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 K'IProjIAcc8Pres/ClRC/DateDue.rndd DEC‘()I\'S"I‘RIIC"I‘ED HIGGSLESS MODELS OF ELECTROWEAK SYMMETRY BREAKING By Baradhwaj Pan-ayancheri-Coleppa A DISSERTATION Submitted to h’lichigan State University in partial fulfillment. of the I'cquirmnvms for the degree of DOCTOR. OF PHILOSOPHY Physics and Astronomy 2009 ABSTRACT DECONSTRUCTED HIGGSLESS MODELS OF ELECTROWEAK SYMMETRY BREAKING By Baradhwa j Panayancheri-Coleppa \Ve study deconstructed Higgsless models of electroweak symmetry breaking. As the name implies. these models break electroweak symmetry without the presence of a scalar Higgs boson in the spectrum. These models are inspired by compaetified extra dimensional models, where the ll’Lll’L scattering amplitude is unitarized by a tower of new. heavy vector bosons in place of the l-liggs. We study a simplified theory with only one set of extra vector bosons and derive the wavefunctions and couplings in this theory. we then extend this model to include a “top-Higgs“ link, so as to separate the top quark mass generzittion from the rest. of electroweak symmetry breaking. which still goes through via a Higgsless mechanism. This enables us to have new, heavy Dirac fermions that are light enough to be (.lisccwered at the LHC. we present. the phenomenoh)gy of these heavy fermions. showing that. they are discoverable at. the 50 level at the LHC for a wide range of masses. Finally. we move on to consider the question of unitarity and the heavy Dirac fermion mass generat ion by investigating the printess H- -—+ W; W"; in a family of deconstructed Higgsless models. and show how the Appehmist-Clianowit7. bound can be substantially weakened for sufficiently light Dirac fermions. DEDICATION I dedicate this thesis to my late grandmother, Ammani l’aatti. iii TABLE OF CONTENTS List of Tables ................................. List of Figures ................................ Introduction . 1.1 Quantum Electrodynzmtics ........................ 1.2 The Standard Model ........................... 1.2.1 Remarks on the Higgs sector ................... 1.3 A Higgsless Standard Model ....................... 1.4 Extra. dimensional theories ........................ 1.4.1 Scalar field in the bulk ...................... 1.4.2 Gauge theory in the bulk ..................... 1.4.3 Fermions in 5-D .......................... 1.5 Deconstruction .............................. 1.6 Deconstructed Higgsless Model ..................... 1.6.1 Gauge sector ........................... 1.6.2 Fern'iion sector .......................... A Three Site Higgsless Model 2.1 A minimal model ............................. 2.2 Masses and Eigenstat es .......................... 2.2.1 Gauge bosons ........................... 2.2.2 Fertilions .............................. 2.3 Couplings ................................. 2.3.1 Ideal fermion (‘lelocalization ................... 2.3.2 Charged currents ......................... 2.3.3 Neutral Currents ......................... 2.4 l’lienomenological bounds ........................ 2.4.1 gzufur and Illurl ......................... 2.4.2 Ap and MI) ............................ 2.5 Remarks .................................. Triangle Moose Model . 3.1 The Model ................................. 3.2 h-lz'tsses and Eigenstates .......................... 3.2.1 Charged Gauge Bosons ...................... 3.2.2 Neutral gauge bosons ....................... 3.3 Fermions and Ideal (Llelocalization .................... vi 49 50 52 (i1 63 (ix-1 (i7 67 (it?) ‘71 3.6 3.7 3.8 3.3.1 Masses and wave functions .................... 71 3.3.2 Ideal fermion delocalization ................... 72 Light Fermion couplings to the gauge bosons .............. 7, 3.4.1 Charged Currents ......................... 7-1 3.4.2 Neutral Currents ......................... 7'6 The Top quark .............................. 78 3.5.1 h-Iasses and wave functions .................... 78 3.5.2 Z bf) and choice of EU ....................... 80 3.5.3 Ap and M D ............................ 80 Heavy fermion phenomenology at. hadron colliders .......... 81 3.6.1 Heavy fermion decay ....................... 82 3.6.2 Heavy quarks at the LHC .................... 84 Pair production: pp —> QC) —> 11"qu ——> llfujj ......... 84 Single production: pp ——+ Qq ——> W’qq' —v I-‘l'qu’ ........ 89 Related Vector Quark Models ...................... 95 Remarks .................................. 99 Unitarity and Bounds on the Scale of Fermion lVIass Generation . 101 4.1 4.2 4.3 4.4 The Appelquist-Chanowitz Bound .................... 104 The n( +2) Site Decrmstructed Higgsless Model ............. 109 4.2.1 Gauge Boson Sector ....................... 109 4.2.2 Fermion Sector .......................... 114 4.2.3 Goldstone Boson Sector ..................... 1.18 4.2.4 Couplings ............................. 119 Unitarity Bounds on If -—> WLH’L .................... 120 Summary ................................. 126 Bibliography..........................l27 V LIST OF TABLES 3.1 The complete set of cuts employed to enhance the signal to backgrtmnd ratio in the process M) —> QQ —> Wqu —+ ff/I/jj. ........... 87 3.2 The complete set of cuts employed to enhance the signal to background ratio in the process pp —:~ Qq ——> W'q'q —, H'Zq’q —‘- [Ill/jj ....... 