bl: :1 I... .2: ) 1.11.!» E .I..‘ ‘ Q. .4 5.33.. 13‘ San-cl 51.1}. V '5 Ifiq id“-I’; ill. 3 I“ d. .. . curt.“ :3! an... 3% u. tat-33.1.2. Jet: It... IH‘I‘IR .‘2'! u. radial .o‘ln‘ifiu ‘ . QV .1592... K‘ 5* “) \ 1x) ‘ :\ Lia???" Michigan. state University This is to certify that the thesis entitled TOPICS IN PATICLE THEORY presented by Stefano Di Chiara has been accepted towards fulfillment of the requirements for the Doctoral degree in Physics & Astronomy [xfg> fluxkaAfig; Major Professor’s Signature ép§M7 Date MSU is an Affirmative Action/Equal Opportunity Employer 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 5108 KzlProlecc8PreleIRCIDateDue.indd TOPICS IN PARTICLE THEORY By Stefano Di Chiara A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Physics & Astronomy 2009 ABSTRACT TOPICS IN PARTICLE THEORY By Stefano Di Chiara We introduce the standard model (SM) of elementary particles and show that it is afflicted by a large fine tuning of the tree level Higgs mass parameter. We solve this problem by eliminating the Higgs scalar from the theory and defining a Higgsless model based on an SU (2)” +1 x U (1) gauge symmetry. We show that the scale of unitarity violation of the new theory is higher than that of the SM without the Higgs, and we show that the new model satisfies the electroweak experimental constraints. We determine model-independent constraints on new physics contributions from precision electroweak measurements. We then test the viability of top-color assisted technicolor models using these constraints, and find that one of the models studied is indeed consistent with the experimental results. We introduce a simple extension of the SM which implements custodial symmetry, that protects electroweak observables from large corrections, in a top-quark mass generation sector of the Lagrangian. We then study the phenomenological viability of the model, and Show that the new physics gives an unusual negative contribution to one of the electroweak parameters. Finally, we study the consequences of a noncommutative space, which is character— ized by a non-zero minimal measurable distance, on the particle physics phenomenol- ogy. We study 6+6“ -—> 777 scattering in noncommutative quantum electrodynamics at future colliders, and extract lower limits on space noncommutativity in function of cross sections relative to that process. DEDICATION A mia madre e mio padre, Annamaria e Gaetano, ed a mia sorella Barbara. iii ACKNOWLEDGMENT I would like to thank my advisor, Prof. R. Sekhar Chivukula, for his guidance during these years at Michigan State University. His deep physical insight and immense knowledge of theoretical particle physics have greatly helped me, through his expla- nations of physics and encouragement, to tackle productively the course of my PhD studies. I would like to thank also my co-advisor, Prof. Elizabeth H. Simmons for her guidance: her deep knowledge of theoretical particle physics and willingness to guide me through it have been very helpful to me. I am particularly thankful to Prof. Wayne W. Repko, whose advice has always been invaluable since the very first day here at Michigan State University. I am thankful also to the other members of my PhD guidance committee, Pro- fessors Pawel Danielewicz, Joey Huston, and Carlo Piermarocchi, for their guidance and advice. I am grateful to Prof. Alberto Devoto for his advice and for giving me the oppor- tunity to have this magnificent and unbelievable experience that these years in the USA have been for me. I would like to thank the postdocs Roshan Foadi, Ken Hsieh, and Neil Christensen, for their help and explanations in answering my physics questions. I am thankful to the secretaries Debbie Barratt and Brenda Wenzlick for their precious help. I thank my officemates Barath Coleppa, Mohammad Hussein, Kai Schmitz, J iang— hao Yu, for the good time in the office. I thank my friends Christine, Angela, Giuseppe, Roshan, Monica, Alfredo, An- gelo, Sebla, and Jeremy, for the great time in Michigan and elsewhere, and for the internationally acclaimed, unexpectedly relaxing opera trips. Last but not least, I thank my parents, Annamaria and Gaetano, and my sister, iv Barbara, because in all these years even if they were far away, I always felt them close. Grazie. TABLE OF CONTENTS List of Figures ................................ Introduction ...................................... The Standard Model and Beyond ....................... 2.1 Gauge symmetry ............................. 2.2 Higgs Mechanism ............................. 2.3 The Standard Model ........................... 2.4 Higgsless Models ............................. 2.5 Noncommutative Quantum Field Theories ............... Higgsless Models in 4-Dimension ....................... 3.1 Introduction ................................ 3.2 The Model and Electroweak Interactions ................ 3.3 Three Site Model ............................. 3.3.1 Boson Masses and Eigenstates .................. Charged Gauge Bosons ...................... Neutral Gauge Bosons ...................... 3.3.2 Fermion Wavefunctions and Ideal Delocalization ........ Fermion masses and wavefunctions ............... Ideal Delocalization ........................ 3.3.3 Fermion couplings to the W boson ............... Left-handed fermion couplings to the W boson ......... Weak mixing angle ........................ Right-handed fermion couplings to the W boson and b —+ 37 . 3.3.4 Fermion couplings to the Z Boson ................ 3.4 One-Loop Corrections to p ........................ 3.5 Experimental Bounds on Fermion Masses ................ 3.6 Conclusions ................................ Electroweak Fits and Technicolor Models ................. 4.1 Introduction ................................ 4.2 Flavor Universal Fits to Electroweak Observables ........... 4.3 Experimental Observables ........................ 4.4 Theoretical Predictions .......................... 4.5 The Fit .................................. 4.6 Topcolor Assisted Technicolor Models .................. 4.6.1 Hypercharge—Universal Topcolor ................. Bounds on M z’ .......................... vi 7 11 13 18 32 38 38 41 48 49 49 51 53 53 55 57 57 59 60 62 63 69 72 74 74 78 80 81 84 87 87 91 UOW> Interpretation using standard electroweak parameters ..... 93 4.6.2 Flavor-Universal and Classical Topcolor Models ........ 93 4.7 Conclusions ................................ 95 Custodial Invariant Top-Mass .......................... 96 5.1 Introduction ................................ 96 5.2 Doublet-Extended Standard Model ................... 98 5.2.1 Custodial Symmetry and Z coupling .............. 98 5.2.2 The Model ............................. 99 5.2.3 Mass Matrices and Eigenstates ................. 102 5.3 Phenomenology . . . . _. ......................... 104 5.3.1 Z coupling to bLbL ........................ 104 5.3.2 Oblique Electroweak Parameters ................. 107 Parameter aT ........................... 108 Parameter aS ........................... 110 5.3.3 Goodness of Fit .......................... 110 5.4 Conclusions ................................ 113 NCQED ......................................... 115 6.1 Introduction ................................ 115 6.2 NcQED Amplitudes ............................ 116 6.3 Cross Section Results ........................... 119 6.3.1 Unpolarized Cross Section .................... 119 6.3.2 Polarized Cross Sections ..................... 123 6.4 Discussion and Conclusions ....................... 125 Experimental Data and SM Predictions .................. 131 Spin Averaged Cross Section .......................... 138 Phase Space ...................................... 143 Squared Modulus of the Helicity Amplitudes ............... 147 Bibliography ..................................................... 149 vii 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 LIST OF FIGURES Feynman diagram, involving a fermion anti-fermion loop, which con- tributes to the Higgs mass ......................... 17 gauge boson contributions to W+W‘ elastic scattering ......... 21 additional gauge contributions to W+W' elastic scattering. ..... 21 Higgs contributions to the W+W‘ elastic scattering. ......... 22 Moose diagram for the sigma model presented in [18]. The solid circles represent SU (2) gauge groups, while the dashed circle represents a U (1) gauge group. the lines connecting two circles represent link fields transforming under the adjacent gauge groups. ............ 24 Moose diagram notation for the Higgsless SM gauge and scalar sector in Eq. (2.67). ............................... 24 The J = 0 partial wave amplitude as a function of ,/3 for, from left to right, the Higgsless SM and the U ( 1) x SU (2)1“1 Higgsless model for N = 1, . . . , 100 with mw1 = 500 GeV [19]. ............... 28 Moose diagram notation for the coupling of fermion fields to the model of Fig. (2.5). The lower (upper) segments represent LH (RH) fermions, while the diagonal dashed lines represent Yukawa couplings of the cor- responding fermions to the intersected sigma field. .......... 30 Moose diagram for the model analyzed in [18]. The solid circles rep- resent SU(2) gauge groups, while the dashed circle represents a U(1) gauge group. The lines connecting two circles represent link fields transforming under the adjacent gauge groups. All gauge couplings and f-constants are arbitrary parameters. ............... 42 viii 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 5.1 5.2 Moose diagram notation for the coupling of fermion fields to the model of Fig. (3.1). The lower (upper) segments represent left-handed (right- handed) fermions, while the diagonal dashed lines represent Yukawa couplings of the corresponding fermions to the intersected link field. 44 Vacuum polarization amplitudes for left-left and left—right gauge currents. 65 Plot of f (N), defined by Eq. (3.109). The continuum limit gives ap- proximately a correction of 17%, with respect to the three-site model. Lower bounds on the mass "1x1 of the lightest among the heavy fermions, as a function of MW], with N varying between 1 and 122 (left). We also plot the same quantity as a function of N, for Mw1=500 GeV (right). In each case a solid line corresponds to Ap < 2.5 - 10‘3, while a dashed line corresponds to Ap < 5 - 10’3. We notice that the three-site model is already a very good approximation for the continuum model, with a difference of just 3%. ........................... Here is the plot of the probability density Pd(x2/d) for d = 1,... ,5. At x2 = 0, P,- ranks j-th in magnitude, and therefore P1(O) has the highest value (00). ............................ Here is the plot of the cumulative probability cPd(X2/d) for d = 1, . . . , 5. The smaller value near the origin corresponds to cP1(:r), and the others follow in the same order of d. ...................... 95%CL in the aS — QT and Ap - ad planes. The origins of the axes are placed at the parameters optimal values for my = 800 GeV. . . . The solid (dashed) curve shows the restriction on model parameter space obtained from a fit of all Z pole and off-pole data listed in the first two tables of the Appendix (A) for an 800 GeV (1500 GeV) reference Higgs mass. The region outside the parabola is excluded at 95%CL. . The solid (dashed) curve shows limits on the Z’ mass as a function of In resulting from a fit of all Z pole and off-pole data listed in the first two tables of the Appendix for an 800 GeV (1500 GeV) reference Higgs mass. Values outside the boomerang shaped region are excluded at 95%CL .................................. The line on the top refers to m7, and the one on the bottom to m E. . Loop level contribution to_gu, in the gaugeless limit; the blob represents the renormalized 7rz —> bb process ..................... ix 67 71 86 91 92 103 105 5.3 5.4 5.5 5.6 5.7 5.8 6.1 6.2 General triangle diagram for the 7rz ——> bf) process at one loop. DESM prediction on gu, in function of the parameter p. The y axis intersects the M axis at the value gill," = —0.42114. The horizontal dashed line corresponds to 92?, = —0.4182, while the two horizontal solid lines show the relative $30 deviations. .............. Vacuum polarization diagram contributing to the oblique electroweak parameters. The indices 2', j = 1, 2, 3, Q, where WQ = A, the EM field, while k refers to a given quark-antiquark pair consistent with the EM charge of W,-. ............................... aT in function of a, defined as the difference of the DESM result with the corresponding SM one with my = 115 GeV. The horizontal lines show the optimal fit value of aT = 0.68 X 10‘3 (dashed line) and the relative :l:30 deviations (solid lines). The origin corresponds to the SM (1T value, which is zero by definition. .................. 08 in function of it, defined as difference of the DESM result with the correspondent SM one with my = 115 GeV. The horizontal lines show the optimal fit value of a3 = 0.34 x 10‘3 (dashed line) and the relative i3a deviations (solid lines). The origin corresponds to the SM aS value, which is zero by definition. .................... Here plotted are the 95%CL regions of the a8 and aT parameters for my = 115, 150, 200, 300, ..., 1000 GeV. Also plotted are the points rel— ative to the DESM with p = 3, 4, ..., 20, 00. We defined the coordinates as AaS(mH) = a3 (my) — aS(mH)mm + as (my = 115GeV) AaT(mH) = aT(mH) —- aT(mH) + aT (mH = 115GeV) min ’ min min' ' ' ' Diagram (a) represents the contributions from abelian interaction terms and diagrams (b) and (c) represent contributions from the non-abelian interaction terms. ............................ In the top panel, the solid line is number of events from NcQED as a function of 45 for the case x/S = 5TeV, ANC = 1TeV, A = 7r/4, L = 500 fb"l and no out on cos 9. The dashed line is the uniform background from QED with no cos 0 cut. The bottom panel shows the effect of imposing the additional cut |cos 9| 5 0.9 ............ 105 107 108 109 111 112 117 122 6.3 6.4 6.5 6.6 Al A2 A3 A4 A5 The 95%CL lower bounds on A No attainable using the left-right asym- metry of the total cross section are illustrated as a function of lumi- nosity at x/S = 2 TeV (top) and as a function of x/S for [I = 500 fb'1 (bottom). The solid lines correspond to A = 0 and the dashed lines to A = 7r / 4. ................................. The polarization asymmetries with respect to the photon energy frac- tion V (top) and the photon angle with respect to the beam axis (bot- tom) are shown. The three shaded regions correspond , in order of magnitude, to center of mass energies of 5, 1.0 and 0.5 TeV. ..... Same as Fig. 6.3 with |cos6| S 0.85. ............. l ...... The 95%CL lower bounds on ANC for 0.5 g |cos gmaxl S 1.0 are shown for x/S = 0.5 TeV (dot-dash-dash), 1.0 TeV (dot-dash), 2.0 TeV (dashed) and 5.0 TeV (solid). Here L = 500 fb’1 and cosA = 1. The bounds scale as [21/ 4 and \/ cos A ................... l-loop fits to Z-pole observables in the Standard Model with M}? = 115 GeV, showing the experimental values, the predicted values, and the pulls ................................... 1-loop fits to LEP II observables at several values of V3 in the Standard Model with M9” = 115 GeV, showing the experimental values, the predicted values, and the pulls. ..................... Fits to Z—pole observables in the Hypercharge-Universal TC2 model introduced in this paper. Ffom left to right, the columns show the experimental values, the l-loop SM values 0,1 "O"? with M2,” = 800 GeV, and the predictions for the TC2 model with their pulls. The T02 model fit assumed f, = 75 GeV and M}?! = 800 GeV ....... Fits to LEP II observables at several values of ([5 in the Hypercharge— Universal TC2 model introduced in this paper. Fiom left to right, the columns show the experimental values, the 1-loop SM values 0,14“? with MI,” = 800 GeV, and the predictions for the TC2 model with their pulls. The T02 model fit assumed f, = 75 GeV and M3! .= 800 GeV ..................................... Fits to Z—pole observables in the Classical [56, 57] and Flavor-Universal TC2 [83, 55] models. From left to right, the columns show the experi- mental values, the 1-loop SM values 0,1400” with MI,” = 800 GeV, and the predictions for the TC2 model with their pulls. The TC2 models fit assumed ft = 75 GeV and MI,” = 800 GeV. ............ xi 126 127 128 129 132 133 134 135 136 A.6 Fits to LEP II observables at several values of ‘5 in the Classical [56, 57] and Flavor-Universal TC2 [83, 55] models. From left to right, the columns show the experimental values, the l—loop SM values 0,1 710°” with M];3f = 800 GeV, and the predictions for the TC2 models with their pulls. The TC2 models fit assumed ft = 75 GeV and ML” = 800 GeV ..................................... 137 xii Chapter 1 Introduction An elementary particle is a physical entity with no substructure, and is defined solely by its kinematical and dynamical properties. The kinematical properties are mass and spin, which determine how a free particle propagates, and the dynamical properties are the quantum numbers, which determine the strength of a particle’s interactions. There are four known interactions: electromagnetic, strong, weak, and gravitational. Gravity in general can be neglected in particle interactions. For a relativistic theory to be consistent, space-time coordinates have to be treated as simple parameters, while the quantum numbers of a particle are carried by a space—time dependent quantum field. The standard model (SM) is the most successful quantum field theory (QF T) in explaining and predicting the current experimental results with a minimal number of parameters. The SM predictions, in fact, fit all the available data very well, both in relative terms and in terms of standard deviations (on average at the 0.1% level with a deviation in most cases smaller than 20), and no measurement rules the model out. At the present time, the SM is, indeed, a very accurate model of particle physics. The SM, though, does have a few open problems. First, the physical mass of a particle is obtained by adding to the bare mass the 1 relative quantum corrections, which, as corrections, are reasonably expected to be comparably small. This is not the case for the mass of the Higgs boson, a spin zero, electromagnetic (EM) neutral particle. If the SM is assumed to describe particle physics up to the Planck scale, where gravity cannot be neglected anymore, the bare mass of the Higgs boson needs to be fine tuned to one part in 1032, to produce a physical Higgs mass m H of the order of the electroweak scalel. Though perfectly possible, this level of fine tuning is highly unlikely to happen by chance. One therefore expects some physics beyond the SM to explain this puzzling result. The fact that the value of my favored by experiment excluding direct search of the Higgs boson is, indeed, much lighter than the lower bound on m” from the direct search at CERN LEPII experiment[1], seems to confirm the hypothesis that the SM is a low energy effective theory, which represents a low energy scale approximation of a more complete model of particle physics. If we require a more reasonable fine tuning on the order of a few percent, new physics underlying the SM should appear at an energy scale of order a TeV, scale that will be soon explored at the CERN Large Hadron Collider (LHC). Second, the SM predicts that most of the mass in the universe is baryonic; this is in striking disagreement with the observations. In order to understand the rotational velocities of stars in galaxies, a neutral, non-relativistic, new form of matter must be postulated, or otherwise the galaxies (including ours) would simply fall apart. The same dark matter is required to explain the measured expansion rate of the universe. The observed properties of dark matter cannot be associated with any particle of the SM. Moreover, the observations suggest that a dark matter candidate should be weakly interacting and with a mass possibly in the hundreds of GeV. If that is indeed the case, any extension of the SM, which is required to accurately describe particle physics up to a few TeV (LHC will eventually reach an energy in the center of mass 1mg of the order of v = 246 GeV is required for the SM to be consistent. 