _ a __ a _ 60‘4Lp—0’AL5—0’ A“ _0 B'Aa _0 12 1.2 5 —M_ ’ 0 -5_ ’ 5 AR’u=0,05AR‘5=0, ,.’L‘=0 ,:r =7rR. 65A1u20, Ai5=0 3 __ 3 _ 8514R’u—0, 44125—0 (3.24) These BCS do not allow for a zero-mode scalar, thus in unitary gauge A355 E A315 E 0. As in example (1), there is a charged KK tower, and a neutral KK tower: Af11< .1 :15 =20 115(1 (1), A L11 1, :51: =20 gLn(l Mznp $1) AR/1( ~73 “75 =ZOfR11($ )W1111(I)1 11%,,(115): 209115(r5)z1.1(x). (3.25) The BCs (3.24) lead to the mass equations cos(2'm.n7rl1’.) = 0, sin(2Mn7rR) = 0 . (3.26) The solutions are n — 1 2 mn: 2R/ ,7221,2,..., n, Mn 2 ~27?- , n = 0,1,2... , (3.27) and the normalized wavefunctions are .— 2 1131) = (/,f—§,cos . 33 SU(2)L SU(2)1. 0 «R 311(2) 0 1rR 21rR Figure 3.1: Equivalence between the 80(4)~8U(2)Lx8U(2)R gauge. theory on a five- dimensional interval, with 80(4) broken down to SU(2ldiagonal at one end, and a single SU(2) with double interval length. - / 2 , fRn(:z:°) = :3]? cos (mn(27rR — 130)) , 2 gLn(:1:5) = W 73—22 cos(]lzfn:r") , an(;1r5) = 5% cos (ll/[11(27r/1’. — $23)) . (3.28) Comparing (3.26)-(3.27) with (3.20)-(3.21), we observe that model (2) can be obtained from model (1) by replacing R with 2R. In fact the boundary 7r]? acts as a mirror: The SU(2)R wavefunctions are just the mirror images of the SU(2)L wavefunctions, as a comparison between (3.28) and (3.22) shows explicitly. Then we arrive at the conclusion that an 8U(2)Lx8U(2)R gauge group in the bulk, with 8U(2)Lx8U(2)R broken down to SU(2ldiagonal at one end of the interval, is equivalent to a single SU(2) in a bulk with double interval length. This is schematically shown in Fig. 3.1 . (3) 0280(4)xU(1)~8U(2)Lx8U(2)RxU(1), BCS breaking SU(2)RXU(1) to U(1) at one end of the interval, and 80(4) to SU(2)diagonal at the other end. Defining the 34 A; fields as in (3.23), the BCS are 65A%H=O,A%5=O, ‘ 1,2 _ 1,2 _ 14:3,” — 3,, = mag/11,35 — 35) = o, 65(gg2Ag‘,” + 9,335) = 0 , 93214:},5 + 9335 = 0 , Ail, = 0, 05A‘l5 = 0, 65Af‘wzo, A1520, ,$5=7rR. (3.29) 85 B” = 0 , B5 = 0 Once again the gauge-field fifth components can be transformed away. As in models (1) and (2) there is a charged—boson KK tower, which is identical to the one of model (2), and a neutral—boson KK tower. The KK expansions are Am .1 1:5 =20 1111(1: 31:0 ) A1561: x5 =ZogL11(r (1,) AEA I $5=ZOfR11(I1111I( ) A115( 1:, m5 =20 9121101: (:r.) B,,( 1: 1:5 =2 h5(1: 255(1: 1:). (3.30) The BCS (3.29) lead to the mass equations cos(2mn7rl1’.) = 0, 2g;2 tan(Mn7rR) = 1 + é’ . (3.31) 95 The solutions are n — 1/2 = ' = 1.2,. .. , mn 2R 7 n . 0 n = 0 Mn: 2% garctan1/1+295 2/g5+n-1) 1121735,,” , (332) 2%; ——arctan(/1+2g5/gs+n) "=2141611-1 and the normalized wavefunctions are 35 115(15): jf—Rcos , 5 9' .fRn(-T )2 :\/17:_R 9595 5) 9L0( x r——' RQgV 5 + 92295 cos(M5,:r5 ) 9L11(33 V— :1? (g 29 5+ 29/2 2cos(Mn1rR) $5) 9595 95 2+2932 cos (mn(27rl? — 15)) , 9120(55 cos (Mn(27rR — 1:5)) 9Rn1 )=‘/W2—2—R 9525:2952 cos (1115(1)? 11 )) 11 _ . (3.33) In all these examples there is an unbroken U(1) symmetry, and therefore a massless 17t21. gauge boson. However, from a purely four-dimensional standpoint, it is not clear yet what symmetry is actually broken down to U(1). A better understanding can be achieved by putting the extra-dimensional interval on a lattice: The corresponding model will then be purely four-dimensional, with a larger symmetry group, but also with a clear symmetry breaking pattern. We will come back to this point in sec. 3.5. Also, we still have to show that symmetry breaking via compactification and BCS is soft (spontaneous), rather than hard (explicit). In other words, we must show that the symmetry-breaking mechanism preserves the special relation between gauge couplings which guarantees the cancellation of the energy-growing terms, in longitudinal gauge boson scattering amplitudes. This will be the subject of the next section. 36 Figure 3.2: Diagrams contributing to the elastic scattering AgAIT’, —> AfiAg: Contact interaction plus s-channel, t-channel, and u-channel exchanges. 3.3 Unitarity in Extra-Dimension In the last section we have seen that BCs may lead to different symmetry breaking patterns. We now want to show that symmetry breaking via compactification and BCS is soft. In other words, we want to show that the terms growing like E4 and E2, in longitudinal gauge boson scattering amplitudes, exactly vanish when the BCS satisfy (3.5). Let us consider the elastic scattering amplitude for the process AfiAg —> AgAg. In general we expect four diagrams, as shown in F ig.3.2: The four—point interaction, plus 3, t, and u exchanges of KK modes. We will assume that the external modes satisfy the same BCS, but we will not assume this for the exchanged modes. At high energy, the scattering amplitude can be expanded in powers of E / mn, where mm is the A: mass: M = MW (3)4 + Mm (ff—)2 + mm) + 0 ((mn/E)2) . (3.34) "In mn It might seem inappropriate to formally expand the amplitude in powers of E / m", when for a given energy E there is an infinite number of cxchanged KK modes whose mass is larger than E , and the series is potentially divergent. On the other hand. 37 imposing a sharp cutoff on the spectrum would explicitly break the gauge invariance. However such a hard breaking would have little effect on the scattering amplitudes, because the couplings with the heavy KK modes are suppressed. This can be seen by considering the higher-order gauge-invariant operators which result from integrating out the KK modes above a spectrum cutoff A. In D dimensions, an example of such operators is I FEWMNP , (335) where we are still using the normalization of (3.1), with —1/(4g2D) multiplying the kinetic term: Then (F M N)3 has dimension 6, whence the AG‘D factor in the denom- inator of (3.35). Alternatively, we can see that this operator contains three gauge fields, so if it comes from loops of heavy KK modes it should contain three powers of the coupling in canonical normalization, and zero powers in the normalization of (3.1). We would like to compare the contribution of the (F M N)3 and (F M N)2 operators to the fig/ll}, ——> AgAg scattering amplitude. The ordinary (F M N)2 operator gives a contribution of order 939mg . (3.36) This can be seen from the four-point interaction: There are two powers of the cou- pling, and four polarization vectors, each carrying a power of E / mn ~ ER. The contribution of (F M N)3 can potentially scale like E6. However this contribu- tion comes from two factors of 6,). V — EVA”. and two factors of the gauge fields. This implies that two of the polarization vectors appear in the combination pfleu — pueu, which, after substitution in the scattering amplitude, turns out to give a contribu- tion proportional to the mass of the external gauge bosons, rather than growing with )3 energy. Therefore, the (F M N term gives a contribution of order 931) 2 A6_DE , (3.37) where two powers of g D come from the fly/1,, — (iv/l), terms, and the remaining two powers come from the quadratic term. The ratio between the (F M N)3 and (F A 1 N)2 38 contributions is 2 91) 94 A6— DE2R4~ (AR)6“D(ER)2 where we scaled 90 as 94 2RD 4. For E >> 1/ R AR > 1, and D < 6, the contribution (3.38) from the large-n KK modes is suppressed. This remains true also when the non-trivial cancellation of the (E R)4 term in (3.36) is taken into account, as long as AR > 1 and D < 6. For a five-dimensional theory AR can be as large as 247r3 [68]. In practice we can therefore extend the series of virtual KK modes to infinity: It is just a simple way to preserve the gauge invariance. Then we obtain the following coefficients for the expansion (3.34): M“) = (33mm,— €329,179.) [( (3 + 6cos9 — cos2 9)fab6fcde +2(3 — cos2 6)fa'cefbde] , (3.39) Ala) : 7—,12—i—facefbde (4937111112 mn- 3 Zgnnkfilk) n _ 57:12 —fab€dee [49mmn7nn_ 3 Zgnnk All; + (1293",,"7713, + 293,”).(3n—zf. — 16mg») cos 6] , k (3.40) where 93mm is the contact-interaction coupling, gunk is the coupling of the external modes to the k-th exchanged KK mode, and M k is the mass of the k-th exchanged KK mode. Notice that the KK indices should be interpreted as double indices, including both KK and color index, 8.9., k —> (k, 6). Since the external modes are assumed to satisfy the same BCS, the KK index n is color-blind. In the expression for M (2), the Jacobi identity on the structure constants has been used. This requires summations like 2k gunk or :1: gfiifiUWf)? to be independent of the color index 6, which will be confirmed below. If 93”,," = 2k yank. M (4) cancels, and M (2) becomes M(2>=L2(4g%nnnm fi—3zginwf)f3gk = 4771?. ("Eda L363) . k (3.46) Using the equations of motion Ugfn = —m%fn and aggk = _A/Ik2‘gk’ integrating by parts, and using the completeness relation, it is not difficult to show that (3.46) is satisfied up to contact terms like [fnf£]6R, [fn 2 ’ gka, and [£1];ng R, which vanish for Dirichlet or von Neumann BCS. Therefore, we arrive at the conclusion that the BCs (3.8)-(3.10) guarantee the cancellation of both E4 and E2 terms in longitudinal gauge boson scattering amplitudes. Notice that mixed BCS, eq. (3.45), only insure 3Notice that (3.10) is a von Neumann BC in unitary gauge, Ag E 0, or for Ag vanishing on the boundaries. 40 the cancellation of the E4 terms, but the E2 terms are in general non-zero. In fact mixed BCS can be obtained by including brane mass terms for the gauge fields, which explicitly break the gauge invariance. A brane localized Higgs field would then be necessary to cancel the bad high energy behavior. The cancellation of the E4 and E2 terms delays unitarity violation to higher energy scales rather than restoring unitarity at all energies. In fact even after the cancellation there is still a logarithmic growth in the partial wave amplitudes, which becomes more and more important as the number of the exchanged KK modes is allowed to increase. The high energy behavior will be analyzed in chapter 4 for a specific model of EWSB. However the scale of unitarity violation can be estimated by taking the extra—dimension to be infinite in size, because the high energy limit corresponds to distances short compared to the interval length. Then the only mass scale is l/gg, and we therefore expect unitarity to be violated at energy scales of order 1/gg times a numerical factor [34] [37] [38]. 3.4 Unitarity and the KK Equivalence Theorem In section 3.3 we have seen that the terms growing like E4 and E2, in longitudinal gauge boson scattering amplitudes, exactly vanish for BCS consistent with the vari- ational principle. This can be seen also by using the Goldstone boson equivalence theorem. In a five-dimensional theory the Goldstone bosons are the KK excitations of the gauge field fifth component. This interacts with the four-dimensional components via cubic and quartic terms in the action (3.1): Sgauge 3 / d4m [NR dsr5 __1_Farpaus 0 29% #0 NR 1 3 / (14:1: [0 (19:5 [fabcag—g (6,,Ag — 35.42) Abf‘Ag 1 +2—7fabcfadeAzAgAd’1AE] . 95 (3.47) 41 A quick dimensional analysis shows that the Goldstone boson scattering amplitudes cannot grow with energy, because each cubic vertex carries only one power of mo- mentum. However we have seen in section 3.3 that in longitudinal gauge boson scattering amplitudes the term growing like B2 only cancels for BCs consistent with the varia- tional principle. (While the term growing like E4 cancels for a broader set of BCS.) On the other hand we have just seen that the Goldstone boson scattering amplitudes cannot grow like E2, by simply power counting. This again proves that symmetry breaking via BCS which are not consistent with with the variational principle is not soft, and the equivalence theorem does not apply. It is not our intention to give here a proof of the KK equivalence theorem, but we want to show that the KK excitations of Ag behave properly as eaten Goldstone bosons when the gauge fields satisfy the BCS (3.8)—(3.10). In the action (3.1) the quadratic term mixing the u-component with the 5-component is R 8(2) = [.143]; dx5[—i265Aa#a,,A:] mixing = —12-.[4a#a Ag]0 "R + f (14:); /OR (1.55 —A“ 3,35/13 95 The contact terms vanish for BCs of the type (3.8)-(3.10), and the last term has the right form to be a gauge-Goldstone quadratic term. For simplicity, we consider BCS which do not depend on the gauge index (1. Then, using the expansions (3.11) leads to M)A?#(x)¢t¢n.(r5>¢n2(:c’ 5) = 6(m5 — :65) , (3.54) which is valid for a broader set of BCS, (3.53) becomes N3, = [”3 c1159- x5)fma¥( 5) = g—g-[fmfZlS’in— f,” (115521m<5>m<'-5>. (3.55) 4Strictly speaking the BCS (3.10) involve a non-trivial dynamics on the boundaries, which requires a more detailed analysis. Here we only demand that (3.10) is trivially satisfied by the choice 8514;: = 0 and Ag '2 0, in which case the contact terms in eq. (3.53) vanish. 43 The contact terms [fmfln] 3R are zero under the same conditions which guarantee that lfm¢nl3R = 0. Therefore, using the equation of motion (3.12) and the normalization condition (3.13) we obtain (3.52). With these results, the KK equivalence theorem for the Ami), ——> AgAg elastic scattering reads M(A%A% —> 442.4%.) = Cmod ((405353 —> 55.55;) + 0 (mu/EV) , (3.56) where the radiative modification factor Cmod = 1 + 0(loop) arises only at one- loop level [15] [16] [17] [18] [19]. We have already noticed that in general the eaten Goldstone bosons are not mass eigenstates. For the 53,59. to be mass eigenstates, with the same masses mm of the KK gauge bosons Ag, (in Feynman-’t Hooft gauge), the condition 552(9) = mnfn(:c5) (3.57) must be satisfied. For instance, this is true for the BCs of an orbifolded circle, eq. (3.7), because the corresponding gauge-boson and Goldstone-boson wavefunctions are cosines and sines, respectively, of the same argument [34]. 3.5 Deconstructed Models The KK expansion is a way to see five-dimensional gauge theories on an interval from a four-dimensional point of view. A different, purely four-dimensional approach is provided by deconstruction, in which the extra-dimensional interval is put on a regular lattice [39] [40] [41]. In the five-dimensional action TTR 1 S = / (1426 / drr5[— 2 _ 4,, ”R ,,5 1 a any 1 a (1)15 _ / d1. /0 (1.1. [—49§F,,,,F —QF#5F ] (3.58) a aM N FMNF ] the $5 coordinate is replaced by a discrete index j, and the integral in c1235 is replaced by a summation over j. 44 3+6!) ................ 4 ja (j + l)a 5 -—> Figure 3.3: Rectangular Wilson loop in the (n, 5) plane, with the fifth-dimensional interval on a lattice. In order to evaluate F115, we consider a rectangular Wilson loop in the (p, 5) plane, as shown in Fig. 3.3. We denote the unitary link operator (or comparator) between the points ($1,115?) and (2:2, 3:3) by U($1,:17§l.7?2, $3), and the position in the latticized fifth dimension by ja, where a, .=_ 1rR/ (N + 1) is the lattice spacing, and N + 2 is the number of points in the lattice (N internal plus the two endpoints). The U (X|Y) operator is defined by its transformation law under the gauge group, - . , . b b U(XIY) _. e55aGaU(X|Y)e—55 (ch , (3.59) where the 00‘s are the group generators, and the a(X)a’s are the transformation parameters. (Here X and Y are points in the five-dimensional space.) Therefore, the trace of the unitary operator U(;1:,j(:.) E U(.17,ju,|:r., (j +1)a) U(:r.(j+1)al:r+ ([1, (j + 1)a.) U(1)? + ([1, (j + 1)a|.'1t + ([1, ja) U(l‘ + (fi,ja|:1‘,ja) , (3.60) where )7. is a unit vector in the y. direction, is invariant under a gauge transformation. For 6 << 1, we can express the link U(zr,ja|:r. + ([1, ja) in terms of the gauge field )1-th 45 component, .. —‘A +5“; +0 3 U(x,ja|:r+ep,]a)=e 16 ”(3E WHO) (6 ), (3.61) where A“ E AfiGa. On the other hand, U($,ja|:r, (j + 1)a) cannot be properly expressed in terms of A5, because a is not an infinitesimal quantity. Then we define the 23(6) field by Zj(r) E U(x,jal:1:, (j + 1)a) . (3.62) Its transformation law can be derived directly from (3.59), magma —mb (5)05 Ej(1:)—>e J Xj(:r)e j+1 (3.63) where (155(1) E a“(;1:,ja,). (The “color” index a should not be confused with the lattice spacing a.) Expanding in powers of c, the operator (3.60) becomes U(m,ja) 2 1+ 622(5) (DA-(270* + 0(8) , (3.64) where [)#Zj 2 0H2]. — iGa/l?