93 vi 1.1 1.2 1.3 1.4 1.5 1.6 LIST OF FIGURES Images in this dissertation are presented in color The potential for the Higgs field takes a. “l\~lexica.n hat" form. The "trough” corresponds to continuous directions in which one can move expending zero energy - these correspond to the Goldstone boson modes. *1 The Feynman diagrams for the longitudinal gauge boson scattering in the Standard Model. The E4 contributions cancel due to gauge invariance. The ['32 contributions only vanish when we include the Higgs exchange. diagrams. . . . . . . . . . . . . . . . . . . . . . . . . It) The Standard Model without. the scalar Higgs boson. The result is an Stft2iL x SU(21R non-linear sigma model with the 81.7(2) x (3(1) part. gauged. 11 The Feynman diagrams for the. longitudinal gauge boson scattering in extra. dimensional model. Unitarity in this process is achieved by the exchange of the heavy vector boson instead of a Higgs. . . . . . . . . 19 A deconstructed picture of the extra dimension. The 4-D gauge groups are connected by means of min-linear sigma. model fields. [0 Cf! A deconstructed Higgsless model derived from a flat extra dimension. All the bulk gauge couplings are the same and so are the decay con- stants. The left and right handed fermions have Yukawa. couplings to the sigma fields and also have a. bulk Dirac mass term. . . . . . . . . 28 vii 2.1 2.3 3.1 3.3 The three site model analyzed in this paper. g0 and f} are the gauge couplings of the SU(2) groups, while the coupling of the U( 1) is rep— resented by g’. The left—handed fermions are denoted by the lower vertical lines (located at sites 0 and 1), and the right-handed fermions are denoted by the upper vertical lines (at sites 1 and 2). The dashed lines correspond to Yukawa couplings, as described in the text. As discussed below. we will take (201) = (212) = V/2 "v. One-loop contributions to Ap arise from vacuum 1;)olarizatitm diagrmns involving two left handed fermionie currents (left) and mixed left and right. handed currents (right). The RR piece is the same as the LL piece. The X and Y indicate the type of fermions in the loop. We compute the leading contribution in the limit 5 L ——-, 0 and g’ —+ 0. Phenoment)logically acceptable values of M D and 111‘,“ in CeV for (1T 2 2.5 x 10‘3 (solid curve) and 5 x 1()—3 (dashed curve). The region bounded by the lines 380 GeV < A IW’ < 1200 GeV and above the appropriate curve is allowed. For a given M D and MW” the value of 5,1? is determined by Eqn. (2.125). The gauge structure of the model in Moose notatitm [25]. g and g, are approximately the SM 8 U ( 2) and hypercharge gauge couplings while f) represents the “bulk” gauge coupling. The left (right) handed light. fermions are mostly localized at. site 0 (2) while their heavy counter- parts are mostly at site 1. The links connecting sites 0 and 1 and sites 1 and 2 are non linear sigma. model fields while the one cmtnecting sites 0 and 2 is the top Higgs field. The decay modes of the heavy quarks in the theory. The decay rate is controlled by the off-diagonal left handed coupling of the vector boson to a. heavy fermion and the corresponding light fermion ( the corresponding right handed coupling vanishes in the limit of massless ligl'it fermions). The plot of the branching ratio of the heavy quark into the charged and neutral gauge bosons. The masses of the W’ and Z ' gauge bosons were taken to be 500 GeV each. (a). Pair production of the heavy quarks occurs through {it} annihilat ion and gluon fusion. viii (i .1 ()4 82 83 3.0 3.8 3.10 3.11 3.12 The cross section for pair production (for one flavor) as a. function of the Dirac mass. As can be seen from the figure, for low values of :1 II), the cross section for the gluon fusion channel is higher than the quark aimihilation process. As M D increases, the quark annihilation process becomes equally important because the pdf of the gluon falls rapidly with increasing parton momentum fraction. .1:. The I] distribution of the outgoing hard jets for the process pp ——> QQ —> 1'1"qu —+ llIz/jj, corresponding to MD 2 700 GeV and MW’ : 500 GeV for a luminosity of 100 f (2‘1. One can see that the events are .in the central region: —2.5 < 1) < 2.5. The slight asymmet ry in the shape of the curve is because we add the distrilgmtions corresponding to the jets from both the Q and the (2 decays. Predicted signal invariant mass distributions AIII- for 11/ D = 300 GeV and M D = 700 GeV for a fixed MW’ 2 500 GeV. The small off peak events arise because we added the. distributions corresponding to the jets from both Q and Q decays. Contour plot of number of events in the pair production case for a fixed integrated luminosity of 100 f b‘l. The shaded region corresponds to MW; > 2.14 D and is non perturbative and is excluded from our analysis. as discussed in the beginning of this section. Feynman diagram for the t channel single productimi of the heavy fermion via the exchange of the. Z and the Z I bosons. Cross section for the t channel single production of the heavy fermion as a function of the Dirac mass 111 D' It is seen to fall more gradiuilly as compared to that. of the pair Infodiictirm case. The transverse mass distribution for the single production of a heavy quark in the model for M D 800 GeV and 1 TeV.fo1 a. fixul 1]” I = 500 Ge\ It is seen that. the signal falls sharph at 11I). Contour plot of the number of signal events for the single production channel for an integrated luminosity of 100 f b‘l. The shaded region is where MW’ > 2M D and is non perturbative. One can see there is a. considerable number of events in the low Allw, region of the parameter space. 