2 of 14 TeV), will have to include such a dark matter candidate, as well as solve the fine tuning problem discussed above. Third, within the SM the neutrinos, weakly interacting, spin one-half particles, are predicted to be massless. This contradicts experimental results from solar neutrinos which measure a mass gap among the three neutrino flavors. Fourth, the SM does not include gravity, and this is clearly a problem. This, though, is not a currently compelling problem, since we expect gravity to be relevant only in black holes (which we expect not to produce at LHC) and during the big bang. It is, however, interesting to study whether gravitational interactions can have some detectable, indirect effects on particle physics at future colliders. Besides the previous problems, the major shortcoming of the SM is that it does not address many fundamental questions, but rather parametrizes our ignorance. An open question is electroweak symmetry breaking (EWSB), the process through which most of the SM particles acquire mass. While the SM accommodates this mech- anism, it does not require EWSB to happen: if the Higgs squared-mass parameter in the Lagrangian were positive, rather than negative, no particle in the SM besides the Higgs would have mass (i.e. no EWSB would occur). Another open question regards the number of fermion families. In the SM there are three fermion families with identical quantum numbers, but different masses: experiment shows that three families exist, and the theory is able to accommodate the three families, but it does not require there to be three. In fact, a slightly modified SM with four families would be still perfectly self-consistent. All these arguments suggest that the SM, though very successful, is only a low energy effective theory. A model beyond the SM is needed if we want to address the problems and open questions mentioned before. In this thesis we will tackle some of these issues, with the goal of gaining a deeper understanding of the theory and possibly finding a model beyond the SM which will accurately predict the upcoming 3 experimental results from LHC. An intuitive solution to the fine tuning problem is to eliminate the Higgs boson from the theory. The SM without the Higgs is the prototype of a Higgsless model; the electroweak precision measurements, though, already require the SM to include a Higgs with a mass of the order of the electroweak scale or lighter. In particular the role of the Higgs boson in the SM is to unitarize the theory, or stated in another way, to cancel from the theory large contributions to scattering amplitudes so that the probability associated with a single scattering event is equal or smaller than one. The unitarizing role of the Higgs in Higgsless models is taken over by heavier copies of the Wi boson, a spin one, EM charged particle, introduced in the theory by additional gauge groups, which represent a particular symmetry of the theory. In principle the theory can be made consistent, according to unitarity criteria, by adding more gauge groups, but at some point the process breaks down, because the extra gauge bosons add more channels to each initial state in a scattering process, and the probability of a single scattering event becomes greater than one at some given energy. Nevertheless the scale of unitarity violation is delayed to energies high enough for Higgsless models to be consistent at LHC. Higgsless models have a very interesting and peculiar phenomenology, most notably because they predict the total absence of any particle (spinless, colorless, neutral and massive) that can be associated with the Higgs boson. Because of this, and the need for a light Higgs for a theory like the SM to be unitary, Higgsless models will be soon decisively tested at LHC. A general approach that allows one to test models that modify the (EW) elec- troweak sector of the SM is to parametrize their new physics contributions to the vacuum polarization corrections (termed “oblique”) to the self-energy of the W*, Z bosons. The optimal values for these parameters, as well as their confidence level (CL) regions, are determined by a x2 minimization. Since the SM predictions fit the EW precision measurements well, Peskin and Takeuchi’s oblique parameters S, T, U 4 [2] are strongly constrained to be small, on the order of a percent. A key test for a model beyond the SM is to arrive at predictions sufficiently close to those of the SM so as to be phenomenologically viable, and therefore to result in oblique parameters which are sufficiently small. Some string theories, which assign a finite dimension to elementary particles, have, among the very many possible vacua, some that correspond, in the low energy limit, to a quantum field theory defined in a noncommutative space. This fact has rekindled the interest in the phenomenological consequences of gravity in particle physics. Noncommutativity is already required by general relativity, which predicts that the minimum measurable distance is equal to the Planck length. A photon having the energy necessary to measure a length shorter than the Planck length would create a black hole and be captured by it, making any such measurement impossible in practice. A minimum measurable length translates into a non-zero commutator of the coordinates, which are promoted to operators. The noncommutativity scale identifies the scale at which gravitational effects on the geometry of the space become non-negligible. Since these effects are not a consequence of a direct gravitational interaction of the particle, the noncommutativity scale might be much larger than the Planck scale and, hopefully, detectable at future colliders. In Ch. (2) we will review the basic principles of quantum field theory and model building. We will start with quantum electrodynamics (QED) and gauge invariance, then introduce the SM Lagrangian, finishing the chapter with an introduction to Higgsless, and noncommutative quantum field theories (NcQFT). In Ch. (3) we will present an N —site Higgsless model,and study in detail its phenomenological viabil- ity. In Ch. (4) we will review the principles of the X2 test, using a general oblique parametrization of new physics contributions to oblique observables, and then we will apply this test to Top—Color Assisted Technicolor models (T C2). In Ch. (5) we Will study a model having a custodial SU(2) invariant top quark-mass sector, and 5 determine its phenomenological viability using S, T constraints from EW precision data. In Ch. (6) we will study the phenomenology of noncommutative quantum elec- trodynamics (N cQED) at a future electron-positron collider, calculating the model’s predictions on the ele’ —-> 777 process, and will extract lower limits on the noncom- mutativity scale in function of cross sections relative to that process. Chapter 2 The Standard Model and Beyond In this introductory chapter we will present the theories that will be studied in the rest of the thesis. We start with an introduction to QFT focusing on gauge symmetry and the Higgs mechanism. Then we introduce the SM, by specifying its gauge symmetry group, the particle content of the model, and the quantum numbers of each particle. At the end of the SM section we study the fine tuning problem. Following that we introduce Higgsless models [3, 4, 5, 6, 7, 8], by specifying their gauge symmetry group, particle content, and by studying their phenomenological viability. The last section will introduce QFT defined in a noncommutative space, and show how to redefine the theory in terms of the ordinary commutative coordinates and their functions. 2.1 Gauge symmetry The Lagrangian for a. free Dirac field tb(:r) is (for an introduction to quantum field theory see for example [9, 10, 11, 12]) L: 2 “WWW ‘ 771159,}: (2-1) 7 where we omit the dependency on the coordinate as. This Lagrangian is invariant under a global phase transformation we) —» W) = exp l-ial was). (2.2) If one, though, requires this transformation to be local, and therefore 1/2(:r) —> 1/2’(:i:) = exp [—z'gl"(a')] 112(1). (2.3) a new term git—J7“8,,I‘ 1b appears in the Lagrangian (2.1) after applying the transforma- tion (2.2). The system of reference in the field configuration space, is called “gauge”, and (2.2) is therefore a gauge transformation. A vector field Au(:1:) must be introduced in the theory to make Eq. (2.1) locally invariant. The Lagrangian for a massless free vector field is 1 .c = —ZF‘“’F,,,,, (2.4) where F” E 8W1" — WA“. (2.5) Eq. 2.4 is invariant under the gauge transformation A,,(:r) —+ A;,(a:) = Ap(:r.) + 6#F(:I:). , (2.6) We are allowed to add an interaction term to Eq. (2.4) without spoiling its gauge invariance, and write the Lagrangian for an interacting vector field as 1 c = 7W3“, — J“A,,, (2.7) 8 where J“ represents an external conserved current interacting with the system de- scribed by Eq. (2.4). The gauge-invariant Lagrangian for a theory containing both a Dirac field ’t/J(.’II) and a vectorial field Ap(:r), therefore, is given by .C = “Effigy/J — mifiw — EFWFW — J“A,,. (2.8) Eq. (2.8) is invariant under the gauge transformations (23,26) only if we identify J“ = {Ii/77W, (2-9) in which case we can write the Lagrangian for an interacting Dirac field ’i/)(.’L‘) and a gauge field A,,(:r) as — — 1 L = iw7"D,,i,D — mww — ZFW’FW, (2.10) where D,, = 8,, + 29A,, (2.11) is the covariant derivative. If one identifies g = Q6, where, for h = c = 1, e = 0.3028 is the positron EM charge, Eq. (2.10) is the Lagrangian describing QED. The Lagrangian for a free complex scalar (spinless) field ¢(:r) is L = (aflesfl are, (2.12) which is invariant under a global phase transformation of ¢(r). Proceeding as we did for a Dirac field, we generalize to the case of a local phase and write 1 L = (D[L¢)f DM¢ — 4 FWFW, (2.13) 9 which is invariant under the gauge transformation (2.6) and etc) -+ err) = exp l-igF(=r)l em. (2.14) For a scalar field 45 with N components a gauge transformation can be defined as ¢,(:z;) -+ 95207?) = U(flilwflfli), (2-15) where U (:r) is a unitary matrix with N components. In general two unitary ma- trices U’ (:r), U ”(22) do not commute with each other, so the relative gauge trans- formation is non—abelian. The one defined by Eq. (2.14) is abelian (U (1)), since U (:r) = exp [—igI‘(:r)], for a given :3, is just a c-number. The gauge transformation for an N -component Dirac field \I1 is similarly defined as \I'(;r) —> \II'(:1:) = U(sr)\I/(:r), (2.16) while the gauge field A,,(:r), which, as described below, is an N x N Hermitian matrix, transforms as Ap(a:) —> A],(:c) = U($)A,,(2:)Ul(:r) + §U($)8,,Ul(:r). (2.17) The covariant derivative is still expressed by Eq. (2.11), while the field strength is generalized by 1W 5 1 [1)“, 0"] = 0M" — em“ +19 [A",/l"] , (2.18) 9 that indeed in the abelian case reproduces Eq. (2.5). In case the transformation is special (determinant one) unitary (S U ( N)) the trans- 10 formation matrix can be written as U(x) = exp [—igF“(:r)T“], (2.19) where the generators T“, in the fundamental representation, are Hermitian, traceless matrices satisfying the equations [T0,Tb] = ifabr'Tc, r111, (TaTb) = é‘ab. (220) The gauge field and field strength are therefore accordingly re—expressed as Au(:r) = Az(:c)T“, FW(;2:) = F:u(x)T“, (2.21) where the field A: is associated with a gauge boson. It is important to notice that the coupling constant 9 in Eq. (2.19) must be all the same for the transformations (2.15,2.16,2.17) for the Lagrangians (210,213) to be gauge invariant. This is not the case for a U ( 1) transformation, and that is why Q is allowed to be different from 1 for different scalars and fermions interacting with the same gauge boson in Eq. (2.11) ®=Qd- 2.2 Higgs Mechanism A gauge field mass term (émz’At‘Ap) in the Lagrangian (2.10) is not allowed because it would actually violate gauge symmetry under Eq. (2.6). However, a scalar field can be introduced in the theory to give mass to the vector field through spontaneous gauge symmetry breaking. We can actually add to Eq. (2.13) the potential 1 we) = 771.2qu ¢ + ZA (¢*¢)2, (2.22) 11 that is invariant under the gauge transformations (26,214) and leaves the theory local and renormalizable. So we write 1: = (12m)l D“¢ — v (e) — EIFWFW. (2.23) If m2 < 0, A > 0, the vacuum state, defined as the state of minimum energy of the system, belongs to the configuration ¢(a:) such that —2m2 122 <0I(p= , =3. (2.24) where v is the vacuum expectation value (vev). A physical system evolves naturally to the state of minimum energy, and therefore we expand around a minimum defined by pa) = g (e + Hus» exp [rm/v]. (2.25) where H(:1:) and x(:r) are real fields. The Lagrangian in Eq. (2.23) is not gauge invariant around the minimum defined by Eqs. (224,225) (the symmetry has been spontaneously broken), since 1) is constant under gauge transformations. The field x(a:) is simply a phase which moves the system to other vacuum states. By performing a gauge transformation, we can choose a specific vacuum and set x(a:) to zero (unitary gauge). The kinetic term in Eq. (2.23), re-written using Eq. (2.25) and x(:r) = 0, reads 1 2 (D,,¢)i 0% = Esuperstar + 93 (v + H)2 A”A,,, (2.26) which shows that the A"(:r) has acquired a mass 111,, = 922, and that 6“X(:r) has been replaced (or “eaten”) by the longitudinal component of A“(a:). Substituting 12 Eq. (2.25) with x(:c) = 0 and m2 2 —¥ in the potential (2.22) one obtains A H 2 V(H) = H2— —- + v , (2.27) 4 2 from which we can see that m H = #2). This process by which the gauge symmetry is spontaneously broken and consequentially a vector field acquires mass is called the “Higgs mechanism”, and the massive, neutral, scalar field H (:r) is the Higgs boson [13]. 2.3 The Standard Model The SM’s Lagrangian is invariant under SU(3)c x SU(2),,, x U (1)y gauge transfor- mations, where the subscripts stand for, respectively, color, weak-isospin, and hy- percharge. All the particles included in the SM have been detected, with the Higgs scalar H being the only exception. These are the quarks u, d, c, s, t, b, the leptons 6, ye, p, u,“ 'r, VT, and the gauge bosons g, Wi, Z, 7. The weak interaction is chiral, therefore it distinguishes between left- and right-handed states. The quarks and the leptons are organized in three families, having the same quan- tum numbers but different masses. The quantum numbers under S U (3)c x S U (2),, x U(1)y, for the first fermion family, are given by QL= “(3,29%)1uRz(3717%)1dRz(3711_%)1 Cir. (2.28) V LL: " ~(1,2,—%),ep~(1,1.—1), 8L where L, R stand for left—handed (LH) and right-handed (RH). Here we wrote explic- itly only the components relative to the S U (2) transformation, otherwise the multiplet 13 Q 1,, for example, would have six components (three color eigenstates times two weak isospin eigenstates). The fermion kinetic terms are given by Acfermion : QLDuA/IJQL + ° - -: (2'29) The covariant derivative in the SM, determined by the model’s gauge symmetry, is l ' A“ a - OJ fa - DI, = 8,, + zc3g,?Gp + 2029—2—14“ + zg'YB“, (2.30) where A“ and a“ are the Cell-Mann and Pauli matrices, respectively, which act in different subspaces, and 03, c2,and Y are the gauge charges of the field on which Du acts. For an SU (N ) multiplet (singlet) (:N = 1 (CN = 0). The hypercharge Y is given in Eqs.(2.28). The quantum numbers of the gauge fields are defined by of, z (8,1,0), w; z (1,3,0), B, z (1,1,0). (2.31) Their kinetic terms are given in function of the relative field strength tensor (2.18) by 1 u a u a u Lgaugc = —Z (F3? Fbe/JV + Flt/6‘ FVVpu + Ff; Full”) ' (2'32) The Higgs field in the SM is a doublet defined by <2: 4"“ z(1,,,2%) (2.33) (150 whose Lagrangian and potential have the usual form: 1 cm... = (0.18" 01,, — v (e). me) = "1% + 1% (W)? - (2.34) 14 If we assume that m2 < 0 and A > 0, (b gets a non—zero vev that we choose to be real and defined by 1 0 —m2 (0|¢|0)——‘/—§ v ,v—2 A . (2.35) By convention, we chose the lower component of 45 to acquire a vev. The electromag- netic charge Q in the SM is then determined by Q = T3 + Y, (2.36) with T3 = 1/2(—1/2) for the upper (lower) component of an S U (2) doublet and zero for a singlet. We expand ¢(:1:) around the ground state defined by eq. (2.35) and redefine qb(:r) q, ___ _1_ flit” , (2.37) fl v+H+iX Because of this shift, the electroweak symmetry S U (2),” X U (1) breaks down to U (1)Q. Once the shifted Higgs in Eq.(2.37) is inserted in the Higgs Lagrangian (2.34), the covariant derivatives acting on the vev give the following gauge boson mass terms 1’2 , 2 . , . ,2 2 11mm = g (9 (w; — Mg) (w; + M“) + (9w; — g'B,,) ) . . (2.38) From this matrix we can extract the mass eigenstates (properly normalized) Z,, = cos 19,014": - sin 6,93“, (2 39) , . / W3 = &5 (IV; IFzWE), tandw = 99—, 15 with the respective eigenvalues 2 1 12 2 2 2 1 2 2 mZ=Z(g +g)v,mwi=ng. (2.40) The massless eigenstate is identified with the photon A“: An = cos OwBfl + sin QwWS. (2.41) The respective coupling is therefore defined equal to e by 9’9 e = —— (2.42) /gIZ +g2 A Dirac mass term like the one written in Eq. (2.10) would explicitly break the S U (2),” gauge symmetry, since (1 3F 75) 11), mibi/J = mtbLt/JR + h.c. . (2.43) [\Dlt—a ll’L/R = The masses of the fermions are a consequence of EWSB as well, through the Yukawa interaction terms £Yukuwa = —Q_L‘yu¢’uR _ Q—Lydfiida — L—LyeCACR + h-C'1 (2-44) where yum, are 3 x 3 matrices in family space 1 and gb = 102 - 45*. In case of a single family the fermion masses would turn out to be 1} ’U ’U mu=,u—,m = —, LC='C—,Tr,,=0. 2.45 ”fl .1 ydfin y\/5 1 ( ) 1in Eq. (2.44) we suppressed the family indices. 16 “HI Figure 2.1: Feynman diagram, involving a fermion anti-fermion loop, which con- tributes to the Higgs mass. The complete Lagrangian of the SM is £51” : fifermion + figauge + £H-1'ggs + ACYchawm (2-46) where the terms in Eq. (2.46) are given, in the same order, by Eqs. (2.29,2.32,2.34,2.44). From these equations we can see that [ISM is gauge invariant and the SM coupling constants in LSM are dimensionless. The fact that the coupling constants are dimen- sionless, together with the gauge invariance of ESM, guarantees the SM to be unitary at high energies. Indeed one might worry about anomalies in the theory, since the weak interactions are chiral, but the SM hypercharge assignments in Eq. (2.28) de- termine all the anomalies to cancel within each fermion family [9, 10]. The squared Higgs mass at tree level is 2 _ 2 mHO — A” /2, (2.47) but it receives large higher order corrections Am” from Feynman loop diagrams. The diagram in Fig. (2.1), involving a fermion f in the loop, gives a contribution 2 Afmi, = £324,131,, + . .. (2.48) where AUV is the ultra-violet scale up to which the SM is required to describe ac- 17 curately particle physics, and the ellipses represent terms, proportional to mfi, that grow at most logarithmically with Auv- In general we can write mi, = min) + 2: Am?“ (2.49) where at leading order in Auv we have 2 Amfi, or Agv. (2.50) Unitarity constraints require my to be of the same order as v = 246 GeV. If one requires the SM to be consistent up to the Planck scale, and therefore defines AUV = Aplanck, from Eqs. (249,250) it results in A771,?! A2UV E _ 1032, (2.51) mi}, mi! which means that the free parameter A must be tuned with a precision of 32 digits to produce the right order of magnitude for m H. 2.4 Higgsless Models An intuitive solution to the fine tuning problem consists in eliminating the Higgs boson from the theory. In doing so though one must still be able to generate the right mass spectrum. One important parameter which relates the wt and Z bosons masses is p [2, 14, 15, 16], which at tree level can be defined by 2 2 = mng0 2 2 1 ngcc (2.52) 18 where ch is the Z coupling to the T3 component of a neutral current, while gm; is the Wi coupling to a charged current. For the SM 931m = 92 + 9”. gcc = 9- (2-53) As we can see from Eq. (2.40), the SM predicts p to be equal to 1 at tree level. Experimentally p is measured equal to 1 within a 0.1% margin [1]. A model without the Higgs boson therefore, to be phenomenologically viable, has still to predict p = 1, at least at tree level. The SM prediction p = 1, indeed, is not determined by the Higgs boson itself but by an accidental, “custodial”, global symmetry, of the Higgs potential. This symmetry is revealed by rewriting the Lagrangian Eq. (2.23) as a function of a matrix Higgs field (I) as v 2 2 £11.99. = tr (D),)l D“ — :- (trqfie — 3) , (2.54) where the field components are defined by 1 ~ 1 7’0 ¢+ 0, A > 0, the potential in Eq. (2.55) will have a minimum at 1 v 0 (0| (I) [0) = — . (2.56) 2 0 Li Therefore we shift (I) by its vev, redefining the field as 1 . (I) : 5(1) + H) E, E = exp (Zflaaa/U), (2.57) 19 where E is a nonlinear sigma model [17] field involving a triplet of pions, which are pseudo-scalars under a parity transformation. Plugging this in the Higgs Lagrangian (2.55), we have 1 1 ,\ H 2 cmgg, = §DLHD“H + 4 (u + H)2 tr (0,23)‘ D"EJ — H2Z (3 + v) (2.58) where 03 12,23 = 6,2: - emu/5072 +1g1>3 2 B,,. (2.59) The field E transforms under SU(2)w x U (1)y symmetry as 0,2 _. ULDflEUb. (2.60) The covariant kinetic term of 2 in Eq. (2.58) is therefore gauge invariant indepen- dently from the Higgs boson. Expanding the covariant derivatives acting on 2 and retaining only terms quadratic in v one obtains the same gauge field mass terms expressed in Eq. (2.38), and therefore the same eigenstates and masses. The unitary gauge is selected by the choice 2 = . (2.61) The field responsible for the emergence of the Z and Wi masses with the right value of p at tree level, therefore, is the sigma field, rather than the Higgs boson. The Higgsless SM Lagrangian is a nonlinear sigma model defined formally by eliminating the Higgs field H: 2 £2 = thr (D,,2)i DMZ. (2.62) 20 wr— w/— W- W— Figure 2.2: gauge boson contributions to W+W‘ elastic scattering. W+ W+ 2,7 W‘ W— Figure 2.3: additional gauge contributions to W+W' elastic scattering. The Yukawa interactions in the Higgsless SM can be written as e 0 e y R +6.6. (2.63) u 0 u _ y R -—LLE 0 0 0 EYukawa = —QLE 0 yd In the limit Y -> 0, we see that Eq. (2.62) is invariant under the global S U (2) L x S U ( 2) R transformation (2.64) 2 —+ ULEUl, nga —+ Ubwgrauz. that reduces to S U (2)v E S U (2) H R after EWSB. The fields W: transform as a triplet under the custodial symmetry S U (2)v, which is broken by U (1)y gauge interactions. In general for a sigma model the combination of custodial symmetry and a massless photon imply p = 1 at tree level [18]. The Higgs boson is important, however, for the unitarity of the SM. Unitarity 21 W+ WVWWV [47+ W+ w+ I I I IH ______ : I I W“ ’V\/\/\)’\/\/\/\./ W" W— W‘ Figure 2.4: Higgs contributions to the W+W‘ elastic scattering. requires that no tree level scattering amplitude exceed 1, since that would correspond to a probability greater than one for a scattering event to happen. The gauge boson contributions to the WfWi' —+ W; W; tree level longitudinal scattering amplitude, represented in Figs. (2.2,2.3) grow like E2 in the high energy regime, where E is the energy of W*. The terms proportional to E4 cancel because of electroweak gauge invariance. The amplitude therefore grows indefinitely for higher energies, violating unitarity for [19] ,/E > 4fiv 2 1.7TeV, (2.65) where s = 4E2 is the square of the energy in the center of mass. A tighter limit is obtained from the scattering of weak isospin boson eigenstates, which produces a bound on \/5 given by [20] «E > \/87rv z 1.2TeV. (2.66) 'This energy scale is well within the range of energies that will be tested at the LHC, which will reach a maximum fi=14 TeV (proton-proton collisions), that probes parton, which is any of the proton constituents, scattering beyond 1 TeV. The Higgs contributions to the process, represented in Fig. (2.4) cancel the diverging terms so that the amplitude approaches a constant in the high energy regime. The key role Of the Higgs is therefore that of unitarizing the SM. For a model without a Higgs to 22 be a viable candidate accurately describing the physics that will be observed at LHC additional contributions are needed to delay unitarity violations to higher energies. A generalization of the non-linear sigma model presented in Eqs. (2.62) which implements an SU (2)1"+1 x U (1) gauge symmetry is defined by [18] N+l N+1 1 la la [U 1 £gauge = _Z Z Fijj I + Z Z fagT" (011233")T Dpzj , (2-67) j=0 j=l where 0,2, = 6,33,- +ig,_1w(’;_,,,,T“2, —ig,::,-14/;;Ta . (2.68) Here T“ = a“ / 2, a = 1, 2, 3, and we indicated with primed fields the gauge eigenstates. Since the last group is a U(1) gauge group, we have W5}, = W23 = 0, and the corresponding field-strength tensor is the usual Abelian one. Under the S U (2)” H x U (1) gauge symmetry the j-th sigma field transforms as 2, _. U,_12,U} . (2.69) The unitary gauge is selected by identifying E,- with the identity matrix. The fields in Eq. (2.67) are represented in the “moose diagram” of Fig. (2.5): the solid circles represent S U (2) gauge groups, while the dashed circle represents a U (1) gauge group. The lines connecting two circles represent sigma fields transforming under the adjacent gauge groups. The Lagrangian in Eq. (2.67) reduces to the Higgsless SM gauge and scalar sector in Eq. (2.62) for N = 0, which is represented by the moose diagram of Fig. (2.6). The N + 1 SU(2) gauge groups and the U (1) gauge group are associated with a. spectrum of N + 1 charged gauge bosons and N + 2 neutral gauge bosons. The minimal extension of the Higgsless SM is obtained for N = 1, and has an SU(2)L x 23 21 2:2 2:3 . . . —« U(1)." » 2n \\\ III Figure 2.5: Moose diagram for the sigma model presented in [18]. The solid circles represent S U (2) gauge groups, while the dashed circle represents a U (1) gauge group. the lines connecting two circles represent link fields transforming under the adjacent gauge groups. Figure 2.6: Moose diagram notation for the Higgsless SM gauge and scalar sector in Eq. (2.67). S U (2) R x U (1) gauge symmetry. It is known in the literature as the three site model [21]. It presents one extra set of Z and W3: gauge bosons. The three site model is studied in detail in Chapter (3). In the limit of N —) 00 the moose in Fig.