_1#2j + iEjGa/l?“ , (3.65) and A- (1‘. E A (:13, ja) is the gauge field corresponding to the j — th rotation, 111 #1 cry-(2:) = a(:r,ja). DMZJ-(x) is the covariant derivativeof 23(3). In fact, from the transformation law of the five—dimensional gauge fields, . b b . 55,,(X)G5 —+ e55 6103”) (A315#(r)Ga + 26),) e 201(I) , (3.67) and D#2j(;r) transforms like Zj(1:). Therefore, Tr ((D#Zj(1:))lD“EJ-(:r.)) is invari- ant, and D#2j(;r) must be proportional to 35(33, ja), the deconstructed version of Fg5(:r, 3:5). The proportionality factor can be found by letting a to be small, so that 2j(1:) can be expressed in terms of A5(X) E Ag(X)Ga: . . 1 22(6) 2 U(x, jalx, (j + 1)a) = e"'“‘A5(“'U+?)‘2. (3.68) 46 Expanding in a, and taking the Ely-derivative, gives , i FS5(I,]CL) = EDpzj-(x). (3.69) Therefore, replacing the integral over 15 with a sum over j, ”R dy —-> Nil a, (3.70) 0 j=0 the Lagrangian of a deconstructed five-dimensional gauge theory becomes 13— "LINEN F55V+fi1§ln((0 sfipus) (3 71) _ 4&2 j=0 JHU j 4 j=1 II J J ’ ° where Film; is the field-strength tensor for the gauge field Ag”. The dimensionless four-dimensional gauge coupling 27 is - _ g5 _ 95 N _ _ \/ +1, 3.72 9 ,/5 7m, ( ) and the dimension-one f constant is 2 2 — 95\/a — gsVnR Notice that both 5 and f grow like VN + 1, but what really enters in the calculation f V N + 1 (3.73) of scattering amplitudes are the effective coupling g/JTV—fi and the effective mass scale f / m , which are independent on the number of sites. This results prove what was claimed in chapter 1, namely that deconstructed five- dimensional gauge theories on a flat background are NLSMs with identical couplings and VEV's. The Ej field can be expressed in terms of the Goldstone boson fields: 2559(5)Ga / f 6 J . Zj(:r) = (3.74) Comparing this equation with (3.68) we see that the Goldstone fields are related to the fifth component of the gauge field, 7135(5) = —§A‘5‘ (x (j + %)a). (3.75) Therefore, in deconstruction the four-dimensional components of the gauge fields are taken at the lattice points, while the fifth component is taken between the points. This picture corresponds precisely to the circles and lines of a moose diagram. 47 SU(2) SU(2) SU(2) SU(2) U(1) Figure 3.4: Moose diagram for the deconstructed SU(2) gauge theory on a five- dimensional interval, with BC’s breaking SU(2) to U(1) at one end of the interval. Notice that: (i) If the five-dimensional theory is invariant under a gauge group G, and BCS break G down to GO and Cl, at 2:5 = 0 and 1:5 = 7TH, respectively, the corresponding deconstructed four-dimensional model is invariant under a larger gauge group, GO >< G'N x G1. (ii) In the five-dimensional theory spontaneous symmetry break- ing is due to compactification, while in the four-dimensional deconstructed model spontaneous symmetry breaking is achieved through the 2 fields VEV, Z? = f - 1, where 1 is the identity operator. (iii) As the mechanism which leads to compacti- fication is not explained in the five—dimensional model, and must be supplied by a UV completion of the theory, so the deconstructed model does not explain the ori- gin of the VEV, and must be UV completed by a more fundamental theory whose low-energy content is described by the N LSM. Deconstruction makes the symmetry breaking pattern more explicit. For example, a five-dimensional SU(2) gauge theory, with BCs which leave the symmetry unbro- ken on both branes, corresponds to an SU(2)N+2 gauge symmetry which is sponta- neously broken to SU(2) by the 2 fields VEV. We now consider the three examples of section 3.2, with symmetry-breaking BCS, and deconstruct the corresponding five- dimensional theories. (1) G=SU(2), and BCS breaking SU(2) down to U(1) at one end of the interval. The deconstructed model is an SU(2)“,+2 NLSM whose SU(2)N+1x U(1) part is gauged, with the U(1) coupling identical to the SU(2)N+1 coupling. With the BCS 48 U(1) SU(2) SU(2) SU(2) 2N+2 2N+1 2N 2N-l Figure 3.5: Moose diagram of the deconstructed SU(2)LXSU(2)R gauge theory on a five-dimensional interval, with BCs breaking SU(2)R to U( 1) at one end of the interval and SU(2)L> 00 models, with the latter corresponding to the extra-dimensional case. 4.1 Higgsless Models on a Warped Background In section 3.2 we considered three examples of Yang—Mills theories on a flat extra- dimensional interval, with BCs breaking the gauge symmetry. These models share some common features: First, the unbroken symmetry is U(1). Second, there is a charged-boson KK tower and a neutral-boson KK tower. Third, the lowest mode of the neutral tower is the massless gauge boson of the unbroken U(1) symmetry. In section 3.5 we considered the corresponding deconstructed versions. Each of them has a symmetry breaking pattern which contains SU(2) x U(1)—>U(1), and is therefore potentially a model of EWSB. Model (1) has an SU(2) gauge symmetry in the bulk, with BCS breaking SU(2) to U(1) at one end of the extra-dimensional interval. We found that the mass of the lightest massive neutral boson is twice as large the mass of the lightest charged boson, and so they cannot be interpreted as the SM W and Z bosons. Therefore, model (1) has obviously no chance of being a realistic model of EWSB. Neither does model (2), which we saw being equivalent to model (1). Model (3) has an 80(4)xU(1) gauge symmetry in the bulk, with BCS breaking 80(4)~8U(2)Lx8U(2)R to SU(2)diag at one end of the interval, and SU(2)RXU(1) to U(1) at the other end [36]. The mass of the KK modes are given by (3.32). For the lightest massive charged and neutral boson we have1 1 mW'O : 74—H- 1 12arctan(/1+2 ’2 2 95 /95 Notice that for g3 = 0 we obtain mWO = mZO’ while for O < 2g?)2 < g?) the neutral boson mass becomes slightly larger than the charged boson mass, as in the 8M. To check whether this can be a realistic model of EWSB, we must also consider the heavy KK modes. For le and le (3.32) gives m W1 2 m WU + 2mWO 1Here we substitute the KK index n of (3.32) with n —— 1, so that the lightest massive modes correspond to n = 0, rather than n. = 1. 52 77121 = mZO + 2mm;0 . (4.2) These values are very low, and can only be realistic if the fermion couplings with the heavy gauge bosons are suppressed. In five dimensions the smallest irreducible representation of the Lorentz group is four-dimensional, thus our fundamental objects are Dirac spinors. In order to obtain a low-energy effective Lagrangian in agreement with the 8M, we introduce two fermion SU(2) doublets, \II L and \IIR, whose Lorentz—Dirac structures are ‘I’L ___ WI. \IIR : X12 XL we To be more specific, 1/2L and XL (11)]; and XR) are doublets under 8U(2)L (8U(2)R), and singlets under 8U(2)R (8U(2)L). The U(1) charge is chosen to be (8 - L)/2 for both W L and ‘11 R1 where B stands for baryon number, and L for lepton number. Then the five-dimensional Lagrangian for one generation (of quarks or leptons) is 6‘5) = i: L115“ (6M — 1.42MT5 — isMg—g—L) 111 L fermion _ , B - L + \I/Rirjw (811,] — i/l‘IIina — i8)” 2 ) ‘I’R . (4.3) The matrices T“ E a“ / 2 are the SU(2) generators, where a“ is the a-th Pauli matrix. The matrices I‘M are the five-dimensional version of the four-dimensional 7" matrices, and are defined by I‘M = (7“, —2'75) [71]. In this Lagrangian we omitted the mass terms, which are not of our concern now. Since the electroweak symmetry is unbroken on the 3:5 = 0 brane, it is natural to try first with fermions which are strictly localized at x5 = 0. Using the BCS (3.29) for the gauge fields, working in unitary gauge, (all gauge-field fifth components equal to zero), and imposing the BCS XL(17, 0) = X1295» 0) = 0 for the fermion fields2, the fermion action becomes fermion 77R r r 5 Sfermion : /d4-T‘/(; d$06(-To)£() - . . . B —- L = [$520 (14.1: [mus/H (0)) — 2.4%“Ta - zBflT) 'de 2Consistent BCS for fermion fields are discussed in Ref. [71]. "ti/912M (0)21. —iBu M(T3+ +B;L))1[;R [T520 (141 [ll—127” (3— '- iByY) I/lL +£32.75! (8,2 - inY) LT’l/iR] (4.4) where we used the 8M relations (B—L)/2 = YL and T3+(B-L)/2 2 YR. Notice that YL is proportional to the 2x2 unit matrix, while YR is diagonal but not proportional to the unit matrix. Writing 1123 in terms of its SU(2) components, 11;}; == (uR,dR), we recognize in (4.4) the electroweak fermion Lagrangian, with the four-dimensional gauge fields replaced by the five-dimensional fields All.“ and Bl), taken at x5 = 0. In order to evaluate the effective couplings of the fermion fields with the KK gauge fields, we must substitute the expansions (3.30) in (4.4). The charged-current and neutral-current Lagrangians are .CCC = i [90C 1,)5/“PL)T+7,/ W +] + h.c. / n=0 fl n“ 1 LNC = Z [1177” (5771:.ICPLT3 + QIQVEQ) d" Zia/1.] a (4-5) where (D E 1le + 1,1113, PL E (1 — 75)/2 is the usual left—handed projection matrix, Ti 5 T1 :1: 3T2 are the isospin raising and lowering matrices, and Q E T3 + Y is the charge matrix. The effective couplings are related to the Wn and Zn wavefunctions at 11:5 = 0: 6.875" = 152(0) . of.” = 1115(0) — 112(0) . anC = 1112(0). (4.6) From (3.33) we obtain (:0 = 95 9" 71R. ’ NC /_ 2 9,, = 7rR 95 sin(Mn7rR), .2 2 ’2 98,? = ———g§—cos(Mn7rR) . (4.7) 77R V952) + 2932 54 From these equations we notice that the fermion couplings with the heavy KK gauge bosons are not suppressed, relative to the couplings with W0 and Z0. Therefore, the values (4.2) for the first heavy KK modes are well below the experimental lower bounds from the direct searches [72] [73]. In order for the extra-dimensional model to be realistic, le and m 21 must be heavier, and this can be achieved by replacing the flat background with a warped R81 metric. The latter is given by d52 : e‘2kxonflyd1‘“dr” — dx5dx5 , (4.8) where, as usual, :55 E [0, 71R], and It measures the Ad85 curvature. With an exponen- tial factor multiplying the Minkowskian metric, the EWSB scale and the Planck scale can be naturally embedded in the same model, for a factor k7rR of order 108(MPlanck /TeV) ~ 30 is sufficient to achieve the goal. The metric (4.8) is often written as (1.92 = (751—)? [7)de”d:r” — (d:)2] , (4.9) where z is defined by e555 , (4.10) and belongs to the interval 1 Elm}? (2h E ‘1‘“) S 3 S (2125 k ) , (4.11) is of order of the Planck scale, and 5,71 is in the TeV range. (The subscripts . 2-1 “here ~h “h” and “v” stand respectively for hidden and visible.) With the warped metric (4.9), the wavefunctions are superpositions of Bessel functions, rather than sines and cosines. Therefore, mass equations and normalization integrals become more complicated than in the flat-metric case [47]. For the gauge boson masses, a perturbative expansion in 1/ log (322/13,) gives, to leading order, 1 1 mw = ————, 21; ~ 10 (35) g “h 2 12 9. +29 1 1 m2 = ‘liQ—Tfé—T—a (4.12) 95+95 ~11 10g(£1;) 55 for the W and the Z boson, while numerical results show that the W1 and Z1 masses are around 1.2 TeV, heavy enough to have evaded detection at the Tevatron, but within the reach of the next generation of colliders. 4.2 Brane Kinetic Terms The main motivation for using the Ad85 geometry is of course the large hierarchy between the TeV scale and the Planck scale. However, since we are only interested in the EWSB scale, it is sufiicient to consider an effective field theory where the high momentum modes - all the way down from the Planck scale to the TeV scale —— are integrated out. In other words, for our purposes it is sufficient to consider an effective theory with a new hidden brane bounding the space at zv > a >> zh, with the requirement for the new theory to reproduce the same physics from the point of view of an observer living at z > a [56] [57] [58]. The resulting five-dimensional model has still the same bulk and TeV brane Lagrangians, due to the conformal invariance, but new kinetic terms localized on the hidden brane. For example, for a free photon field the coefficient of the localized kinetic term runs like [57] where 1/e2(zh) is the coefficient of the initial kinetic term localized on the Planck brane. If a large slice of Ad85 is integrated out — that is, if a/zh >> 1 — the corre- sponding extra-dimensional interval will be approximately flat, as shown in Fig. 4.1, and the integrated-out region will generate large kinetic terms on the UV brane. These results lead us to believe that the 80(4) x U(1) model with a flat background, and large kinetic terms on the brane where the electroweak symmetry is unbroken, can potentially be realistic. Of course a model like this is only an effective field theory with a cutoff in the TeV range, and an unknown UV completion. However this does 56 Planck TeV (a) Figure 4.1: Background of a Randall-Sundrum model (a). The same model after integrating out a large slice of AdS5 near the Planck brane (b). The new model has a smaller radius, an approximately flat background, and a large brane kinetic term on the UV brane. not bother us. The corresponding action, for the gauge sector, is3 7I'R 1 1 = m 651—2A2.221M~—222M.22MN 0 495 495 1 MN 6(135) a and! 601:5) nu ZEBMNB "' VALIIVAL — —4g—QB“VB ] , (4.14) With the BCS A255,,(0) = 0 .Ai’aw) = 8,.(0) . Ai#(TrR) = 0 ,35 1#(7FR)= 0 . (4.15) From (4.15) we see that the last term could have been equally written as 6(175) 3 3,141/ _WAR#VAR . This model was first introduced in Ref. [49]. Notice that the presence of 6—functions on the 3:5 = 0 brane generates discontinuities in the 85-derivatives of the gauge fields. 3Here and in the following we omit the gauge-fixing terms. We always assume BCs which do not allow for a scalar zero mode, and work in unitary gauge, where all gauge-field fifth components are set to zero. 57 This can be seen by pushing the brane kinetic terms slightly away from the boundary, to make them part of the bulk. Then the equations of motion for Ail], 14%. grand Bu become “gig-(0” ALpu "‘ EabCALpl/AL (’5 Lu) + $25035 - 5) (0“ ALW " 5m AL)“; A L“) = 01 i (0"41251/ _ 5.3554,,WAR 5 — 53.4%,”) + 5,6(55 — 1) (MARW— 5355.412,” 5") = 0, 91? (8118“,, B2B")+ +9126($5 — {)6} Bull 2 0 ’ (4'16) where 0 < 6 << 1, and cube is the SU(2) antisymmetric tensor. An integration around the delta functions picks the discontinuity of 65/13], 65422“, and 85B“: 1 _ bc . _g—2la5AL1/lc_ + 7 (aflAipu— Ea Altai/A Lp)e : 0 ’ 10 —;};[65AR,,]:+ 9,2 ~1—2(8"ARW— e3bcARWAj,*‘) :0, + 1 —E(653.]:_ + F (658...), = 0 . (4.17) With the brane terms as part of the bulk, the BCS 05A‘Izl#(0) = 0 and 65(gg2A%# + ggBu(0)) = 0 should be imposed, since these are the BCs which leave the electroweak symmetry unbroken on the 2:5 = 0 boundary (see eq. (3.29)). Then, taking the limit 6 —> 0, and using the bulk equations of motion, (4.17) gives g2 lim (6% g], — 9735/13) 2 5 x5—>0+ I2 lim 8.2013 +B )- L8v(g5 A3 +gr 2B ,2)) =0. (4.18) These equations, together with the BCS (4.15), give rise to non-trivial mass spec- tra, for both charged sector and neutral sector. We do not show here the solutions, since our focus will be on a simpler model, which will be introduced in the next section. However, it is clear that with a flat background the wavefunctions are super- positions of sines and cosines, rather than Bessel functions. This makes this class of models considerably simpler than the warped extra-dimension scenario. 