86 90 91 92 91 3.13 3.14 4.1 4.2 4-3 4.4 The SM backgroimd for the single production channel, pp ———+ 11/ij —+ jjlz/ll. calculated by summing over the u. d. c, s and gluon jets and the first two families of leptons, and with the cuts in Table 3.4 imposed. The bin size is 20 GeV. Luminosity required for a 5 0 discovery of the heavy vector fermions at the LHC in the single (blue) and pair (red) production channels. The shaded portion is non perturbative and not included in the study. It is seen that the two channels are complementary to one another and allow almost the entire region to be covered in 300 fifl'. The diagrams that contribute to the process t+f+ —+ W211"; in the Higgsless SM. There are analogous diagrams for the process I.-t_ ——> 11’311’5. Each diagram has an amplitude that. grows linearly with \/§ for all energies. However, most (but not all) of this linear VG growth cancels when the diagrams are summed. The remaining piece that grows linearly with \/3 comes from the t channel diagram, and it eventually surpasses the unitarity bound. In the SM, this unitarity 90 97 violation is eliminated by the contribution of the Higgs in the s channel.105 The diagram that contributes linear growth in V3 to the process 15+ fir. ——+ 77+ 7r" in the Higgsless SM. where we have used the equivalence theo- rem to replace the longitudinally polarized gauge-boson by the corre- sponding “eaten" Goldstone Bosons. There is an analogous diagram for the process L]: ——+ n+7r- This diagram. corresponding to s-channel Z-boson exchange in the equivalcnce-theorem limit. does not contribute to the J z 0 1:)artial wave scattering amplitude for the process t+f+ ——> ”n+7” in the Hig- gsless SM. Moose [25] diagram of the n( +2) site model. Each solid (dashed) circle represents an SU(2) (U ( 1)) gauge. group. Each horizontal line is a. non-linear sigma model. Vertical lines are fermions, and diagonal lines represent Yukawa couplings. the process {+17 + —+ 117211"; in the 'n(+—2) site Higgslcss model. There are analogous diagrams for the process L7- —+ lit/+11"! . As in the SM. most of the linear growth in V: will cancel. All the persisting linear growth in \/s_. comes from the 1' channel diagrams. . V. 107 108 109 4.6 2‘“ «i Diagrams contributing to unitarity violation at high energies in the process t+t+ -> 7r+7r". There are analogous diagrams for the process 1---?" —+ 77+7r—. The top diagram grows linearly with x/z for all en- ergies, whereas the bottom diagrams only grow with \/._9 up to Alp/fl, after which they fall off as l/\/§. The scale where unitarity breaks down in the helicity nonconserving channel in the n(+‘2) site model. Unitarity is valid in the region below and to the left of a. given curve. The bottom-most. curve. is for n. = (l and is the AC- bound. The line directly above the bottom one is for n. = 1 and corresponds to the. Three Site Model. The line directly above that is for 'n = 2 and so on until 72 = 10. The line above that is for n =2 ‘20. the line to the right of that is for n = 30 and the line to the right of that is the continuum limit (71. —.» DC). We find that unitarity breaks down if either \/s is large or A I F1 is large. If .1 I F1 is large. then unitarity breaks down for \/5 very close to the AC bound. ()1) the other hand, if M171 5 4.5TcV, unitarity can be valid. in this process to very high energies. with the precise value depending on the number of sites 12. Expanded view of low \/E region of Figure 4.7. . xi 1‘21 1225 1‘34 Chapter 1 Introduction There are [our basic kinds of forces in the world: the strong nuclear force, the weak nu- clear force. electromagnetism, and gravity. These. are distinguished from one another based on their strengths and range. For example. the gravitational force between two ol_)je(;-.ts is proportional to the product of their masses and hence is relevant only when the objects involved are very massive. But gravity has an infinite range and hence plays the dtmiinant role in determining the large scale structure of the universe. On the other hand. in the sub-atomic world, the strong, weak and electrtnnagnetic forces dmninatxe. The quantum theory that explains the nature of these sub—atomic forces is called the Standard Model of Particle Physics (SM for short), while the nature of gravity is so far explained by the classical theory of General Relativity. The SM is a gauge theory and incorporates two different classes of particles - the matter content (fermions) and the force carriers (the gauge bosons). The force ex- terted 11)); one particle on another is transmitted via the gauge bosons. The range of the force is dictated by the mass of the gauge boson involved - for example, electro— magnetic forces that have infinite range are transmitted by massless photons while the short range weak nuclear force is transmitted by the heavy W and Z bosons. The bez-inty of the SM lies in the fact that it explains these two different kinds of forces in one unified framework. i.c.. as an clectro—wcz-ik theory [1].Of course. the strong nuclear forces are also built into the SM as a “color” gauge theory and the complete gauge group of the SM is SU(3)C X SU (2)”; x U(1)y, where the. subscripts C, ll", and Y stand for color. weak, and hypercharge respectively. In this thesis. we will be largely concentrating on the electro—weak sector, i.e.. the $1.112)”; ix t."