(2.5) represents a fifth dimension in theory space, where the internal sites belong to the “bulk” of the extra dimension and the two external sites are the “branes” delimiting this dimension [22]. A flat extra dimension corresponds, in the deconstructed model, to a “flat” profile for the coupling constants relative to the groups in the “bulk” of the moose and for the f]- constants [15, 19]. We make the assumption of a flat profile in the bulk and define: 9) = .6, yo = g, 9N+l = g’ fj = f. (2-70) Since experimental limits require extra gauge bosons to be heavy, we make the as- sumption (2.71) H Ill tailie /\ /\ ...J Expressing the field strengths in Eq. (2.67) as functions of the gauge fields, and using the previous identities, the charged gauge boson squared-mass matrix in the I/Véj‘,...,W,’\,i basis can be written as {2:2 —:r 0 -- 0 \ —:r 2 —1 0 if 0 _1 2 s . (2.72) 3 —1 K 0 0 —1 2 j where v is defined by f E \/N—+1v.(2-73) The j-th eigenvector is obtained by solving the eigenvector equations recursively, and can be expressed as [19] 610,- = N (lsin((N +1)wj),sin(ij),sin((N — 1)wj) , ...,sin(wj)), (2.74) (I? where j = 0, 1,2, ..., N; N = normalization factor. (2.75) The W 1* mass eigenstate is defined in the basis of the gauge eigenstates W;i by N we = Z emj,w;i, (2.76) .7 i=0 where ewij is the j-th component of the vector defined in Eq.(2.74). The w]- are the solutions of the characteristic equation 46.12%) sin((N 4.1)...) = 2:2[sin((N+1)w) — sin(Nw)]. (2.77) 25 From this one calculates the (2.72) matrix eigenvalues, defined by 2 _ ~2 2 - 2 “’2' mwj -— 29 v sm (g) . (2.78) The resulting squared masses for the lightest eigenstate W and the N heavy states W'jare: 2 2 2 v ( 2N(2N+1)) mw=— 1—:1:————- .., N 1 6( + )3,” M (2.79) 2 _~22 -2 . 2 2 ij—gt (N+1)sm (2(N+1))+2mwcos (2(N+1))+”" where the ellipses stand for corrections of order C(34). We can see that the mass mw at leading order has the SM form: this is a consequence of the choice :1: << 1, which localizes the Wi mostly on the external site 0, with the contribution from the bulk gauge bosons being negligible at leading order. In the same limit the heavy states W].i are much heavier than the SM Wt, which is in agreement with the experimental bounds [1]. The neutral gauge boson squared-mass matrix has additional entries from the U (1) group, and can be written in the basis W63, . . . , W163 +1 as f 1:2 —.1? 0 0 \ —x 2 —1 ' 0 9:21): 0 —1 2 5 , (2-80) ' ’y K 0 0 —y y2 j where y is simply the U (1) gauge coupling 9’ divided by the bulk coupling 5“): (2.81) ‘6 || {QIICQ 26 The j-th eigenvector can be expressed as [19] 62,- = N (:— sin((N +1)wj + $1) ,Sin (New, + (151),...,sin(wj + d5) , 58in (453)), (2.82) where N is a normalization factor. The phases a), and ¢j are determined by 4sin2(-;—J) sin((N +1)w) = (2:2 + 312) [sin((N +1)w) — sin(Nw)] + 2323/2 sin(Nw), 2 tan (cbj)tan (%) = y2y_ 2 . (2.83) These equations have the trivial solution w = 0, ct) = g which produces the massless eigenstate components maven...” ). (2.84) 1 y We identify this state with the photon by defining (ii/V)“2 = 615 + iv- + — = i. (2.85) The squared masses for the lightest state Z and the heavy states Z, are obtained by m2zj = 262122 sin2(;) , (2.86) and turn out to be 2 r2 2 2 (9 +9)v 2 2 N(2N+1) =—————— 1— _— . m2 4 ( (”3 +y) 6(N+1) ’ 2 ___~22 N 1 -2 __j_7r_ 2 2 2 __._77r__ mzj gv( + )srn (2(N+1) + mzcos 2(N+1) + , (2.87) 3' = 1,2, N. 27 a0 Illlllllllllllll LillllllLlllllllllllllll llll o o )_ F 5 10 15 5 (TeV) N 0 Figure 2.7: The J = 0 partial wave amplitude as a function of J5 for, from left to right, the Higgsless SM and the U(1) x SU(2)N+1 Higgsless model for N = 1, . . . , 100 with mw1 = 500 GeV [19]. From the mass eigenstates Eqs. (274,282) and eigenvalues Eqs. (279,287) the WE‘W; —> WE" W; scattering amplitude can be calculated. The term proportional to E4 cancels because of the gauge symmetry of the theory. The term proportional to E2, instead, causes the theory to become non-unitary at some given energy [20, 23]. The presence of the extra Wi bosons, though, delays the scale of unitarity violation. In Figure (2.7) is shown the J = 0 partial wave amplitude as a function of \/E for the Higgsless SM and the SU(2)NJr1 X U(1) Higgsless model for N = 1,. . . , 100 with mw1 = 500 GeV [19]. As we can see unitarity violation is delayed to \fs— % 2.6 TeV for the three site model, compared to \/§ E’ 1.6 TeV for the SM. For large N the unitarity violation scale is \/§ E 19 TeV. All the Higgsless models beyond the SM will therefore be consistent, by unitarity criteria, at the energy scale that will be tested at LHC. 28 In [18, 24] it has been shown that to satisfy EW constraints, SM fermion wavefunc— tions need to be “delocalized” on the entire bulk, by receiving contributions from fields at different sites. Such a delocalization is achieved by introducing the S U (2)" x U (1) gauge invariant Yukawa interaction terms given by [25, 26] N N—l £fermion = — E :mjwleij_ E : fj+1yj+1 wleEj+le+l)R j=1 j=0 u yN+1 0 — fN+1 IDA/LEN“ ¢EN+1m + h.c., (2.88) 0 yilv+1 which is a generalization to N sites of the Yukawa interaction terms in Eq. (2.63). In Eq. (2.88) [L and 11233 are SU(2) doublets, with the only exception of ¢EN+1)R’ which should be interpreted as two S U (2) singlets written in a two-component notation. The mass terms written in Eq. (2.88) were not allowed in the SM because they would break SU(2),” gauge invariance. Notice that in order to obtain the appropriate hypercharge interactions for the light fermions, all doublets must be charged under the U( 1) gauge group at the end, with the U ( 1) charge given by the SM hypercharge of the corresponding LH fermion. The U (1) charges of the RH singlets, "(1)/Hm and d[N+1)R, are as in the SM. In Fig. (2.8) we show the moose diagram for the N -site Higgsless model given by Eqs. (267,288). The lower (upper) segments represent LH (RH) fermions, while the diagonal dashed lines represent Yukawa couplings of the corresponding fermions to the intersected sigma field. The rest of the fermion Lagrangian is given by the ordinary gauge invariant kinetic 29 Figure 2.8: Moose diagram notation for the coupling of fermion fields to the model of Fig. (2.5). The lower (upper) segments represent LH (RH) fermions, while the diagonal dashed lines represent Yukawa couplings of the corresponding fermions to the intersected sigma field. terms. The covariant derivative for a doublet 1/23. is Dal/’3' = (0,, _ z..(IJ'Tal’I/{a ‘iYg~+1W(f’v+1),.) "(’31 (2-89) .711— where Y is the 1p;- hypercharge, fixed for each doublet by the definition Q = T3 + Y, since both Q and T 3 are given. As we have done for the gauge and sigma kinetic Lagrangian, we assume that the Yukawa couplings in the bulk are much stronger than those on the external sites and that the profile of the bulk is flat: foyo E MEL, fN+1ylix+1 E MEXR, ijj E mj E M, 6L,€X <<1, (2.90) where X indicates either the up (u) or down (d) components of a fermion doublet. The choice 6L, ex << 1 is necessary to satisfy the experimental lower bounds on the mass of extra fermions. The fermion mass terms of Eq. (2.88) in unitary gauge ()3,- = Identity) can be written as a matrix product as Cmass = ‘i’LM‘I’R, ‘I’L = ($01.), - - - , 10m.) , ‘I’R = (111112), - . - ,11’(N+1)R) - (2-91) 30 The LH fermion squared-mass matrix in the basis 1601,, . . . ADM. therefore is given by (6% 6L 0 0 \ MMT=M2 0 1 (2.92) 1 \0 011+eiflj The squared—mass matrix for the RH fermions is obtained by switching 6L and 6X3. Solving the recurrent eigenvector equations we obtain the components of the j-th LH fermion mass eigenstate: 1 eXJ-L = N (a sin (Q53) ,sin(wj + (251),...,sin((N —1)wj + 9%) ,sin (ij + (93)). (2.93) The mass eigenstate xJ-L can be expressed as a function of the gauge eigenstates X21, N ij. = Z erz‘LXlL- (2.94) i=0 The phases wj, (15 are determined by the characteristic equations 4 5562(5) cos ((N +341) = cos ((N 331051533- 2 sin (‘1’) sin (No.2) (5% + 533) , E 2.95 2 < ) 2 W' EL tan ((9,) tan (31) = 6% _ 2 . The matrix (2.92) eigenvalues are determined by 2 _ 2 - 2 fl mxj _ 4])! sm ( 2). (2.96) 31 From this we obtain the masses for the SM fermion X0 and the heavy fermion X]- MW”, (1 _ 1+ (N -— 1) 53,, + (2N2 __ 3N +1)/65‘}R_1\_f&_2) (2.97) m0=—-—— 2 L ,/1+N5§R (1+sta) 2 _ N—j+1 si+€§n . .= I —— 1 —— ..., =1,2,...,N. m9 Msm( 2N+1 7T)(+2N+1 J where the ellipses indicate corrections of order C(62). The first fermion mass eigen- state, identifiable with the corresponding SM fermion, is indeed much lighter than the other states, as expected in the limits we have chosen for the mass term coefficients and for the Yukawa couplings. It has been proven in [18] that Ap is exactly equal to one at tree-level, for arbitrary values of the model parameters in Eqs. (267,288). This is a consequence of the approximate SU(2) custodial symmetry of the model, which becomes exact when the hypercharge, and the Yukawa interactions involving the U (1) site are turned off. Moreover, with an appropriate fermion delocalization, that is, with an appropriate choice of the coefficients 6X01, three of the four leading oblique parameters defined by Barbieri et.al. [27] vanish. This occurs when the left-handed light fermion profile is related to the profile of the electroweak bosons, in this case the light fermions become orthogonal to the heavy vector bosons, and therefore decouple from them. From a study of the mass spectrum and of the new physics contributions to the oblique parameters, we see that Higgsless models, which solve in an intuitive way the fine tuning problem of the SM, are phenomenologically viable models. The extra W bosons in the theory need to be lighter than about 1 TeV, for the theory to be consistent by unitarity criteria, and therefore Higgsless model phenomenology will be relevant at LHC. In Ch. (3) we will study in detail the minimal (3-site) Higgsless model: we will determine a profile, known in literature as ideal delocalization, of the fermion wave function that makes the oblique parameter S = 0 at tree level. Also, 32 we will calculate the one loop corrections to p, and show that the result is consistent with experimental bounds. We will then extend these results to the N -site Higgsless model. 2.5 Noncommutative Quantum Field Theories A clear shortcoming of the SM, besides the fine tuning of the /\ parameter, is that the model does not include gravity. Indeed, a fully consistent, predictive quantum theory of gravity is still missing. The possibility of observing direct gravitational interac- tions at the particle level seems, at least for now, completely negligible. Still, some indirect consequences of gravity might be observable at future particle accelerators. An appealing and already testable hypothesis is that, because of gravity, space-time coordinates become noncommutative at some high energy scale A NC. Coordinates x are therefore promoted from simple numbers to operators, which in this section we will indicate with a hat (5:). Their commutator can be defined as a constant, non-zero tensor 6”: [2223'] = 2'6” , detH‘j = 11,326.. (2.98) The implementation of noncommutativity in QFT is not straightforward (for a com- plete study of the subject see [28, 29]), since any function of the coordinates, including particle fields, is now promoted to the status of operator. The physical observables though are not the operators but rather their eigenvalues. We therefore make contact with ordinary coordinates and their functions by defining (2.99) l The operator f (:2?) can be defined as a function of f (2:), where :c = :1: ,...:c" is the position vector in n dimensionsz, by using the Weyl’s quantization procedure: . 1 . - I?) = n dnk 8X iji‘] k , 2.100 f() (2762/ p( )m < > with the Fourier transform given by .. 1 . k = n d".r ex —z'kJ-r’ :1: . 2.101 m (2767/ p( M) < ) Substituting Eq.(299) in Eq.(298) we see that the operator f(i‘) can be expressed “as a function” of f (2:) That is true though only if the resulting integral does indeed exist. In any case one can check that f(i’) satisfies Eq. (2.9%) by expanding the exponential and then using Eq. (2.99a). In general, given two functions f (:r), g(y), fame) Ix> = lr) 4(2) (2.102) 75 |$)f($)9(1?) , (2-103) assuming that [23) is an eigenstate of both f(r) and 9(2). Using Eq. (2.100) we have that few) II) = L, / dnkrp exp (1k.:2i)exp(ip.i:j) Rage) lx>- (2.104) (2”) This expression can be simplified by combining the two exponentials and then using 2T his is actually a necessary generalization, since divergent integrals in QFT are regularized in a gauge invariant way by dimensional regularization. 34 Eq. (2.98) exp (ikiii) exp (ipJ-ij) = exp (206,5:i + p31?» exp (—%k,pj[5:i, EH) + . .. This allows us to substitute the operator (2'3 with the eigenvalue 2:1 and write A A A '2" 7: = T 1 nae/(91> I~> (,,,), iieiji - ' '~ = (2)6289 69—17, / d"kd"pe'kixz+pjyjf(k)§(19) -. .x'_i .~ / “d"pe“’v+”2’ ’ 2k:9”"27p

(277 y—ox éio’iji = ll“) 6’ 8‘”: ‘99] “1199(9) y—ez = [513) h (1‘) (2106) We therefore see that h(:1:) exists and define it by the Moyal product [30] z' 3 -- (9 = --—-.61J _T 2-1 f .. p exp (2 ,,,, W) f(x)9(y) .4. < 07) The Moyal *-product allows one to redefine the interaction terms, given in a noncom- mutative quantum field theory (NcQFT) Lagrangian by a product of field operators, as functions of the observable ordinary fields, their momenta, and the tensor 6'7. An ordinary QFT is promoted to NcQFT simply by applying the substitution f (117)9(117) -* f (337)9(5) = f * 9 (2-108) to any product of coordinate functions. The presence in f at g of the exponential, which is an operator, indeed complicates the structure of the gauge transformations, affecting the corresponding symmetry. By 35 expanding the gauge transformation 21,,(2) _. 21;,(2) = 0(2)A,(2)U*(2) — 30(2)a,01(2). (2.109) we obtain the infinitesimal change in A“: M, = 8,1“ + z'[f‘, 21,]. (2.110) Using the equations 71,,(1) = Agar“, 11(2) = f‘“(:z‘:)T°, (2.111) the group commutator [F, A] can be written as a sum of commutators and anticom- mutators of the matrices T‘: 1 [1“, A) = (fr/i8 + A51")[T", T’] + gig-(PAS — A56”){T", T3}. (2.112) [\D In the commutative ease the second term is zero and the first is proportional to a - “I A _ “I ~ . generator, so the gauge transformed field can still be expressed as A ,,(x) — A:($)T“. Here, however, F’ and A” do not commute. As a consequence only U (N) groups are closed under gauge transformations, since the trace would give the identity matrix for an S U (N ) generator, which instead is traceless. For a U (1) gauge invariant Lagrangian in noncommutative space for a massless Dirac fermion 19(2) we have 2 . .. 1 . .. L = 913,786 — ZFWF’W’ (2.113) 36 where, F,,,,(:i:) = g [15,,(2), [242)] (2.114) D,,(i~)—+D’(§:) = exp ]—z'gf‘(2)] D,,(2), (2.115) the Lagrangian (2.113) is invariant only if Q = 1, since the two “abelian” transfor- mations do not commute: HI 3’ :2 Q: E I Q: a S’ H) v ‘Hx 0 [0(2), U’(2)] (2.116) Therefore noncommutative quantum electrodynamics (NcQED) admits only charges Q = 1,0, —1, where the negative charge particle is just the antiparticle of the Q = 1 particle. Also, it is interesting to notice that the field A,,(:E) interacts with itself, since [Mir/11(2)] 2 0. (2.117) NcQED is in all respects, because of the coordinate operators noncommutativity, a non-abelian gauge theory. In Ch. (6) we will study the phenomenology of NcQED. We identify the Q = 1,0, —1 fields respectively with the positron, the photon and the electron, and study the phenomenological signatures of noncommutativity for the process 6 e‘ —» 37 at future 8+6— colliders. 37 Chapter 3 Higgsless Models in 4—Dimension 3.1 Introduction Higgsless models [3] are effective field theories which break the electroweak symmetry without a scalar Higgs boson.1 The most popular among these models are based on an SU(2) XSU(2) x U(1) gauge theory on a slice of AdS five-dimensional space [31] [32], where the electroweak symmetry is broken by an appropriate choice of boundary con- ditions. The five-dimensional gauge fields can be expanded in Kaluza—Klein (KK) towers of four-dimensional charged and neutral vector bosons, where the lightest modes correspond to the ordinary electroweak gauge bosons (including the massless photon). Longitudinal W and Z boson scattering amplitudes are then unitarized through exchanges of the massive modes [6] [20] [23] [33], where a non-zero back— ground warping factor insures that the mass of the lightest of these heavy modes is pushed above the current lower bounds imposed by direct searches [32]. Other extra- dimensional models employ flat backgrounds, with brane kinetic terms simulating the effects of warping [15] [19]. Higgsless models can also be studied in a four-dimensional context by using the technique of deconstruction [34] [35], or without referring to 1In this chapter Sections (3.1,3.2,3.4,3.5,3.6) are based on [26], while Section (3.3) is based on [21] 38 extra-dimensions, by constructing the most general chain of non-linear sigma models with arbitrary gauge couplings and f-constants [5] [8] [16] [18] [24] [27] [36] [37] [38]. Most of the recent efforts on Higgsless physics have been focused on the tension between unitarity, which demands the new vector bosons to be relatively light, and electroweak precision data, which instead favor heavy vector bosons. The corrections to electroweak observables depend crucially on the way matter fields are coupled to the gauge sector of the model. The simplest choice is to have fermions strictly local- ized at the ends of the extra-dimensional interval. With this choice no extra fermions are introduced into the model, but it turns out to be impossible to simultaneously sat- isfy the experimental constraints on the Peskin-Takeuchi S and T parameters [5] [19] and unitarity. It is therefore necessary to delocalize fermions, or, in other words, to have five-dimensional matter fields propagating into the bulk of the extra dimension. As with the gauge fields, this introduces towers of four-dimensional fermions, with the lowest mode of each tower corresponding to a standard model fermion. The latter couples not only with the gauge fields at the interval ends, but also with the bulk gauge fields. However, if the profile of the left-handed light fermions is adjusted to be related to the profile of the standard model W boson, then the former will be “orthogonal” to the heavy charged vector bosons, and the corresponding couplings may vanish, decoupling the light fermions from the new physics. This has been proved to be possible in a large class of four-dimensional Higgsless models, consisting of an SU(2)N+1xU(1) chain of non-linear sigma models with arbitrary parameters, where three of the four leading zero—momentum electroweak parameters defined by Barbieri et.al [27] can be simultaneously adjusted to exactly vanish [18]. In models from extra dimension, an exact vanishing of all the electroweak parameters may not possible, since the profile of left-handed fermions cannot be shaped arbitrarily. How- ever the S parameter can be tuned to zero, and all other parameters are naturally suppressed [24] [37] [38]. 39 In this chapter we focus on the contribution to one of the electroweak parameters, namely the p parameter, defined as the ratio between the strengths of the isotriplet neutral current and charged current interactions at zero momentum. At tree level this computation has been done for the SU(2)N+1x U(1) model with arbitrary parameters, for which it has been proved in an elegant way that p = 1 exactly, regardless of fermion delocalization [18]. In fact this is achieved quite naturally thanks to an approximate custodial isospin symmetry, which becomes exact when hypercharge and Yukawa interactions are turned off. One-loop contributions to p in a simple three-site model (corresponding to N =1) are calculated in [21] for fermionic loops, and in [39] for loops with gauge and Goldstone bosons. The latter give cutoff dependent contributions, reflecting the fact that Higgsless models are non- renormalizable effective theories of electroweak symmetry breaking. However the fermionic loop contributions are free of infinities and provide strong constraints on the fermion spectrum. We therefore focus on these fermionic contributions by extending the corresponding analysis to models with an arbitrary number of sites and a “flat background”. By this we mean that couplings and vacuum expectation values do not depend on the particular copy of extra SM gauge group or sigma field. In these computations we only consider loops from the third generation of quarks, since these are the only fermion sector of the theory expected to give non-negligible contributions. We observe that the new-physics contribution to Ap E p — 1 rapidly increases to quickly stabilize, as the number N of internal sites increases. The experimental upper bounds on Ap translate into lower bounds for the mass of the heavy fermions. These, however, turn out to be very weakly correlated to N. The bounds from Ap turn out to be stronger than the ones imposed by the top quark mass and the decay b -—2 s + '7, which were considered in previous works [21] [25]. This chapter is organized as follows. In section 3.2 we briefly review the most general SU(2)N+1>< U(1) Higgsless model, with arbitrary gauge couplings, f -constants, 40 Yukawa couplings, and Dirac mass terms. We give formal expressions for the tree- level low energy effective Lagrangians in terms of propagator matrix elements, and derive an expression for the p parameter. In section 3.3 we study in detail the case N = 1, deriving explicitly masses and eigenstates, and the relative coupling constants. In section 3.4 we calculate p including radiative corrections, and compute one-loop fermionic contributions from the top and bottom KK modes in the models with a flat background. We show analytical results for N = 1, and numerical results for arbitrary N, arguing that the infinities cancel in each case. In section 3.5 we compare these results with the experimental bounds on Ap, which translate into lower bounds on the heavy fermion masses. Finally, in section 3.6 we offer our conclusions. 