4.3 A Minimal Higgsless Model In section 4.2 we introduced an 80(4) xU(1) gauge theory on a. flat extra-dimensional interval, with large localized kinetic terms on the brane where the electroweak sym- metry is unbroken. The deconstructed model is represented by the moose diagram of Fig. 1.2 (a), where the couplings of the SU(2) and U(1) groups corresponding to 21:5 = 0 are the gauge couplings of the brane fields. The same moose diagram can be unfolded to a single chain of SU(2) groups followed by a chain of U(1) groups, as shown in Fig. 1.2 (b). A simpler model can be obtained by eliminating all U(1) sites, with the only exception of the first one, which corresponds to the U( 1) gauge group on the 2:5 = 0 brane. What is left is an SU(2)N+2 NLSM whose SU(2)xSU(2)NxU(1) part is gauged, and the corresponding moose diagram is shown in Fig. 1.3. In this section we study the gauge sector of this model for N = 1, arbitrary N, and N —> 00, where the latter corresponds to the continuum limit [42]. We find the unitarity bounds of longitudinal gauge boson scattering amplitudes for each case, where our analysis is restricted to unitarity of the WEWE —> WZWE scattering. We will work in tree-level approximation throughout the rest of this dissertation. 4.3.1 The SU(2)0xSU(2)1XU(1) Model We begin by studying the simplest Higgsless extension of the SM model, namely an SU(2)0xSU(2)1xSU(2)2 NLSM whose SU(2)0xSU(2)1xU(1) part is gauged. The corresponding nioosc diagram is shown in Fig. 4.2. The NLSM fields, 21(;1-)=1g2i”f‘(I>T“'/f1 , 29(;1~)=112"'”‘2"($)T“/f2 (4.19) consist. of two SU(2) triplets, which are coupled to the. gauge fields by the covariant derivatives4 01121 = (91131 -1th“141'5',,21 + igzlrau'fi, . 4In this section and in the next. one we will work in canonical normalization. where the coefficient of the gauge kinetic terms is -1/4. 59 SU(2) SU(2) U(1) Figure 4.2: Moose diagram for a global SU(2)0xSU(2)1xSU(2)2 NLSM whose SU(2)0xSU(2)1xU(1) part is gauged. All parameters are taken to be independent quantities. 0,22 = aRzg —1'gTan,,22 +1g’22T3BR . (4.20) Notice that, in order to maintain generality, we take the VEVs of the 2 fields to be independent parameters. The Lagrangian for this model is 1 , 1 , 1 ‘C : ”Zng/l’l’gm "ZM'fuum/flpl/‘ZBWBW 2 2 1 + %Tr((DR21)TD”21)+54llr((0#22)10"22) , (421) where we only kept the lowest dimension opertors. After the 2 fields acquire the VEV, < 2,- >= 1, the SU(2)0xSU(2)1xU(1) gauge symmetry breaks down to U(1), and the last two terms in the Lagrangian become mass terms for the gauge fields. The mass spectrum consists of a neutral massless gauge boson, which will be identified with the photon, a tower of two charged gauge bosons, and a tower of two neutral gauge bosons. The light modes of these towers will be identified with the SM W and Z bosons. The heavy modes are two new particles, which will be denoted as W’ and Z ’ . There are overall five independent parameters: g, g, 9’, f1, f2. we can trade three of these for the electromagnetic coupling, e, and the W and Z boson masses, mw and m Z- The remaining two parameters can be expressed in terms of the W' and Z ’ masses, mW/ and "’2’- The gauge eigenstates, W61, W1", and B can be expanded in terms of the mass eigenstates. We have ”bi = a00Wt + 001””i 1 60 W 5+ $1 5+ 7+2 + Z, 21 fl 17, Figure 4.3: Tree—level diagrams for the 142' W; ——> 14’; WE elastic scattering, in the SU(2)OXSU(2)1> W; H L scattering. In addition to the SM exchanges of virtual photons and Z bosons, there are exchanges of virtual Z’ bosons, in the s- and t-channel, as shown in Fig. 4.3. The amplitude is an easy generalization of (2.10): 2 gr ,' f f . M = ”—“4M [p2E2(—2+6cosl9) — 15491129] m”; l. 2 2 2 1 e 9 1 . g , ,. , + —4 — + H H g + WWZ2' (-4p2(p2 — 3E2)2) C080 m”: S S — 7712 8 — m2, 1 e2 9 g ,r I + _+_ HHZ+ ”VIZ 111%,, t I. — mQZ I — 111%,, h x [ — 4E2 (p2 + (E2 — 2112) cos 6)2 —2p2(1 + cos 9) (2E2 —- p2 — E2 cos 9)2] . (4.25) The quartic and cubic couplings are obtained by inserting the expansions (4.22), 62 10001 1 1 ~ 1 / 800 ’ f2 / 600 '1 / / 400 ~ / fl ¥ f (GeV) 200 ’ 0.01 0.1 l 10 100 mzr -mw1 (GeV) Figure 4.5: The quantities f1 (blue) and f2 (red) as a function of the Z’ and W' mass difference, with mW/ = 500 GeV fixed. The vertical line indicates the position where 77122, _ ma” = 771% —' may. (4.23) into the gauge kinetic terms of the Lagrangian (4.21). This gives gwwz = 9 01210500 + £7 aioblo , gWWZ’ 9 01210501 + g afobll 1 Ham/WW 92 030 + 62 Clio - (426) At high energy the amplitude can be expanded in powers of E / mw. The term proportional to (E /mW)4 exactly vanishes due to gauge invariance. The leading contribution is then proportional to (E/mW)2: 2 E 1 , 6 M = ( ) + COS K + 0 ((mW/E)0) , (4.27) mW 2 where 1 _ 2 _ 3 2 2 , 2 2 K — 4!]“rwyflfv‘; m‘f‘f (TIIZ'WZ + IIIZIQ‘%I”_Z’) . (4.28) Using the formulae in appendix A, we can treat K E K (mWI, m Z’) as a function of mwl and m2]. 63 mzr—mw1 (GeV) Figure 4.6: The coupling constants 09 = 92/47r (blue), 0 I = g’2/47r (red), and 9 of] = 52/471 (green), as a function of the Z' and W’ mass difference, with mW/ = 500 GeV fixed. The vertical line indicates the position where 171%, — ma” = 77122 - may. In Fig. 4.4 we plot K as a function of the mass difference, m Z’ - mWr, for mW/ = 500 GeV fixed. As a comparison we also plot the same quantity in the SM without the Higgs boson. Notice that K is significantly suppressed for 771%, — Inf/V, 2 17122 — may. When this relation holds, the value of K is reduced by almost precisely a factor of 1 / 4, a result which does not depend on the particular value of mW” This indicates that the unitarity violation that occurs in the SM without the Higgs boson would be postponed to higher energy in this model. We also plot in Fig. 4.5 the scales f1 and f2, and in Fig. 4.6 the couplings constants ag = 92/4713 agr = 9'2/47r, and ag = 52/471, as a function of the Z’ and W' mass difference, with mW/ = 500 GeV fixed. We notice that the relation m2Z, — 771%,, 2 77122 - ma, also corresponds to f1 2 f2 and § >> 9, g’. In fact when this relation holds the couplings are given to a good approximation by g = e/ sin 6W, 9' = 19/ cos 6W, and 2 w' ' definition of cos 6W = mw /m Z.) Thus, the SU(2)0 and the U(1) act approximately 1} = (mW1/2mw)g, up to corrections of order may / m (We have used the tree level 64 0.4? , g 0.3 » 514/ i 0.1 I / 5 0 500 1000 1500 2000 2500 3000 \5 (GeV) Figure 4.7: The J = 0 partial wave amplitude as a function of \/E for the SM without a Higgs boson (red) and the SU(2)0xSU(2)1xU(1) model (blue) with mW/ = 500 GeV and mzz, — ma” = m22 — may. like the SU(2)L and U(1)y of the SM, while the intervening SU(2)1 has the effect of softening the unitarity violation of the SM WZWI: —+ WZ'WE scattering. We can observe the effect of the delayed unitary violation by plotting the J = 0 partial wave amplitude as a function of \/§ = 2E. This is shown in Fig. 4.7 for both the SM without a Higgs boson and in the SU(2)0xSU(2)1x U(1) model with mW’ = 500 GeV and mQZ, — ma” = 77222 — mgv. Since unitarity requires |Re (10] < 1 / 2, we can use this figure to infer that unitarity violation in this amplitude has been postponed from a scale of J3 2 1.6 TeV in the SM without a Higgs boson to J3 2 2.65 TeV in the SU(2)0xSU(2)1xU(1) model with this choice of parameters. We have found that the behavior of the W3 WE —> Z LZ L amplitude to be essen- tially identical to that for W; ”I: —-> WZ' WE . In particular the corresponding value of K, the coefficient of the leading E2/mgv term in that amplitude, is reduced by the same factor of 1/4 when 211%., — 771%,, 2 77122 - may. 65 SU(2) SU(2) SU(2) SU(2) U(1) Figure 4.8: Moose diagram for the SU(2)xSU(2)N> mu, — —_4(N+1) 1+O()\) , 2 I2 2 m2 = W0+0020 , 2 2 ,, _ 7171' 7171' "Lg/1,7,1 : g2f2 (SID. m) +2',Tn%t (COS m) (1+O()\2)) , 2 2 2 — *2 2 ' _"_’7r_ 2 l— 2 ng — gf (51n2(N+1)) +2mz(cos2(N+1)> (1+O(A)). (4.35) It is easy to check that for N = 1 this gives mQZ, — ma” 2 "’22 — mar, and g = (mWI /2mW)g, up to corrections of order "lar/ m as found in section 4.3.1. 1 2 w{’ The scattering of longitudinal W's is easily generalized from the N = 1 case, since the exchanges of a single Z ' in the s- and the t-channel are replaced by echanges of N 67 I 17+ 11 L w; + I. + 7' + Z ‘J’ \.’ \/ \.' 7+Z Figure 4.9: Tree-level diagrams for the W3 W; —+ WIT ”I: elastic scattering, in the SU(2)xSU(2)NxU(1) model. heavy neutral bosons, ans shown in Fig. 4.9. The amplitude for WE W; —> W; WI: is M 2 .(l 7 ,r r r W [p252(_2 + 6c056) — E4 sin2 a] mw ' 2 2 N g 1 e2 9 z ’ __4 _ + —” W? + E —WW§" (—4p2(p2 — 3E2)2) 0056 771W 8 S—‘mz nzls-mzh ' 2 2 N 9 1 62 9 , , WWZ’ 2 4 — + —-u “g + E ——2" [— 4E2 (p2 + (E2 — 2p?) cos 6) mW - t —- m2 ”:1 t — ng 2 —2p2(1 + cos 6) (2E2 — p2 — E2 cos 6) J , (4.36) where the cubic and quartic couplings are N ng/VZ = g a80b00 + g :1 agobjo 9. J: 2 ~ N 2 9“,”,72; g a’OObOn 'l' 9 32—2 ajObjn a 2 2 4 ~2 N 4 ngWI'W/W' = g Cl(_)0 + g .721 (1ij . (4.37) 68 0.0 ....1....1....1.h.. O 5 10 15 20 V; (TeV) Figure 4.10: The J = 0 partial wave amplitude as a function of \/E for the SM without a Higgs boson (red) and the SU(2)xSU(2)NxU(1) model (blue) for N = 1 to 100 with mva = 500 GeV. 1 Then the coefficient of the leading (E / mW)2 term, defined by (4.27), is N _ 2 _ 3 2 2 2 2 K — 4gVVI/VW'I'V m2 ( Zgl/VW'Z + Z mZthWZ’ ) (438) W n=l n In Appendix B we obtain for this model 2 A, = W+O(l\2) = “If/+81??? + 0(A2) . (4.39) where the corrections also fall off as (N + 1)‘2. As expected, this agrees with the results of the previous section for N = 1. In Fig. 4.10 we plot the J = 0 partial wave amplitude as a function of \/E for both the SM without a Higgs boson and in the SU(2)xSU(2)N xU(1) model with mWi = 500 GeV for N = 1 to 100. For large N in this model the unitarity violation is delayed to an energy of about \/5 = 19 TeV. Thus, we may expect that the effective 69 theory with a KK tower of vector bosons should be reliable up to about this scale.5 At high energies and large N, the partial wave amplitude asymptotes to 1 r s g2 + 4572 (10 s 1) a z — __ _ _ _ 0 327r _4m%,, (N + 1)2 N +1 g A2 2 F 2 1 2 4 2 mw’ 1 z -—-— 82 9 2 + Jay—24 (log '12 — —) , (4.40) 3277 4mW (N + 1) 7r mW A 2 where A is a scale on the order of a few times mW" 1 4.3.3 The N —> oo Limit From the analysis of section 3.5 we know that the limit N ——> 00 gives a gauge theory on an extra-dimensional interval, as long as g and f grow like N + 1. The five-dimensional gauge coupling and the interval length are then given by 7 NR = lim 2(A~+1), N—mo gf 2~ 2 - 9 95 = Nil—131007. (4.41) Since the gauge couplings of the edge sites are different, the five-dimensional action has localized kinetic terms on both branes, NR 1 1 8 , 2 [d4 / d _ vva rwra [VIN _ (S w'alWapI/ 907198 I 0 y 493”]? [MIN (30492 In 1 —6(7rR—y)4—gfiW3,/W3W , (4.42) where the dimensionless five-dimensional coupling 95 is defined by figflR = 9%, and the five-dimensional coordinate 1:5 has been renamed y. As explained in section 4.2, the 6-functions should be intended as slightly pushed inside the bulk, with the BCS leaving SU(2) unbroken at y = 0, and breaking SU(2) to U(1) at y = 7TB. These are the same BCS we met in the toy model (1) of section 3.2, which allow us to set W5“ E 0, in unitary gauge. Then, pushing the delta functions 5A coupled-channel analysis, as considered in Ref. [34], would give a lower energy scale for unitarity violation. 7O back to the branes, we obtain 2 0 r2 0 A " a: _ Ell—1331+ (Us W” '— m()5wfl) —— 0 , - 2 3 A” 3 ylfl— (ash/”+fia5wp) = 0, WlflnR) = 0 , (4.43) for the behavior of the gauge fields near the boundaries, where A2 E 92/93, and 4’2 E 9’2/93- All of the results found in appendix B have a well-defined limit as N —> 00, with the discrete label j becoming the continuous extra-dimensional variable y, and the vector expansions (4.32), (4.33) becoming KK expansions. An alternative method for deriving the solutions consists in working directly with the continuum model, as in section 3.1, and KK expanding the five-dimensional gauge fields, :i: 00 "l- W “(2% y) = Z fn(y)WrT“ (1C) . 7u=0 W3"(x,y) = eAM+§gn23] , A 07;, [coshflznm — ":6" Sin(mZny)] , (4.46) gn(y) where the masses mwn, m Zn solve the transcendental equations rhWntanrhWn = A2, 2 I2 2 2 (Iiizn— - )tanfiign = /\ +/\’ , (4.47) mg” 71 with fn, E mIrR. The normalization constants Fn, Gn are determined by requiring a canonical orthonormalization of the mass eigenstates: 1r}? 1 1 - f0 dy T‘— + 975W) fn(y)fnl(y) = 07m! . ggwR NR (1 'l /0 ., 1 1 1 - _R @7561) + 35““? — 31)] g"(y)9n'(?/) = 51m" 9&4 (4.48) Notice that the integrands in (4.48), for equal values of n and 71', should be interpreted as position probability densities in the extra-dimensional interval. The presence of 6-function terms tells us that the charged gauge bosons have non-zero probability of being exactly localized at y = 0, and the neutral gauge bosons have non-zero probability of being exactly localized at y = 0 and y = 7rR. Using the mass equations, we obtain r - . -1/2 1 sm2mw F = *, _ 1 ___n " 9°12( + 2771M. )l ’ - o A A2 . A -1/2 1 sm 2mg 1 2 m Z 811127712 G = "r — 1— n —_ o A n 1 n , n 9" 2( 27th )+,\2sm mzn+ 2A4 ( + 27542,, (4.49) for the normalization constants. Once the normalized wavefunctions are found, the cubic and quartic gauge coup- ings can be computed using the formulas git/,sz, = ("Edy Eff—Efiz—ay) f1(y)fm(y)9n(y). “NR : 1 1 : ngWIWmWn = /0 dy 5m+gjfly> fk(y)fz(y)fm(y)fn(y), 7TH - 1 1 2 ywkwlzmzn = [0 dy Egfig+gj5tyl .fklylfl(?/).(lm.(?/).‘In(?/)- (4.50) These formulas are also valid for couplings involving the photons, for it is sufficient to replace the gn,(y)’s with the photon wavefunction. 97(y) = 6. Therefore, using also 72 0.5} V 0.4,1 f//’ 3 0.3 0.35 0.2 a 01 V . a, olgw 0.2 0.4 0.6 0.8 1 l/R(TeV) Figure 4.11: The coupling constants 09 = Q2/47T (blue), agI = g’2/47r (red), and rigs = 63/42 (green), as a function of l/R. the normalization conditions (4.48), we obtain gl/an‘lfn‘y : 66m" ’ _ 2 gtt'mwm ‘ e 5m" . 91(1qu 2717 Z egll"lll"m,Zn - (4-51) In the examples with a finite number of sites we saw that the largest postponement of unitarity violation occurs for g2 ”29 << 92 / (N + 1), which in the continuum model corresponds to 92, 9'2 << (352,. Therefore, we can find the solutions pcrturbatively for A2, 2V2 << 1. For the masses, we obtain 2 2 A2 A2 7””? E "ll/V0 : —_(7TR)2 1 —' _3 + O(/\ 4)] a A2 + 1'2 A2 + A’? /\_:__/\’2 2 _ 2 _ 4 for the SM gauge bosons, and n. 2 A2 mavn : (E) [1+ 2(—7r n)2 + O(/\4)] Ii=1,2,.... , 73 0'8 i fWu(y) A 0.6 1 3) ‘“ 0.4» 1 fwz (y) fW, (.V) . // 0 v 0 2 4 6 y (T W") Figure 4.12: Probability density for the position of the W boson (red), W1 (green), and W'g (blue) in the extra-dimensional interval, for 1 / R =500 GeV, with 6(y) re- placed by a narrow Gaussian. ‘ 2 A2 A12 2 ’1 l I l 4 1122 1 _ 1 2 on.» 40 for the heavy gauge bosons. Inserting these equations in (4.49) gives 2 F0 = 9—[1+’\—+0(A4)], A 6 fig A2 4 F = —— 1——— 0 =1.2....., n A 2(7171’)2 + (A ) n ' ' A A4 + 2121/2 — 2x4 Co = ——g-— - +004) . 09+»? 