(1)y part. To understand how the SM operates. let. us begin by describing the simpler theory of Quantum Electrodynamics (QED). 1.1 Quantum Electrodynamics Let us start by writing down the Dirac Lagrangian for a free electron of mass m: E = Eiyflapxr — 'IIIEQJ, (1.1) where \I/ is a. four component Dirac spinor. This Lagrangian is invariant: under a phase transformation: ‘11 -> exp (tell) \II, (1.2) where c is the electric charge. The parameter H is independent of space-time and correspt)I'irlingly, the transformation is termed "global”. l\loreover, since this is a one parameter group. the. above Lagrangian is said be invariant under global (2(1) transformatitins, where U tells us that this is a unitary group. But suppose we insist that the parameter H depend on sl')ace~t.inie. i.e.. H —+ 9(1) (i.e.. a local or gouge transformationi). Then, of course. Eon. (1.1) is no longer invariant. C —> Tier“ (9M1! —— mfiq! — (5(7),.071771“ \II. (1.3) 2 It is clear that. the theory of a free electron cannot be invariant, under local trnas— formations. If we demand that this theory still be invariant, we are forced to add another ingredient whose [7(1) transformation would cancel the extra piece in Eon. (1.3). From :\"Iaxwell"s classical electromagnetic theory. we know that the photon field transforms inhoinogenously under gauge transformations as follows: {1” ‘4’ 4” "" If)“0(.'.f). (1.4) This suggests that. we add the photon field. A”. in such a fashion as to cancel the extra term in Eqn. (1.3). Thus, we write down the interaction term: —- )—f\/ l t l" £11”; -— (.‘I’ ‘1/ W41”. (1.)) It. can be verified that adding this term to the Dirac Lagrangian makes it": invariant under local 1" (1) transformations. The (5(1) gauge symmetry allows a kinetic energy term for the photon that. takes the form: 1 my £= _1FIUIF , (1D) where F)“, = (9),.41, — fit/.4”. Note that a mass term for the photon of the form marl/1:1” is not allowed since this is not (3(1) invariant. Thus. the complete QED Lagrangian. is given by (restricting ourselves to terms of dimension 4): ._ It ~ 1 '1 ‘#1/ — — EQED 21(1),? Dliqj— ELM/[v —m.\IJ\I’, (14) where 1))[7 (9,1— ILA/1. (1.8) is the covariant derivative. Promoting the ordinary (‘lerixj'ative to a. covariant derivative in the form of Eqn. (1.8) to make the Lagrangian gauge im'ariant is called the "';\Iinirnal coupling” prescription. Thus. we see that the principle of local gauge invariance determines the structure of the Lagrangian and also naturally introduces a vector boson into the tl‘icory. Next, we turn to the full Standard Model. 1.2 The Standard Model The Standard Model Lagrangian can be constructed by extending the [)I‘i'lltfiplC‘S of the last section for the full group SU(3)C x .S'I.I'(2)W x U(1)y. The matter content of the SM (quarks and leptons) come in three families (or gem-nations). Both the quark and lepton families have electroweak interactitms and hence transform under the .S'1_."(2)W X I.="(1)y part of the SM gauge group. However, strong interactions (mediated by gluons) are only felt by the quarks. and thus only the quarks and gluons have SI} (3) charges (The non-Abelian nature of the gauge group permits self cotuiling of gluous, as opposed to electromagnetism). we give the quantum numbers of the quark and lepton fields under the SM gauge group below: Q = ”L ~(32+1.) 1"~(3’1—1) "~(‘3‘1+?) .L .,6.(1? .,3, UR ..3. (IL v 1 . LL: L ~(1.2,—§), (J‘R~(1,1,1). (1-9) ({L The L and R stand for left. and right. handed helicit y states. based on the Lorentz tra—msforniation properties of the fermion. The gauge interactions of the. quarks can now be written down by extending the covariant derivative in Eqn. (1.8) to include the eight. gluons (Cl/Ii)” the three weak gauge bosons (IV/ii). and the hypercharge 4 gauge l‘_)oson ( l)’ )1). [:EEWFQU” 3/1*‘igaégaif—fglojwif ""5038!“ QL + "5%?“ pay — magi-Cf? fig-”2191i “it +13%?” f8” -'i932\7;6'}f - igg‘zB/ti it Him” e),,t_.g9192‘—4Hff+ 'iérrznw 14L f i757” l3” - 'ir12Bul 6'??- (171”) Here. 93. 91 and 92 are the SU(3)C SU('2)W and U (1)y couplings respectively and the A‘A’s and (IA’s are the Cells-farm and Pauli matrices for the SII(3) and 51(2) gauge groups. As explained in the previous section. invariance under local gauge transforma- tions demands that the associated gauge bosons be massless. But we. know from their short range interactions that the weak gauge bosons do indeed have. a mass. Thus. we conclude that in order to have massive gauge bosons. the symmetry must somelum be ln‘oken. i.e.. the vacmun state must not respect the symmetries the Lagrangian does. This phenomenon. wherein the Lagrangian is symmetric under certain transfor- rmitions while the ground state is not. is called “Spontaneous Symmetry Breaking" (8813). In the SM. the breaking of the electroweak symmetry is engineered by in- trodtu-ing a scalar Iliggs field [2], which has the following quantum numbers under 311(2)”: and U(1)y: (pic a: ~(1.2.