3.2 The Model and Electroweak Interactions The Higgsless theories we consider in this note are SU(2)N+2 non—linear sigma models invariant under SU(2)N+1>] J12. 1 , " §J¢va (3.8) and the charged current effective Lagrangian N 1 1300 = —5 [2 21302;, ()IV,+w,.-)] J:J_,, , (3.9) i,j=0 where (Wf'WJf’) denotes the coefficient of —ig*“’ in the inverse of the mass matrix3. This Lagrangian is also valid at one-loop order if we neglect vertex and box correc- tions, which is a good approximation if the loops involve new-physics heavy parti- cles [41]. In terms of weak and electromagnetic currents, the neutral current La- 3The q“q" term gives negligible contributions, for external light fermions. 45 grangian of Eq. (3.8) is ENC = 1 N N —2 [2: 01.200340 — 2 Z a§0(WJ3WR,+1)+(Wg+1WI-3+l> .1513” i,j=0 j=0 N 1 _ [Z (130 JélJQn '- 5(Wg+ll4/§,+1)J5JQ,, i=0 (3.10) We assume that the profile of the Higgsless model is flat, with large “brane kinetic terms”, and define gi=§,fi=f=\/1V+1’U,i=1,...,N, (3.11) where v is the vacuum expectation value. The gauge—sector Lagrangian for a flat profile is 1 1 N 1 1..-: — 1‘4’62.W6““”--Z 22.20“” — .22. Wm: 4j=---1 f—2 N+1 + 4 12(D,,,**,2:)*D2:, (3.12) j= —l where 92, g’2 << g2 / (N + 1). The coefficients aJ-n and (2,,, of Eq. (3.3) can be calculated perturbatively in 2:2 E g2/§2. The SU(2) and U(1) gauge groups at the chain ends act approximately as the standard model SU(2)LXU( 1) gauge group, and the internal SU(2) groups act approximately as the new physics. Then the numerical values of g and 9’ will be close to the corresponding standard model values [19]. 46 The fermion sector Lagrangian, for the mass and Yukawa terms, is N N—1 £fermion = —M 5L1/j'0L211/jin + Z w’jijR + Z wleEj+li/’(j+l)R Eu}; 0 0 51112 where 5%, 8:3 << 1/ (N + 1). The coefficients a?" and 1);, of Eq. (3.6) can be calculated perturbatively in these parameters. With this construction the fermions localized at the chain ends act approximately as the standard model fermions, while the fermions coupled to the internal SU(2) groups are mainly superpositions of the new heavy fermions. Notice that for 5;, = 0, both 110 and do, in the expansions of Eq. (3.6), are massless, since the corresponding mass matrices have zero determinant. Similarly for EXR = 0,0nly X0 is massless. Therefore we expect mx OC Il/IELEXR , (3.14) 0 to leading order in 51, and 5X3. It is therefore a different value of 5x3 within an SU(2) doublet, a“); 75 am, which encodes the violation of weak isospin. Moreover, to the extent that we can neglect mm, the corresponding value of 5x}? can be neglected as well. As a result the standard model and new physics contributions to the p parameter are mainly due to loops with top modes, tk, and bottom modes, bk, where k = 0,1, 2, ...N, and to, bo are the standard model top and bottom quarks, respectively. To underscore this correspondence, from now on we will drop the zero subscript from the mass eigenstates. The charged current coupling constants are calculated directly from the eigenstates 47 expression, and are formally given by un d un d k] 2910‘ij (“JO 1 . INT—2.1123 1.131030 1 (3.15) for the left-handed and right-handed couplings of uk and d; to the W’ boson, and N Zr )1 9L k l = Zajkajdgjbjo — 9N+1b(N+1)0)1 j=0 2): x . k ’— £33.. .1 (22-0 —g~.1b(~.11o> (3.16) j: 1 for the left-handed and right-handed T3 couplings of X). and x, to the Z boson. The last, negative contribution is obtained substituting Y —2 Q — T 3. The last term in Eq.(3.16) can be simplified using the normalization and orthogonality of the eigenstates. These expressions can be used to find the couplings perturbatively in the small parameters. In the next section we will study a Higgsless model for N = 1, the three site model. 3.3 Three Site Model The methods to derive the mass eigenstates and relative coupling constants formally defined in the previous section are general and do not depend on the particular number of sites N. In this section we therefore study the simplest case N = 1. The three site model [21], though it is a minimal version of Higgsless model, implements all the qualitative features of more complex Higgsless models. The gauge-sector Lagrangian for the three site model is 1 In W I u/ cgauge = __ Z (id/(lzuf/lew'l' Wllzuwl I +W2iuwl3l ) 2 ‘l' :- (LT (0,121)] DMEI + LT (D;LEQ)] 0,122) i (3'17) 48 while the fermion sector Lagrangian, for the mass and Yukawa terms, is Efermion = M ELI/3,01,211vZ’lRW—IEIIL’i/Jlli _ 511R 0 I 0 Ed]; where 531,53“; << 1. In the next subsection we will work out the masses and eigenstates defined by Eq.(3.17). 3.3.1 Boson Masses and Eigenstates This section reviews the mass eigenvalues and the wavefunctions of the gauge bosons of the three-site model. We define I 9 C 9 and calculate the results perturbatively in .2:. Charged Gauge Bosons The charged gauge-boson mass-squared matrix may be written in terms of the small parameter a: as ~2'U2 1.2 _1. 9—2— (3.20) —x 2 Diagonalizing this matrix perturbatively in x, we find the light eigenvalue 2 2 2 6 g 1) 2: :1: M 2, = —— 1 — — — . .. . 14 4 4 + 64 + , (3 21) and the corresponding eigenstate W” = 230114;“ + 2'10 wk 1:2 51‘4 , :1: 2:3 92:5 = ___._ W11 - ___— W’“ 3.22 (1 + ) 0+(2+16 256+ ) 1’ ( l where W6", are the charged gauge bosons associated with the S U (2) groups at sites 0 and 1. Note that the light W is primarily located at site 0. The heavy eigenstate has an eigenvector orthogonal to that in Eq. (3.22) and a mass 2 4 ,2 ___ ~2 2 33__ f; [WWI gv 1+4+16+... , (3.23) Comparing Eqs. (3.21) and (3.23), we find May 2:2 2:4 :56 = — — — — 3.24 1113,, 4 8 + 64 + ’ ( l or, equivalently, 2 2 2 2 2 3 9 _ 2 MW MW MW — = = 4 2 .. . .2 (g) . (M21) +8 (1113.1) + 80131.1) + ’ ‘3 5’ which confirms that the W1 boson is heavy in the limit of small 2:. The gauge-eigenstates in the mass-eigenstates basis are obtained straightforwardly from Eq. (3.22) by using the orthogonality of the eigenstates and their normalization to one: Iu_ u_ I I 11 W0 —a00W"+a10W —a00W"+a10H/l , W1,” = (101 W” + 0.11 ”,1" = —(1’10 W," + 0160 ”/1“. (3.26) 50 Neutral Gauge Bosons The neutral bosons’ mass-squared matrix is $2 -.’L‘ 0 ~2v2 ‘92— —:r: 2 —a:t , (3'27) 0 —xt. 11:21.2 where t E tan9 = s/c. This matrix has a zero eigenvalue, corresponding to the massless photon, with an eigenstate which may be written AP=EW#+§WF+%W?, (3%) 9 9 where Wéf‘l are the neutral gauge bosons associated with the S' U (2) groups at sites 0 and 1, the W5" is the gauge boson associated with the U ( 1) group at site 2, and the electric charge 6 satisfies 1 1 1 1 The light neutral gauge boson, which we associate with the Z, has a mass 2 2 2 2 2 6 2 2 4 gv 13(02—3) :r(c-s) M2 = — 1 — —— —-——-—— . Z 48 4 c2 +64 & '+ ’ “BM with a corresponding eigenvector Z‘u = bgowg“ + b’le'“ + b’20W2’“ , (3.31) I 1132 3 2 b00=c(1+3(21—C§-2+C))+.H, (3.32) :cc(1—t2) ar3c3(1—t2)3 b’ = ... . 10 2 + 16 + , (3 33) I $2 4 The heavy neutral boson has a mass 2 4 .22 2 ~22 .2: :1:(1——I.) —- +—+———+... 3.35 M21 91) 1 c2 16 , ( ) with the corresponding eigenvector Z1” = bglwg‘ + b’qu'“ + ball/V5“ , (3.36) :r x3(1 — 3t2) ' = —— — —— . 7 x2(1+t2) '11: --—8—-+..., (3.38) t x3t(3 — t2) b’ = —‘”— ___— . 21 2 + 16 + (3 39) The gauge—eigenstates expansion in the mass-eigenstates basis is obtained inverting the matrix of coefficients defining the gauge eigenstates: W5" = booz" + (>012? + SA". (3.40) 2:2 3 2 b00=C<1+-2—('4—c§—2+C))+..., (3.41) 2 301:4”- 1+£—(1—3t2) +...; (3.42) 2 8 W,” = 3102” + bHZf + EA“ , (3.43) ate 1 :cc 2 z. =_1_.2 -(_ -..2) . 10 2( )(1+2 2(1 ) + (344) b $2 = 1 - —- ' .45 52 W2“u = (9202“ + (321le + $14“ , (3-46) 2 33 4 b20=—S(1—§C—2(1—4C))+..., (3.47) xt 1122 2 3.3.2 Fermion Wavefunctions and Ideal Delocalization This section analyzes the fermion sector of the three-site model and implements ideal fermion delocalization explicitly. Fermion masses and wavefunctions Consider the fermion mass matrix EL 0 m 0 M1”) = M a . (3.49) I 1 5:“ng M mud The notation introduced at the far right is used to emphasize the “see-saw” form of the mass matrix. In what follows, we will largely be interested in the top- and bottom-quarks, and therefore in em and 5),}; (or, equivalently, in mQ/M and mg/M). Diagonalizing the top-quark seesaw-style mass matrix perturbatively in EL, we find the light eigenvalue MELEtR [ 8% mt=-—— 1— +... , (3.50) 4/1 +5153 2(5223+1)2 mm; (3.51) “’W Note that this is precisely the same form as found in [25]. For the bottom-quark, we find the same expression with 5m —-> 5m, and therefore (neglecting higher order 53 terms in 523) m5 5513 — z — 1 + 52 3.52 m. m m ( ) The heavy eigenstate t1 corresponding to the top-quark has a mass 2 E mg1=M 1+€?R[1+‘2T'€2;LT]3§+...] , (3.53) t 3 (MP + mg? (3.54) and similarly for the heavy eigenstate corresponding to the bottom-quark ((21) with am —> 553 (or, equivalently, m; -—> mg). The left- and right-handed light mass eigen- states of the top quark are 2 2 4 _ ll. I It I _ 51 (85tR _ 3)EL I tL — aOO tOL + 010 tlL — ( 1+ 2(1 + 522192 8(52 +1)4 + tOL EL (2538 _ ”5% ) I + —+———+... t , (3.55) (1 + 53,, 20533 +1)3 1L 2 5tR EtRE t: Ittl +IBIttl = _ + L +... t! R 10 113 20 2R m (1+ Eggs/2 112 1 53R 5% I + —— + ——-——’-— + . .. t , 3.56 ((7—14. 53. (1 + ear/2 ( ) and similarly for the left- and right-handed b-quarks with em —) em. Clearly the smaller the value of EL (5212), the more strongly the left-handed (right-handed) eigen- state will be concentrated at site 0 (site 2). Note that the relative phase of the eigenvectors t L and t}; is set by the eigenstate condition Mtlltb) = 7mm.) . (3.57) 54 The left- and right-handed heavy fermion mass eigenstates are the orthogonal com- binations 5L (2512R_1)5%+ I t — t t — - t 11,001 (”+0 1L_( 1+Et2R 2(Et2R+1)3 0L 52 (862 —3)64 — 1— L — ‘R L t’, 3.53 ( 2<1+erfl>2 8(.,2,,+1)4 + ) 1L ( ) 1 5,23 5% +. t’”, th = ititlm + 1323323 = " 5 2 ‘/1+EER _¢2(1+5 nl/ 5112 EH? 8L I _ ___. — ————————-— + . .. t , 3.59 ( /'—"1+'Et23 (1+5t2R)5/2 . ) 2R ( ) and similarly for the left- and right-handed heavy B quarks with am —> 51,3. Anal- ogous results follow for the other ordinary fermions and their heavy partners, with the appropriate 5x3 substituted for at}; in the expressions above. The left- and right- handed gauge-eigenstates in the basis of the mass-eigenstates are obtained straight- forwardly from Eqs. (356,355) by using the orthogonality of the eigenstates and their normalization to one: I t t .It It is}; = 550153 + 13:0 $112 = I330 1513 + 311M113 3 til, 2 (.161 h, + 0:11.11, = —a"1t0f.L + (1301,”, , t’m = 63, t3 + 6:, tm = —x31‘0 tn + 630 tln. (3.60) Ideal Delocalization As shown in [18] it is possible to minimize precision electroweak corrections due to the light fermions by appropriate (“ideal”) delocalization of the light fermions along the moose. Essentially, if we recall that the W is orthogonal to its own heavy KK 55 modes (the W1 in the three-site model), then it is clear that relating the fermion profile along the moose appropriately to the W profile can ensure that the W1 will be unable to couple to the fermions. Specifically, at site 2' we require the couplings and wavefunctions of the ideally delocalized fermion Xj and the W boson to be related as 93(afol2 0C 040 (3.61) In the three—site model, if we write the wavefunction of a delocalized left—handed fermion as x31, = afinL + afjle, then ideal delocalization imposes the following condition (having taken the ratio of the separate constraints for 2' = 0 and i = 1): (03(0)2 61-00 ~ = __ . 3.62 9(afo)2 ‘110 ( ) ‘Q Based on our general expressions for fermion mass eigenstates (Eqs. (3.55) and (3.56)) and the W mass eigenstate (3.22), it is clear that (3.62) relates the flavor-independent quantities :r and EL to the flavor-specific 5x3. Hence, if we construe this as an equation for 5L and solve perturbatively in the small quantity as, we find4 2 2 4 6 2 2 2 I 1 5x8 4 55x33: 1 -- — — —— —— . . . . 3.63 5L"(+5x6)[2+(8 2)a:+ 8 + ( ) Regardless of the precise value of 5x3 involved, it is immediately clear that ideal delocalization implies 51, = 0(32). Since :1: << 1, this justifies the expansions used above in diagonalizing the fermion mass matrix. The value of 51, that yields precisely ideal delocalization for a given fermion species depends on 524R and therefore (3.52) on the fermion’s mass. For example, the value of EL that ideally delocalizes the b depends on am. As we will see below, however, bounds on the right-handed Wtb 4In the three-site model, this choice of 5"]: is equivalent to a choice of the parameter b in [42] to make 83 or aS vanish. 56 coupling will yield the constraint 5),]; S 1.4 x 10‘2; when Eq. (3.63) is applied to the b quark and this constraint is imposed, terms proportional to 8),}; become negligible. As all other fermions (except the top) are even lighter, the associated values of Ex}; will be even smaller. In practice, therefore, we may neglect all terms proportional to EXR in Eq. (3.63), and the condition for ideal mixing is essentially the same for all fermions except the top-quark: 2 4 2 2 2 2 3 2 :1: :r 8 MW MW MW 5 =—+—+O(:z:)=2( )+6( )+22( )+..., (3.64) L 8 M3,, May, M3,, where the second equality follows from Eq. (3.25). This is the value of EL we will henceforth use for all fermions in our analysis. As discussed in [43], we expect that the value of a: will be bounded by constraints on the WWZ vertex when the light fermions are ideally delocalized. 3.3.3 Fermion couplings to the W boson Left-handed fermion couplings to the W boson We begin with the left—handed Wtb coupling, assuming ideal mixing for the b—quark in the am -> 0 limit. Because the W wavefunction receives contributions from sites 0 and 1 only, the W f f’ coupling is the sum of the overlap between the W and fermion wavefunctions on those two sites: Wtb _ ,t b ~ I b , 9L — 90000006100 + 9010010010 , (3-65) for which we find 9W“, = g (1 _ 35‘?” + 45,213 + 3 2:2 35,8” + 165st + 505?” + 8:33,, +15 2:4 + L 8(sz +1)2 128(ch +1)4 "' ' 57 The corresponding equation for the coupling of standard model fermions other than the top-quark to the W may be obtained by taking em —2 O in the equation above, yielding 3 15 w 2 4 = 1 — — — o o o a 3067 Combining this with Eqs. (3.19), (3.21), (3.29), and (3.30) we find g)” = —-i— [1 + (’?(.92 24)] , (3.68) M2 1 _ MZ which shows that the W-fermion couplings (for fermions other than tep) are of very nearly standard model form, as consistent with ideal delocalization. Eqn. (3.67) corresponds to a value of CF (91352 1 1‘2 1174 20 = =— 1—— — , 3.69 f F 4M3, 1.22 + 4 + ( ) and the relation 2 2 6 4 2 Wtb w 5:12 2 EIR(35IR + 85m + 45412 +10) 4 = 1+———:1: — 17+... . 3.70 W 9L ( 4633,, +11)2 32(efR +1)4 ( ) The W also couples to the heavy partners of the ordinary fermions. Here, we quote the results for the t1 and b1 heavy fermions; analogous results follow for other generations when a”; is replaced by the appropriate 5X3. There is a diagonal Wt1b1 coupling of the form Wt b 2 9L 1 1 = 9051031000 +90i10i1010 : (3-71) 4 2 9 54R - 65:12 " 5 2 ) =— 1— 33+... (3.72) 2 ( 8(€?R +1)2 w 4 2 9L EIR+65IR+4 2 > = — 1 + a: + . .. . 3.73 2 ( 4(5fR +1)’2 ( ) 58 There are also smaller off-diagonal couplings involving one heavy and one ordinary fermion 913th = 90310130000 + fiaiiaioalo , (3-74) = 23%;?” (:r + 033)) , (3.75) and git/[bl = 9030031000 + fiaioaglalo , (3.76) _ 9(1+25?R) T 2'3 _ 2\/2(sz +1) (.. + O(., )) , (3.77) which play an important role in radiative corrections. Weak mixing angle From Eqs. (3.29), (3.30) and (3.69) we can calculate the“Z standard” weak mixing angle5 Qty ] Z: 2 322522 E ___—e 4¢20FM§ 1 = 32C2 + 32(c2 — 32) (c2 — Z) 2:2 + 0(x4), (3.78) where .32 E sin 9wlz and cg E cos 9wlz- The relationship between the weak mixing angle awlz and the angle 6 defined in Eq. (3.19) is expressed as follows: 5; = .92 + A, cg = c2 — A, (3.79) A a 32 (c2 — i) :62 + 0(54). (3.30) 5See also the BESS results [42, 44]. 59 In other words, 32 and 822 differ by corrections of order 2:2. Right-handed fermion couplings to the W boson and b —> 37 The right-handed fermions exist only on sites 1 and 2 while the W is limited to sites 0 and 1; hence, the right-handed coupling comes entirely from the overlap at site 1. For the tb doublet we find 9);“) =§I3io filo 040 (3.81) _g 57.722 EbR 2 — — 1+ 0 a: 3.82 2 ~9_m_b 5m __ 3.83 ~2 m, 1+ 5,23 ’ ( ) where reaching the last line requires use of Eq. (3.52). It is interesting to note that this expression is precisely analogous to the related expression in the continuum model (see Eq. (4.17) of [25]). The right-handed Wtb coupling can yield potentially large contributions to b -+ 37. As shown in [45], agreement with the experimental upper limit on this process requires u w 9—}? l—V < 4 x 10’3 (3.84) 91. Combining this bound with our expressions for gL (3. 67) and thb (3.83), recalling :1: << 1, and using m, = 175 GeV, mb = 4.5 GeV, yields the constraint 5”; < 0.67 . (3.85) As we shall see below, this constraint will automatically be satisfied for M > 1.8 TeV — a mass limit that will be shown to be required for consistency with top-quark mass 60 generation and limits on Q. Finally, combining Eqs. (3.85) and (3.52), reveals that am < 1.4 x 10‘2 , (3.86) as referred to earlier. Again, this confirms that the same value of 5L can produce nearly perfect ideal delocalization for the b and all of the lighter fermions. The W also has right-handed couplings to £1 and b1, for which we compute the diagonal coupling Wt b - 93 1 1=gflilfiifialo (3.37) 4 2 =—— 1+ :17 + . . . 3.88 2\/1+ 5,5,, ( 8(54212 +1)2 ( ) w 4 2 9L EtR+35tR+1 2 ) =— 1+ :1: + . .. , 3.39 2,/1+e,§,, ( 2(efR + 1)2 ( l and the off-diagonal coupling ww ~ 912 1 =gfiioflflalo (3.90) 4 2 _ = {1543 (1 + EtR +225m 33:2 + . . ) (3'91) 2 1+ 5?), 8(5,R +1)2 w 2 2 9L EtR 523(5m + 2) 2 > =———— 1+————x+... . 3.92 2\/1+ 8,2,, ( 2(efR +1)2 ( ) As in the case of 92"”, the right-handed coupling 9);,“ turns out to be proportional to 51,3, and is therefore very small. Other right-handed WXin couplings involving the light standard fermions are straightforward to deduce from Eq. (3.82) and clearly sup- pressed by the small values of 5243- Similarly, the off-diagonal ggxxl are proportional to small 5X3. The diagonal 923(le are analogous in form to (3.89). 61 3.3.4 Fermion couplings to the Z Boson In this section we calculate explicitely the Z couplings to the top mass eigenstates. The bottom mass eigenstates coupling to the Z boson are obtained immediately from the correspondent top’s result by taking am —-> 0. These are the only states that give a non-negligible contribution to the physics beyond the standard model. We work out the couplings using Eq. (3.16), appropriately simplified for each specific fermion pair: 917:“ = 91203960)? + 91910011702 ‘ 9,520 = 69 (3-93) 13 2 4 2 95123(2+5123) 2 32 $2 4 —— 3 6t —t — 1—— 1-4 SC g( + )3: + 4(:(1+€,2R)2 I +gc 8(:2( c) ’ 951“ = (fiblo — g'b26)(/3§0)2 (394) = ——"€‘2’1 1 + ”3‘5?” + 1>2 + 8365(sz + 2) — Met. +1)2x2 26(5123 + 1) 8c2(5,2R + 1)2 ’ .. 1 9,?" = gb00((r[),)2 + gblo(at11)2 — g'bgo = —§(:g (I.2 — 1) (3.95) cg (4 (t2 +1) — c2 (sz +1)2(t2 — 1)3) 82 2:2 + 2 x2+g— 1——(1—4c4) , 16 (5,2,, + 1) c 84'? 95” = (9510 - 9’b20)(1311)2 (396) = g + g(—3(s?,, +1)2 + 8c2(€fR + 353,, +1) — 464(sz + 1)?)1:2 245,23 + 1) 16c3(e,2,, + 1)3 ’ ZtT _ t t ~ ,t ,t gL — 91100000001 + 9111007100,, (3.97) = 9 a: + 9((5123 +1)2 + C2(5?R + 65123 - 3) — 404(5123 +1)2)$3 2454sz + 1) 16¢§c3(s§,, + 1)3 ’ 935T = (fibio — .0'b2o)(lzfill (3.98) = gem + 95112 (—3(sz + 1)2 + 462(28):}; + 55?), +1) —- 4C4(£t23 +1)2)r2 2453,, + 1) 16c3(5,R + 1)3 ' 62 3.4 One-Loop Corrections to p In this section we calculate the p parameter and test the phenomenological viability of the Higgsless model, for a generic value of N. The p parameter is the ratio of the isotriplet neutral current and charged current interactions at zero momentum: 25:0 0112001320 (WEWf’) - 2 2;:0 ago (”Claw/3+1) + ,0 = N _ 213:0 01120030 (Wit/l3 > (3.99) It has been proven in [18] that this quantity is exactly equal to one at tree-level, for arbitrary values of the model parameters. This is a consequence of the approximate SU(2) custodial symmetry of the model, which becomes exact when the hypercharge, and the Yukawa interactions involving the U(1) site are turned off. Moreover, with an appropriate fermion delocalization, that is, with an appropriate choice of the coef- ficients 013-0, three of the four leading zero-momentum parameters defined by Barbieri et.al. [27] vanish. This occurs when the left-handed light fermion profile is related to the profile of the electroweak bosons, because in such case the light fermions become orthogonal to the heavy vector bosons, and therefore decouple from them. Having established that the p parameter is exactly one at tree-level, it is new interesting to compute one-loop corrections. To leading order we can assume that the light left-handed fermions are exactly localized at the j = 0 site, aJ-O —-1 63-0. Then Eq. (3.99) becomes 2 4461131 — 2 + (113.31.» (warm) (3.100) 63 To leading order we can also take :1: —> 0, in which case 92 W8 = 814 + ———2 ,2 Z V 9 + 9 3 9’2 W = 6A —— ———-Z N+1 r——92 + 9,2 W0i = 9W 2 2 mw g = — , 3.101 m2z 92 + 9’2 ( ) where A, Z, and W are the ordinary electroweak bosons. Inserting these expressions in Eq. (3.100), we see that the photon contribution vanishes, as it must. Then, expanding the W and Z propagators, we obtain, for Ap E p — 1, = nwwm) _ 1122(0) 2 2 ’ 771W mz Ap (3.102) where wa and 1122 are the coefficients of ig,W in the 1PI W and Z functions, respectively. Notice that this equation for Ap contains loops in the W and Z boson propagators only, and thus corresponds to the Peskin-Takeuchi aT parameter. In fact we are considering the leading order term in an expansion in 232, which amounts to ignoring the small contribution from the heavy boson propagators. Including higher modes in the expansions of Eq. (3.101), and considering that the coupling of the heavy vector bosons to the heavy fermions is of order 9', it can be shown that the heavy modes give corrections of order 0(34) to Eq. (3.102). Therefore, to order 0(x2) we have Ap = aT in this model [16] [18] [36]. Notice that since we take q2 = 0, only‘the isospin part contributes in II 2 Z. We define H L L(X , Y; q2) as the coefficient of 39,“, in the vacuum polarization amplitude with left-handed currents only, and fermions X and Y in the loop. In a similar way we define Hug, as shown in Fig. (3.3), while it can be easily proved that HR); = TILL. (Trivially, HRL = IILR.) At zero momentum 64 X L X L L R HLL(X1Y;Q2) = HLR(X1Y;q2) = q q Y Y Figure 3.3: Vacuum polarization amplitudes for left-left and left-right gauge currents. these functions are [46] l HLL(0) = 16—79- [(mg( + m§)E — 2 (mfib1(mx,my; 0) + m§b1(my, mx; 0))] 1 111.3(0) m {-Q'mmeE + 2mX'mYb0(mX: my; 0)] 3 (3-103) where l 2 _ 2 — - 2 b0(memY;q2) = / d2: log (I mX + (1 xiTZY 55(1 :12)q ) 0 l. l 2 _. _ bl(mx,my;q2) = / dz: xlog (x mx + (1 IZZY_ $0 ”(12) .