6(A2+A’2) 2 2 l2 0,, = [94 l—gA—H‘5—+O(A4) n=1,2,...., (4.54) rm 2 (mr) for the normalization constants. Perturbative expansions for the couplings (4.50) are given in appendix C. This model has four parameters: 9, 9', Q5, and the compactification scale R. We can trade 9, g', 55 for e, mw, mg, leaving R as the only parameter beyond the SM. In Fig. 4.11 we show the behavior of ag = 92/47T,0’gl = 9’2/47r, and 0&5 = fig/4w as 74 gn(y) Figure 4.13: Probability density for the position of the photon (dashed), the Z boson (red), 21 (green), and 22 (blue) in the extra-dimensional interval, for 1 / R =500 GeV, with 6 (y) and 6(7rR — y) replaced by narrow Gaussians. functions of 1/ R, which, as shown by (4.53), is approximately equal to the mass of the W1 and Z 1 bosons. We see that large values of 1 / R correspond to small values of A2 and X2. This can be seen directly from the relations (4.52), which fix the W and Z mass, and is in agreement with the results we found for the deconstructed model. Inserting the normalization factors (4.54) and the masses (4.52), (4.53) in (4.46), we obtain that near y = 0 and y = 77R the wavefunctions of the SM gauge bosons are of order of the electroweak couplings, and are not suppressed by any power of A. On the other hand, the wavefunctions of the heavy modes are suppressed by one power of A. Since the position probability densities have 6-functions at y = 0 and y = 7rR, with coefficients 1 //\2 and 1/A'2, respectively (see eq. (4.48)), it follows that all gauge bosons have non-zero probability of being exactly localized on the two branes, but this probability is much larger for the SM gauge bosons than for the heavy KK modes. In Fig 4.12 we show the probability density for the position of the W boson, W1, and W2 in the extra-dimensional interval, for 1 / R =500 GeV, where the 6-functions have 20 . - - 0 0.5 1 1.5 2 2.5 3 1/R (T 6V) Figure 4.14: Unitarity violation curve for the W3 WI: —> W3 W1: scattering, in the (fi, 1 / R) plane. For a given value of l/R, unitarity is satisfied for values of \/3 below the curve. been replaced with narrow Gaussians. In Fig. 4.13 we show the probability density for the photon, the Z boson, Z1, and Z2. We see that the SM gauge bosons spend indeed more time than the heavy modes near the branes. Therefore, in accordance to what we had found in the deconstructed models, the brane SU(2) and U(1) groups act approximately as the SM SU(2)L and U(1)y. In the limit A ,A’ —> O we must also have R —’ 0, and the heavy modes decouple. Therefore, the A ,A' -’ 0 limit corresponds to a four-dimensional theory, namely the SM without the Higgs boson. With the model fully set up, we can calculate the unitarity bounds from longitu- dinal gauge boson scattering amplitudes. The W3 W1: —+ WEWE scattering is given by the diagrams of Fig. 4.9, and the amplitude by (4.36) (without the primes on the KK gauge bosons), with N replaced by 00, and the gauge couplings given by (4.50). The coefficient of the (E / mw)2 term, in the high energy expansion, is easily found to be zero by taking the N —> oo limit in (4.39). Alternatively we can prove this directly from the extra-dimensional model, as we did in section 3.3, with the difference that 76 the gauge couplings receive contributions from the 6—function terms. Since R is the only free parameter, the amplitude M depends on E, 6, and R, M = M (E ,0, R). Therefore, the J = 0 partial wave amplitude depends on E and R, (1.0 = a0(E, R). In Fig 4.14 we show the (L0 = 1/2 curve in the (J3, l/R) plane: unitarity is satisfied below the curve, where a0 < 1 / 2. We observe that for a given energy, unitarity sets an upper bound on 1 / R. This was expected, since the heavy neutral bosons must come into play early enough to unitarize the amplitude. The asymptotic behavior of a0 can be found by taking the limit N -—+ 00 in (4.40), which gives 2 m2 a0 z 237:; [:LQm—g; (log/i32— — $>J , (4.55) where A is a scale of the order of a few times mwl. Using (4.52) and (4.53), we notice that, to leading order in A2, the coefficient of the logarithmic term in (4.55) is fig times a numerical factor. Therefore, the unitarity violation scale is approximately given by the reciprocal of the dimensionful coupling 9% times a numerical factor, in agreement with the discussion at the end of section 3.3 [34] [37] [38]. 77 Chapter 5 Coupling to Matter Fields and Experimental Constraints In chapter 4 we have introduced the gauge sector of an SU(2) xSU(2)N xU(1) N LSM, where the N —+ 00 limit corresponds to an SU(2) gauge theory on an five-dimensional interval, with BCs breaking SU(2) to U(1) at one of the boundaries, and localized kinetic terms on the two branes. In this chapter we will add matter fields to this model. Since the SU(2) and U(1) gauge groups localized at the endpoints of the interval act approximately as the SM SU(2)L and U(1)y, the simplest choice is to have the SM fermions charged under these groups only, with the usual quantum numbers. We will show that this manner of coupling fermions leads to tension between the constraints imposed by the EWP data and the unitarity constraints. In order to release this tension, we let the fermion fields to have some leakage into the bulk — in a fashion similar to the gauge sector setup — with the left-handed fermions peaked at the boundary where SU(2) is unbroken, and the right-handed fermions peaked at the other boundary. We show that the correction to the SM electroweak observables can be tuned to zero by imposing a relation between the amount of leakage into the bulk of the gauge fields and the left-handed fermion fields. We also show that delocalized fermions in this model naturally allow for multiple generations and fermion mixings. 78 As the N ——> 00 model has been proved to be computationally easier than the finite-N NLSMs, we will only present our results in this limit. However we should not rely upon the extra-dimensional interpretation, because this would leave little freedom for model building: Gauge-fermion interactions would be forced to be local, in the five—dimensional interval, and “Yukawa” interactions between fermions and the gauge field fifth-component would be forced to have the same strength of the ordinary gauge interactions. We will then interpret the variable y of section 4.3.3 as a continuum index, and the interval from which y picks its values as a TS interval, rather than an extra-dimensional interval. We will take advantage of this interpretation in both this chapter and the next one. 5.1 Model I In section 4.3.3 we considered an SU(2) gauge theory on a [0,1rR] extra—dimensional interval, with BCS breaking SU(2) at y = 7rR, and localized kinetic terms on both branes. The action, for the gauge sector, is given by (4.42). Fig. 4.11 shows that, as 1 / R grows, the dimensionless bulk coupling 9?, becomes larger than the brane couplings, g2 and 9’2. Fig. 4.12 and Fig. 4.13 show that, for small values of gz/yg and ga/yg, the SM gauge bosons are much more peaked on the two branes than the heavy KK gauge bosons. (The W boson is only peaked on the y = 0 brane, since SU(2) is broken to U(1) at y = 7rR.) Therefore, as previously stressed, the SU(2) and U(1) gauge groups on the two branes act approximately as the SM SU(2)L and U(1)y. It is therefore reasonable to try first coupling the SM fermions to the SU(2) and U(1) brane fields only, in exactly the same way they are coupled to the electroweak gauge bosons in the GW S theory. With this choice, the action for one generation of fermions is (I) , ”I? - Sfermion = /(l4:1'0 (1y [0(y)wL2'y“DM/)L +6(TTR — y) ('flRi’yi’lDuuR + JRiA/“DudRH (5.1) 79 where if”, = (uL,dL) is an SU(2) doublet, and 113, dR are SU(2) singlets. The covariant derivatives are 19pr = (8,,—2‘T“W§(y)—iYLW3(7rR))z/2L, DWI? = (an-iYRWBOJD UR, DfldR = (8#,—iYRl/l/3(y))d1;. (5.2) Notice that the left-handed field 1,!) L lives at y = 0 but couples also to the gauge field W 3 at y = 7rR. This is not allowed in a five-dimensional Yang-Mills theory, but is perfectly reasonable in continuum TS. The fermion action could be made local by folding the SU(2) gauge group at y = 7rR, as in example (2) of section 3.2, and coupling all fermions at y = 0. However, a mass term for the fermions can only arise in this model from a Wilson line connecting the two branes. This is again a non- local operator in 5D, but is fine from a four-dimensional standpoint. Therefore, as previously argued, we see that giving up on the five-dimensional interpretation opens new possibilities for model building, and our choice of labelling the x5 coordinate as y is to emphasize this point. The four-dimensional charged-current and neutral-current Lagrangians are 1 oo -gccu> _ LEX)? = Z __L\n/§ we/ll'PLT+IL‘ l/l/fiZ-l-ha] , n=0 _ 1 °° '- N01 N01 55,}, = Z M(gLn ( )PLT3+an( )Q)wzn,.] . (5.3) a II o where W0, and 20 are the SM IV and Z boson, respectively, and w = 1121, + @013. The couplings are 92?”) = we), 92?“) = yam—mm), 932?”) = gnch). (5.4) where fn(y) and gn(y) are given by (4.46). 80 5.2 Experimental Constraints on Model I In this section we study the most immediate phenomenological implications of model I. We will first briefly consider the direct constraints from producing the heavy gauge bosons at colliders, and then consider the indirect constraints from the EVV P data. 5.2.1 Direct Constraints on Heavy Boson Production The most significant bounds on the W1 and 21 masses come from the Tevatron and LEP II, respectively. The Tevatron (CDF) limit on a W' that couples with SM strength is presented in Fig. 2 of Ref. [72]. In our case, the ratio 0(qri —» W1 ——> (u)/o(q(i —> W ——> 81/) is suppressed by the small value of the W1 wavefunction on the boundary where the fermions are localized. The coupling of SM fermions with W1 is gEOCU) = [1(0), which gives the suppression factor (f1(0)/g)2. Using (4.46), together with (4.53) and (4.54), this gives 2 (5.5) ‘ ’ 1 - 7 M 0qu —* ”"1 -+ (my )= 2mgV [U(qqa I41 —’ Eu) (S ) le 0(qij —* W —> 131/) U(qij —’ W —» (JV) By rescaling the cross sections shown in the figure, we estimate that the corresponding limits in our case would be about mw1 > 500 GeV. The LEP II bound on new four fermion contact interactions are presented (for the case of strong coupling) in Ref. [73] by making fits to 0(e+e‘ -—+ ff). This can be translated to a bound on mg1 since a heavy Z' effectively induces a four fermion contact interaction. Extracting the relevant contact interactions induced in our model, and comparing to the results of the LEP II analysis, we estimate that the mass bound is about 77121 > 480 GeV. 5.2.2 Indirect Constraints on the Low Energy Fermion La- grangians The fermion couplings with the gauge boson are given by the values of the correspond- ing wavefunctions at the two interval ends, as shown by (5.4). In section 4.3.3 we have 81 argued that the wavefunctions of the heavy modes are suppressed by one power of A at y = 0 and y = FRI This is shown explicitly in Fig. 4.12 and Fig. 4.13. Therefore, four—fermion operators arising from exchanges of heavy gauge bosons are suppressed by A4, with two powers of A coming from the couplings, and two powers from the large mass in the gauge boson propagator (1 / R ~ mw / A, see (4.52)). The couplings of the SM gauge bosons with the heavy gauge bosons are also suppressed by one power of A, due to the little overlap of the corresponding wavefunctions. This means that dimension five operators, with two SM fermions and two SM gauge bosons, only arise at A4 order. 'As a consequence, at order A2 the new-physics corrections to the low-energy ob- servables are purely oblique, and are therefore entirely parametrized by the Peskin- Takeuchi S, T, and U parameters, where S = T = U = 0 corresponds to the SM [43] [44] [59]. The way these quantities enter in the charged-current and neutral- current effective Lagrangians depends on the chosen set of input observables. It is customary to take the electromagnetic coupling strength at the Z-pole, the Z boson mass, and the Fermi constant, because these observables have been measured with a high level of precision. The current estimates are [53] a'(mZ)_1 = 128.80:l:0.12, mg = 91.1876i0.0021, GF = (1.16637i0.00001)x10—5GeV_2. (5.6) However here we take mW instead of CF, even though mW is not known as precisely as m Z: mW = 80.425 :l: 0.038 . (5.7) This choice is in fact useful, because it is independent of the fermion profiles, and will pay off in section 5.3, where we compare this model, in which fermions are localized, with a different model, in which fermions are delocalized. Taking (1, mg, and mW as input observables, and defining s by, 2 s2 E 1 — TL2L , (5.8) ”‘2 82 with c E V1 — .92, we find the following expressions for the low-energy charged-current and neutral-current Lagrangians: qCC _ [:00 = ‘L2 d27upLT+1/)l/l/;+h.c., ENC = 1132/“ (giYCPLT3+ggCQ) 11' Z). , (59) where CC _ 6 1 + as c207” _ (c2 — 82)(xU gL — s 432 2.32 854 ’ NC _ _e_ 1+ 95 _ (C2 - .92)OT _ ( 2 — 52)aU ‘ gL — SC 482 232 854 ’ NC 68 aT 0U] = —— 1 — — .1 9Q c i 232 834 (5 0) This equations show that T is related to the p parameter by asz—l. (5.11) This relation is however only valid as long as the new-physics contribution to the low—energy interactions is purely oblique. Inserting (4.46) in (5.4), and using the perturbative expansions (4.52) and (4.54) for the masses and the normalization factors, respectively, leads to the following tree— level expressions for the couplings in model I: 950”) E [1 + 9/6 + 00.4)] , 9150(1) = i [1 + A2/6 + 094)] , 930”) = J33 [1 + 0(A4)] . (5.12) C Comparing these equations with (5.10), we obtain for this theory 0:8 = 2.3/\2/3, 0T 2 0 , crU = 0 . (5.13) The fact that T = 0 at order 0(A2) is an expected result, because this model has an approximate custodial symmetry: This is most easily seen in the deconstructed 83 version, from the choice of coupling 8,, as the T3 component of a global SU(2). U is usually expected to differ from T by a percent, and is accordingly approximately zero in this model. However S is not zero at order 0(A2). This is in agreement with a general result found in Ref. [55]: In an arbitrary SU(2)0xSU(2)N> 0(1). In our model T is naturally suppressed, thus the previous relation reads S > 0(1), in agreement with (5.13), since (1 ~‘ A2. Recent experimental constraints on S and T can be found in Ref. [78], where the limits are given as a function of the Higgs boson mass. In principle, its contributions must be subtracted from the above S and T parameters, since there is no Higgs boson in our model. However, given that the dependence on m H is not too large, we can still obtain an estimate of how these constraints impact our model. For m H = 600 GeV with the constraint S 2 0, and using Bayesian statistics, the limit on S is S S 0.14. This result corresponds to le > 3 TeV. Unfortunately, for models in which le is so large, unitarity will be violated even before the scale of le is reached, as shown by Fig. 4.14. Therefore, it appears that the method used in this section to incorporate matter fields into the model is not viable. 5.2.3 Indirect Constraints on the Low Energy Gauge La- grangian Although we have already proved that brane localized fermions violate the bounds imposed by unitarity and the EWP data, we press on and consider the indirect con- straints of this model on the low—energy gauge interactions. The results we find here will be useful later in this chapter. Additional constraints on the W1 mass can be found from the analysis of anoma- lous couplings in the WWZ vertex. To leading order, in the absence of CP-violation, the triple gauge boson vertices may be written in the Hagiwara-Peccei-Zeppenfeld- Hikasa notation [74], cgiLge = —ie:—‘: (1 + AKZ) ninja” — ie (1 + A..,) ujW; AW 84 — ie:—Z (1+ A912) (WWW; — W—ijm Z — ie(W+“"I/V’f — I4«"WW,;“)A,, , (5.14) where the two-index Lorentz tensors denote the U(1)Q-invariant field strength tensors of the corresponding field, and the “Z standard” weak mixing angle is defined in terms of 6, mg, and GF by 82 4&6”?ng , 322022 (5.15) where, as usual 5% + C22 2 1. In the SM, AK Z 2 Ana, 2 Ang = 0. In our Higgsless model, like in any vector- resonance model, the interactions (5.14) come from re-expressing the nonabelian cou- plings of the original Lagrangian in terms of the mass eigenstates, in which case one obtains equal contributions to the deviation from the SM in the first and the third terms, and in the second and the fourth terms [75]. Moreover, the contribution to the fourth term is fixed by electromagnetic gauge invariance. Therefore, we obtain AKZ = A912 An, = 0 . (5.16) In order to express the WW’ Z vertex in terms of .92, rather than .9 E mw / m Z, we must find an expression for 01:. To order A2, we simply have 00(1) 2 (a. > mW because, as previously noticed, the heavy KK exchanges only contribute at order A4. 'L'sing (5.10) and (5.13), we find ...|m a. = _1 2] .fi] 4\/2ma,8 232 1 (m.w1rR)2 4'\/2771%V S [ + 3 Ice MM + O ((mWn'R)4)] , (5.18) where we used (4.52) to express A in terms of mW and R. Inserting this equation in (5.15) gives 8:821 + 1 0S] 1— .S2Z/C2Z 4522 85 1 (mwer)2 = 8 1+ Z 1—82Z/C22 6 + O ((mWnR)4)] (5.19) The WWZ vertex can be found in appendix C. Expressing it in terms of e, 3Z2 and (mw7rR) gives CZ — (rnW7rR)2 1 a8 4 ,. = e—1+——'————————+O mr7rR 9” Vl’ Z 5‘Z _ 12C2Z 022 - 522 4322 (( W ) ) F 2 62 (m '7rR) 2 1 = e.