+%). (1.11) CI! The Lagrangian for the Higgs field can written as (restricting ourselves to terms of dimension four or less): £Higgs = éTr [IN‘HpTIJHcD] — é-ringrfiDl‘o) — $Tr(olo)2. (1.12) where the. covariant derivative is given by: [We = 0/" + sugar/i“ — 7%”? B“ d). (1.13) The lliggs also has \"11k2,iwa. couplings to the matter fields as follows: ['Yukawa == 6.1/11. em I? + @g/(f'iagd)* (IR + IygcfieR ”l“ [1.0. (1.14) where the y's are the ’ukawa couplings. The quark and lepton fields should be written with a. generatioru-rl index ((22) to accomodate the three families - we are sluipressing these indices here. The potential for the Higgs field in Eqn. (1.12) takes the form of a. “l\'1exica.n hat”, as shown in Figure 1.1. The minimum of the potential does not. lie at q) z: 0, but rather lies on a continuous 31/(2) manifold along the “trough” of the Mexican hat. One could do perturbation theory around any one of these minima. The choice of a particular vacuum breaks the gauge symmetry as it corresponds to choosing a particular direction in, the. SU(‘2) space, so the vacuum is no longer invariant under SUB) rotations. Vt’riting the Higgs doublet in a form that separates the Goldstone bosons (denoted 7r(.7:)) from the Higgs boson, 0 c+h(:r) v72 we) — Catt-swim Figure 1.1: The potential for the Higgs field takes a “Mexican hat” form. The “trough” corresponds to continuous directions in which one can move expending zero energy — these correspond to the Goldstone boson modes. we can write the vacuum expectation value (vev) of the Higgs as: 0 <0|¢10) = 1.» . (1.16) fl where v = (l—mi/A. The gauge interactions of the Higgs, Eqn. (1.12), now give rise to mass terms for the gauge bosons when we insert the vev 0f the Higgs. For the neutral gauge bosons, we find , 71gg¥v2 ‘i9192v2 3" E: 13,) H3 - (1-17) p. _ 1 2 1 2 2 will 191921) 2:92“ 3 We can diagonalize this matrix by unitary transformation using the matrix: c086 sine U = w w (1.18) —Sin 6w C0861”, where tan 0),: = (11/.‘12- we can identify the two mass eigenstates as: .4” = cos ()mBu + sin 071,113”. Z“ = — sin 9153):. + cos 6101173“. with masses mi 2 0. 1 f r) 111% = :4—( f + g§)v2 The charged gauge bosons also acquire a mass: where 1 . 1 wt = —— (W1 ; 2211/3) . #- \/§ 11 u (1.19) (1.20) (1.21) (1.22) We started with n'iassless gauge bosons and a complex Higgs field with four real scalar degrees of freedom. We see that three of the four degrees of freedom of the f'figgs have now become the longitudinal components of three gauge bosons. niz‘tking them massive. However. there is one scalar pl‘iysical degree of freedom that remains, which we identify as the Higgs boson. Wit-h 1118533 III/2' == 2Au2. (1.23) Expressing the gauge eigenstates in Eqn. (1.10) in terms of the mass eigenstates, we can write down the charged and neutral current interactions of the fermions. e .C _ = ————- (71‘ ’j/Hl'l'r—(I +7 “fill/lbw ) + hc. (1.24 CC «25111 0w L )1 L L 11. L > ) £NC = {713" - Q f sin2 011:)77"pr + (0 f7?” .4111} (135) Sill (In) COS 0-“? Here. Y1; = i1 / ‘2 is the third component of weak-isospin of the left—handed fermion IL (Is;f = t) for [3). and Qf r: 71; + Yf. The electric charge. c. is defined as: . (' (9 —. e = 92 sin 911: = 91 cos 6w = ———J—U;—— (1.20) / 2 2' .01 + 92 The Yukmva lIlllPI‘a('l-10’ns of the Higgs. Eqn. (1.14), now turn into mass terms for the fermions. and the Yukawa couplings are chosen so as to reproduce the correct fermion mass. Thus. we see. that the phenomenon of 3813 gives rise to mass terms for both the gauge and the fermionic sector of the SM. 1.2.1 Remarks on the Higgs sector The SM is a phenomeno11;)gically successful theory whose predictions have been borne out by various experiments. But. two facts still remain: the Higgs boson has not been found in collider experiments and, more importantly. the SM does not offer an explanation for why Electroweak Symmetry Breaking (EWSB) occurs in nature (the Higgs only engineer‘s EVVSB). These considerations motivate us to build models that go beytmd the SM. Before we move on to present one such alternative. let us remark on one more purpose the Higgs serves in the SM. All calculations within the SM are performed as perturbative expansions in the small coumings. \Vhen one performs a computation for the. cross-section of a particular process. it is ii’imortant to check that the l_)1'ol.)211l)ility that the process occurs is less than one, so the results make 9 L _ +. + _ 7+2 W W _- flag Figure 1.2: The Feynman diagrams for the ltiingitudinal gauge boson scattering in 1 L . . . . _ . ‘) the Standard Model. The. E 1 contributions cancel due to gauge 11'1var1ance. The 1?" contributions only vanish when we include. the Higgs exchange diagrams. physical sense. In other words. the theory has to be unitary. This is crucial because absence of unitarity indicates that the perturbation theory has broken down. and thus. self-(:1insistency of perturbation theory requires the probability be btmnded by 0110. As an example. let us look at the longitudinal gauge boson see-uterine; i.e., the process ll'L WL —+ H’LH'L in the SM - the reason for the. choice is that the longitudinal components are the ones acquired by the gauge bosons by eating the Goldstone fields. and hence are most. closely asscociated with the Higgs mechanism. The Feynman diagrams that. contribute to this process are shown in Figure 1.2. W