(3.104) 0 Here E is the divergent part of the loop diagram from dimensional regularization , E = g- - 7 + log(47r) — log(p2) (e = 4 — d), and p is the renormalization mass scale. The 1PI functions can be computed by 3 Wt.b 2 Wt b 2 wa(0) = Z§[(9L l‘ 1) HLL(tkabl;0)+ (QR kl) HLL(tk,bz;0) k,l b ) + 29Wlqw‘k’lnm(1,.,b,;0)] (3.105) 65 for the W boson, and 3 Zt t Zt t 2 1122(0) = 224(9), “)2 HLL(tkatl§O)+(ngl) HLL(tkatl;0) 1 k 20 b + (qLZb WI) HLL ([)k,b[,0)+ (QR k [)2 HLL(I)k,b(;O) z: 1 Zt 1 Zb b Zb b 291 k [913 k lHLRUka t1; 0) + 291, k 1911 k lHLRU’k: bl; W]- (3-106) + for the Z boson. In these expressions the factor 3 takes into account the different color contributions, the factor 1/2 in IIWW comes from 1/\/2 in the Lagrangian, and the factor 1/4 in H22 is the product of isospin quantum numbers. Inserting Eqs. (3.105) and (3.106) in Eq. (3.102), and using Eq. (3.103), gives the following expression for the infinite part of Ap: Wt,b 2 Wt,b 2 (91. Al) +(gn u) m2 +7713! mkblglqutkbl ‘k 9L 2 — mt Tnb g": mw 2 ma, k l (thktl)2 + (QZtktl)2Tn2 +m2 Ztktl Ztktl m _ L R ‘k ‘1 + 91. 912 m‘k t1 m2 4 m2 2 z 2 211,3 2 Zb b 2 (9L k I) + (QR k 1) mg + mi 23ka 23ka mb m1, _ k 1 9L 9R k 1 2 + 2 . (3.107) mg 4 mz 2 This can be proven to be exactly zero, for a: —-> 0, to all orders in 5L, 5m, and am, by using recurrence and completeness relations for the expansion coefficients of the top and bottom towers.6 The next step is to calculate the finite part of Ap. We do it analytically for N = 1, and show numerical results for arbitrary values of N. Since we are only interested in the leading order new-physics contribution, we take 5L —1 0, since a finite 5L would 6This was independently proved, in private communications, by R. S. Chivukula for the decon- structed model, and by one of the authors of this note (R. Foadi) for the continuum model. 66 1.25 1.2K 1.1 - 1.05» f (N) 0 2'0 40 6O 80 100 120 N Figure 3.4: Plot of f (N), defined by Eq. (3.109). The continuum limit gives approx- imately a correction of 17%, with respect to the three-site model. give, to leading order, the ordinary standard model contribution. To see why, we first neglect the bottom mass, which in our language amounts to setting em = 0. Then the standard model contribution to Ap is proportional to m? and thus, by Eq. (3.14), to M zsisz. On the other hand, in the 5;, —+ 0 limit the heavy top and bottom modes are not degenerate (since m, k 71$ mbk for 513 7e 51,3), and give a non-zero contribution to Ap. The latter is therefore the leading order new-physics contribution [21]. Using the results in sec.(3.3), it can be shown explicitely that Eq. (3.107) is indeed satisfied for N = 1, and the infinite part is canceled out. The finite part may be ob- tained by inserting the masses and coupling constants into the expressions for wa(0) and IIZZ(0), Eq. (3.105) and Eq. (3.106), respectively, and these into Eq. (3.102). Ex- panding in am, the new-physics leading contribution to Ap, for N = 1, is found to be 1 E4 M2 A10(1) — ——'R _ 1W 1,22 (3.108) The methods used in Sec.(3.3) to calculate the fermion and light gauge boson masses and coupling constants are general, and so the corresponding results for an 67 arbitrary N are obtained. The N-site model, for arbitrary values of N, indeed, involves complicated expressions, especially in the fermion sector. We therefore opt for a numerical computation of Ap. As for the three-site model, we do this in the limit 5L,:c —+ 0, since this gives the leading order new-physics contribution. Based on general arguments [21], and on the three-site model Ap expression, we expect a result of the form 4 2 f (N) EtRM : 167r2 v2 Ap (N) , (3.109) to leading order in 5,3, where f (N ) is the quantity we set out to find. Since in a numerical calculation are included not only the leading term but also higher order corrections, we need 5%, x2, and 5,23 to be much smaller than one, in order to make the non-leading order contributions negligible, and recover the analytical results for the three-site model and the continuum model. Also, since we work in the limit 5L, :1: —» 0, we take 5;, and a: much smaller than 513- Therefore, we arbitrarily choose the values x,eL ~ 10’5, em ~ 10‘3, and calculate Ap. Dividing the result by 5‘,‘RM2/161r""u2 gives f (N) To get an estimate of the error of this, we performed also a semi-analytical calculation of f (N), for N = 3,4,..., 10, by approximating irrational numbers with rationals having fifty significant figures. The error on f (N ) for the numerical calculation compared to the semi-analytical one was approximately constant and equal to 0.1%. We show our results in Fig. (3.4), for N between 1 and 122.7 For the three-site model we obtain f (1) = 1, in agreement with the analytical result found in the last section. Moreover we find f(122) = 1.177, which is very close to the value f (N ——> oo) = 1.1724 obtained in [26] for the continuum model. 7The reason to stop at N = 122 is simply that the percentage increase in time to complete the calculation was becoming much larger than the very small percentage change in f (N). 68 3.5 Experimental Bounds on Fermion Masses With these results, the experimental upper bounds on Ap translate into lower bounds for the mass of the heavy fermions. These can be derived once a relation between 5% and 2:2 is established. In the N = 1 model such relation is imposed by ideal delocalization, in which case three of the four leading electroweak parameters intro- duced by Barbieri et.al. [27] exactly vanish at tree level. In the model with arbitrary N, ideal delocalization is not possible, since we have already imposed translational invariance on the “bulk parameters”. However we can require that the S parameter vanishes. This is phenomenologically sufficient, since the terms parametrizing low energy four-fermion interactions are naturally suppressed. Taking a, m z, and mw as fundamental input parameters, a fermion’s coupling to the W boson, as a function of the S, T and U parameters, is [24] w e = - 1 — - — , 3.110 91’ s + 452 2s2 832 ( ) where c E mw/mz and 32 E \/1 — 02. Since at tree-level T, U = 0(234), we can obtain the leading order expression for S by just computing g)". Including corrections of order C(29) and 0(5i), this is N(2N+1) N W 1 ...2__ 2 . . 9L —g 12(N 1) ’1: 2 EL (3111) Expressing e and s in terms of the input parameters (see [19]), we find that S vanishes if 1N+2 2 2 ___—_- EL—3N+1:c . (3.112) Notice that this gives the correct expressions for the three—site model [21] and the continuum model [24]. To turn Eqs. (3.111), (3.112) into a bound on m f, we need to gather some additional piece of information. First we need expressions for the W 69 and W1 masses, which can be found in [19]. To leading order in 332: m2 = Jiff— M2. = §2f2sin2 ——”— (3113) W 4(N+1)’ ”1 2(N+1) ' ' To leading order in 5L and a”; the top mass is Tn; : A! EL EN? . (3.114) The heavy fermions are all approximately degenerate. To leading order in EL and 5va the mass le of the lightest of these heavy modes is 7er1 = 2M sin( (3.115) MIL—1))“ Recalling that x E 9/5}, and using Eqs. (3.112) - (3.115), Eq. (3.109) gives m — :3— 31n(—2(2N+1)) f(N) m? M3“ (3116) x — ‘/ . . 1 877 (N + 2) sin2(——2(§+l)) AP 7)me We see explicitly that the upper bounds on Ap become, for fixed values of MWI and N, lower bounds on mm. The experimental bounds on the p parameter depend on the value of the reference Higgs mass, mfif. For mi, >> m?” the Higgs contribution to Ap is (Ap) - 30 log 3’ (3 117) mm 161w2 m2”, ' ' In our Higgsless model the contribution to Ap from the W1 boson, for (”121/1 >> mfv, has the same form, with exactly the same coefficient [39]: 3 A12, 0 1 ‘31. (3.118) 0 2 167rc mw (Ap)w1 = _ 70 17.5 4 15 3.75 3.5/f A 12.5 A E 10, % 3.25 b b 3 x- _________ i 7.5 {(275_ E 5 E ° 2.51 2.5’ 2.25. 0.4 0.6 0.8 1 1.2 0 10 2A0 3A0 40 mw,(TeV) N Figure 3.5: Lower bounds on the mass mX1 of the lightest among the heavy fermions, as a function of MWI, with N varying between 1 and 122 (left). We also plot the same quantity as a function of N, for Mw1=500 GeV (right). In each case a solid line corresponds to Ap < 2.5 - 10'3, while a dashed line corresponds to Ap < 5 . 10‘3. We notice that the three-site model is already a very good approximation for the continuum model, with a difference of just 3%. We therefore interpret the phenomenological bounds on Ap, extracted for a given value of mfif, as bounds extracted for the same value of the W1 mass, MWI = mfisf. Current bounds (see for example Langacker and Erler [47]) yield approximately Ap < 2.5-10‘3, at 90% CL, assuming a moderately heavy (340 GeV) Higgs boson, and Ap < 5 - 10'3 in the case of a heavy (1000 GeV) Higgs boson. In Fig. (3.5) (left) we show the corresponding lower bounds on mx1 as a function of Mm, with N varying from 1 to 122. In Fig. (3.5) (right) we plot the lower bounds on mx1 as a function of N, for Mw1=500 GeV. In each case we add (3a/167rc2) log(Ma,1/(m§$f)2) to the experimental upper bound on Ap, in order to take into account the difference between the W1 and the Higgs contribtuions in Eqs.(3.117,3.118): this gives an appreciably weaker bound for mx1 only for mfif=340 GeV. We notice that the three-site model is already a very good approximation for the continuum limit: the 17% difference in 71 f (N) between N = 1 and N —> 00 is reduced to approximately 3% for mm. This is because the factor in front of \ff—(T), in Eq. (3.116), behaves approximately as the inverse of mm itself, for relatively large values of N. We also notice that, for values of MWl within the unitarity bounds, the fermion mass scale is approximately one order of magnitude larger than the gauge mass scale [21]. This implies that even the lightest among the heavy fermions is probably well beyond the reach of LHC. 3.6 Conclusions Higgsless models solve the SM fine tuning problem while being phenomenologically vi- able models. The mass spectrum satisfies the experimental constraints on the masses of extra gauge bosons and fermions thanks to large gauge couplings, Yukawa cou- plings, and mass terms, at the internal sites of the moose. The oblique electroweak parameters are all defined equal to, or can be adjusted to zero at leading order in the model’s small parameters [18]. This allows the Higgsless models to satisfy, at least at tree level, the EW precision measurements constraints. In this chapter we computed the fermionic one-loop correction to the p parameter in the SU(2) xSU(2)N >fiv Figure 4.1: Here is the plot of the probability density Pd(X2/d) for d = 1, . . . ,5. At X2 = 0, P, ranks j-th in magnitude, and therefore P1(0) has the highest value (00). under study is actually described by the model used to generate the predictions, one would expect to obtain a value of $0 = X31111” / d such that Pd(x0) is large, in the optimal case equal to the maximum of the function Pd(z). If Pd(:1:0) is much smaller than the 75 maximum, it is likely that the probability distribution is actually different from the theoretical one, and therefore that the theoretical model does not describe correctly the system under study. Since in general 0 < Pd(2:) < 00, it is difficult to interpret Pd(:c) quantitatively, so the cumulative probability cPd(z) = fz°° Pd(z’)dx’ is rather used: it formally rep— resents the probability that X2 turn out to be greater than d - 1:. A model which describes correctly the system on average would have an associated value of X2 such that cPd(.1') E 0.5, since that corresponds to the maximum of the distribution Pd(rc). A plot of cPd(x), for d = 1,. . . ,5, is presented in Fig.(4.2). The smaller value near the origin corresponds to cP1(:r), and the others follow in the same order of d. The de 1.0 _— 333, 0.8 —— 25 a.|>1, Figure 4.2: Here is the plot of the cumulative probability cPd(x2/d) for d = 1, . . . ,5. The smaller value near the origin corresponds to cP1(:c), and the others follow in the same order of d. smaller, for a given model, CPd(.T0) is the less likely that model describes correctly the system under study. A generally accepted limit to reject a model on the basis of the experimental evidence is cPd(:::0) < 0.05, which means that the probability 76 to obtain a value of x2 larger or equal to xfm-n is just 5%. Such a low probability suggests that the X2 probability distribution associated with the model under test is indeed different from the theoretical one, and therefore the model is not correct. This is usually stated as “the model is ruled out at the 95% confidence level” (95%CL). On the other hand, there is no value of cPd(2:0) for which the model can be accepted as correct once and for all. Besides the viability of a model, the X2 probability distribution can be used to make indirect constraints on the value of observables not yet measured. Taking the example of a single observable, which in general is a function of the input parameters and therefore can be defined as a parameter itself, say (11, one defines the quantity Ax2 = x2 — xfnin, and finds the value 1126 such that cP1(:I:()) = 0.05. The two values am, am such that AX2(ala,b,a’2, . . . ,a2) = 9:6 (around the minimum x2 is approx- imately a quadratic function of the input parameters) are the limits of the 95%CL interval within which a1 is expected to be found by direct measurement. Until now we assumed the observables to be totally uncorrelated. In the more general case of correlated observables one has to calculate the correlation matrix, whose element pub is defined as a = W , 4.2 M 0an ( ) where the measured value of the a—th observable is 0,? = a, which is simply the average of the N data points a,. The correlation matrix pub, as the experimental values and the relative errors, is determined by the experiment. The general expression defining x2 is a simple scalar product of the vector having as its elements the pulls Ac = (Off — O;")/0a with pub: X2 : AaloabZSba (4.3) 77 where the summation over observables is implicit. In the rest of the chapter we will apply the x2 test, using precise electroweak data from experiments, first to a class of elementary particle models, which have a rather general extension of the SM electroweak sector, and then we will focus on the topcolor assisted technicolor models (TC2). Among these, the hypercharge-universal TC2 [48] produces encouraging results compared to the SM, when a heavy Higgs is assumed, and therefore we will study in detail the indirect constraints on the lightest extra SM particles of its spectrum. A discussion of the results will conclude the chapter. 4.2 Flavor Universal Fits to Electroweak Observ- ables The x2 test becomes more stringent as more precise measurements are included in the study. The most precisely measured electroweak data come from Stanford Linear Accelerator (SLAC), the Large Electron Positron Collider (LEP) at CERN, and the Tevatron at FNAL (Fermi National Accelerator Laboratory). The Standard Model (SM) is the most successful minimal model in fitting the data, where by minimal we mean having the minimum number of input parameters. We focus our attention on the electroweak sector of the Lagrangian defining the model. Because of the high precision required by the experiment, the SM predictions have been successfully tested up to one, or even two loops. Therefore any electroweak correction coming from extensions of the SM must be largely flavor universal and of the same size or smaller of the SM loop contributions. It is convenient to parametrize these flavor universal corrections to the SM in a model independent way, implementing them as numerical parameters in the observables’ prediction. A realization of this parameterization for a rather general model in the case of 2 -—> 2 scattering processes is given by a tree 78 level neutral-current scattering amplitude of the form [16] _MNC ___ €2Q_Q' + (T T3 — s2Q>(T'3 — W) (4 4) 2 . P2 (32- r-e.)P2+7§G—(1—QT+—‘§:z) + $0.83— :2T3T’3+4\/_GF(AP— aT)(Q- T3)(Q’-T’3), and of charged-current processes of the form T+T" + T"T’+ 2 06 T+T" + T'T'+ _MCC= 2 é )/ +\/—GF22( 2 ), _ 2 a6 (62— 1671’) P + 45216}? (1+ 482;) (4.5) at energies of 0(mw, m Z), where the quantum numbers of the final state are primed, P2 is a Euclidean momentum-squared, and c2 = 1 — 32. It can be demonstrated that S and T are the well-known Peskin-Takeuchi [2] oblique electroweak parameters. We did not include U in the parametrization since the relative correction is negligible compared to that of the other parameters. The parameters Ap and a6 in Eqs. (4.4,4.5) allow for contributions from additional SU(2) and U (1) gauge bosons, besides the SM W*, Z, A. Here the tree level contributions of the additional bosons, which are constrained by direct searches to be much heavier than W and Z, are approximated by contact interactions in the last terms of Eqs. (4.4,4.5). From Eqs. (4.4,4.5) one has 06 = 0 when there are no additional 8 U (2) groups, while Ap = aT when there are no extra U (1) groups. In general the expressions of the parameters aS, aT, (16, Ap (where a(Mz) = 1/128), as a function of the parameters of a specific model, can be calculated directly in terms of the couplings and masses of the model. Later in the chapter we work out aS, aT, ad, Ap for a specific model. 79 4.3 Experimental Observables We included in our fit the most precise electroweak measurements available, that can be divided in two sets, depending on whether they are measured on or off the Z and W:t boson poles. In the first set there are, assuming flavor universality, fourteen observables; those measured at LEPI and SLAC are I‘z,ah, Rh, Rb, RC, A’LR, A943, A213, A'FB, A23, A323, A}. The first one is the total decay rate of Z, while 0;, rep- resents the cross section, evaluated at the Z pole, for the process e+e' —’ hadrons. The term Al” is the left-right asymmetry coefficient, measured in Z —+ 1+1“ decays, and given by the ratio (I‘L — I‘R) / (I‘L + PR) (where FLU?) is the decay rate of Z to left-handed (right-handed) 1+1", and lepton universality is assumed). The forward- backward asymmetry coefficient Alva: instead, refers to scattering processes, and is obtained calculating (0p -— aB)/(ap + 03), where 0,:(3) represents the e+e‘ -> 1+1- cross section for the lepton l‘ to travel forward (backward) with respect to the 6‘ direction. The meaning of the corresponding quantities for the bottom and charm quarks follow automatically. The T lepton polarization A}, is the only polarization pre- cisely measurable at LEP experiments; though the SM predicts it to be equal to AER, experimentally the first one measures the asymmetry between helicity states (which refers to the kinematics), while the second refers to chiral states (which determine the kind of interaction, and for the T in the SM at high energies, it is the same of the helicity state). Rh gives the ratio I‘thadmm/I‘Zfifir, while Rb is Fzflbg/I‘thadrmw Two additional observables from LEPII and Tevatron are the total decay rate of the W boson I‘w and its mass MW. We will see that the theoretical predictions for all of these quantities can be obtained using just the expression for the general decay rate Fz—qfa and for the A LR again relative to a generic fermion-antifermion pair. Since Ap is related to the new physics corrections off the Z pole, we included in our analysrs the cross sections ”fir—x117." ae+e__w+p_ and ae+e__.r+r_ , measured 80 at LEPII for the set of energies (in GeV) E =189, 192, 196, 200, 202, 205, 207; for a total of twenty one experimental values. The SM input parameters M 2, Mg, and a, are precisely measured, and, consa quently, a minimization of X2 produces optimal values of those parameters close to their measured value [49]. Rather than minimizing on M 2, Mt, and as, we use their experimental result and do not include these observables in our set. This choice does not increase substantially the value of x2 compared to its global minimum. We did not minimize in the Higgs mass m H, either, but rather we imposed the experimental lower limit m H > 114.4 GeV, from direct searches at LEPI, and applied the X2 test on various models for values of my up to 3 TeV, to study the likelihood of such configurations. 4.4 Theoretical Predictions The scattering amplitudes in Eqs. (4.4, 4.5) refer to 2 —+ 2 processes. Studying Eqs. (4.4, 4.5) we can extract masses and coupling constants that can be used to calculate the theoretical predictions for all of the LEPI/SLAC/Tevatron observables of our set. The mass of a particle is defined by the value of the momentum corresponding to the pole in its propagator. Looking at the scattering amplitudes we can write immediately the equations 2 6 (15 a5 A12 = _ 1 __ _ T __ Z 432C2\/§GF ( + 482C2 (1 + 43202) ’ (4.6) 62 OS (16 M2 = —— 1 — _— W 452\/§GF ( + 432 + 4328), where a5 is just S multiplied by the fine structure constant a = 62/(471’). The coupling constants of the fermions to the W and Z can be written as a function of 81 the electric charge Q and the third component of the weak isospin T 3: e T3—52Q e T;t 92 = —(———)-. gw = -——, (4.7) SC 1_ OS 3 1_ OS 48262 4:2- where T3 = a3 / 2, Ti = (a1 i 02) / 2, and the specific coupling for a fermion pair to the Z (Wi) are obtained multiplying T3 (Ti) on the right— and the left-hand side by the proper weak isospin doublets. Both T3, T2t charges are zero for a weak isospin singlet. The quantity sin 62 is defined as 7ra —————. 4.8 sin2 oz cos2 62 = Plugging in the first Eqs. (4.6), and solving the second order equation in 52, we get 2 2 a8 — 4s§c§aT + 016 = _ , 4.9 S 5" 4 (s: - c3) ( ) where we used the abbreviations sin 62 E 50, cos 62 E Co. (4.10) By using Eqs. (4.4, 4.5, 4.6, 4.7, 4.9), we can calculate the predictions for the experimental observables mentioned in Sec. (4.3). To match the precision required by experimental tests, we have to use the renormalized SM coupling constants, calculated at least at one-loop, while the tree level corrections from new physics are sufficient, since they have to be small compared to SM tree level contributions. The new physics contributions can then be conveniently approximated by a linear expansion in the 82 electroweak parameters [14] S, T, 6, Ap, and cast in the form: 60tree l-loo new 0m = (05M P) (1+ Thee) , SM 1‘ tree (60,—4—2'”) = 5,-0.9 + tiozT + diaé + riAp, (4-11) 0s; .