— 1— W12 ( 2 2 — -2— +O((mW7rR)4) , SZ _ CZ ‘— SZ CZ (5.20) whence, comparing with (5.14), , R 2 2 1 12 CZ ‘ Sz CZ The 95% (3.1.. upper limit from LEP-II is IAgIZ | < 0.028 [76]. Using the experimental results (5.6), (5.7) and eq. (5.15), the upper bound on Aglz translates into the lower bound 1/ R 2 mwl > 682 GeV, which is considerably stronger than the direct-search bound found in section 5.2.1. 5.3 Model II Drawing on the analogy of the gauge action (4.42), which has S U (2) and U (1) kinetic terms peaked at the two ends of the interval and connected through the bulk kinetic term, we now consider a theory with left-handed and right—handed fermion kinetic terms peaked at the two ends of the interval and connected through a bulk fermion kinetic term [45],[50]. The fermion action is (11) _ 4 ”R Sfermion _ _/ d a: /0 dy 1 .— . 1 _ . - . +5(y)t7wLW”Du¢L + 5(7TR - y) (ig—unziflDuuR + TdRZV’mudR) L , 1 1 _. A’I _ I f I L I I h. C I / 1’) The five-dimensional Dirac matrices were introduced in section 4.1, and are defined in terms of the four-dimensional ones by PM = (7”, —i75). The five-dimensional 86 fermion is equivalent to a four-dimensional Dirac fermion, 1/2 = 1/2 L + #13, where 11) L and 111R are S U (2) doublets, 11]le 7119B: dL d1; We have written the action for one doublet, consisting of an up and a down quark. We will discuss the possibility of more generations and mixing in Section 5.3.2. We can assume the bulk mass M to be real in (5.22) without any loss of generality. In fact any imaginary part of M can be removed by the replacement 1/) —’ eulmfi'lylw. The sign of M, however, is physical. In analogy with the gauge brane kinetic terms (see section 4.2), the fermion brane kinetic term at y = 0 is defined by interpreting the 6-function as 6(y — c) for t. —+ 0+ with the boundary condition 1113 = 0 at y = 0. Similarly, the boundary term at y = 7rR is defined by interpreting the 6-function as 6(7rR — y + c) with the boundary condition 1/2 L = 0 at y = 7rR. The general treatment of possible fermion boundary conditions can be found in Ref. [71]. The covariant derivative in (5.22) is DM¢ = (3M — iTaWXMU) —iYLWii1(7TR)) 1b, (523) where YL is the VI’L hypercharge. At the interval ends the four-dimensional part of the covariant derivative (5.23) becomes: (upwnyzo = (0,. — irawgw) — (Yin/Sore) we , (Dptyp),:,,p = (BM—iT3WE(7rR)—iYLW3(7rR)) 11’]; = (Bu—iYRll/gUrRDwR, (5.24) where the 11.er hypercharge, YR, is related to YL by YR = T3 + Y , as in the SM. Note that YR is a 2x2 diagonal matrix, with the u R hypercharge on the upper left, and the LIE hypercharge on the lower right. Therefore, at y = 7rR the covariant derivative term, J37”D,,,wR, splits into two separately gauge invariant terms, 17. Ryf‘Dflu R and dRyflDfldR, as in (5.22). Note also that in the limit of small tL, tuR, and th the action SUI) fermion describes massless left-handed fermions gauged under an SU(2) x U(1) 87 group living on the left end of the fifth-dimensional interval, and massless right- handed fermions gauged under a U(1) living on the right end of the interval, exactly as in model I. It is the presence of the bulk fields which allow these light states to communicate with each other, supplying the analog of the Yukawa coupling of the SM, and giving mass to the fermions. 5.3.1 Fermion Masses and Wave Functions In order to find the KK eigenstates for all fermion fields, we must diagonalize the free action. Let x denote either u, the up-type fermions, or d, the down-type fermions. Turning off the gauge couplings, the action of (5.22) becomes 8280) + 8(9) , where 8(0) 2 /d4:1:/07ery 1 _ _ _ _ — 2 (XR05XL - XLasxR + h.C.) — 1” (XRXL + XLXR)) 1 _ . _ . . FE (XLZ'Y/‘auXL + momma - 1 _ _ 1 _ . , + 0(3/ - Elgar/“Bum + 50?}? — 6 * y)t—2-XRW”3;1XR] - (520) L XR In (5.25) we have explicitly included a finite c to push the delta—function terms slightly away from the interval ends, allowing us to unambiguously impose the BCS XR(0) = 0 , XL(7rR) = 0 - (536) The field equations in the bulk can be obtained by variation of 8]?) . Integrating these equations around the 6-functions, taking the limit 6 —+ 0, and using the boundary conditions (5.26), leads to alternative expressions for the boundary limits: 1 lim 2 ——i ”8 0 , y—»0+ X1? ti 7 [LXL( ) . 1 . hm _ XL 2 —-t-§—27“8#XR(7TR) . (5.27) Comparing (5.26) with (5.27), we see that X R has a discontinuity at y = 0, but X L is continuous. Similarly, XL is discontinuous at y = 71R, but X R is continuous. The fermion fields can be expanded in a tower of four—dimensional KK states: xdaxy) = Z axn(y)XnL($)a 11:0 88 XR :20 6X71(y an-W 15°) (528) The four dimensional fields Xn L and X71}? are the left-handed and right-handed pro- jection, respectively, of a mass-mu Dirac fermion, Xn = Xn L + an. Wavefunctions and mass equations are obtained by diagonalizing (5.25). It is most convenient to treat the action as finite in the bulk (0 < y < 7rR) with (5.27) as BCS. In this case the bulk equations of motion become a Tl + (”Chm _ mnfixn : 0 , 3;,” — 111.3,,” + mnaXn = 0 , (5.29) with the BCs 1h li1n+)3)m(0 ) = _7220‘Xn(0) , y-+0+ L lim _ aXn(7rR) = —:;n 3xn(7"R) . (5.30) y—‘VITR XR (Recall that masses with a hat are expressed in units of (71R)—1; i.8., 711. E m7rR.) The wavefunctions must also satisfy the orthonormalization conditions 11R _ 1 d _ 0 y _71R WRd [ 1 1 —5 R— . — + 71' /0 y 71R t—2 XR ( y) + (1265)] a...a.., = 6..., . 'L fiXn(y),3n/(y) = 6n”; , (5.31) where the integrands, for equal 11 and n’ , should be interpreted as position probability densities for the left-handed and the right-handed fermions, respectively. With the five-dimensional fermion fields propagating into the bulk, the four- dimensional charged-current and neutral-current Lagrangians involve not only KK gauge bosons, but also KK fermions. For one generation of quarks (or leptons), we have (11) 1 _ . CC CC ,+ CC = gut/.5 emu (gLn,(uk.d1)PL+ani(U},~~d1)PR)dl 11,,,,+he , (11) 3 NC 3 £NC = Z 2 >211“ (1mm (”FLT +912" (\1 “#3117" +983. CQ) sznp k. l. n x: —u. d (5.32) 89 where the coupling constants are given by CC 11R 1 9Ln,(uk,dl) : [Jim/0 dy a 915...,1,/d4/Riy—a,,(ymd,ad, , L 91m (\1 xz) =/d4:r [0R d:[;r1§ + L]%)]a exp(y)ax,(y)(gn(y) -gn(7rR)) . 93% (X, Xl)_ /d4rr [07mg— 1725),),(y)13xl(y)(gn(y) - 9110113)) , anC = gn(7rR) . (5.33) Perturbative expressions for the fermion couplings can be found in appendix C for M = 0. The solutions to the mass equations are simplest in the case of zero bulk mass M. We study this case first and then look at the numerical solutions for nonzero M. (i) M = 0. With no bulk mass, the solutions of (5.29) are 711. . An [cos(mny) — :23 Sin(mny)] , am 2 L 751T; . ,3)”, = —An [tTcos(mny)+sm(mny)] . (5.34) L Applying the boundary conditions given in (5.30) to these solutions leads to an equa- tion for the fermion masses, (15% -+- ti mn tan mn + 111% =tLt2 (5.35) R) XR The lowest mass state of the KK tower corresponds to a standard model fermion. This light mass can be easily obtained in a perturbation expansion if we assume t% to be small: 2 4 v1+th 2(1+t§R) If we also assume 1,2 X1? to be small. the heavy state masses are ti + C(11) . (5.36) 1 t2 +t2 mn— _ it (11 — 2) 1+ —L——"‘—B—,— + 0(14) n = 1, 2, . (5.37) 7.25-1) 90 The normalization factor An can be fixed by requiring the KK states to be canon- ically normalized in the four-dimensional Lagrangian. We obtain . . . . - —1/2 1 2 1 2 2 For t% small this gives 2 4 1+tXR+tXR/3 2(1+t2 1% + 0%) , (5.39) m) A0=tL1-— for the lightest state, and for both t% and ti R small this gives “(n-i) 2525-5)? for the heavy states. + 0(t4) n = 1,2, (5.40) From (5.36) we see that the lightest fermion mass is suppressed by the factor t LtX R' For small values of these parameters this lightest Dirac fermion lies mainly on the branes, with small contribution from the bulk. The left-handed bulk wave function a0(y) goes to zero as t L —+ 0, and the right-handed bulk wave function 50(3)) goes to zero as t R ——+ 0. Since the fermion masses arise from the 65-terms, which mix left-handed and right-handed wave functions, it follows that m0 goes to zero, as either tL ——> 0 or i}; —+ 0. Notice also that (5.36) and (5.37) are symmetric in t L and tXR. This was expected, since the mass equation is t L — tXR symmetric. However, we shall treat t L and tX R differently. For starters, t L is an SU(2) invariant parameter, whereas tX R can have different values for the up and down fermions. We shall take this distinction further by assuming that t L is universal for all quarks and leptons, and that the different particle masses are determined by tXR. We shall find in section 5.4 that if t L is of order A, it can be used to cancel the positive contribution to the S—parameter that comes from the gauge sector. To have an idea of the orders of magnitude involved, 'let us assume R"1 ~ 1 TeV and tL ~ /\ ~ 10"]. Then tXR ranges from ~ 10—11 for the lightest neutrino, to ~ 10‘2 for the charm quark. 91 0.8 ' 0600') 0.6 * an()’) 0.4 ’ y (TeV") Figure 5.1: Probability density for the position of the left-handed electron (red), and its KK resonances, 6L1 (green), and e L2 (blue) in the extra-dimensional (or T8) interval, for l/R =500 GeV, and tL = A/\/3, with 6(y) and 6(7rR — y) replaced by narrow Gaussians. All these features are made explicit in Fig. 5.1 and Fig. 5.2, where we show the position probability density in the [0,7rR] interval for the left-handed and the right-handed electron, respectively, together with the first two KK resonances, for 1/ R =500 GeV, and t L = A/x/3. As we did previously for the gauge boson probabil- ity densities, we replaced 6 (y) and 6(7rR — y) with narrow Gaussians (with the same width, for a faithful comparison). We can directly observe the suppression in the bulk of ae(y), and the very large suppression of fie(y). Correspondingly, the left-handed electron spends most of its time near the y = O brane, while the right-handed electron spends virtually 100% of its time in proximity of the y = 7rR brane. Notice that these considerations apply only to light fermions. We shall return to the issue of the third generation in chapter 6. 92 )' 0.8 l My) 1 l A 0.6 i 3: : “1 0.4 E . - 13m) ’84-") y (TeV") Figure 5.2: Probability density for the position of the right-handed electron (red), and its KK resonances, (231 (green), and 632 (blue) in the extra-dimensional (or TS) interval, for l/R =500 GeV, and tL = /\/\/3, with 6(y) and 6(7rR — y) replaced by narrow Gaussians. (ii) M ¢ 0. For nonzero bulk mass, the analysis is similar; the equations are just a bit longer. The equation for the fermionumasses becomes [(mfi + (itiR) M + (1% + (ER) mg] T(mn) = {iii}? — m3, , (5.41) where the function T(mn) depends on the relation between mn and M: r m tan (#53, — A712 for mn > |M| , n_ . T(mn) = 4 (5.42) 1 tanh (M712 — m2 for IN] > m... . r 2 A 2 TI For a large and positive bulk mass M > 0, M >> 1, there is one light solution to this mass equation, given approximately by mg m (it? e—W . (5.43) 93 M (TeV) ' Figure 5.3: Masses of fermions, as a function of the bulk mass, for t L = 10"], I. R = 1, and l/R = 500 GeV. For a large and negative bulk mass M < 0, IM | >> 1, there are two light solutions, which are asymptotically given by (for t L < tXR) Th leIt% #40 ON 22 22 m 2|M)t§R . (5.44) This behavior is displayed in Fig. 5.3, where we plot the mass states as a function of the bulk mass M, with the other parameters fixed at Q; = 10'1, tXR = 1, and l/R = 500 GeV. The transformation M —) -M can be shown to be equivalent to a reflection in the fifth dimension. Since the boundary conditions (5.26) that we have imposed are asymmetric in this reflection, we obtain the asymmetric behavior in A! —> —M of Fig. 5.3, even in the absence of the brane kinetic terms. The heavier modes are less asymmetric, because they are less affected by the boundaries. 94 5.3.2 Generation Mixings It is not difficult to implement multiple generations in our fermion model II. In general the bulk mass, M, and the normalizations of the brane kinetic terms, ti2, $12,, and tag, would be independent 3 x 3 matrices for both the leptons and the quarks. However, this proliferation of mixing matrices would open the door to large flavor- changing neutral currents, which must somehow be avoided. The simplest way to achieve this is to restrict all of the flavor physics to the right brane, and impose a global U(3)quark x U(3)]epton symmetry on the quark and lepton doublets in the bulk and on the left brane. This flavor symmetry would only be broken by the kinetic terms on the right brane (which, incidentally, is also the only place where the SU(2) weak gauge symmetry is broken). The generalization of the fermion action in (5.22) is fermion 7TH R _. . _. . 8‘”) —/d“a:/O7r @[i (éWiI‘MDMt/Iz+ h.c.—Mz/»'(/)') 3 17., .- _ _. . I... . + 5(ylt—z2‘wll"/“DW'L + 0W? - y) (131ng WWW}, + J’Rhffnl‘Dpdfi) L (5.45) where 11 and j are generation indices, and there is an equivalent contribution for leptons. In principle the t L and M parameters, as well as the K matrices, can be different for the lepton and quark sectors. The five-dimensional fermion fields u’fi’s can be considered four-dimensional Dirac fermions, which are also SU(2) doublets: w = 62 + 1;}; = I; + I} (5.46) The quark sector matrices K u and K d are arbitrary Hermitian matrices; however, we can exploit the U(3)quark symmetry of the quark fields in the bulk and on the left brane to reduce the number of real physical parameters to 9 + 9 - (9 — 1) = 10, where we have taken into account the fact that the U(1) part of U(3) is just an overall phase symmetry. We can identify these 10 parameters as the six quark masses, and the four physical parameters of the CKM matrix. To see how this works, we first perform an SU(3) transformation on the vi to diagonalize Ku. Thus, without loss of generality, 95 we can assume K 17)] = 0,7596”. We can also assume that K? is diagonalized by a unitary matrix V, so that K}? = Vik(tgk2R)(Vl)kj . We now relate the (primed) gauge eigenstates to the (unprimed) mass eigenstates by the redefinition (1% R = Vijdjf..R' The action now becomes R 1 1 _. . _(_ _ 5542“ = f07r ill/(14$ [77R- (EWH‘MDMIZ’z + be — MWW) 1 —- , ,' 1 _- . ' 1 ' . f + 6(y)?2—2/2'Lw"Dpw?L + 6(7TR — y) (t2 quz'yf‘Dflufi + tTJlRWHD/xdhfl , 'L “4R am (5.47) where (Ii 2 u_L . , d2}? = u}? . . (5.48) V‘JdJL wad}, The unitary matrix V corresponds precisely to the CKM matrix in the SM, only arising in terms that involve the exchange of charged SU(2) gauge bosons. Just as for the CKM matrix, it can be reduced to three real parameters and one phase, via five independent phase redefinitions of the u L, R and d L, R fields. It is not difficult to see that any implementations of the SM can be mapped into this picture. In the lepton sector, for example, we could induce a see-saw mechanism by including a Majorana mass term for the neutrino, at y = 7rR. In that case the matrix V would contain two more physical parameters, corresponding to the Majorana phases of the MNS matrix. Alternatively, we could have a zero-mass neutrino, by imposing the boundary condition VR 2 0 at y = 7rR. In that case the number of physical parameters would be 9-(9-3)=3, corresponding to the three lepton masses. 5.4 Experimental Constraints on Model II We would like now to study the experimental constraints on model 11, starting from the four-dimensional Lagrangians (5.32), and the coupling strengths given in ap- pendix C. We consider first the direct constraints on W1 and Z1 production, and then the indirect constraints of EWP data on the low energy effective Lagrangians. 