hen we compute the amplitude for the entire process W], H"); -+ ”7].. W L, and look at the large energy behavior. i.e., in the limit E / ””W >> 1. we find that the pieces of the amplitude that grows like It"1 cancel between the contact interaction and the photon and Z exchange diagrams. (This is due to gauge invariance which guarantees the relation e = 92 sin 0m). However, for the pieces that grows like [52 to cancel, we have. to include the lliggs exchange diagrams. Thus. we find that the Higgs not only serves to give masses to the gauge. bosons and the fermions. but also serves to regulate the 10 SU(2) % U( 1) Figure 1.3: The Standard Model without. the scalar Higgs boson. The result is an .S'I..5(‘21) L X SIJ'(2)R non—linear sigma model with the Str’(‘2) X If ( 1) part gauged. had high energy behavior of the theory. If we are to formulate a theory that. goes beyond the SM and does not have a Higgs, we. have. to make sure the theory does not violate unitarity. 1.3 A Higgsless Standard Model Though we established in the last. section that the SM without. the Higgs would not. be unitary. it is still instructive to ask what the theory would look like if we do not include /l(.’1,') in the theory. Let. us start by reproducing Eqn. (1.15). 0 i'+h.(:r.) t/‘Z ' (I. (I. .‘ (:3:ch (7/11 The above form clearly separates out the three Goldstone bosons (7r”) that become the ltnigitudinal components of the gauge bosons from the 1;)hysical Higgs boson. When the Higgs boson is eliminated, what; remains is a non-linear .5'1..-"(‘2) L x 81 (2)1? sigma model of which the SI."(‘2) x U (1) part is gauged. We show this in pictorial notation in Figure 1.3. To derive the Lagrangian for this low energy cil'ective theory, we plug in Eqn. (1.15) in the Higgs Lagrangian, Eqn. (1.12). 1 . - 1 . .- l r i /\ , , p : Z(i,'—t~-Ii.)‘)Tr [(12,127) (0/12)]+'2‘(()/th)(duh1*37”}2,(1‘+h)2":fi(""+hl4- (1.28) 11 In the limit In}, ~4 00, we can read off the effective Lagrangian from the above equation by simply disregarding the Higgs field and it is given by: n2 £Goldstone = TTr [(DflET) (””2” - (1.29) and contains only the eaten pious. This picture is called a “non—linear sigma nuidel". The Goldstone boson equivalence theorem [3,, 4] tells us that at high energies. the amplitude for absorption or emission of longitudinal gauge bosons is the same as the one. for the corresponding eaten pion. We can determine the Feynman rules for 17 — 7r . . . . . brand/1' . q . scattering 111 this model by pluggmg in E : e . 1n Eqn. (1.2.1) and expanding in powers of TF/‘t’. At, tree level. there is only a contact, interaction term and the amplitude for this is given by: 1 - :6 E ‘2 ‘ . .ll(7r+7r_ —» 7r+7r") = g2 (lg—(1L) . (1.30) In”: where 9 is the scattering angle. It is hardly sm‘prising that the 1?“ growth does not cancel. as there is no physical Higgs boson in the spectrum. But the question we would like to address is whether it is possible to extend a theory of this kind by including additional. particles to retain unitarity. in place of a Higgs. How would one (i'tmstruct an electroweak symmetry breaking sector without. a scalar particle? A glimpse to an answer to this question is provided by QCD. Consider QCD with two flavors - the up and down quarks. Let. us. for the moment, assume that. these are. nn-issless — the u and d quarks are light compared to the QCD scale, AQCD‘: 300 :\IeV, and hence this is a good approximation. Then. the Lagrangian of QCD can be. written down (with \I/ z (u. (1)) as: 12 and is seen to possess a global 81," (2)1, X .91.."(2) R symmetry, the chiral symme— try. When the running QCD coupling constant. becomes large at. the scale of QCD (AQCD), the strong interactions bind quark anti—quark pairs into a composite spin—(l object: ((ll‘TJ t11(0) - this is analogous to the formation of Cooper pairs in the theory of superconductivity. This, like the Higgs in the SM, develops a. vacuum expec- tation value (\TNII) :2 A2201), thus spontaneously breaking the 8L" (2) L x .S'I'.f('2)R chiral synnnetry down to the diagonal subgroup, SI/'(2)V. Each ferrnimi lield has a mass dimension 3/ 2, and thus the condensate has a mass dimension 3. Typically, spontaneously breaking a continuous symmetry generates massless Goldstone bosons. But. the three QCD pious will not; be Inassless, as we started with an approximate symmetry (i.e., valid only in the limit mud. —> 0). For this reason, the QCD pi- ous are really pseudo Goldtone bosons. Now, if we were to describe EW’SB using this picture, we would let these three pious be eaten by the W: and the Z , thus making the. gauge bosons massive. Unfortunately, the scale characterizing the gauge boson masses would be wrong - the pion decay constant that. sets the scale in this model is ffl = 93h’1eV. but we know that the electroweak scale that sets the. scale of the llv' and Z bosons is v 2246 GeV. Thus, QCD, though successful in achiexdng the correct. symmetry breaking pattern, cannot reproduce the correct gauge boson masses. Hmvever, one could construct a. “scaled up” version of QCD, called Techni- color [5, 6. 