- where 05:00", 037;; are the SM predictions at one loop and tree level, and 05,255 is the new physics tree level prediction. Note here that, implicitly, the SM (for the given Higgs mass) corresponds to (18 = aT = a6 = Ap = 0. Written in this form, the four expressions needed to calculate the numerical predictions for the LEPI/SLAC/Tevatron observables are I. I-wSM 1_ 1 (gA—gv)9vg_ Z”: Z” 2 2 93+93 23% ALR_ASM(1_ 2C3 2gA—9V9Ag 232(052 _ Oz-+-T 06 _)), (,,,—s,9 gv gA+gV 490% 439% 106 M =MSM 1—-——— S— 2 . W W( 4(a3—ss>("‘ CeaT+2cs))’ (412) 1 3 1+c§ Fa=1VISM 1——— -S— 2T 6 ... ....( 2(c3—s5>(2“ + SW) where gA, gv are the axial and vectorial tree level couplings, which depend simply on the fermion quantum numbers and so. In I‘wfid u, d refer to the upper and lower weak isospin component. The remaining Z pole observables AfaB and 0,, are defined in terms of the quantities of Eq. (4.12): 3 e I‘e Ph ALB = Z LRAiR’ 0’}; = 127Tm.(4.13) The LEPII predictions (as functions of (13, aT, a6, Ap) can be obtained by plugging 83 Eq. (4.9) in Eq. (4.4) and then working out the cross sections in the form of Eq. (4.11). 4.5 The Fit The value of the oblique parameters 05', GT, (16, Ap is determined minimizing the function x2 (015, (IT, 06, Ap) defined explicitly by 11 02h _ 02:1: 0th _ 061: X2 = 2 b b 61: pab a,b=l a a? , (4.14) where of” is the experimental error of the a-th observable and pub the correlation coefficient of the a—th and b-th observables. The input parameters and the experimental results are taken from [49] [50] [51]. The reference mass of the Higgs constitutes an additional parameter, that in most of the fits we took equal to 115 or 800 GeV. The numerical values of the axial and vectorial coupling constants 9A and gv are determined by the particle’s quantum numbers, while sin2 62 = 0.2312. The SM prediction at one loop is obtained using ZFITTER [52] [53], for the off-pole observables, and SMATASY [54] (which is actually based on ZFITTER), for the on-pole ones. We calculated x2 (aS,aT,a6,Ap) as defined in Eq. (4.14), and then minimized it with respect to its parameters. The result is xfnin /d.o. f. = 43.48/31 = 1.40, for my = 800 GeV, corresponding to a 6.8% probability to get a larger value of film-n. This has to be compared to the SM fit, defined by 015' = (IT = 06 = Ap = 0 (which therefore has four more degrees of freedom), with my = 115 GeV, that produces xgm-n/do f. = 51.58/35 = 1.47, corresponding to a 3.5% probability, and to the SM fit with my = 800 GeV, that gives Xian/(10f. = 139.6/35 = 3.99, yelding an approximately zero probability. The set of values of the electoweak parameters minimizing X2: for m H = 800 GeV, 84 and their correlation matrix K. are 025' = (0.63 i 0.40) 10-3 f 1 —0.63 0.65 0.10 ) aT = (0.05 :1: 0.21) 10-3 —O.63 1 —0.87 —0.78 , K. = . (4.15) 66 = (—0.85 a: 0.22)10-3 0.65 -0.87 1 0.62 Ap = (0.23 a: 0.29) 10-3 K 0.10 —O.78 0.62 1 ) As a check the corresponding fit performed on the experimental data quoted in [27] reproduced the results presented by Barbieri et al. in that same article. The results in Eqs. (4.15) are derived for a reference Higgs mass 771'}?! = 115 GeV. It is necessary to specify my! because the one-loop SM predictions depend on m”, and they must be subtracted in order for the oblique parameters to quantify the contributions from new physics. The optimal values of C13, aT, Ap are consistent with zero within 20, for m H = 800 GeV. In particular the smallness of QT = (0.05 :t 0.21) 10‘3, Ap = (0.23 :l: 0.29) 10’3, which can be assumed equal well within the margin of error, suggests a negligible neu- tral boson contribution related to extra U (1) gauge symmetries. On the contrary, the optimal value (16 = (—0.85 :t 0.22) 10'3, with a deviation from zero equal to almost 40, requires a charged boson contribution related to extra SU (2) gauge symmetries. In displaying the 95%CL region, since a 3-D plot is not easy to read (and possibly a 4-D plot even less), we simplify our task limiting the parameter space to the aS—aT and Ap—a6 planes intersecting the point that minimizes X2- For the aS—aT plane, for example, we have to solve the following equations 2 AXE,“ (as, 6T) = x2 ((13, OT, 66', M) — x3“... r (g, AXQW) /r (g) = 1 — 0.95, (4.16) with respect to one of the two variables, where Axfmx is obtained solving the second of Eqs. (4.16), k is the number of parameters (in this case two), and l" (2:, y) is the 85 incomplete gamma function. The primed parameters represent the Optimal values. Plotting both of the solutions on the same plane, and limiting the range of variation of the abscissa by the requirement that those solutions be real, we get the 95%CL ellipsis for the 015, aT parameters, which we show together with the one for 06, Ap in Fig. (4.3). These limits have to be considered conservative, since a minimization 10-3x GT 0.7 r Tfi T T T I T I j 10‘3xAp Figure 4.3: 95%CL in the £13 — QT and Ap — C16 planes. The origins of the axes are placed at the parameters optimal values for m H = 800 GeV. of X2 performed also on the SM parameters M Z, M), Aafgd, as, and in particular m”, and including also the direct measurements of the first three observables and sin2 0(Mz) in the analysis, would have likely produced a slightly greater probability. 86 4.6 Topcolor Assisted Technicolor Models A great advantage of the parametrization given in Eqs. (45,44) is that it approxi- mates 3 large class of extensions of the SM electroweak sector. For a model whose electroweak sector is a family universal extension of the SM one, the calculation of x2 in function of the specific model’s parameters, starting from X2 (05, 0T, (16, Ap) is rather straightforward. It requires one to determine the oblique electroweak param- eters by comparing tree-level model results to Eqs. (4.4, 4.5), and then to substitute the results in x2 (025', (1T, (16, Ap). We apply this prescription to study the viability of the Hypercharge-Universal Topcolor model in Subsec. (4.6.1). In Subsec. (4.6.2) we briefly discuss the results of the x2 test on the Flavor-Universal [55] and Classic Topcolor models [56, 57] , whose calculation, since the hypercharge couplings are actually non-universal, required a slight generalization of the previous prescription. 4.6.1 Hypercharge—Universal Topcolor Technicolor Models (TCM) postulate a strong coupling interaction drives some fermions to form scalar condensates, which are ultimately responsible for the electroweak sym- metry breaking and the emergence of masses. These models, though theoretically very appealing, suffer from tight experimental constraints. The problem arises be- cause of the need to generate a large mass for the top quark. The top mass is inversely proportional to that of the boson responsible for the (extended) strong interaction, which, for a universal technicolor model, couples the third generation to the other two, leading to various problematic predictions. In a Topcolor-assisted Technicolor (TC2) model the third generation is charged under an additional color gauge group, different from that of the other two generations. This allows for an extra parameter to adjust the top mass to the right value. The Hypercharge—Universal TC2 model be- longs, indeed, to the TC2 class of models since the third quark generation is charged 87 under an SU(3) gauge group, while the first two generations are charged under an- other S U (3) group, as we schematically show in Tab. (4.1). The charges under weak isospin and hypercharge gauge groups (SU (2)w and U (1)) instead are universal. The Hypercharge-Universal Topcolor SU(3)1 SU(3l2 SU(2)W U(1)1 U(1)2 I - SM SM SM - II - SM SM SM - III SM - SM SM — Table 4.1: Gauge charge assignments for fermions of generations I, II and III in the hyperchargeuniversal topcolor model introduced in this paper. The entry “SM” indicates a charge assignment corresponding to that in the Standard Model. Hypercharge—Universal TC2 model has been presented in [48], and we refer to that paper for the theoretical description of the model and for the derivation of the cou- pling constants and the masses of its physical states. Here we present a goodness of fit study of the TC2 model. We call Z’ the extra neutral boson associated with the additional U (1) gauge group in the theory. The strength of the additional U (1) coupling is parametrized by rt]. Extra fermions are charged under a technicolor gauge group SU (N )TC. Because of this strong interaction this “technifermions” eventually form a condensate, neutral under the technicolor and weak groups, which breaks dynamically the SU(3)1 x SU(3)2 x U(1)1 >< U(1)2 symmetry to SU(3)C x U(1)y at a scale u, which is greater than the EW scale 1). The free parameters of the model are pu and ml, which are related to the Z’ coupling and mass by the equations 042/ = pu‘ / 1‘61 (631 + City) ,gZ; = \/47I'I‘£1Y. (4.17) 1 In Eqs. (4.17) 01,» = 0.01, u is related to the scale at which the two U (1) interactions break to ordinary hypercharge, and p is the U ( 1) charge of a scalar quark-antiquark 88 condensate. An additional parameter, ca, determines the relative contribution of the technifermion condensate to the EW scale 1) = 246 GeV. To work out the predictions given by Eqs.(4.12) in function of the model’s parameters, we introduce an alternative parametrization, given by 2 v2 2 “I 1‘ = —§—2 << 1, C¢ = ___. (4.18) p u K1+ 02y The masses and coupling constants [48] are then defined to be: gz = i (T303 — Y (53 — 3:234, (cg — Ci))), 2 __ eYc¢ M2 _ 82112 (4'19) g2, _ C9 ’ Z, — 4x2c93¢c¢’ 2 2 2 e v 9w——,0w=—,9 I=0, 483 W where all the expressions are approximated to the 2:2 order, and , as usual, Y = Q—T3. Here .99 is the SM sin 62, while the other angles are parameters of the model. By isolating the individual contributions, at zero momentum P2, to the scattering matrix elements, we can immediately read off from Eqs. (4.4, 4.5) the following system of four equations, involving implicitly, through the Eqs. (4.19), the parameters 2:2, 0 and (b 2 49—2- : (T3 — SZQ) 4&0}? Mg 1 — 0T 2 g I :2 = (72. — (2)2 4\/§GF (Ap — aT). (4.20) Mz’ 2___(T3_32Q)2 2 _ Ii 92 — 122—Tm” — ‘E—T’ e — E? :2 “ r67 in which (16 = 0 since, as it can be noticed from 9w’ = 0, there is no extra SU (2) group in this model. Solving Eqs. (4.20) with respect to the remaining variables a5, 89 aT, Ap and 32, we obtain the expressions a8 = 4cgci2:2 (6,2, — cg) , aT = 2:2 (0,2, — cg), (4.21) a} Ap = $26 2__ 2 222 2 2 s —30+:rc(,c¢(ca—c¢), to be substituted in X2- The number of input parameters can be further reduced using the Pagel-Stokar formula [58], that determines cos2 a ”E 0.91. We have performed a two—parameter fit in 3:2 and cos2 (b for the set of data shown in App. (A). Note that the fit is linear in .22, but due to the occurrence of cos4 (b in the theoretical expressions for the observables, non-linear in cos2 (b. The search for the global minimum of )(2(2:2,cos2 d2) in the physically allowed region of the two parameters (a:2 2 0 and 0 g coszqfi S 1) reveals that the minimum is actually on the boundary at cos2 ()5 = 0 and x2 = 0.0035 corresponding to a Xr2nin = 48.88. For cos2 0‘) so near to zero, we can treat the fit as linear in cos2 (b for purposes of statis- tical interpretation of the results. Since we have 35 observables and do a fit in two parameters, we have xfnm/dof. = 48.88/33 = 1.48, corresponding to a 3.7% proba- bility of obtaining a xfnm of at least this size if our model is the correct description of the data. We have used a reference Higgs mass of 800 GeV in order to reflect the dynamical origin of electroweak symmetry breaking in TC2. However, as mentioned earlier, the fit of the SM with an 800 GeV Higgs to this data is quite poor. For purposes of comparison, we note that when we fit the same data set to the SM, with a reference Higgs mass of 115 GeV, we obtain nearly the same probability: 3.5% (xim/dof. = 51.58/35 = 1.47), while a fit to the SM with a reference Higgs mass of 800 GeV yields a probability of essentially zero (xfmn/dof. = 139.6/ 35 = 3.99). To aid in interpretation, we may also rephrase our results in terms of the variables 90 ’61 and pu. As figure 4.4 shows, n1 is restricted to a value less than 1.2 x 10’3 at 95%CL. It is worth recalling that the coupling of the Z ’ boson to fermions has strength W Y (see Eq. (4.17)); this is of order .05 for :61 = .001. At the same time, pu must lie between about 3.48 TeV and 4.86 TeV. Values of pa this large are clearly in accord with our prior assumption that 1152 << 1; in fact, we find 12 3 0.005 at 95%CL. Note that the presence of an upper bound on pu is actually consistent with the fact that our expressions for the observables reduce to their SM (with my = 800 GeV) values as u —+ 00 (rather than simply when m --—> 0), which is ruled out. 1.2 -' 1’. ..., 0.8} mm-3 C O\ 3.25 3.5 3.75 4 4.25 4.5 4.75 pUITeV] Figure 4.4: The solid (dashed) curve shows the restriction on model parameter space obtained from a fit of all Z pole and off-pole data listed in the first two tables of the Appendix (A) for an 800 GeV (1500 GeV) reference Higgs mass. The region outside the parabola is excluded at 95%CL. Bounds on M z’ Translating the constraint on pu into a limit on the Z’ boson mass as a function of K31 using Eq. (4.17) yields the results shown in Fig. 4.5. We observe that Z’ masses less than 2.08 TeV (2.12 TeV) are excluded at a confidence level of 95% when using a Higgs mass of 800 GeV ( 1500 GeV). The data do not provide an upper bound on 91 M 2’ because the best-fit value of K1 is zero and M z’ or 1/\/I€—1 for rel-values small compared to ay z 0.01 (from Eq. (4.17)). Precision electroweak data clearly provide 6110-31 Mz’ [T6Vl Figure 4.5: The solid (dashed) curve shows limits on the 2’ mass as a function of m resulting from a fit of all Z pole and off-pole data listed in the first two tables of the Appendix for an 800 GeV (1500 GeV) reference Higgs mass. Values outside the boomerang shaped region are excluded at 95%CL. a stronger lower bound on M 2’ than direct searches for Z’ bosons. The present CDF limit on a “sequential” Z ’ boson with standard-model-strength couplings to fermions is M z’ > 923 GeV [59]; the CDF limit on our Z’ would be even weaker since the couplings between the hypercharge-universal Z’ and fermions are smaller than those for a sequential Z’ by factors ranging from 1.5 for right-handed electrons to 15 for left-handed quarks. We estimate (a - B)T(,~2 z .24(a - 8),”, implying that the direct lower bound on the 2’ mass is of order 800 GeV. By way of comparison, we note that the limit lies between the CDF limit on the E6 Z’ bosons Z} and 2;, as might be expected from a comparison of the various Z’ bosons’ couplings along the lines discussed in [60]. 92 Interpretation using standard electroweak parameters Our bounds may also be understood in terms of the standard parametrization of the electroweak corrections given in Eq. (4.21). Since the best fit value for cos2 6 is zero, Ap z QT and S a: 0. Fixing cosza = 0.91, gives T m 0.4 for the best fit value x2 = 0.0035. The experimental bounds on S and T are commonly illustrated as an ellipse in the S—T plane for a given reference Higgs mass. Some of these plots (See, e.g., Fig. E.2 in ref. [49].) also show how contributions from a very large Higgs mass would pull the Standard Model prediction outside the region defined by the experimental bounds. For a (reference) Higgs mass of around 1 TeV new physics adding ~ 0.4 to the value of T is required to move the theoretical prediction back into the experimentally-favored ellipse. This is why the presence, in our model, of new physics that persists in the limit cos2 cf) -—> 0 is important. Overall, we conclude that the Hypercharge—Universal TC2 model is consistent with the data at hand, and the Z’ should have a mass and coupling that render it accessible to LHC experiments. 4.6.2 Flavor-Universal and Classical Topcolor Models For comparison, we briefly look at the Flavor-Universal and Classical TC2 models, which differ from the Hypercharge—Universal TC2 and between themselves in the fermion charges, as it can be seen from Tables (4.1,4.2). Though the S U (3) charges are different in the Flavor-Universal (this name because of the universality of the strong sector) and Classical TC2 models, the electroweak sector charges are identical, and, since we take in account only electroweak observables in our analysis, we can study both models at the same time. The models have different U (1) charges for the third from the first two generations, and so the Z’ boson coupling is non-universal. Electroweak precision limits on these models were previously obtained in [61]; at that 93 Classic Topcolor SU(3)1 SU(3)2 SU(2lw U(1)1 U(1)2 I - SM SM - SM II - SM SM - SM III SM - SM SM — Flavor-Universal Topcolor SU(3)1 SU(3)2 SU(2)w U(1)1 U(1)2 I SM - SM - SM II SM - SM - SM 111 SM - SM SM - Table 4.2: Gauge charge assignments for fermions of generations I, II and III in the classic [56, 57] and Flavor-Universal [55] Topcolor models. The entry “SM” indicates a charge assignment corresponding to that in the Standard Model. A dash indicates that the fermion is not charged under the gauge group. time, there was still a narrow window of parameter space in which a Z’ mass below 1 TeV was possible. Because of the non-universaility of the two models, we recalculated x2 in Eq. (4.14) using two different sets of oblique parameters (one for the third generation, and one for the other two) to express the predictions in Eqs. (4.12) in function of the model’s parameters; note that we must now employ the data that does not, a priori, assume generation universality. Searching for the global minimum of X2 in the physical region where 1:2 2 0 and 0 g cos2¢ S 1 (and setting f, z 75 GeV), we find xfnin/dof. = 106/39 = 2.71, corresponding to a probability of order 10’s. The best-fit value of cos2 0‘) is close to 0, which forces sin2 ¢ to be of order 1, thereby increasing the difference between the Z boson’s couplings to fermions in the third generation and those in the first or second generations; given the degree of lepton universality displayed by the Z—pole data, it is not surprising that a poor fit results. These TC2 models are simply not concordant with the precision electroweak data. 94 4.7 Conclusions In this chapter we reviewed the general x2 techniques to test a model of experimental data. We defined a rather general parameterization for deviations from the standard model coming from extensions of the SM electroweak sector, and derived expressions for the observables as functions of the electroweak oblique parameters. These ob- servables include experimental results from LEPI, SLAC, LEPII, and Tevatron. To calculate the SM part of the predictions we used the program ZFITTER [52] [53], which guarantees a high degree of precision since it includes corrections at least of one loop order. Our analysis of the Hypercharge—Universal TC2 model shows that precision elec- troweak measurements provide tight constraints on this model. In contrast, we find that precision electroweak constraints now exclude the Classical and Flavor-Universal models [57] [55]. The Hypercharge-Universal TC2 model can fit the electroweak data as well as the SM with a light Higgs boson. However, the topcolor-symmetry-breaking scale is driven to the value pu z 4 TeV, which means that m, is much lighter than the scale of the dynamics by which it is generated and implies a need for fine tuning of m. The Z’ coupling factor 161 is, indeed, of order .001 or smaller (the Z’ couples to fermions as JIM—n] Y). The Z’ mass must therefore lie above 1.6 — 2 TeV: too heavy for direct production at the Tevatron, but within reach of LHC. 95 Chapter 5 Custodial Invariant Top-Mass 5. 1 Introduction The standard model (SM) is the most successful elementary particle model in ex- plaining and predicting the current experimental results with a minimal number of parameters. The SM, though, is believed to be an effective theory. This conjecture is justified mostly by the extremely large fine tuning (about one part in 1032) of the bare mass of the Higgs boson, the last missing particle of the model, once the scale up to which the SM is considered to be accurate is set equal to the Planck scale. The fact that the optimal value of the SM Higgs mass (mg) is 773% GeV [1], well below the current lower limit m H > 114 GeV from direct search at LEPII, seems to confirm that conjecture. A much more reasonable fine tuning, in the order of a percent, results from as- suming the SM is accurate up to the TeV scale. The energy in the center of mass of proton-proton collisions at LHC will reach 14 TeV, and, if one assumes the SM to be a low energy effective theory, it is reasonable to expect new physics to be observed. Any extra SM particle, though, is constrained to be heavy, in the range of hundreds of GeV and beyond, for a weak coupling strength, otherwise its effects would have 96 been already become apparent in precision electroweak (EW) tests. The Z boson coupling to the left-handed (LH) bottom quark-antiquark pair (9“,), in particular, is measured with a 0.35% precision [50], and therefore imposes a strong constraint on new physics. The SM prediction on 91.3 is about 20 off from the experimental value [50]. It has been shown by Agashe et al. [62] that the custodial symmetry S U (2)V of the SM Higgs sector, once extended to the Yukawa sector of an extended SM, can be used to protect g“, from corrections, which can be large if extra SM quarks or gauge bosons contributing to the Z -—> bL5L process are light. A softly broken custodial invariance of the top-mass Yukawa sector allows one to reduce the size of the new physics contributions to Z —> 5151,. In this chapter we illustrate this principle in an SU (2);, x U (1)y gauge invariant extension of the SM, that implements custodial symmetry in the Yukawa sector by the introduction of an extra S U (2) L quark doublet. This extra doublet, together with the LH tL, bL quarks, makes up a bi-doublet under an SU(2)L x SU(2)); transformation. An SU(2)L invariant Dirac mass term for the extra doublet breaks the custodial symmetry, giving a non—zero loop correction to g“, (6ng). This mass term for the extra doublet allows one to satisfy the EW experimental constraints. Besides being interesting in its own sake, this study can give an indication of the viability of more complicated models implementing custodial symmetry in the Yukawa sector, and having the doublet extended standard model (DESM) as a low energy effective theory. One such example could be a model defined in five dimensions whose Kaluza-Klein excited states are significantly heavier than the one identifiable with the extra doublet in the DESM. In Sec. (5.2) we review the use of custodial symmetry to protect ng from large cor- rections, then present the DESM, and work out its mass eigenvalues and eigenstates. In Sec. (5.3) we calculate 9“, at one loop order, and compare it with the relative exper- 97 imental measurement 92%: gm, turns out to be in good agreement with gii, especially for a relatively light extra doublet. In Sec.