96 5.4.1 Direct Constraints on Heavy Boson Production With the fermions in the bulk, the overlap of the light fermion wavefunctions, with the heavy gauge boson wavefunctions is enhanced, relative to model I, by a factor of order 1//\2. However the probability for the light fermions to be in the bulk is suppressed by a factor of order t2 , and thus we expect the coupling of light fermions to heavy gauge bosons to be of the same size in model I and model II (for f% ~ A2). However the cross sections for W1 and Z 1 production change by numerical factors. The relation between the ratio U(qri —* W1 ——> €1/)/0(q(7 ——> W —+ 61/) in the SM and in model II is now [U(qq —» w1 .4 411)] (1’ > _ 2 ma, (1_ 2,2L )2 [0m _. w, _, M] (SM) 0(qq -—+ H' —-> (V) m?“ ? 0(qq —+ W’ --* €11) , (5.49) for M = 0. In accordance with the results of the next section, we set t L = A/\/3, so that last expression becomes [U(qu —& 1471 —> [IA] (11) Ema; [U(qq- —> W71 —> (11)] (SM) (550) 0((15 —+ W' —> (V) = 9 le’I/l U(qq —* W —* 61/) The suppression factor is nine times smaller than in model I (see eq. (5.5)). Rescaling the cross section shown in Fig. 2 of Ref. [72], we find mw1 > 350 GeV, which is significantly weaker than the corresponding bound in model I. It can be shown that numerical factors also significantly weaken the bounds on m Z1, relative to the model with localized fermions. 5.4.2 Indirect Constraints on the Low Energy Fermion La- grangians In section 5.4.1 we argued that the couplings of light fermions (and SM gauge bosons) to heavy gauge bosons are also suppressed in model II as they were in model 1. Therefore, dimension-five and dimension-six operators do not arise at A2 order, in the low-energy effective Lagrangian. Also, we observe that although the delocalization of fermion fields give rise to anomalous right-handed couplings, these must vanish as (X R —> 0, because in this limit the right-handed fields become exactly localized on the 97 y = 7rR boundary, where the five-dimensional Witt, y) is zero, and W3(:r,y) only couple with YR E Q 124 Since tX R is negligibly small for light fermions, no anomalous interactions are relevant at low energy. Therefore, the new-physics constribution to the low-energy interactions is still oblique, with the charged-current and neutral-current Lagrangians given by (5.9). The couplings are now = g$C<1>/d4r/"”dy {-1- 351104404 < 53—5 y) R 0) ’ NC(II) _ 4 "R 16y) 90(y)- 90(7TR) 9L — 0(1)] (1 {/0 (13/ 7+ Tngaufiz/Mdfiylm 0_) 90W?) . 430“” = :35 , (5.5.) where we used (5.4) and the normalization conditions (5.31). The ratios in (5.51) are positive and less than one, Mzgo(y)~go(7r3)zl_i <1 f0(0) 90(0) - 90(7TR) 5R — ’ and the suppression factors for go Cand gN Care identical to leading order in A2. (5.52) Evaluating the integrals, we obtain CC 11 CC I 9L ( ) _ 9L ( )(1_At:124), NC(II) _ gsz0(1)(1_At%), 9L — N 11 NC] 4,,“ l = .44 (l. (5.53) where (5.54) 47’8in AA, — —1.—(1—e-MM) . M 2M In the limit M ——> 0 we find A ——> 1 / 2 [45]. By allowing the fermions to extend into the bulk, as in model II, one can cancel the effects of S in electroweak measurements. Comparing (5.53) with (5.12), we see that S can effectively be set to zero (while retaining T = U = 0) by the choice {-2 = — . (5.55) 98 5.4.3 Indirect Constraints on the Low Energy Gauge La- grangian In section 5.2.3 we analyzed in the constraints imposed by the EWP data on the WWZ vertex in model I. The results we found there can be easily adapted to model II. In fact, in equations (5.18), (5.19), and (5.20), the first line is also applicable to model II, because it is generically written in terms of the S parameter. We have just seen that the latter can be adjusted to zero in model II by setting t L = A/\/3, for M = 0. Therefore, with this choice we obtain 1 82 4 for the Fermi constant, .9 = 82 [1 + 0 ((mWnR)4)] , (5.57) for the relation between the sin Hw’s defined by (5.8) and (5.15), and __ CZ (TT'IWWR)2 4 - gWWZ - 8'8; [1 + ——1-ZCTZ— + 0 (Gnu/FR) ) . (0.58) for the coupling in the W WZ vertex. Comparing last equation with (5.14) gives ("1W4”)2 2 12CZ With this result, the 95% CL. upper limit IAglz I < 0.028 from LEP-II [76] translates A912 = > 0 . (559) into the lower bound le > 498 GeV for the W1 mass. This is weaker than the corresponding bound with localized fermions, but stronger than the direct search bound we found in section 5.4.1. The analyses of direct and indirect constraints in model 11 show that delocaliz- ing fermions can always relieve the tension between unitarity and experimental data. However we have not yet considered the constraints imposed by the top quark phe- nomenologv, which might be severe. In fact ttR may not be small enough to suppress anomalous right-handed couplings. Worse, there might not even be a QR which gives a realistic top mass. We will see that this is indeed the case, and offer a solution to this problem in the next chapter. 99 Chapter 6 The Top Sector In Section 5.3.1 it was shown that. the fermion masses are suppressed by the factor tLtXR. In Section 5.4.2, we saw that we could choose t L to cancel gauge sector contributions to the S parameter, thereby relating t L to A by (5.55). Thus, we are left with tX R as the final degree of freedom to fit the fermion masses. In this chapter we will see that this works well for all of the light fermions, except the top quark. In fact, accommodating a realistic top mass would require the increase of the value of 1 / R beyond the bounds imposed by unitarity of the WZ'WI: ——+ Will/VI: scattering. We will then show that this problem can be solved by breaking the five-dimensional Lorentz symmetry, which leads to two independent compactification radii, or mass scales: One for the gauge sector, and one for the fermion sector. Upper bounds on the latter are then shown to arise from the (I —+ W2" W1: scattering, and lower bounds from EVV P data on the tbll" vertex. 6.1 The Top Mass in Theory Space In chapters 4 and 5 we have presented a Higgsless model of EWSB from continuum TS, whose full action is given by the sum of the gauge action (4.42), and the fermion action of model 11, given by (5.22) (or (5.45), for multiple generations). In this model the overall mass scale is set by 1/ R — which is required to be less than about a TeV in order to sufficiently delay unitarity in If"; 1171': -—+ I'VE ”E scattering — and the other 100 50 40* 30* mo (GeV) 20* 10* — 1000 —500 0 500 1000 M (GeV) Figure 6.1: Mass of the lightest fermion, as a function of the bulk mass, for t L chosen to adjust the S-parameter to zero, and 1 / R = 500 GeV. The curves correspond to several values of tXR, from 10”1 to infinity. independent parameter is t L (which measures the amount of leakage of left-handed fermions into the bulk) while the tXR’s (which measure the amount of leakage of right-handed fermions into the bulk) are in a one-to—one correspondence with the SM fermion masses. In section 5.4.2 we saw that we could choose t L to cancel gauge sector contributions to the S parameter, by relating t L to A by (5.55). Therefore, 1/ R is left as the only free parameter beyond the SM. With this setup, however, it is impossible to obtain a realistic top quark mass of 175 GeV. For example, for 1/R = 500 GeV, M = 0, and tL fixed by (5.55), the lightest fermion mass solution to (5.41) has a maximum value of about 45 GeV. Even if we allow the bulk mass M to be nonzero, we cannot do much better, since (5.55) involves M in a dramatic way. In particular, when M -—> —00, t L tends exponentially 101 to zero, and so does m0. In the other limit of M —> 00, the solution for ma itself is exponentially suppressed, as shown in (5.43). Thus we find a peak near M = 0 in the curve for mg as a function of M (for fixed tXR), as shown in Fig. 6.1. From this curve with 1/ R = 500 GeV, we find that the maximum possible quark mass for any value of M is about 47 GeV, which occurs for tX R set to infinity. One possible way to solve this problem is to allow a different t L for the third generation of quarks. This approach might be viable, since the constraints on the S parameter do not directly involve the third generation fermions. However, we find it unattractive, since universality of t L (and M) was the simplest way to avoid any dangerous flavor-changing neutral currents. Moreover, large values of t L for the top-bottom doublet could lead to a violation of the experimental constraints on the Z bf) vertex. An alternate solution that we prefer is to allow a different size of R for the gauge sector and the fermion sector. This possibility had been suggested in the context of the warped-space model in Ref. [46]. It is even more sensible in the context of a theory space, since there is no reason for the coefficients of (HWDfli/J and 113F5D511’7 to be identical, in the bulk sector of (5.22). In terms of the deconstructed version of the theory, this just corresponds to allowing the gauge couplings and the Yukawa couplings to be independent of each other [20]. The most general extension of model II, with y-independent parameters, is described by the action Sgauge +Sf(II)’ ermion ’ where Sgauge is just the gauge sector action given in (4.42, and 11 “R 1 7. 1 7. 7 5543.201. 2 [0 dy / 44:4[m (Mr/1045 + 4. (5541751954 + h.C.) — MW) 7 1 — 1 I -. + 6(y)71111L7“D#wL + 6(7TR — y) (TifiR'yf‘DpuR + TidRVMD/rdR) tL tuR th (6.1) where K- is a new parameter. Notice that from an extra-dimensional point of view, this action corresponds to a theory with a microscopic breaking of the Lorentz invari- ance along the fifth dimension, in addition to the macroscopic breaking due to the (I)’ cornpactification. Rescaling the parameter y by y —-> y, = y/n, the action ngrmion 102 (11) becomes identical to S , with R replaced by R/ 75.1 Therefore, the gauge sector mass scale, 1/Rg E 1 / R, and the fermion sector mass scale, 1/ R f E n/ R, are inde- pendent quantities in theory space: setting R f = R9 is an unnecessary and arbitrary choice. I) With the action ngrm’ion replacing 8g!) mi on, the fermion masses are (for M = 0), tt 1+t2 +t4 /3 ‘ — ~ L“? 1— "R XR 72L+0(t‘},) , (6.2) m0 — I»? 2 \/1+th 2(1+t§R) 2 2 1 t +t 771" = mr (77—5) 1+—-£—fl—+O(t4) n=1,2,.... (6.3) 72 (n - 92 Now we can account for mt by simply increasing 75. Of course K, cannot be too large, due to unitarity constraints similar to those which give bounds on 1/ R9. In the case of 1 / R f the limits come from scattering processes such as H. —+ WZ' W I: . We shall investigate these unitarity bounds in the next section. 6.2 Unitarity of Fermion Scattering Amplitudes Here we shall restrict ourselves to considering the unitarity bounds coming from the it —+ W 3 WI: scattering process. General constraints on couplings in Higgsless models from this and related processes have been considered previously in Ref. [79]. In the SM, the tree-level tt- —+ W; WI: scattering amplitude is given by the four diagrams of Fig. 6.2(a). If t and t- have opposite helicities, the 7- and Z-exchange diagrams produce quadratically divergent terms, in the high-energy limit, which are cancelled by the b-exchange diagram [11]. The Higgs boson is not involved in this cancellation, which is confined to the J = 1 partial wave, so there is no quadratic growth of the amplitude with energy, regardless of the Higgs boson mass. If t and 5 have the same helicity, the b-exchange diagram produces a linearly divergent high—energy term in the J = 0 channel, which is cancelled by the Higgs boson exchange diagram. In our Higgsless model the Higgs boson exchange diagram, of course, does not occur. The 7-, Z-, and b—exchange diagrams are supplemented by corresponding dia- 1The only difference is the interaction term with Wg, which is zero in unitary gauge anyway. 103 t ll“ ' f w ‘ \”i + b +\>— 53$) /. 7+2 / / 11%,, f (a) t W $1“ ‘ N51“ \, + . b + 23:1“ 7 + Z ”a 4- v30 Z /’ (”l/1 ‘ “71:1 71 w; W,- (b) Figure 6.2: (a) Diagrams contributing to the tt_ —+ W'Ij’WI: tree-level scattering amplitude in the SM. (b) Same process, in our Higgsless model. grams with exchange of heavy Zn’s and bn’s, as shown in Fig. 6.2(b). As 1 / R f ——> 00, these heavy Zn’s and bn’s are removed from the theory, which becomes equivalent to the SM without the Higgs boson. Thus, it is reasonable to expect that the cancella- tion that occurs for opposite helicity t and f in the SM also occurs in our Higgsless model, and that the amplitude does not display quadratic energy growth at any scale. We have directly verified this in our model. However, if the t and t- have the same helicity, the linear growth in energy, that was cancelled by Higgs boson exchange in the SM, now must be cancelled by some other sector of the theory. In our Higgsless model this cancellation occurs through the bn-exchange diagrams. In this respect, the heavy b—quarks play the role of the SM Higgs boson for this scattering process. In section 4.3.3 we used the quadratic growth in energy of the WELWI: —> W; WI: scattering amplitude to place approximate bounds on the scale 1 / Rg, where the heavy vector states come in to restore unitarity. We can now do the same here, using the (I —> W 2" WE process to place approximate bounds on 1/Rf. Note that, since the fermion amplitude only shows linear growth with energy in the high-energy limit, the 104 corresponding limits on the heavy fermion states will be significantly weaker. For left—handed t and f, the J -- 0 partial wave amplitude is given by _ 1 00 1 + ._ _ “0W“ 7 WL ”1 ) _ 6'47 2 P2+k2+mgn 71:0 ((16323)? + (1455)?) (mtkgcn) + 2%p122hcn) — (7147(5)) W mb +m_§hg.€h%? (41713256.) — 224215:2 — mfv)f(€n))] . (6.4) where 212k E , 6.5 {n p2 + k2 + m2 ( ) E and p are the t (or 1) energy and momentum, respectively, k is the W; (or WI: ) momentum, and mbn is the bn mass, with mbo E mb. The functions f (2:), g(a:), and h(:r) are 7(4) = but: = 50—53:), 5(4) = $(1—lgxx2lnii3. (6.6) Using the notation of eq. (5.33), the couplings are 11%,? E 933000 b") and 111%? _=_ gg€(t0.bn)' For M = 0, to leading order in ti, th, and A2, the formulas of appendix C give thC = g[1+0(t2)], CC _ fitL (-1)"+1 2&1; 2 _ th _ 91+th ”201—5), ”301—”, [1+O(t)] n—1,2,..., . tt £5 11%? = g—M— [1 + 002)] , 2,/1+t,2R fl ft]; g7r2 (n— %)2,/1+th In the high energy limit, (6.4) becomes 5,95," = [1+O(t2)] n=1,2,.... (6.7) a (t ? __) M1,,'+I,V—) ~ 1 mtE 00 (hCC)2 + (hCC)2 _ 2771b” hCChCC (6 8) 0 L L L L — 327, mfv 2:0 Ln Rn —mt Ln Rn - . , n: 105 It is straightforward to show that this vanishes, to leading order in t2, th, and A2, using the couplings given in (6.7) and the masses given in (6.2) and (6.3), applied to mt and mbn. respectively. In fact, using the completeness relations 0° F 1 5(11) , - 12;) 577? t% labn(y)0'bn(y) = <>(y- y’) . 0° ' 1 6 _ , 2 5+ 7%”) 46411753447) = 414-17) . (6.9) 71:0 _ I bR as well as the equations of motion, (5.29), and the boundary conditions, (5.30), for the t. and the bn’s, it can be shown that °° CC 2 CC 2 'mb CC CC Z [(thi) + (hRn) — 2 n th hRn E 0 ' (6-10) 7120 mt Therefore, this cancellation is exact in this model for any values of the couplings, and the linear growth in energy at high energies does not occur. 