7], wherein technicolor interactions (assumed to be confining. like QCD) bind teclnii—(piark techni—antiquark pairs into a (‘i’TC‘PTC> condeusate. The scale of technicolor interactions (i.e., the scale at which technicolor interz-tctions become strong and form comlensates) can be tuned to reproduce the correct gauge boson masses. To get fermion masses, this picture has to be extended. and the resulting theory, called "Extended Technicolor” (ETC) is described in [T 8. 9, 10, ll]. Theories like the one described above are stronrrly interactin r. and. thus, cannot be h . treated as perturbative quantum field theories. One has to develop lat tice calcula-rtirms 13 and other non-permirbative tools in order to be able to compute in such theories. Ht’m-‘ever. recently. there has emerged a special[correspondence that relates strongly interacting four dimensional theories to weakly interacting five dimensional ones - the AdS—CF'I‘ correspontlence [19, 2t). 21. 22] first. arose in the context of string themies describing the duality between type IIB string theory and classical supergravit y. Later works have established that such a duality exists more generally and that many strttmgly interacting theories have a dual description in an extra dimensiom-il context. Higgsless models in an extra. dimension have thus emerged as viable theories of EV‘VSB that. are the analogues of technicolor theories. We turn our attention to these extra (lirncnsimial models in the next sections. 1.4 Extra dimensional theories It is possible that our universe may have dimensions other than the customary :1- D spa-ice—time [12. 13, 14. 15. 16. 17. 18]. These extra dimensions must have to be compact... so we dont realize their existence in everyday life. This compact fifth dimension can be thought of as an interval. without loss of generality. For example. if the extra dimension is a circle of radius R. one could map it onto an interval [(1.27r1{[ with periodic boundary (.;on(_lit.ions. li' a rcllcction symmetry (a 22 symmetry) is imposed on top of the circle, we could map it onto the interval [0. 7TH[. The fields can have odd or even t.ransformation properties under the Z2 symmetry and the extra dimensitm is said to be "(i-oinpactified” on an interval. A circle is a 1-D surface. 51 - the process of imposing the Z2 symmetry on top of this surface is called “()rbifolding“. in particular. 51/23 orbifold. As we will see below. this picture naturally offers a rich spectrum of new. heavy particles that would be (.)l")serva.ble at energies greater than the inverse t‘OII’lpElCtil‘lt?£ltlt)ll radius. i.e.. E > 1/H. where It is the radius or size of the extra dimmsion. Let us first try to understand the features of such a theory by 14 having the extra dimension populated by a scalar field. for simplicity. 1.4.1 Scalar field in the bulk Consider a rnassless ('oinplex smlzu‘ field living in 5-D. The action for Sl'lt'll a theory is given by: S : [(15I£(:E. z), (1.32) where is the fifth ('lilnension co—ordinate and the Lagrangian is given by: We will let 5; run from 0 to 27rH, with the points 2 == 0 and z = ‘27? H. identified. This n'ieans that. we are cori‘ipnct.ifyi11g the extra dimension on a (*irt'le. and thus. we can expand (l) as a Fourier series in the following way: +30 _ on: 7:) = Z oln‘)(1~)e(W/m. (1.34) nz—oo In this form, it is clear that (Mi, 0) : (I)(i’,27rR). Plugging hack the solution. Eqn. (1.34) in the action, Eqn. (1.32), and integrating over 2. we get: +00 E = Z (i9aq)(n))l(aa<1>("))— 777%(;;1“~)“ ., (1.37) where “j; ,N = 8A,,Ajt — 6N4}, + Who/431 Aft. (1.38) and 95 is the five dimensional gauge. coupling and the second term in Eqn. (1.37) is the gauge. fixing term. The form of the gauge fixing term is so chosen that it cancels . . . , ¢ 3 . . the nnxnig between the. gauge and Goldstone helds, (9;.4fidlylfi0, that arises from the PL"; Fm”. The variation of the action Eqn. (1.37) leads to the equations of motion: (7A1 I'wQAIU _ ./.(lb(.,"b."11!//‘E\II + Edi/(104;; __ ()I/£)Z1g : 0‘ (1.39) a". .312 — f”’"‘F”zA"” + 8;, (in/1”" - {8:211}; = t). (1.41)) The requirement. that. the boundary piece \A'anishes leads to the condition: ,a , (11/ r (m -: a - (1 NR __ '- [13,251 + ((10.4 — 5193.45) 0.45]0 o. (1.41) The behavior of the fields at the boundaries of the extra dimension (the boundary 16 conditions) now have to be chosen. There are three choices that respect 4-D Lorentz invariance: r1?! = 0, A” = const.. (1.42) .4?) = 0, 83.43 = 0, (1.43) F/(iz = 0. 12 = const.. (1.44) The choice of the boundary condition determines the pattern of symmetry breaking. Once we choose a particular breaking pattern. we can expand the gauge fields in Ix'K modes like in the last section: .1;1(.i", 3, = Zfr1(3)/1iip($) It ago. 2) = Z,0n(:)7r;',j(r). (1.453) 7?. Thus. we see that the 5-D gauge. field is decomposed into a vector and a 5—D scalar. In a r talistic model. one lets the bulk gauge group be 911(2) and the. boundary conditions have to be chosen so that we have a zero mode that represents the W and the Z bosons. plus a tower of vector bosons. The Higgs mechanism still operates: the .15 fields become the longitudinal components of these KK vector bosons. making them massive. Thus. the spectrum consists of the SM particles and their heavy copies. Let us briefly discuss a toy model with these f‘atures. 'We will let the bulk gauge group be SU( 2) and let the following boundary ctmditions break 51(2) down to U(1) at. one end of the interval: At :5 = ()2 (“)Aj’, = 0. (1.41;) .43 = t). (1.47) 17 and at = 7tl1’: A}? = 0, 05/43 = 0 (1.48) 83.11/32 =0, .43 =0. (1.49) We will work in the unitary gauge. Ag 2 t). The KK expansions are: .