(5.3) we calculate the oblique electroweak parameters (15 and aT in the DESM, which give actually the tighter constraints on the model’s extra free parameter. The optimal configuration for the DESM, given by a goodness of fit study on the Z pole observables and LEPII cross sections, turns out to be the SM. In Sec. (5.4) we summarize our results and offer our conclusions. 5.2 Doublet-Extended Standard Model 5.2.1 Custodial Symmetry and Z coupling The Higgs sector of the SM Lagrangian is invariant under a global S U (2) L x S U (2);; transformation (the subscripts L and R refer to the fact that in a bi-doublet the first transformation acts on the left and the second on the right, and are not related to a particle’s handedness). After EWSB the symmetry breaks down to the diagonal S U (2),; group, which provides the “custodial symmetry” that guarantees the value of the p parameter to be one at tree level, plus a discrete symmetry PM, that switches the labels L and R in the quantum numbers. The SU(2)V symmetry of the Higgs sector is accidental, and it is actually broken by the SM Yukawa sector and the hypercharge gauge interaction. Following Agashe et al. in [62], we show how the custodial symmetry can be used to protect ng from loop corrections. The EWSB determines the breaking pattern SU(2)L x SU(2)); —> SU(2)V x PM in the Higgs sector. If the Yukawa sector is extended in such a way that it does not break the residual custodial symmetry, the charge Q?) associated with this symmetry is conserved in the gaugeless limit. Therefore the loop correction 6?; to the tree level value of Q3 is zero. A gauge eigenstate having quantum numbers TL = T R, T 2 = T3 is an even parity eigenstate of the symmetry Pug, which guarantees Q3 = Q1}, also at loop level. Since Q?) = 98 1+ (2%, one has 6623 = 6Q3 = 6623/2 = O. The Z coupling to LH fields 1,0,,th at tree level is defined by - e 911:)” = 9291.11), Qz “ 3 - 2 — —— , = — 6 , 5.1 where Q is the electromagnetic (EM) charge. Since Q is conserved, the assignments TL = TR, T3 = T ,3; guarantee that the loop correction to gm), is (5ng = 0. Once gauge interactions are turned on, custodial symmetry is broken by the different hypercharge of LH and RH isospin doublets and, therefore, gm), receives non-zero corrections from loops. These corrections, though, are small compared to those coming from Yukawa interactions, which, as we just showed, can be made zero by custodial symmetry. We will study here the implementation of this idea in a toy model. 5.2.2 The Model The toy model we study here must implement a global custodial symmetry SU (2),, x S U (2) R in the Yukawa sector, which is softly broken by a mass term. The gauge group of the model is the standard SU (2),, x U(1)y. The SM Higgs field (I), here defined as a bi-doublet under SU(2)L x SU(2)R, breaks spontaneously SU(2)), x U (1)y down to U (1),,,,, by acquiring a vacuum expectation value (vev) Since only the vectorial combination of generators commutes with (1)0, SU(2)), x S U (2) R —+ S U (2)v = S U (2) H 3 after EWSB. The Yukawa couplings for the first two generations are very weak, so we focus our attention on the third quark generation. To protect g“, from large corrections we make the assignments T 1. = TR = 1/2, T3 = T ,3; = —1/2 for the LH bottom-quark b1, (T2 is constrained by experi- 99 ment). The EM charge is fitted by introducing an extra U (1)x group and defining Q=T2+Tg+X=—1/3, (5.3) that fixes X = 2/ 3. The LH top-quark, belonging to the same SU(2)), doublet, automatically has T L = T R = 1/2, T 2 = —T;°§ = 1/2. The quantum numbers are obtained by the transformation law of a bi-doublet (2, 2)2/3 = QL —> U L QLU ; under SU(2)), x SU(2)); x U(1)x. The right handed (RH) top and bottom quarks t3, b3 experimentally have T f: = 0. For their Yukawa interaction terms to be invariant under U (1)EM: t3 and b3 must also have X = 2/3. From this and Eq. (5.3) one gets T,3, = 0 for 153, and T13; = —1 for ()3. Therefore one can simply have t); = (1,1)2/3 or can embed tR in a triplet (1, 3)2/3, part of a multiplet (3,1)2/3 GB (1, 3)2/3, while b3, if present, has to be embedded in a triplet (1, 3)2/3. Here we study the simplest case: t3 = (1,1)2/3, and ignore ()3, which, because of its small mass, would give higher order corrections to Q“. The quark bi-doublet and the Higgs field can be defined, together with their symmetry properties, as it. EL 9150 73¢— QL = = (212)2/31 ti? = (111)2/3 a q) = . = (22)0 ' (5'4) bL TL 145+ Q55 We write the S U (2) L x S U (2) n X U (1) x invariant extension of the Yukawa sector as _‘CYukawa = AtTI' (QL ° (P) 5}; + h.C. . (5.5) Since (5ng 75 0 from the experiment, and a massless extra S U (2) L doublet is clearly not phenomenologically viable, the custodial symmetry needs to be softly broken by a new mass term, so that SU(2)), >< SU(2)); x U(1)x ——> SU(2)L x U(1)y, the last 100 being the gauge group of the DESM, as well as of the SM. The hypercharge Y is defined by Y = TE; + X. (5.6) We split Q into two S U (2) L doublets and introduce RH fields for E, T, expressed with their quantum numbers under an S U (2) L x U (1),» transformation as EL,R t1, ‘I’L,R = = 27/6» 'l/H. = = 21/5. (5.7) Tun bL The mass terms are written as —£m=M\-pL°\pR+h.C.. (5.8) The terms in Eq. (5.8) softly break custodial symmetry, and, therefore, give a non—zero loop correction (5ng to 9“,. The mass parameter M allows one to fit the experimental value of ng. The RH component \IIR of the Dirac field \I! is needed to make the model anomaly free. The kinetic terms for the Dirac heavy doublet and for the SM chiral fields are written as: £16m = ‘i’ 2241 + 15L pl/JL +512 pin +512 131111- (5-9) The covariant derivative is defined as D“ = a" + z' (lgaaW’: + Yg’B"), (5.10) where the weak isospin l = 1/2 for a doublet and l = 0 for a singlet, and the hypercharge Y, determined by Eqs. (53,56), is given in Eq. (5.7) for the SU(2) doublets, while Y = Q for t 3, b3. The whole Lagrangian follows the EWSB pattern 101 SU(2)), x U(1)y —» U(1)EM once (I) acquires the vev (1)0 in Eq (5 2) 5.2.3 Mass Matrices and Eigenstates To find the t T mass matrix, we make the substitution $0 —> 71-(1) + H + i7rz) 1n Eq. (5.9) and define Aw 1v_1 . 5.11 fl u= m ( ) m: to obtain: 1 0 AILT = m (5.12) 1 u The remaining 5 E mass terms are linearly independent, with the bottom assumed to be massless, and m); = pm. Defining u2 = \/4 + 114, the masses for t,T can be 2_ 2 mt=m]/1+# 2V, m7~=m (5.13) while the correspondmg RH and LH gauge eigenstates can be expressed 1n functlon of the mass eigenstates as written as V2 (5.14) 1 p2 — 2 p2 — 2 (TR)gauge ——2- ( 1+ U2 TR + 1 — V2 53) , (TL) )gauge = :(\/1+_”2TL + \/—t h)- The parameters )1, m are related by Eq. (5.13) and the experimental input m.) 172 GeV The plot of the masses mT, mg in Fig. (5.1) shows that m, < mg < m) These masses have minima (m E) min = 243 GeV, (mflmm = 415 GeV, for p = 1.41, 0, 102 m7" "'5 (TeV) 1.5L Figure 5.1: The line on the top refers to "12‘, and the one on the bottom to mg. respectively. In the limit 12 >> 1 (that is M >> m = Atv/2) at leading order the masses reduce to mt g m, m’]‘ g M, (5.15) while the mass eigenstates reduce to the relative gauge eigenstate: (taalgauge 9 tan, (TL,R)gauge E“ Tan; (5.16) which is what we expected since the heavy doublet decouples and the Lagrangian reproduces the SM one. In the opposite limit 11 ——> 0, assuming m to be finite, we have the (experimentally unviable) masses mt = 0, mr = x/2m, (5.17) 103 while the gauge eigenstates are given by "L _ TL {'L + TL (tR)gau9€ = TR’ (TR)yauge = t3? (tL)gaugc = fl 1 (TL)gauge = fl . (5.18) 5.3 Phenomenology 5.3.1 Z coupling to bL5L In the SM the dominant contribution to the one loop correction 591.6, as obtained from the Z —+ bL5L amplitude, comes from triangle diagrams involving the top quark in the loop and is proportional to 771?. It is convenient to work in the “gaugeless” limit [63, 64, 65, 66], treating the Z boson as an external field (meaning that we neglect its kinetic terms so that the Z does not propagate) that couples to the current ng — jg sin2 91, with strength g2. Specifically, our aim is to calculate (5ng defined by 1 1 , 2 91.6 = —5 + 3 sm 91. + 591.6 (5.19) In the t’Hooft gauge, the pion 7rz couples to bL5L through an effective action operator of the form A 8,7125147‘74. (5.20) Because of EWSB the third component of the isospin current acquires a new term: "‘ = “4‘ + 36 5 21 .731. .731. 2 117IZ v f ' l where 35’]. is the third component of the isospin current in unitary gauge. From Eq. (5.21) the operator in Eq. (5.20) contributes to (5ng through the diagram in 104 Figure 5.2: Loop level contribution to g“, in the gaugeless limit; the blob represents the renormalized 7rz ——> bb process. Figure 5.3: General triangle diagram for the fig —+ b5 process at one loop. Fig. (5.2) by an amount 6,,ng = A. (5.22) MIC The blob in Fig. (5.2) represents a renormalized vertex. At one loop level the leading order contribution to (5ng in the SM is represented by the diagram in Fig. (5.3), with tm- = t, the top quark, and is given by [63, 64] 6 SN! __ "1‘22 (5 23) 9L!) (4WU)2 ' ' In the DESM there are additional contributions coming from the T quark, there- fore tisj = t,T. We use the analytical mass eigenstates given in Eq. (5.13) to 105 extract the coupling constants from Eq. (5.5) as functions of )1. By defining ai = \/ 2 + #4 :t 1121/2, flit = V2 :1: p2 :1: V2 (where the first :t on the left-hand side cor- responds to the first :t on the right-hand side), we can write the analytical result in a compact form as 2 , 4 2 2 2 4 2 2 2 _ _ + _. 3_ ... _ Tilt (2&1; (0+ O_) )3__\/a [.1. l/ / +\/ZY+ [1. l/ 6 = _ 9"” 32n2v2/33J4 ran—11+- (Si-1:»- Substituting Eq. (5.13) in Eq. (5.24), one can check that, in the limit where u —> 0 (5.24) for finite m, 6g“, = 0, as expected since the custodial symmetry is intact for M = 0. m2 In the limit 11 —* 00 one, instead, recovers the SM result ng = —L2. For u >> 1 (41m) Eq. (5.24) simplifies to m2 log a 1 6( = t (I — + —) . 5.25 .lLb (471-1102 #2 #2 ( ) We compare this result with the experimental value 92?, = -—0.4182 :1: 0.0015 [50]. The DESM prediction including the SM loop corrections is obtained by subtracting from Eq. (5.24) the SM limit £15232, at leading order in rm, and adding the SM prediction gig" calculated at one loop order. To calculate 9%” we use ZFITTER [52, 53] with a reference Higgs mass my = 115 GeV, and‘obtain gig” = —0.42114 (which matches the result given in [50]), whose deviation from 927, is 1.960. The value of 93,85“, evaluated for v = 246 GeV and mt = 172 GeV, is plotted in Fig. (5.4), where the origin has been set at u = 0, 93]” = —0.42114. The horizontal dashed line corresponds to 92f, = —0.4182, while the two horizontal solid lines show the relative :L-30 deviations. We can see from Fig. (5.4) that the DESM satisfies the Z —> b5 experimental constraints on g“, for [u > 0.55, and reproduces the 91.5 SM prediction 106 ng —0.414 -0.416 -0.418 -0.420 —0.422 r Figure 5.4: DESM prediction on ya, in function of the parameter ,u. The y axis intersects the 9“, axis at the value gig” = —0.42114. The horizontal dashed line corresponds to 92”,”, = -—0.4182, while the two horizontal solid lines show the relative :t30 deviations. approximately for u > 2.2. 5.3.2 Oblique Electroweak Parameters The flavor-universal corrections from new physics beyond the SM can be parametrized in a model independent way using Peskin and Takeuchi’s oblique electroweak param- eters S, T, U [2]. This parametrization allows a direct comparison among different models and with the experiment through a goodness of fit calculation. We take as ori- gin of the STU space the m H = 115 GeV SM coordinates, so that only new physics contributes to the oblique parameters. The only contributions to S, T, U in the DESM come from heavy fermion loops in the vacuum polarization diagrams. In this case U is suppressed relatively to T roughly by a factor mag/MELT; since the limit on g“, tells us that M ET >250 GeV, this is a significant suppression. Also, the correc- tions given by S and T are typically small, on the order of a few percent, and since 107 Figure 5.5: Vacuum polarization diagram contributing to the oblique electroweak parameters. The indices 1', j = 1, 2, 3, Q, where WQ = A, the EM field, while k refers to a given quark-antiquark pair consistent with the EM charge of W,. U is smaller by at least one order of magnitude, we will neglect U and calculate only Sand T. We also perform a goodness of fit calculation on the Z pole and LEPII data in function of the parameters 05, (17", using ZFITTER [52, 53] to calculate the SM predictions at one or two loop order, and MATHEMATICA to perform the fit to the S and T parameters. Parameter aT The parameter aT describes the amount of custodial symmetry breaking, and is defined by 62 T = II 0 — H 0 5.26 a sin2 9;, cos2 6an2z [ ”( ) 33( )1’ ( ) where II,,- is proportional to the sum over the thk pairs of the diagram in Fig. (5.5). In the DESM, for W1 we have tklik = t5, T 5, X t—, X T, t5; while for W39, tkEk = if, TT, tT, T t—, X X , b5. The analytical result for (IT cannot be written in compact form, therefore here we write only the result approximated for )1 >> 1: 3 log ,u 22 2 T = 1— — . f. a (47rv)2 ( 8 #2 + 3H2) m, (o 27) 108 -0.5 —1.0} -1.5} Figure 5.6: aT in function of )1, defined as the difference of the DESM result with the corresponding SM one with m H = 115 GeV. The horizontal lines show the optimal fit value of QT = 0.68 x 10‘3 (dashed line) and the relative i30 deviations (solid lines). The origin corresponds to the SM aT value, which is zero by definition. One can see that, for p —> 00, Eq. (5.27) reproduces the SM result aT = (ax—31077"? [2], as expected. It is interesting to notice, also, that in [67], the authors find that embedding the SM SU(2)L singlet top in the same 80(5) multiplet as the extra doublet is disfavored since, for the correct value of the top quark mass, this leads to a large negative contribution to the (IT parameter at one loop. Taking into account that their 30(5) multiplet in 4D generates an 80(4) = SU(2)L x S U (2) R bi-doublet, their result for aT seems to confirm ours. The analytical expression of (IT, defined as the difference of the DESM result with the corresponding SM one with my = 115 GeV, is plotted in Fig. (5.6). The horizontal dashed line corresponds to the optimal fit value of QT = 0.68 x 10‘3, while the two horizontal solid lines show the relative :l:3a deviations. As one can see from Figs. (56,54) the experimental constraints on aT favor large values of p, while the constraints on (5ng favor smaller values of ,u. 109 Parameter 0:5 The parameter S can be defined by 2 I I where the prime refers to the derivative with respect to the boson momentum q. The complete expression of S cannot be written in compact form, so here we write explicitly only the result for p >> 1: 1 7711, 8 = — — — 2—l .2 S 67r (3 + 210g mt + #2 ( 0g 11.)) , (5 9) where we had to reintroduce a non-zero mass for the b quark to cutoff a divergence in the integral over the momentum in the fermion loop. One can check that Eq. (5.29) reproduces the SM result [2] for p —> 00. We then redefine S as the difference between the DESM complete result and the SM result at my = 115 GeV, and plot the result in Fig. (5.7), where also the optimal fit value a8 = 0.34 x 10‘3 and the 21:30 relative deviations are shown. From Fig. (5.7) one can see that (13 matches the experiment for p = 2.4. 5.3.3 Goodness of Fit We do a goodness of fit study using the results and methods described in the previous chapter. We set (IT 2 Ap in the observables expressions, because there is no extra U (1) gauge group in the DESM, and 6 = 0, because there is no extra S U (2) gauge group. Since the 6g“, correction is non-universal, we modified the model independent corrections involving the coupling 9“, to include 69“,. The minimum value of x2 per degree of freedom is 49.2/33=1.49, with aS = 3.4 x 10'3 and aT = 6.7 x 10’3. The value of x2 per degree of freedom for the SM ((15 = 0, 0T = 0) is 51.1/35=1.46. To 110 aIleO3 1.5 l 1.0; 0.5 1 Figure 5.7: (13 in function of [1, defined as difference of the DESM result with the correspondent SM one with m H = 115 GeV. The horizontal lines show the optimal fit value of 08 = 0.34 x 10'3 (dashed line) and the relative 21:30 deviations (solid lines). The origin corresponds to the SM 05 value, which is zero by definition. fit the DESM predictions to the EW precision measurements we expressed ()8, QT, and (5ng as functions of [1. These functions, in the limit it >> 1, are given by Eqs. (525,527,529), from which one has to subtract the SM result. The value of x2 for the DESM is minimum for u = 00, independently of m H, therefore the experiment clearly favors the SM limit, in which the extra doublet becomes infinitely massive and decouples from the rest of the theory. This is caused by a large negative contribution to aT from the extra doublet. To get the 95% confidence level (CL) limit on p, since it depends on the value of m”, we plot on the same 03 — aT plane the 95%CL ellipses for a set of 11 values of my (77).” = 115, 150, 200,300, ..., 1000 GeV), each centered at the relative minimum. On the same plane we also plot the parametric function having coordinates a8 = 01501),ch = aT(u) (where, for a >> 1, Eqs. (527,5.29) can be used), for p = 3,4, ..., 20, 00. From Fig. (5.8) one can see that the 95%CL lower limit on p for 111 AarTx103 Aan l 03 loofi=3 Figure 5.8: Here plotted are the 95%CL regions of the (1S and HT parame- ters for mH = 115,150,200,300,...,1000 GeV. Also plotted are the points rel- ative to the DESM with p = 3,4,...,20,oo. We defined the coordinates as AaS(m”) = aS(m”) — ozS (m1,)mi“ + aS(m,,=115GeV) AaT(mH) = aT(mH) — aT(mH) + aT(mH =115GeV) min 1 min min‘ 112 m 1., = 115 GeV is about 20, while for any larger value of m” the DESM with p S 20 is excluded at 95%CL. The bound [1 Z 20 gives, at 95%CL, mggw 2 [1 mt 3‘ 3.4 TeV, that are too heavy to be possibly detected at LHC. 5.4 Conclusions we studied a simple extension of the SM, defined DESM, having an additional S U (2) L heavy fermion doublet, which illustrates a custodially-symmetric top-mass Yukawa sector. This symmetry is softly broken by the heavy doublet Dirac mass term, whose coefficient M parametrizes the amount of breaking. The bL quark is embedded in a bi-doublet under global SU(2)], x SU (2);; symmetry, and has quantum numbers T3 = T3 = 1 / 2. For M = 0 this assignments determine 591.1), the one loop correction to ng, to be zero [62], while for M = 00 we recover the SM result. The experimental constraints on ng require the parameter a, roughly equal to M / mt, to be greater than 0.55 at the 30 level. The parameter aT receives, for small values of p, a large negative correction to the SM value. The new physics contribution to the SM 05 result is mostly positive, and, for the same value of u, much smaller in magnitude than aT. We did a goodness of fit study of the DESM using experimental results of EW Z and W pole observables from LEPI, SLD, Tevatron, and cross sections from LEPII. We found the DESM to be consistent with experiment only for p, > 20 at 95%CL, with a Higgs mass my 2 115 GeV. The bound on u translates into the 95%CL limits on the masses of the extra quarks E, T, mgr 2 pm, E 3.4 TeV, that are out of reach at LHC. We conclude that the model is viable for ,u > 20, though the quarks E, T are constrained to be very heavy and therefore the DESM phenomenology at LHC is undistinguishable from that of the SM. It would be interesting to see whether a larger extension of the SM, still imple- 113 menting custodial symmetry in the Yukawa sector, but placing t R in a (3, 1) EB (1,3) multiplet under SU(2)L >< SU(2)R, rather than in a singlet, can be used to improve the gm, fit to the experiment for light extra quarks, and if such an improvement is allowed by the QT experimental constraints. 114 Chapter 6 NcQED 6. 1 Introduction The idea of formulating field theories on noncommutative spaces goes back some time [68].1 Interest has been revived recently with the realization that noncommutative quantum field theories emerge in the low energy limit of some string theories [70, 71, 7 2]. This has led to numerous investigations of the phenomenological implications of noncommutative QED [73, 74]. In noncommutative geometries, the coordinates x" obey the commutation rela- tions [as",:z:"] = 729"”, (6.1) where 6“” = —9"“. The extension of quantum field theories from ordinary space- time to noncommutative space-time is achieved replacing the ordinary products with Moyal :1: products, defined by (f * g) (2:) = exp (aewaaayu) f (x) g (y) I.-. - (6.2) 1This chapter is based on [69] 115 Here, in order to ensure the S matrix unitarity, we assume that 6"" = 6‘0 = 0, z' = 1, 2, 3, and study the non-zero contribution generated by space coordinate non- commutativity. In the following, we study the effect of noncommutative geometry on the pro- cess e+e‘ ——i 777 using noncommutative quantum electrodynamics, NcQED. As discussed, for example, in [75], NcQED has as its Lagrangian — 1 c = w * (M — m) w - a?“ (F... * F'”) + 29...... + 59...... (63) where .Cgauge and .Cghost denote the gauge fixing and ghost terms. The corresponding Feynman rules for phenomenological calculations can be derived from Eq. (6.3) [75]. Since the scale at which noncommutative effects are likely to occur is large [73, 74], we focus on the energy scales typically associated with future linear colliders. Our calculations are performed in the center of mass of the colliding electron and positron. In the next section, we outline the calculation of the squared amplitudes. This is followed by a summary of the cross section computations and a discussion of the results. Details of the calculation are presented in the Appendices. 