6.3 Bounds on the Model Parameters Of course, the cancellation of the term that grows with energy is not a sufficient condition for the unitarization of the amplitude (6.4): The latter could stop growing after unitarity is already violated. The heavy b—quarks should come into play early enough to cancel the bad high-energy behavior, and this is only possible if 1 / R f is not too large. Enforcing (5.55), to keep S fixed at zero, and setting M = 0, the only parameters that are not fixed by the light SM fermions and bosons are R9 and R f, which set the scale for the heavy vector bosons and the heavy fermions, respectively. We can put some reasonable constraints on these two parameters by requiring that the t I] L —+ WEWI: and the WEI/VI: —> Will/VI: scattering amplitudes remain unitary up to some value of the center-of-mass energy J3. As an example, in Fig. 6.3 we display the region in the (1/Rg, 1 / R f) plane that is allowed by the requirement that a0 < 1/2 up to fl =10 TeV or \/§ =5 TeV for both scattering amplitudes. As expected, we see that 1 / R f can indeed be much larger than 1/Rg. If we require the theory to respect unitarity up to J: =10 TeV in both amplitudes, we find 1/ R9 < 106 1/Rr (TeV) 0.4 0.5 0.6 0.7 0.8 0.9 l l/Rg mm Figure 6.3: Bounds imposed by unitarity constraints on the if —1 W2“ WE scattering at J3 =10 TeV (upper curve), and the WEWL— —+ WEWI: scattering at J3 =10 TeV and J3 =5 TeV (vertical lines), in the (l/Rg, 1 / R f) plane. Specifically, we have assumed the requirement of (10 < 1 / 2 for both scattering processes The ti— —> W; W; scattering at J3 =5 TeV imposes no bound, since at this energy the no Higgs boson is required to unitarize the amplitude. The two curves on the bottom correspond to the minimum value of 1 / R f which allows a top mass of 175 GeV to be a solution of the mass equation for M = 0 (lower), and the minimum value of 1 / R f which gives a th right-handed coupling in agreement with the experimental constraint (upper). The vertical curve on the left corresponds to the experimental bound on mW1 from analysis of the WWZ vertex. 570 GeV and 1 / R f < 32 TeV. If we use the weaker requirement that the theory only respect unitarity up to J3 =5 TeV. then we find 1/Rg < 720 GeV, while there is no constraint on 1 / R f, since the t L? L —7 Will/II: scattering amplitude does not violate unitarity at this energy even in the SM without a Higgs boson. Of course, any upper bounds on l/Rg and 1 / R f depend on the somewhat arbitrary scale choice for J3, 107 0.004 gEffm/(e/s) 0.002 . 3 4 5 6 7 8 9 10 1/Rr(T<3V) Figure 6.4: Right-handed th coupling, in units of 63/5, for 1/ R9 = 550 GeV, as a function of 1 / R f. The horizontal line corresponds to the experimental bound of Ref. [80]. where the low energy Higgsless theory has broken down. Lower bounds on 1 / R9 and 1 / R f can be obtained from experimental results. In section 5.4.3 we found the lower bound l/Rg > 498 GeV from indirect constraints on the WWZ vertex, while the corresponding direct search bounds were estimated in section 5.4.1 to be weaker. For the case of 1 / R f, a minimal requirement is that it is large enough to accommodate a top quark mass of 175 GeV. This is displayed in the lower curve on the bottom of Fig. 6.3. It gives a lower bound of 1/ R f > 1-3 TeV, with the dependence on l/Rg entering through the condition imposed by (5.55). However, this curve corresponds to an infinite value of “12’ which is not viable. Tighter constraints can be obtained by limits on the right-handed th and 108 ttZ couplings, which in appendix C are evaluated to lowest order in ttR, tb to be R CC __ g 2 2 ggym) — gtthbR [1+0(A ,t )] , NC 9 2 2 2 9R,(t,t) = Q—CttR [1+0()\ at )] - (6.11) For example, in Ref. [80] it is estimated, using experimental results on the b —+ 317 process, that ggfim/g S 0.4-10‘2, at the 20 level. The corresponding bound on 1 / R f is displayed in the upper curve on the bottom of Fig. 6.3. For the particular value of l/Rg = 550 GeV, we can see how the coupling ggca, b, Wi) / g (where we have used g E e/s) varies with 1 / R f in Fig. 6.4. The experimental bound is satisfied for this value of 1/ R9 by 1 / R f 3.6 TeV, which corresponds to K. 6.5. An even stronger bound might be obtainable from limits on the right-handed neutral current coupling, since it is quadratic in the parameter ttR; however, the extraction of this coupling requires more detailed analysis of higher order effects at the Z-pole in our model. Notice, however, that there is no tree-level constraint on 1/ R; coming from the right-handed Z175 coupling, because 1.), R is a negligibly small quantity. 109 Chapter 7 Conclusions In this dissertation we have built a phenomenologically viable Higgsless model from theory space, with inspiration from the physics of one compactified extra-dimension. It is a well known fact that a gauge theory on an extra-dimensional interval corre- sponds to a four-dimensional theory with an enhanced gauge symmetry. This large symmetry structure has the important property of unitarizing the longitudinal gauge boson scattering amplitudes. The unitarization occurs through exchanges of virtual Kaluza—Klein modes, which ensure the cancellation of the terms growing like E4 and E 2, playing in this way the role which is played by the Higgs boson in the Standard Model and its most common extensions. This is only true, however, for boundary con- ditions on the five-dimensional gauge fields which are consistent with the variational principle. Moreover, rather than restoring unitarity at (almost) all energies, as in the Standard Model, the Kaluza-Klein modes lead to a delay of unitarity violation to en- ergy scales higher than the customary limits of Dicus-Mathur or Lee-Quigg-Thacker. Therefore, any Higgsless model should be regarded as an effective field theory, valid up to the energy scale of unitarity violation. Our model contains three features, which are crucial to any viable Higgsless model of electroweak symmetry breaking. First, it contains a tower of vector bosons which delay the unitarity violation in the WLWL —) WLWL and WLZ L ——+ WL Z L scattering amplitudes, while giving the correct mass for the standard model W and Z (and photon) as the lightest states in the tower. Thus, it can extend the applicability 110 of the effective Higgsless theory up to a higher scale in the 5-10 TeV range. This is accomplished using an S U (2) gauge symmetry on a theory-space interval, broken down to U (1) at the right end of the interval, and with gauge kinetic terms on each end of the interval. The normalization (A, A’) of the gauge kinetic terms on the boundaries are easily arranged to give the correct mass for the SM W and Z bosons. Second, it incorporates a cancellation of the large vector boson contributions to the S parameter, which generically occur in Higgsless models. This cancellation is ob- tained by allowing the light fermion wave functions to leak away from the ends of the interval. In our model this leakage arises through boundary conditions and boundary kinetic terms for the fermions, where the light left-handed fields are predominantly located at the left end of the interval and the right-handed fields are predominantly located at the right end of the interval. The leakage of the left-handed fields into the bulk can be made to cancel the gauge boson contributions to S, while keeping the T and U parameters naturally suppressed, by tuning the normalization (t L) of the universal left-handed fermion kinetic term on the left boundary. Meanwhile the nor- malization (I. X R) of the right-handed fermion kinetic terms on the right boundary can be used to give the correct mass for each of the light fermions. Furthermore, multiple generations and fermion mixings are implemented in the model, without introducing flavor-changing neutral currents, by confining all flavor physics to the right-handed fermion brane kinetic terms, and imposing a global U (3%,,an x U (3)191),on symmetry on the bulk and left brane. Third, it has a realistic top qriark mass and small nonstandard right-handed top and bottom couplings. To obtain this goal, while maintaining the good unitarity properties of the WLWL scattering, it was necessary to separate the overall gauge sector scale (1 / R9) from the overall fermion sector scale (1 / R f). This requires an explicit breaking of the five-dimensional Lorentz symmetry, which is theoretically allowed, since such symmetry is already broken by compactification and brane kinetic terms. In fact, within a theory-space model it can be considered natural, since the difference in the size of the scales is analogous to having different sizes of gauge and Yukawa couplings. By making 1 / R f larger than 1/Rg, it is possible to obtain the 111 top quark mass. It is also possible to suppress any nonstandard right-handed top and bottom couplings, since for a fixed fermion mass, an increase in 1/Rf requires a compensatory decrease in tX R’ leading to a decrease in right-handed couplings. In this way, we have constructed a viable Higgsless model with only three unde- termined parameters, l/Rg, 1 / R f, and the bulk fermion mass M. Since the bulk fermion mass does not seem to add any qualitatively new features to the model, it is reasonable to set M = 0, leaving us with a two-parameter model. The parameter 1/ R9 sets the scale of the vector boson excitations, and the parameter 1 / R f sets the scale of the fermionic excitations. Just as the scale 1/ R9 cannot be too large and still effectively delay unitarity violation in WLWL —-+ WLWL scattering, the scale 1 / R f cannot be too large and still effectively delay unitarity violation in t? —> WL WL scattering. Thus, both of these scales are bounded from above, the exact bounds depending on the energy scale at which the effective Higgsless theory must be re- placed by a more complete theory. A reasonable upper bound for 1/Rg is in the 570-720 GeV range, while the upper bound for 1 / R f is much weaker, of order 30 TeV or more. Experimental lower limits on 1 / R f from right-handed th couplings are in the range of 2-4 TeV. Precise experimental lower limits on 1/Rg require further investigation, although given the small couplings between the light fermions and the heavy W’ and Z ' states, there appears to be a reasonable range for this parameter that is still allowed. In conclusion, we have presented an existence proof of a viable Higgsless model, that can satisfy all current experimental constraints, as far as we know. It is certainly not the only Higgsless model that may work, and it is probably too simplistic in many regards, but it has all of the features that any Higgsless model must have. Thus, it offers a concrete example for use to explore the phenomenology of Higgsless models at the Tevatron and the LHC. In particular, it is worthwhile to further investigate its most relevant phenomenological aspects, with careful attention to those features which are general, rather than characteristic of any particular model. 112 Appendix A Solutions for the SU(2)0X SU(2)1XU(1) Model In this model there are five independent parameters: g,§,g’, f1, [2. We can express these in terms of the SM parameters 6, mw, m Z4 and the masses of the heavy vector bosons, mW’1mZ’: 2 2 l2 _ 2 mZmZ’ g _ e 2 2 a meW, 2 2 2 2 2 2 2 2 2 2 Q2 — ’2 (mnr + mnw’)(mZ ‘l’ mZ, "' mW - mm”) + mI/V’nluf’ — mZmZ, (7,122 + "1.22, - 771%»! — ma/q) 3 2 g : 2 2 2 2 2 2 2 2 2 2 2 2 9,2 m‘WmW,((mH2; + mg,,)(m2z + mg, —' my; — ”lg/VI) +2mw/mu2,’ — mZmZ,) (m2 — mZI)(mZI " mWI)(mZI " mw)(mwz - m2) 2 2 16 mufm ,rl 2 ll [1 = ~ , 9292 f3 4 f22 = 97207122 + 77122, —— may — ma”) . (A.1) The charged boson mixing matrix, defined in (4.22) is given by 0.00 = all = cosd> and —a01 = (110 = sin (15, where 2 "bf/V’Unfv’ — 777.22)(m2z, — mew) cos “b = 2 2 2 2 2 2 2 2 2 2 . mW/(mWI ‘ mg)(mZ/ - WWI) ‘l' mw(mZI - mwllmz — mw) Sing _ mfvlmggr - "15207122 “ mic) mar/(marl - 77122) ("122! — may!) + 771%,;(m2z, ‘ m%;)(m22 _ mfv) 113 The neutral boson mixing matrix, defined in (4.23), is given by , l 2 b mfi;m%,,(m2z, — m%,)(m2Z, — mad) / 00 = i _ mQZ(m2Z, — mQZ)1W4 " 2 2 2 2 1/2 _ m22(m22, — m2ZM'I4 W W Z , - 1/2 b _ _ < 22- meme, — mg) 20 — 2 2 _ 2 , mz(mzl m2) - 1/2 b _ "Ia/marl (m2Z _ m%/)(mr24/I — mQZ) 01 — _ 1712(m2 —m2)M4 ’ . Z Z’ Z 2 2 2 2 1/2 b“ = (mg, - mW)(mZ; — mu”) (mg/I + ma, _ "122) mQZ(mQZ, — 7'122Z)M4 M 1/2 (772,22, — ma’xmzzl —’ mall) [’21 2 _ 2 2 2 1 (A3) mZ,(mZ, - 7712) where M4 = (mar + 771%”)(m22 + 77122; - ma; - ma”) + marmaxr - m2Zm2z’ . (A4) 114 Appendix B Solutions for the SU(2)xSU(2)N> 1 where D(n) is a normalization constant and the eigenvalues are 2 ~2 2 - 2 ”(71) m2); = g f sm -2—. (BIG) 117 The characteristic equation for this system is 2 A2+A’2 p A2x\’2 8111 58111 (j\7+ 1);) = m 2Sln1’Vp, (8.11) [sin (A7 + 1)p—sin IVp] +m which has N + 2 solutions, pm). The phase constant aim) satisfies . p(n) _ x2 tan@(n)tan—2— — A’2—2(N+1) . (B.12) Using (B.11) and (B.12), we obtain for the normalization constant D = r ' —1/2 A +1 + sm [(N+1)P(n)]003[(N +1)p(n) + 2d)(n)i] (13.13) 2 2 sin' pm) There is one trivial solution to (8.11) and (B.12), which corresponds to the photon solution: ph) = 0, (12(7) = 7r /2. In section 4.3.2, the mixing matrix elements for the photons were shown to be constant, and equal to the U(1)Q coupling 6. Identifying the standard model Z 5 Z6, we obtain for the the remaining neutral boson masses N(2N+1)+ A2A’2 N 6(N+1)2 A2+X2N+1 2 2 ~2 2 ‘_ 7171' 2 71,77 2 7,121; = (} f (Sll’l m) + 27722 (COS m) (1+ O()\ )) . m2 = (92 +9'2)f2 Z 4(N + 1) 1—(A2+A’2) + out] , (13.14) The elements of the charged boson mixing matrix are 9 2 r2 N(2N+1) boo = ——[1—(,\ +,\ )— /g2+g/2 12(N+1)2 A/4 N 4 +2(/\2+/\’2)N+1+O()‘ )i’ A2 lV-i-l—j /\’2 j 3 bjO : i 32-. ‘_'_32+O(A), W(N+1)/ WWHH I _ __2__ _ 2 ,2 11:21:12 biNJ'llo _ 92 +912 1 (A +/\ )12(N+1)2 A4 N 4 /\ +2(A2+A'2).N+1+O( )i’ / 2 sin 7mN/(N + 1) 3 b 2 —A + O A , 0n N + 1 45in27r'n/2(N+1) ( ) 118 ‘ _ 2 . 7T(N+1—j)n 2 by", — ‘/N+lsm N+1 +C’)()\), 2 SiIlTl’Tl/(N + 1) 3 b , = —,\"/ , 0 ,\ , 13.15 (A'Hln 1V +14sin27rn/2(N + 1) + ( ) ( ) where j and n run from 1 to N. Finally, we can use the characteristic equations, (B4), (8.11), and (B.12), along with the orthonormality of the rows of the Z’ mixing matrix, to obtain a simple expression for the leading Ez/mfv, term in the W’fiWT’f —§ W,',+I/li’,’,_ scattering n amplitude, which is the generalization of K in (4.38). We find 4 mil/7,1 3 sin [2(N + 1)w(n)l Sin [4(N + 1)“J(n)l KW) = C(n) [5 + — N 1 2( + sin ”(71) 43in 2w(n) (B.16) It is interesting to note that this quantity is exactly independent of g’, and it falls off as (N + 1)‘2 for large N. Setting n = 0, we obtain the result for W+W‘ scattering in this model which, to first non-zero order in A2, is K = —g—— . (8.17) 119 Appendix C Coupling Constants in Model II C.1 Gauge Boson Couplings The Feynman rules for the cubic and quartic vertices, in the continuum TS gauge model of section 4.3.3, are shown in Fig. C.1, where W0 and Z0 correspond to the 14' and Z boson, respectively. The coupling constants are 7 2 2 4 . , , . = 1-—/\ +0 /\ gll"0,W0.H’0.l/l“0 g i 15 ( ) 26¢2(-12 gl/l'k.W0,WO.WO: 92 A— k37r3——1— )k[1+0()‘ )l ’ 2 gu}.14-',.H’O.WO= 923“ 212,, —1—52) kl 8(—1)k+l kl sin[(k—l)1r] 2 2 ._ 1 _ —— 1 +(192—12)27r2 (k-m l +0” )l ’ 2 (_1)k+l+m ( 1 1 1 —) 6 = ( — — — '— ng.Wl,Wm,Wo 1 (15—— 2\/27r l+m l m “+7" +( 1 1 1)5 +(_ 1 1 1)as m+k m k 1m” k+l k z m-k“ 16klm1-sin[( (k-l—m )7r]m/( (i—l—m k + l + m k _1_ x1— sin[(k + l — m)7r ]/(k + l — m.)7r1— sin[(k — ml + m)7r]/(k — l + m)7r k +1 - m k — l + m i x [1+0(12)] , 2 A2(_1)k+l+m+n - - gll'k,Hr’l.l/Vm.ll’n : 95 2 [0k+l.m+n ‘l' 6k+m,l+n + 0k+n,1+m 120 ~ 2 ’2k,l+m+n - 6l,m.+n+k ’ 6m.n+k+l — 6n.k+l+m] [1 + O(’\ )] 9 2 _ g4 _ 1492 + 279’2 —18g’4/g2 — 9’6/g4A2 + 004) gufowozozo “ 92 + 9,2 , 30(92+9’2) - 2\/§(—1)’c 2 _ 2 _ L :2 2 2 gliv’kJVOZOZO - (39 +(1+2( 1) )9 )A k3,,3 i1+0(’\ )l = 3 2 2 —1m '2 2 2—1k g2. , :(9+(+( ))9)9)\2 V153) [1+0()\2)], l’l'0.ll0.Zm,Z0 ¢92+g0 k 71' 2 _ 1g4—g2g’2+g’4 92+g’2 gwkwbzozo _ 3 92+ 9/2 — —"'2k2,,2 kJ +8((—1)’“+’g2+g’2) kl 1_sin[(k-l)7r] 2 [1+0(A2)] (k2 — l2)27r2 (k — m ’ 2 = g i _ g’f _ 92 + 9’2 5 gli’kfl’ozmzo W 3 6 —2k2,,2 km 4(—1)k+m (292 + (1+(“1)k+m)9’2) k m. 1 _ sin[(k - m)7r] 2 I (k2 — 771,2)2712 (k — m)1r x [1+ 002)] , + 2 _ 2 l__i_ 5 gW’osl’Vo-ZmZn —g (3 2m27r2 mm m n - 2 +8(—1) + mn (1_ sm[(m—n)7r]) ] x[1+O(A2)] ’ (m2 - 712)27r2 (m — n)7r k+l+m 2 _ / 2 [2 A (_1) ( 1 1 1 > v 7 — + r _ — _ — 6 . g” k-ll'l-Zm-ZO g g 9,) 2\/27r [ l + m l m “+7" +(1 —1-1)5 +(1_l_1)5 m+k m k ”"2”“ k+l k z ”"1"” +92+ (-—1)k'l‘l+mg’2 16 k I m 1—-sin[(k—l-m)7r]/(k—l—m)7r g2+g'2 k+l+m k—l—m x 1 — sin[(k +1 — m)7r]/(k +1 — m)7r1— sin[(k — l + m)7r]/(k — l + m)7r k+l—m k—l+m x [1+ (909)] , gll"A..l’l"0.Zm.Zn _ 9 9" 2\/§7r (m + n _ E _ H) ’“m” 1 1 1 1 1 1 +( ‘ ‘ ' 1;) 5min“: + (m ‘ z ‘ “l 522% n + k n m 16 k m n 1— sin[(k — m - n)7r]/(k — m — n.)7r k -+- m + n k - m - n 121 Z” : igll’plt’zzn [9110(k - Q)!» + gal/(q _ 1))/1. + 91m“) — ’00] (b) Figure (3.1: Feynman rules for gauge interactions. ><1—— sin[(k + m — n)7r]/(k + m. — n)7r1— sin[(k — m + n)7r]/(k — m + n)7r k + m — n k — m + n x [1 + 002)] , 2 _ A2(_1)k+l+m+n 5 + 5 + 5 g“,'k.wrl‘zqun — 95 2 k+l,m+n k+m,l+n k+n,l+m 2 ’6k.l+m+n - 6Lm+n+k — 6m,n+k+l — 6n.k+1+m] [1 + 0(’\ )] s (C-1) ‘ for the quartic vertices, and 92 [1 _ g4 +2g2g’2 _ 9/4 A2 4 9W W. = ———/——— — +00 )] , 57%2 + 9’2) 4 ‘2V§1$::g02R1—1—1W)+cxxfl], gwkM'OzO 2x592 4 9%,“,0‘2" —_- _ 72323 [(1—(—1)")+O(A )] , _ \/.q2+g’2 k1( ”Mfg sin((k+l)7r/2) 2 sin((k—l)7r/2) 2 ng’WbZo — 2 ' ‘ (k+l)7r/2 (k—l)7r/2 92- ’2 2 +6. ———-— 1+ 0 /\ . “2Q2+¢2[ <)]. 122 g [kn(—1)k+"7r2 (sin ((1: + n)7r/2)>2 (sin ((1: - TOW/2))2 QW’M’O-Zn 2 (k + n)7r/2 (k — n)7r/2 +25%] [1+ om] , _ (—1)"n§5 sin ((1: + l + n)7r/2) Sin ((1: + l - n)7r/2) 9”");«W1-3n _ fin (k +1 + n)/2 (k + l — n)/2 sin((k—l+n)7r/2)sin((k—l—n.)7r/2) 2 — (Mm/2 (MW lbw >1 , <22 for the cubic vertices. (In these expressions, and in the expressions below, whenever numerator and denominator vanish, the right formula can be obtained by taking the limit.) Notice that the vertices involving one or two photon lines can be obtained from (C.1), (C.2) by using the relations (4.51). C.2 Fermion Couplings The Feynman rules for the charged-current and neutral-current vertices in model II are shown in Fig. C.2. The corresponding coupling constants, for M = 0, are CC 22 ’i 4 9L0,(u0,d0) = 9[1--6—-—2-+0(t )] , gffiukao) = fi(:—_1;k):l:2tL [1+O(t2)] , 2%?me = fi(:-_1)%k):lflg2tL[1+O(t2)] , = %[(‘°’i“&.":§)2}¥2)2- (Si“(1‘:t’_z)?}‘fl)2 [NW 212320,, = ”7:5 [<1 — (—1)">r-% + (—1)"A2 + 00.4)] , 8 n (-1)k+1§5tL CC _ _ 2 gLn,(Uk’d0) — 7r1+4k(k — 1) —4n2 [1 +0“ )l ’ CC _ § 71 (--1)"‘+1 95 tr 2 gLMuodk) _ 7r 1+ 4k(k — 1) — 4122 [1+ 00 )l ’ (ICC _—_ n [)5 sin((k—l+n)7r/2)sin((k—l—n)7r/2) -Ln,(uk,d1) \/27r (k—l+n)/2 (k—l—n)/2 _sin((k+ l + n —1)7r/2)sin((k +1 — n — 1)7r/2) ( (At+l+n—1)/2 k+l—n—1)/2 li1+0(’")l» 123 ”7:: i 79L“ (uk d,)PL + 91?" uk dz) )PR (a) —l Xk Z" _ —2 921261“ M) )PL + 91?" (XI: X1) )PR] + 2.7/£962" Q (b ) X1 Figure C.2: Feynman rules for charge-current and neutral-current interactions. CC 921112 ‘d 91201210 do) = 2 R [1+0(t2)] . fig t CC “R 2 g = —— 1+O(t ) , R0(uk =dO) (k-%)27r2[ ] fl 9 id , R 2 g u. = ——1+0(z), RO,( Odk) (k _ %)27T2 [ ] CC : (—1)k+'-1 g sin((k-— 1W2) 2 _ sin((k+l—1)7r/2) 2 9120011: dz) 2 (k — 1W2 (k +1 — 1W2 x [1+O(t2)] , CC _ 8 n (-1)k 95 th _ 2 NM ,(U-~k d0) — 7r 1 + 4k(k — 1) — 4712 [1+ 00 )] ’ 8 n (-1)k £151,223 CC _ _ 2 'an’WOvdk) — 7r1+4k(k—1)—4n2[1+0(( )l ’ (CC _ 7,.(—1)k+l-"-195 sin((k—I+n)7r/2)sin((k—l—n)7r/2) -’Rn~ " fin (k—l+n)/2 (k—l—n)/2 sin((k +1 + n —-1)7r/2)sin((k +l — n —1)7r/2) 2 — (k+l+n—1)/2 (k+l—n—1)/2 J“ 0U”? 124 for the charged-current interactions, and NC 9L0,(x0,x0) NC 9L0 (Xk 110) NC 9L0 (11 x1) NC gL"'(XOaX0) NC 9L" (x1: x0) NC 9147113112,le NC 9R0 (x0 X0) NC 912011120) NC ”No (x1; x1) ANTC gR’MXOAO) NC 9R" (X1; X0) NC gnn (x1. 2,) 4 2’2 / .9 "9.9 +9 L 4 92+gl2[1- 2 2 ,9 F_—2‘+O(I)]s 9 (9 +9“) fi(—1)k‘/2+ ’21 ‘ (1-52,: 114,021 , Mg? + 9'2 (sin ((1: — l)7r/2))2 _ (sin ((k +1 — 1)7r/2))2[ 2 (k — 1W2 (k +1—1)7r/2 x [1 +0(12)] , ——‘—"/37j [(1 — <—1)">ti + (—1)" 8 “M 1)k+195tL 2 «1+4k( (k—1)— 4n 2[1 +0“ )l’ n 95 @[singh - l + n )7r/2) sin((k — l — n)7r/2) ( (( 2 n. I2 —1 g (k-l+n)/2 k—l—n)/2 k+l+n—1)7r/2)31n((l..+l—n—1)7r/2) k+l+n—1)/2 (k+l—n—1)/2 g2+2g’2t2 Rt[1(+0 ),] fiVg2+gl22t”[1+O(t (12)], (k- 222)” (_______—1)k+l 1g sin((k—l)7r/2) 2 sin((k:+l—1)7r/2) 2 2 ( (k—l)7r/2 )‘( (k+l—1)7r/2 ) x [1+ C(12)] , SlIl [ [1+0(12)[ , ~ 2 Lug: ‘12 [(1 — (—1)") + C(12)] . 8 Tl (_1)k .05 I’dR F1+4k(k— 1) —4n.2 n (—1)k+l-n-1 g5 sin ((k — I + n)7.—/2) sin ((k — z — n)7r/2) fin (k-l+n)/2 (k—l—n)/2 sin((k+l+n—1)7r/2)sin((ls'+l-n—1)7r/2) 2 — (k+l+n—1)/2 (k+l—-n—1)/2 ][1+0(")l’ 294 _ 2929I2_ 9/4 A2 4 92(92+9’2)g 7+0” 4 ’ [1 +0(12)] , I2 -_9._ [1+ [/92 +9]? 125 7 [2 fl /\ 93,? = -g— [112003)] . gmr for the neutral-current interactions. 126 Bibliography [1] P. W. Higgs, “Broken Symmetries And The Masses Of Gauge Bosons,” Phys. Rev. Lett. 13, 508 (1964). [2] G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, “Global Conservation Laws And Massless Particles,” Phys. Rev. Lett. 13, 585 (1964). [3] F. Englert and R. Brout, “Broken Symmetry And The Mass Of Gauge Vector Mesons,” Phys. Rev. Lett. 13, 321 (1964). [4] S. L. Glashow, “Partial Symmetries Of Weak Interactions,” Nucl. Phys. 22, 579 (1961) [5] S. Weinberg, “A Model Of Leptons,” Phys. Rev. Lett. 19, 1264 (1967). [6] C. H. Llewellyn Smith, “High-Energy Behavior And Gauge Symmetry,” Phys. Lett. B 46, 233 (1973). [7] D. A. Dicus and V. S. Mathur, “Mass Differences In A Unified Theory Of Weak And Electromagnetic Interactions,” Phys. Rev. D 7, 525 (1973). [8] J. M. Cornwall, D. N. Levin and G. Tiktopoulos, “Uniqueness Of Spontaneously Broken Gauge Theories,” Phys. Rev. Lett. 30, 1268 (1973) [Erratum-ibid. 31, 572 (1973)]. [9] J. M. Cornwall, D. N. Levin and G. Tiktopoulos, “Derivation Of Gauge Invari- ance From High-Energy Unitarity Bounds On The S - Matrix,” Phys. Rev. D 10, 1145 (1974) [Erratum-ibid. D 11, 972 (1975)]. [10] B. W. Lee, C. Quigg and H. B. Thacker, “The Strength Of Weak Interactions At Very High-Energies And The Higgs Boson Mass,” Phys. Rev. Lett. 38, 883 (1977) [11] B. W. Lee, C. Quigg and H. B. Thacker, “Weak Interactions At Very High— Energies: The Role Of The Higgs Boson Mass,” Phys. Rev. D 16, 1519 (1977). [12] M. J. G. Veltman, “Second Threshold In Weak Interactions,” Acta Phys. Polon. B 8, 475 (1977). [13] C. E. Vayonakis, “Born Helicity Amplitudes And Cross-Sections In Nonabelian Gauge Theories,” Lett. Nuovo Cim. 17, 383 (1976). 127 [14] M. S. Chanowitz and M. K. Gaillard, “The Tev Physics Of Strongly Interacting W’s And Z’s,” Nucl. Phys. B 261, 379 (1985). [15] Y. P. Yao and C. P. Yuan, “Modification Of The Equivalence Theorem Due To Loop Corrections,” Phys. Rev. D 38, 2237 (1988). [16] J. Bagger and C. Schmidt, “Equivalence Theorem Redux,” Phys. Rev. D 41, 264 (1990). [17] H. J. He, Y. P. Kuang and X. y. Li, “On the precise formulation of equivalence theorem,” Phys. Rev. Lett. 69, 2619 (1992). [18] H. J. He, Y. P. Kuang and X. y. Li, “Further investigation on the precise formu- lation of the equivalence theorem,” Phys. Rev. D 49, 4842 (1994). [19] H. J. He and W. B. Kilgore, “The equivalence theorem and its radiative correction-free formulation for all R(xi) gauges,” Phys. Rev. D 55, 1515 (1997) [arXiv:hep—ph/9609326]. [20] H. Georgi, “Chiral fermion delocalization in deconstructed Higgsless theories,” arXiv:hep-ph/0508014. [21] I. Antoniadis, “A Possible New Dimension At A Few Tev,” Phys. Lett. B 246, 377 (1990). [22] I. Antoniadis, C. Munoz and M. Quiros, “Dynamical supersymmetry breaking with a large internal dimension,” Nucl. Phys. B 397, 515 (1993) [arXiv:hep- ph/9211309]. [23] J. D. Lykken, “Weak Scale Superstrings,” Phys. Rev. D 54, 3693 (1996) [arXiv:hep—th / 9603133] . [24] I. Antoniadis and M. Quiros, “Large radii and string unification,” Phys. Lett. B 392, 61 (1997) [arXiv:hep—th/9609209]. [25] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “The hierarchy problem and new dimensions at a millimeter,” Phys. Lett. B 429, 263 (1998) [arXiv:hep- ph/9803315]. ‘ [26] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “New dimen- sions at a millimeter to a Fermi and superstrings at a TeV,” Phys. Lett. B 436, 257 (1998) [arXiv:hep—ph/9804398]. [27] G. Shiu and S. H. H. Tye, “TeV scale superstring and extra dimensions,” Phys. Rev. D 58, 106007 (1998) [arXiv:hep—th/9805157]. [28] K. R. Dienes, E. Dudas and T. Gherghetta, “Extra spacetime dimensions and unification,” Phys. Lett. B 436, 55 (1998) [arXiv:hep-ph/ 9803466]. [29] K. R. Dienes, E. Dudas and T. Gherghetta, “Grand unification at intermediate mass scales through extra dimensions,” N ucl. Phys. B 537, 47 (1999) [arXiv:hep- ph / 9806292] . 128 [30] A. Pomarol and M. Quiros, “The standard model from extra dimensions,” Phys. Lett. B 438, 255 (1998) [arXiv:hep—ph/ 9806263]. [31] H. C. Cheng, B. A. Dobrescu and C. T. Hill, “Gauge coupling unification with extra dimensions and gravitational scale effects,” Nucl. Phys. B 573, 597 (2000) [arXiv:hep—ph / 9906327] . [32] J. D. Lykken and S. Nandi, “Asymmetrical large extra dimensions,” Phys. Lett. B 485, 224 (2000) [arXiv:hep-ph/9908505]. [33] R. Sundrum, “To the fifth dimension and back. (TASI 2004),” arXiv:11ep— th/0508134. [34] R. Sekhar Chivukula, D. A. Dicus and H. J. He, “Unitarity of compactified five dimensional Yang—Mills theory,” Phys. Lett. B 525, 175 (2002) [arXiv:hep- ph/0111016] [35] R. S. Chivukula, D. A. Dicus, H. J. He and S. Nandi, “Unitarity of the higher dimensional standard model,” Phys. Lett. B 562, 109 (2003) [arXiv:hep- ph/0302263] [36] C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, “Gauge theories on an interval: Unitarity without a Higgs,” Phys. Rev. D 69, 055006 (2004) [arXiv:hep-ph/0305237]. [37] M. Soldate, “Partial Wave Unitarity And Closed String Amplitudes,” Phys. Lett. B 186, 321 (1987). [38] M. Chaichian and J. Fischer, “Higher Dimensional Space-Time And Unitarity Bound On The Scattering Amplitude,” Nucl. Phys. B 303, 557 (1988). [39] H. C. Cheng, C. T. Hill, S. Pokorski and J. Wang, “The standard model in the latticized bulk,” Phys. Rev. D 64, 065007 (2001) [arXiv:hep—th/0104179]. [40] N. Arkani-Hamed, A. G. Cohen and H. Georgi, “(De)constructing dimensions,” Phys. Rev. Lett. 86, 4757 (2001) [arXiv:hep—th/0104005]. [41] R. S. Chivukula and H. J. He, “Unitarity of deconstructed five-dimensional Yang- Mills theory,” Phys. Lett. B 532, 121 (2002) [arXiv:hep-ph/0201164]. [42] R. Foadi, S. Gopalakrishna and C. Schmidt, “Higgsless electroweak symmetry breaking from theory space,” JHEP 0403, 042 (2004) [arXiv:hep—ph/ 0312324]. [43] M. E. Peskin and T. Takeuchi, “A New Constraint On A Strongly Interacting Higgs Sector,” Phys. Rev. Lett. 65, 964 (1990). [44] M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections,” Phys. Rev. D 46, 381 (1992). [45] R. Foadi, S. Gopalakrishna and C. Schmidt, “Effects of fermion localization in Higgsless theories and electroweak constraints,” Phys. Lett. B 606, 157 (2005) [arXiv:hep-ph / 0409266] . 129 [46] G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, “Curing the ills of Hig- gsless models: The S parameter and unitarity,” Phys. Rev. D 71, 035015 (2005) [arXiv:hep—ph/0409126] . [47] C. Csaki, C. Grojean, L. Pilo and J. Terning, “Towards a realistic model of Higgsless electroweak symmetry breaking,” Phys. Rev. Lett. 92, 101802 (2004) [arXivz hep-ph / 0308038] . [48] Y. Nomura, “Higgsless theory of electroweak symmetry breaking from warped space,” JHEP 0311, 050 (2003) [arXiv:hep-ph/0309189]. [49] R. Barbieri, A. Pomarol and R. Rattazzi, “Weakly coupled Higgsless theories and precision electroweak tests,” Phys. Lett. B 591, 141 (2004) [arXiv:hep- ph/0310285] [50] R. Foadi and C. Schmidt, “An effective Higgsless theory: Satisfying electroweak constraints and a heavy top quark,” Phys. Rev. D 73, 075011 (2006) [arXiv:hep- ph/ 0509071]. [51] L. Randall and R. Sundrum, “A large mass hierarchy from a small extra dimen- sion,” Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221]. [52] G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, “Oblique corrections from Higgsless models in warped space,” Phys. Rev. D 70, 075014 (2004) [arXiv:hep- ph/0401160]. [53] S. Eidelman et al. [Particle Data Group], “Review of particle physics,” Phys. Lett. B 592, 1 (2004). [54] P. Sikivie, L. Susskind, M. B. Voloshin and V. I. Zakharov, “Isospin Breaking In Technicolor Models,” Nucl. Phys. B 173, 189 (1980). [55] R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi. “The structure of corrections to electroweak interactions in Higgsless models,” Phys. Rev. D 70, 075008 (2004) [arXiv:hep—ph/ 0406077]. [56] A. Lewandowski, M. J. May and R. Sundrum, “Running with the radius in RSI,” Phys. Rev. D 67, 024036 (2003) [arXiv:hep—th/0209050]. [57] A. Lewandowski and M. Redi, “Spin and a running radius in R81,” Phys. Rev. D 68, 044012 (2003) [arXiv:hep-th/0305013]. [58] A. Lewandowski, “The Wilsonian renormalization group in Randall-Sundrum. I,” Phys. Rev. D 71, 024006 (2005) [arXiv:hep-th/ 0409192]. [59] C. P. Burgess, S. Godfrey, H. Konig, D. London and I. Maksymyk, “Model independent global constraints on new physics,” Phys. Rev. D 49, 6115 (1994) [arXiv:hep-ph/9312291]. [60] C. Csaki, J. Erlich and J. Terning, “The effective Lagrangian in the Randall— Sundrum model and electroweak physics,” Phys. Rev. D 66, 064021 (2002) [arXiv:hep- ph / 0203034] . 130 [61] R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, “Universal non-oblique corrections in Higgsless models and beyond,” Phys. Lett. B 603, 210 (2004) [arXiv:hep-ph/0408262]. [62] R. Sekhar Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi. “Electroweak corrections and unitarity in linear moose models,” Phys. Rev. D 71, 035007 (2005) [arXiv:hep—ph/0410154]. [63] R. Sekhar Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, “Ideal fermion delocalization in Higgsless models,” Phys. Rev. D 72, 015008 (2005) [arXiv:hep—ph / 05041 14]. [64] R. Sekhar Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, “Ideal fermion delocalization in five dimensional gauge theories,” Phys. Rev. D 72, 095013 (2005) [arXiv:hep—ph/0509110]. [65] M. E. Peskin and D. V. Schroeder, “An Introduction To Quantum Field Theory,” [66] Y. Kawamura, “Gauge symmetry reduction from the extra space S(1)/Z(2),” Prog. Theor. Phys. 103, 613 (2000) [arXiv:hep—ph/ 9902423]. [67] A. Hebecker and J. March-Russell, “A minimal S(1)/(Z(2) x Z’(2)) orbifold GUT,” Nucl. Phys. B 613, 3 (2001) [arXiv:hep—ph/0106166]. [68] Z. Chacko, M. A. Luty, A. E. Nelson and E. Ponton, “Gaugino mediated super- symmetry breaking,” JHEP 0001, 003 (2000) [arXiv:hep—ph/9911323]. [69] M. Carena, T. M. P. Tait and C. E. M. Wagner, “Branes and orbifolds are opaque,” Acta Phys. Polon. B 33, 2355 (2002) [arXiv:hep-ph/0207056]. [70] M. Carena, E. Ponton, T. M. P. Tait and C. E. M. Wagner, “Opaque branes in warped backgrounds,” Phys. Rev. D 67, 096006 (2003) [arXiv:hep—ph/ 0212307]. [71] C. Csaki, C. Grojean, J. Hubisz, Y. Shirman and J. Terning, “Fermions on an interval: Quark and lepton masses without a Higgs,” Phys. Rev. D 70, 015012 (2004) [arXiv:hep—ph/0310355]. [72] A. A. Affolder et al. [CDF Collaboration], “Search for quark lepton compositeness and a heavy W’ boson using the e nu channel in p anti-p collisions at s**(1/2) = 1.8-TeV,” Phys. Rev. Lett. 87, 231803 (2001) [arXiv:hep-ex/0107008]. [73] [ALEPH Collaboration], “A combination of preliminary electroweak measure- ments and constraints on the standard model. ((B)),” arXiv:hep—ex/ 0212036. [74] K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, “Probing The Weak Boson Sector In E+ E— ——> W+ W~,” Nucl. Phys. B 282, 253 (1987). [75] R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, “Multi- gauge-boson vertices and chiral Lagrangian parameters in higgsless models with ideal fermion delocalization,” Phys. Rev. D 72, 075012 (2005) [arXiv:hep- ph/0508147] [76] The LEP Collaborations ALEPH, DELPHI, L3, OPAL and the LEP TGC Work- ing Group. LEPEWWG/TC/2005-01; June 8, 2005. 131 [77] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” 5th edition, Academic Press (1994). [78] K. Hagiwara et al. [Particle Data Group Collaboration], “Review Of Particle Physics,” Phys. Rev. D 66, 010001 (2002). [79] C. Schwinn, “Unitarity constraints on top quark signatures of Higgsless models,” Phys. Rev. D 71, 113005 (2005) [arXiv:hep—ph/0504240]. [80] F. Larios, M. A. Perez and C. P. Yuan, “Analysis of t b W and t t Z couplings from CLEO and LEP/SLC data,” Phys. Lett. B 457, 334 (1999) [arXiv:hep- ph/9903394] 132 1111111171111][1]]