6.2 NcQED Amplitudes Typical Feynman graphs contributing to e+e‘ —> “/77 at leading order are shown in Fig. 6.1. Diagrams in which the fermion line is connected to the final photons by a single photon propagator vanish. The complete set is obtained by permuting the photons, which gives a total of twelve diagrams. We have calculated specific helicity amplitudes and it is therefore unnecessary to include ghost contributions [76]. The amplitudes were calculated using the Feynman rules in [75]. We computed the helicity amplitudes with the aid of the symbolic manipulation program FORM and simplified 116 p1 «AA—RN» kau 164.0 mi p1 __f—‘WVWV k1”! vp3=p1—k1 p1 = .. p3=pi-k1 «vvvvvx, 192,12 k V kw! ‘P4=k3’P'2 lpgzkll—m 2, P'z—‘fl p2 I'\/V\/VV\/ k3,/) p2 ——<——4'\/\/ 777 cross section is very sharply peaked in the forward and backward directions. The result of the cut |cos€| g 0.9 is shown in the bottom panel of Fig (6.2). Clearly any observation of a dependency of the production rate of 37 on (0 would represent a violation of Lorentz invariance, since the physics of the process would change after a rotation around the axis of the e‘e beam, which defines the orientation of the laboratory in space. Because of this, and assuming the direction of maximum noncommutativity to be fixed in space, the production rate for a given 03, which is measured relatively to the laboratory, would change with the time of the day consequentially to the rotation of the earth. Since this time-dependency of the production rate, if not unique, is rather unusual, the result presented in Fig (6.2) shows that if noncommutativity has some non-negligible effect on particle phenomenology at collider energies, its signal should be clearly distinguishable from that generated by Lorentz-invariant new physics. 121 p----——-'-—--—------——--———-—-—---—-—--q 720 700 11' 211' Figure 6.2: In the top panel, the solid line is number of events from NcQED as a function of 45 for the case x/S = 5TeV, ANC = 1TeV, A = 1r/4, .C = 500 fb‘l and no cut on cos 0. The dashed line is the uniform background from QED with no cos6 cut. The bottom panel shows the effect of imposing the additional cut |cos 6’] S 0.9. 122 We also checked the N eQED corrections to the QED energy and polar angle dis- tributions of one of the photons. While there are some differences in the shapes of the NcQED distributions relative to their QED counterparts, particularly in the en- ergy distribution, using these differences as a test of N cQED appears difficult because of their 22 and energy dependence. The search for qb—dependence remains the best possibility if the 6+ and e‘ beams are unpolarized. There are, however, CP violating terms linear in z in the individual helicity am- plitudes, as can be seen in Eq. (6.11) or in Appendix D. To probe these terms it is necessary to consider polarization effects. 6.3.2 Polarized Cross Sections We will confine our discussion to cross sections arising from longitudinally polarized beams. In this case, a typical cross section can be written [80] [(1 + Pe_)(1+ Pd") ORR +(1 — P€_)(l - 136+) ULL AIH 0P P = 6‘ 8+ +(1+ P.—)(1 - P+) m +(1— P.—)(1 + 13.0014] . (6.23) C where, for example, 03;, denotes the cross section when the e‘ beam has pure right- handed polarization (Pr = 1) and the 8+ beam has pure left-handed polarization (Pat = —1). The remaining cross sections are defined similarly. For the process e+e‘ -> 777, amplitudes with A = A vanish, and we can express the polarized cross section as [80] 0RL+0LR P-"P'l' (”JR—URL 0 = l—P—P _— 1— e e Pe' Pe+ ( C 6+) 4 1 '— Pe—Pe+ ULR + URL = (1 _ Pe— Pe+)0unpol[1_ PeffALR] , (6.24) 123 VS TeV 00.9 -0.9 fb 0—0909 fb 1.0 4315.4 4316.9 2.0 1187.9 1189.3 3.0 557.1 558.6 4.0 325.1 326.6 5.0 213.9 215.4 Table 6.1: The N cQED cross sections for 90% left-right and right—left polarized beams are shown for ANC = 1TeV and A = 0. In pure QED 01,3 and 03;, should be indentical. where the effective polarization P85 and the left-right asymmetry A LR are P_—P+ R. = —e—e—— , .2 ‘f 1 — PC- 11+ (6 5) Am = M . (6.26) 01.12 + URL The left-right asymmetry can be obtained using the squared amplitudes in Appendix D and Eq.(6.18), which results in ,,,2 E3 003A («(3) _ % + 477) ALR = '— 2 E . ANC [(106%) — 1)2(log(w) — 1) + 3] (6.27) For the process 6+6” ——1 777, the NcQED correction is the main source of a left- right asymmetry. Competing standard model sources of left-right asymmetry such as Z exchange in Mbller scattering [73] are suppressed because they involve loops. Taking ANC = 1.0 TeV, Table 6.1 shows the cross section values for the cases [73] PC- = —Pe+ = i0.9 and several values of x/S and cos A = 1. As the numbers in the Table 6.1 indicate, the left-right asymmetry generated by noncommutativity, though non-zero, is rather small compared to the large left-right symmetric QED 6+6— —+ 777 cross section. To obtain a sense of the range of values of ANC that can be probed 124 polarized cross sections, we examined the signal to square root of background ratio R = £0709 —0.9 - 0—0909) _ (6.28) \/£(00.9 —0.9 + 0-0.90.9) The numerator of Eq. (6.3) in QED is simply the experimental observation, while the denominator, assuming a Poisson distribution, gives the standard deviation. Requir- ing R 2 3 implies the 95%CL lower bounds attainable on ANC illustrated in Fig. 6.3 as function of luminosity L and collider energy \/S. The constraints on A NC obtainable from the polarized total cross section suggest that cuts on the polarization asymmetries in distributions such as (dO’LR — dO'RL)/dl/ (dO'LR - dURL)/dCOS0 01‘ (dO'LR+d0’RL)/dl/ (dO’LR+d0’RL)/dCOS9 (6.29) could improve the 95%CL lower bounds on A NC- These distributions are shown in Fig. 6.4. While both distributions show a distinct left-right asymmetry, the cos6 distribution is the most promising from the experimental point of view in that it can be rather large — ~ few % — over a substantial region of cos 9. By imposing cuts on c036 it is possible substantially increase the lower bound on ANC obtained using Eq. (6.28). The largest lower bound is obtained by restricting e056 as |cos€| S 0.85, which is illustrated in Fig. (6.5). The behavior of the bound on ANC as the cut on c080 varies from |cos6| S 0.5 to |cos€| g 1.0 is shown in Fig. 6.6. 6.4 Discussion and Conclusions To summarize, we have computed the noncommutative contributions to the angular and energy photon distributions as well as to the total cross section for the process + e e‘ —> 777 assuming both the unpolarized and polarized electron and positron beams. Because we are dealing with a three particle final state, it is possible to include 125 rrl V's—=2 TeV o.a~.v--,r v v 0.7 Y V l T V b 06- O b Am (TeV) l A A A l A A A l A A A l A A A 0'2 A 200 400 600 500 1000 Luminosity (I‘D—1) 0.8 V l V V V l V V V l I V V ‘l Luminosity-500 :5" Ana (T917) l A A A l A A L 0.2" . . 1 z 3 Vs— (TeV) Figure 6.3: The 95%CL lower bounds on A No attainable using the left-right asymme- try of the total cross section are illustrated as a function of luminosity at x/S = 2 TeV (top) and as a function of \/S for .C = 500 fb'l (bottom). The solid lines correspond to A = 0 and the dashed lines to A = 7r/4. 126 0.0125 0.0100 0.0076 0.0060 0.0026 —d(0m—0m)/dv/d(0m+0n)/du 0.0000 ' 0. 0.04 —d(0m—0m)/dc039/d(0m+0m)/dcos0 -1.0 -0.5 0.0 0.5 1.0 0059 Figure 6.4: The polarization asymmetries with respect to the photon energy fraction V (top) and the photon angle with respect to the beam axis (bottom) are shown. The three shaded regions correspond , in order of magnitude, to center of mass energies of 5, 1.0 and 0.5 TeV. 127 Am (TeV) l ' ' I r t ' l ' V ' l ' ’ ‘1 1.50- xii-216v ~ 1.25 f; a . v 1.00 g . < 0.75 0.50 mi...1...11..1... 200 400 500 000 1000 Luminosity (flu-1) -Tf-.,...,..., . 1.50 - Luminosity-500 15" - 1.25 1.00 0.75 0.50 _ h vs- (T:V) Figure 6.5: Same as Fig. 6.3 with |cos€| S 0.85. 128 1.50 1.25 1.00 Am (TeV) 0.75 0.50 r . 0'25 . . . l a . . I . . mL L m A l . A . ‘5 0.5 0.6 0.7 0.5 0.9 1.0 l 0089111.: I Figure 6.6: The 95%CL lower bounds on ANC for 0.5 S | cos Own] 3 1.0 are shown for 4/3' = 0.5TeV (dot-dash—dash), 1.0 TeV (dot-dash), 2.0 TeV (dashed) and 5.0 TeV (solid). Here [I = 500 fb"l and cos A = 1. The bounds scale as .8”4 and V cos A. these corrections using only the space-space portion of the tensor 6“”. This enables us to avoid the use of the space-time terms 0"“ and thereby satisfy the requirement of unitarity [81]. The use of space-time terms cannot be always avoided in 2 —> 2 processes e.g. e7 —-1 e7, 6+6“ —-» 77 or e+e‘ —» 6+6“, whose new physics contributions are proportional to 00‘, and this tends to complicate their interpretation. The cross sections and distributions depend on the angle A between the beam direction and the non-commutativity vector ff. In the unpolarized case, the noncommutative effects are second order in the ratio 2 = S / Afvc» whereas in polarized case, the noncommutative effects are leading order in 2. In the unpolarized case, the shapes of the QED and NcQED energy distributions are quite different but it is the dependence of the cross section on the azimuthal angle (b which offers the best opportunity to detect noncommutative effects. The observation of any variation of the cross section with respect to (b is a clear violation of Lorentz 129 symmetry. It is possible to introduce reasonable cuts that significantly enhance this signature of non-commutativity. The possibility to observe such a signature depends clearly on the relative magnitudes of \/S and A No. Further, the use of polarized beams makes it possible to probe the order z CP violating terms in the helicity amplitudes by measuring the left-right asymmetry. In contrast to Méller scattering, where Z exchange introduces a large standard model left-right asymmetry which competes with the NcQED asymmetry, the NcQED left- right asymmetry in the 37 final state is the dominant source of asymmetry, since it occurs at tree level, with standard model contributions being suppressed by 100ps. Even without cuts on the polarized cross section, the 95%CL lower bounds attainable on ANC are competitive with those obtained in pair annihilation [73]. Imposition of cuts on cos 6, the angle between one of the photons and the beam direction, extends the reach on A NC to the TeV range. Accumulating enough data to reach these bounds will require monitoring the (unknown) direction of the non-commutativity vector 5. Techniques for doing this were proposed by Hewett, Petriello and Rizzo [73] and implemented by the OPAL collaboration [82]. Currently, the experimental lower bound on A NC is 140 GeV [82], and the calcula- tions of Ref [73] indicate that A NC scales of 1.7 TeV can be probed in Méiller scattering at a 500 Gev e+e' collider. Our results at the same x/S set a 95%CL lower bound on ANC of 0.5 TeV, which increases nonlinearly with x/S to AM; > 1.5 TeV at a 5 Tev 6+6“ collider. Like Méiller scattering, the NcQED contribution to e+e’ —» 777 can be parameterized solely in terms of the unitarity preserving space-space components of 6“”. This, together with its NcQED dominant left-right asymmetry signature, makes the three photon process a promising candidate in the experimental search for noncommutative effects. 130 Appendix A Experimental Data and SM Predictions This appendix shows the observables used to assess each model, along with the ex- perimentally measured values and the predicted values we obtained. The first pair of tables shows Z—pole and LEP II observables as calculated in the SM with a light Higgs boson. The second pair presents our fit for the Hypercharge-Universal TC2 model; the third pair contrasts the results for the Classical and Flavor-Universal TC2 models. 131 Experimental Value SM Fit Pull Fz=2.4952:1:0.0023 GeV 2.49 GeV 0h=4l.54:1:0.037 nb 41.56 nb Rb=20.767:1:0.025 20.76 - Rb=0.21629:t0.00066 0.216 Rc=0.l721:1:0.003 0.1726 Ap,T=0. 14648 10.00325 0.1476 ALR,.=0.1513 10.00207 0.1476 ALR’b=0.92310.02 0.9363 ALR,C=0.67:1:0.027 0.6686 Apg,e=0.01714i0.00095 0.01626 Ap3,b=0.0992:t0.0016 0.103 - Ap3,c=0.0707i0.0035 0.0735 = Mw=80.398:t0.025 GeV 80.4 GeV Fw=2.14:1:0.06 GeV 2.09 GeV 0 l 2 3 Figure A.1: l-loop fits to Z-pole observables in the Standard Model with MI,” = 115 GeV, showing the experimental values, the predicted values, and the pulls. 132 Experimental Value (pb) SM Fit Pull 0'qq(189)=22.47:l:0.24 22.2 0'#+#—(189)=3.12310.076 3.21 0',+ .-(189)=3.2io.1 3.21 aqq(192)=22.05¢0.53 21.2 6,. ”-(192)=2.9210.18 3.1 6,. .-(192)=2.81¢0.23 3.1 O'qq(l96)=20.53 21:0.34 20.1 071+ ,,-(196)=2.9410.11 2.96 07+ T-(196)=2.94¢0. 14 2.96 O'qq(200)=19.25:l:0.32 19.1 ap.,,-(200)=3.02:0.11 2.83 6,. ,- (200)=2.9:l:0. 14 2.83 O'qq(202)=19.07io.44 18.6 a”. #. (202)=2.58iO.14 2.77 0'1.+ ,- (202)=Z.79:t0.2 2.77 aqq(205)=18.17¢0.31 17.8 071+ ,,(205)=2.45¢0.1 2.67 01+ T—(205)=2.78:l:0. 14 2.67 0'qq(207)=l7.49:t0.26 _ 17.4 0'fl+#-(207)=2.595:t0.088 2.62 6,. ,- (207)=2.53 10.1 1 2.62 Figure A.2: l-loop fits to LEP II observables at several values of \/E in the Standard Model with M}?! = 115 GeV, showing the experimental values, the predicted values, and the pulls. 133 Hypercharge-Universal Experimental Value SM Fit TC2 Fit Pull Fz=2.4952:1:0.0023 GeV 2.49 2.5 0'h=4l.54:1:0.037 nb 41.5 41.5 Rh=20.767;t0.025 20.7 20.7 Rb=0.21629:1:0.00066 0.216 0.216 R; =0.1721 10.003 0.172 0.172 Ap,r=0. 14648 $000325 0.14 0.146 ALR,e=0-1513 10.00207 0.14 0.146 ALR’b =0.923 10.02 0.934 0.935 ALR,¢=0.67:I:0.027 0.665 0.668 Ap3,c=0.01714:t0.00095 0.0146 0.016 Ap3,b=0.0992:t0.0016 0.0978 0.103 AFB,C=0.0707:1:0.0035 0.0696 0.0733 Mw=80.398:t0.025 GeV 80.2 80.4 Fw=2.14:1:0.06 GeV 2.08 2.09 0 l 2 3 Figure A.3: Fits to Z-pole observables in the Hypercharge-Universal TC2 model introduced in this paper. From left to right, the columns show the experimental values, the 1—loop SM values 011-100;; with M3” = 800 GeV, and the predictions for the TC2 model with their pulls. The T02 model fit assumed ft = 75 GeV and M2,” = 800 GeV. 134 Hypercharge—Universal Experimental Value SM Fit TC2 Fit Pull 0'qq(189)=22.47i0.24 17.3 22.2 011+ fl-(189)=3.123:t0.076 2.62 3.21 07+ T-(189)=3.2:t0.1 2.62 3.21 O'qq(192)=22.05 10.53 17.3 21.3 a“. ”-(192)=2.9210.18 2.62 3.1 6,. ,-(192)=2.81io.23 2.62 3.1 aqq(196)=20.53 $0.34 17.3 20.2 a”. p—(l96)=2.94:t0.ll 2.62 2.96 6,. ,-(196)=2.94¢0.14 2.62 2.96 aqq(200)=19.25¢0.32 17.3 19.1 a”. ”-(200)=3.0210.11 2.62 2.83 0,..-(200)=2.9io.14 2.62 2.83 O'qq(202)=19.07i0.44 17.3 18.6 a”. #-(202)=2.58¢0.14 2.62 2.77 0'1.+.,—(202)=2.79:t0.2 2.62 2.77 O'qq(205)=18.l7:t0.31 17.3 17.8 au.,,-(205)=2.453:0.1 2.62 2.67 6,. ,. (205)=2.78:I:0. 14 2.62 2.67 O'qq(207)=17.49:t0.26 17.3 17.4 0',,+ fl- (207)=2.595:I:0.088 2.62 2.62 01+ ,— (207)=2.53 10.1 1 2.62 2.62 0123 Figure A.4: Fits to LEP II observables at several values of J5 in the Hypercharge- Universal T02 model introduced in this paper. From left to right, the columns Show the experimental values, the 1-loop SM values 0,1"le with M I,” = 800 GeV, and the predictions for the TC2 model with their pulls. The TC2 model fit assumed ft = 75 GeV and 111;,” = 800 GeV. 135 F lavor—Universal Experimental Value SM Fit TC2 Fit Pull Fz=2.495210.0023 GeV 2.49 2.49 - 0'h=41.54l 10.037 nb 41.5 41.5 - Rc1=20.80410.05 20.7 20.7 - R,,=20.78510.033 20.7 20.7 RT=20.76410.045 20.8 20.8 I AFB,61=0.014510.0013 0.0146 0.0163 AFB,,,=0.016910.0017 0.0146 0.0163 I AFB,T=0.018810.0033 0.0146 0.0151 - ALR,cl=0-15 1610.0021 0.14 0.148 ALR,,,=O.14210.015 0.14 0.148 ALRJ=0.13610.015 0.14 0.142 Ap,,=0. 1465 10.0033 0.14 0.142 Rb=0.2162910.00066 0.216 0.2163 Rch=0.1721 10.003 0.172 0.172 Am,b=0.099210.0016 0.0978 0.09943 Am,ch=0.070710.0035 0.0696 0.074 ALR’b=0.92310.02 0.934 0.934 ALRch=0.6710.027 0.665 0.6683 Mw=80.39810.025 GeV 80.2 80.3 Fw=2.1410.06 GeV 2.08 2.08 Figure A.5: Fits to Z-pole observables in the Classical [56, 57] and Flavor-Universal T02 [83, 55] models. From left to right, the columns show the experimental values, the 1-100p SM values 0140016 with M}?! = 800 GeV, and the predictions for the T02 model with their pulls. The TC2 models fit assumed ft = 75 GeV and My] = 800 GeV. 136 F lavor-Universal Experimental Value SM Fit TC2 Fit Pull 0'qq(189)=22.4710.24 17.3 21.9 0",. fl-(189)=3.12310.076 2.62 3.12 01+ ,-(189)=3.210.1 2.62 3.2 O'qq(l92)=22.05 10.53 17.3 21 0”.» ,,-(l92)=2.9210.18 2.62 3.01 0'.,+ T—(192)=2.8110.23 2.62 3.09 - O'q q(196)=20.5310.34 17.3 19.9 011* ”—(196)=2.9410.11 2.62 2.87 0',+ T-(196)=2.9410.l4 2.62 2.96 O'qq(200)=19.25:1:0.32 17.3 18.8 0',,+ ,1- (200)=3.0210.1 1 2.62 2.75 - 0'7.» T-(200)=2.910.14 2.62 2.83 0'qq(202)=l9.0710.44 17.3 18.3 031+ ,,—(202)=2.5810.14 2.62 2.68 07+ ,-(202)=2.7910.2 2.62 2.77 0'qq(205)=18.1710.31 17.3 17.6 (If “- (205)=2.4510.1 2.62 2.59 - 0'74» T-(205)=2.7810. 14 2.62 2.67 O'qq(207)=17.49:tO.26 17.3 17.2 03,4. ,,-(207)=2.59510.088 2.62 2.54 0'.,+ T— (207)=2.5310.11 2.62 2.62 0 1 2 3 Figure A.6: Fits to LEP II observables at several values of \fs' in the Classical [56, 57] and F lavor-Universal T02 [83, 55] models. From left to right, the columns show the experimental values, the 1-loop SM values 011—160;; with MI,” = 800 GeV, and the predictions for the T02 models with their pulls. The T02 models fit assumed ft = 75 GeV and 11.1,? = 800 GeV. 137 Appendix B Spin Averaged Cross Section The tensor 6,-1- can be parameterized in terms of a unit vector 6 and a noncommuta- tivity scale ANC as 1 925 = AT'Ez'jk NC 9". (8.1) To define the coordinates, we fix the origin at the center of mass, choose the z axis parallel to pi and take 5 in the plane z—z plane. In this system, k: is defined by its polar angle a1, its azimuthal angle 61 and its energy E. Similarly k; is defined by its energy E2, its polar angle a2 and, for convenience, an azimuthal angle 31 + fig. The phase space integration is given in detail in Appendix C, where it is shown that, in addition to the variables mentioned above, it is necessary to introduce a minimum photon energy Em“, to control the infrared singularities. Introducing the dimensionless variables (2' = 1, 2) Emin ’2 6: E n=\/1—%, (B.2) the terms in Eq.(6.12) can be expressed as p=u1(1—nc1)q=§l/2(l—TLC2) T:-§-(2—V1(1—ncl)-V2(1_nc2)) ““103 INC!) u1(1+ncl) t= §V2(1+n02) u:§(2—u1(1+nc1)—u2(1+nc2)) (8.3) S: The total cross section is expressible in terms of these variables as 1 1-6 1 1 2—u1—c _6 (470,) / dill/1;”1 dV2+/1—ch1./e (ll/2 2 d 2" Me+e—-+777 /. 61/? (162/0 (1’31 c)(c 73 (8'4) _ 2 2.. Oe+e _ #777 X where 61(V1+V2—1):F\/(1-Ci)(1-V1)(1-V2)(V1+V2—1). (13.5) Ci=C1—2 Using the symmetry of Eq.(6.12) under a permutation of 151,163,163, and neglecting m2 in the numerator of Eq. (6.12), the commutative contribution to the cross section becomes 1- -e 2—u1—e l c+ (7qu - Egg/2 du1/: (ll/2+ /: (1111/ dI/g/ dcl / dC2 e e ~777 1 --ul 1 c 1 c_ — V1 - V2)2 + 712(011/1 + C2V2)2 (1 — ”2(3) (1 — n23) \/< 2, the expression for the three body phase space in the electron- positron center of mass is (270“ 4 63151 63k; 63k; (11‘ = __5 + —k —k —k 3!2(4E2) (p1 W 1 2 3) (27032131 (27032192 (27032173 1 4 .. 4 (1356316363133 = _— 2E—E —E — 3 —— .1 360705“ ‘ 2 E3” (k1+k2+k3) ElEzEa (C ) where we have already introduced the symmetry factor 1 / 3!. In spherical coordinates the components of k: and k; can be defined by k1 = E1{sinalcosflhsinalsin/31,cosa1} k2 = E2 {sin a2 cos ([31 + 62), sin a2 sin (61 + 62), cos a2} . (0.2) The integration over 63(163 + k; + k3) gives VIE—32: (/(k]+k3)2 _—_ \/E12+ 117122 + 2E1E2 (cos a; cos a2 + cos 62 sin a1 sin (12) (03) E3 143 where we have used the definitions (0.2). The phase space integral Eq.(C.1) can then be written (11‘ _ 6(2E — E1 — E2 — E3) 63/516316; _ 38 (47r)5 E1E2E3 6(2E — E1 — E2 — E3) E1E2 3S (4705 dEldEgd cos ald cos C!2d/31d,82 E3 (0.4) with E3 given by Eq.(0.3). The remaining integration over 6(2E — E1 - E2 — E3) leads to E32 = E? + E; + 2E1E2 (cos (11 cos 02 + cos 62 sin a1 sin a2) = (2E — E1 — E2)2 . (0.5) Solving Eq.(C.5) for fig results in two solutions fiz = 212520 (0.6) with 1 _ ._ _ 520 = arccos E1E2( cos (11 cos oz) .2E(E1 + E2 E) . (0.7) E1E2 sm a1 Sln (12 The derivative of the argument of the 6 with respect to ,82, d(2E—E1—E2—E3) E1E2 , , . = ——sma sma S1113 . 0.8 dfiz E3 1 2 [2 ( ) then gives 1 6(2E — E1 — E2 — E3) = ‘ (5032 — 520) + (5((32 + (320)) 7E E' . . . i321 S111 011 sm 0'2 8111 620 (09) 144 Replacing the Eq.(0.9) in Eq.(0.4) we get 6 ”3 — ’3 6 ' 3 dr = “—l—dEldEgdcosaldcosazdfiidfiz (F2. ’20? + ((127420) 35 (4")5 sm a1 sm a2 8111 1320 In the squared amplitude, )82 is present only in the noncommutative factor 1172. As can be seen from Eq.(B.3), the other variables are independent of 62. The integration over ,82 produces 11120320) and w2(—[320), but these contributions are equal after the subsequent 61 integration. Hence, for both the QED and NCQED terms, we can write 2 dEl (IE2 dcosal dcos 012 dfll dF = .1 35' (471)5 sin (11 sin (12 sin [32 ’ (C 0) where for simplicity we have dropped the zero on 62. The limits of integration on, say, cos a2 are constrained by Eq. (0.7). Using this equation to solve for sin a1 sin a2 sin 132, we find, in the notation of Eq. (8.2), sin a1 sin a2 sin 1’32 = \/(c+ — C2)(C2 — c_) (0.11) with (V1+V2-1)q:\/(1—cf)(1—V1)(1—V2)(V1+V2—1). (C12) Ci = Cl — 2C1 The limits of integration on the energies are determined by solving the relations E,,,,,, S E, S E E1+ E2 + E3 = 2E, (0.13) with j = 1, 2, 3, for E1 and E2. Elimination of E3 yields the additional inequalities E2 2 E — E1, 132 3 2E — E1 — E,,,,,,. (C14) 145 The limits of integration on E2 depend on whether Em,n 5 E1 3 E — Em," or E — Em," S E1 3 E. In the former case, E—ElsEQSE, (0.15) while in the latter Emin S E2 S 2E _ E'1 " E[min - (C16) 146 Appendix D Squared Modulus of the Helicity Amplitudes The helicity amplitudes with permuted photon helicities can be derived by the cor- responding permutation of the variables p, q,r and s, t,u. Amplitudes with every helicity reversed just change the sign of 0P-breaking term, while amplitudes in which the electron and positron are exchanged can be derived changing the sign of the an- tisymmetric term and exchanging the variables p,q,r with s, t,u. The amplitudes {1,1; A1, A2, A3} and {A,:\;1,1,1} are zero. The twelve non-zero helicity ampli- tudes are given in the following table. The common factor At in all the entries in Table D1 is Aj: -S + 2 bc ac ab pq+(p+8)(q+t)+st _pr+(p+8)(7‘+U)+8u_qr+(q+t)(r+U)+ 2 2 2 2 .2 2 : 266[S—4w2(3 a(p +s)+b(q +t)+c(7 +11) c b a 1 1 1 It 41111) (— + — + ‘) E(k1,k2,p1,p2)):l . a b c 147 tn) /\1/\;/\1, )12,/\3 [Nix/1919213 [2 )1, N A1, A2, A3 [0471111129312 2 2 +1-;+1+a— pgst‘A" +1_;—1—1+ piggy/4- 2 ,2 +1-;+1-)+ #:A- +1_;—1+1— prsuA" . :3. . s2 +1—1_a+1+ qrtu - +1_1+a_1_ qrtu - . 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