(tiflf. z) = Z fn(z)1t".,.,ifl(.r) (1.50) 1?. {12(f,z) = Zr].,7(:)Zn/,(:r). (1.51) n. The eigeiil'unctions [(2) and g(:) are combinations of sines and cosines. Using the BCs. we can derive the following mass equations: cos (iii/$7111) = 0, (1.52) sin (it/2715’) = 0 (1.53) where 11/)? and 111,3: are. the masses of the neutral and charged gauge boson towers respectively. The solutions are given by: _ . — 1/2 11,:r = "—17; n, = 1.2, (1.54) .112 = 3% n, = 0. 1.2. (1.55) (1.51;) we see that. the lowest mode of the charged tower is a massive particle, which we can identify with the SM W boson. The n. = 0 state of the. neutral tower corresponds to the massless photon. and the n 2 1 state can be identified with the Z boson. ()1. course. the precise W and Z mass relation does not come out - but our purpose here 18 Wt WL = + + 7+2 1+2 “’1 "’1 + + M Zn Z n Figure 1.4: The Feynman diagrams for the ltmgitudinal gauge boson scattering in extra dimensional model. Unitarity in this process is achieved by the exchange of the heavy vector boson instead of a Higgs. is to develop a toy model that. has the essential features of extra dimensional theories. Thus. we see that even in a toy model. one could choose the trioundary conditions appropriately to get. a. rich particle spectrum in which the lowest. KK modes can be identified with the SM particles. while the higher modes represent the KK resomuices. These KK resonances serve an in'iportant purpose. As explained in the beginning of the section. we have to ensure that the WI -— WL scattering amplitude is unitary. and the unitarization is carried out by the exchange of these hmvy vector bosons in place of a Higgs in these theories. The Feynman diagrams for llll'Ll't’L —.~ 11111714 is shown in Figure 1.4. The amplitut‘le can be written in a. generic form: "4 _. 52 If... .1. [1(21L A = AW _. . 11;; 11,-; 19 1 . '. ‘) . . The expressions for All) and 21f”) can be derived to be: “111) = 1: (1127177171 __ Z ggnk [fabt’f(‘(1k(3 _+_ GCOS H __ €052 6) .+ 2(3 _ (.052 H)j'(l.(l(’f()d(f k (1.58) l 11/2 ‘2' ”II p j . . ' . . _ ,: .} 3’ . - H .. . . . 11(2) _. __ 4.012117771111115 __ 3 Z “(linkj‘lkz jute] )d( _ sm2 __ l‘dbf’j ((18 . (1.5”) k If the masses and couplings of the KK modes satisfy the. following two sum rules: ”grin-n : 2.072111]; (1110) k 2 .2 3 2 4. 2 . 97111717111111 = I Zgnnk‘l'lk‘ (1.b1) k . . . m, 4) .. . the pleces of the amplitude that grow as [31 and 19“ are cancelled. (In the expressions for the. amplitudes above. the first sum rule has already been used to simplify the form of 11(2)). Thus. we see that it. is possible to maintain unitarity the exchange of new heavy vector states in a model with no physical scalar particle. 1.4.3 Fermions in 5-D Now that we have seen a toy model to generate gauge boson masses in an extra dimei’isional theory. we will now investigate the problem of fermions living in an extra dimension. A 5—D Dirac spinor deconmoses under the 4-D Lorentz subgroup into two tee-compel1m1t spinors: 11: = . (1.62) In 5-D, the Dirac matrices read: _ U n” 5 _ 1 U . , F”: , I" =2 (1.623) a“ () 0 1. where the as are the usual Pauli matrices. Now let us inmost-2 the ZQ orbifold pro— jection. : --> ~:. In order to leave the 5—D Dirac etpiation invariant... ‘11 has to satisfy: - 3' \ll(—z) = —’2F’\ll(:), (1.64) i.e.. “”3“” and 'e”(-r)=—dr(:)- (1.0.5) This suggests that. X and. I...” can be written as: .x» n: .. . \(2:. 2) = 2 cos (7) \((")(2‘), (1.06) 1120 x n. 1,9(1. 2;) = 2 sin (n7?) '9';'(”)(.r) r (1.67) 7120 Thus. we see that only ,\ has a le’. zero mode. "7‘. \r' 10W .v 'etovr 1's ”s . s'nv‘ : 1e " ineva” 2 roar: . ie ._)- 23.1 “c \1111 tr” 1( (Itll 1( ult 111 Otl ‘nt 1 la)1 h Tl L'D l(l1)Il for \11 reads: H : /(‘f5.1' [% (\llll‘luiiliu‘li — ()Mxi; 1‘1” \11) _ milky] . (1.08) where the last term is a “bulk mass”. The above action can be recast in 4—D compo- IlCIllS 'clSI ' .— _ ea +——+_ _ S = / (1%: [am/‘0,“ — tern/‘10,, a") + (W); t —— {4% c) + m (m t W0] . (1159) 21 where ('9; = 2(83 — 32). The variatnm of thls action leads to the equations of mot ion: — tar/WM — 6315+ ‘m'l; = 0 (1.70) — iaf‘aflg—f + 82x + my = t). (1.71) Requiring that. the variation of the action at the boundary vanishes gives the condi- tion: —\(6u) + link + (5);:- —- for?) = 0. (1.72) We also have to impose a boundary eondition for \11 in the form f()(. 13') == 0 at. the two bmmdaries of the interval, and this. along with the equations of motion, will fix all the arbitrary coefficients in the complete solution to the spinor equation. For instance. we can require that the spinor U" vanishes on both boundaries. This would lead to: ((72 + 7!?)Ylo‘wR = 0. (.173) Solving the er nations of motion with these boundary conditions would result in a e) l . zero mode for [\3 but not 2;). As in the ease of gauge fields. we can expand the spinm‘s in KK modes. Performing this KK deconmosition gives us: x = Zantznntr) (1.7.1) 77. = Z .rntziwne). (1.75) n, The fermions obey the Dirac equation: — 7'6/"(‘3;1..\’(1") + mntfln) = 0 (L76) _ .mflawfifnl + mnxtl'll = ()_ (1.77) 22 Substituting the KK (ilecomposition into these equations gives: (1;; + mfin - "Inf" = 0 (1.78) ff: “ mfn + 777-11971. = 0- (1.71)) \Ve can combine these two coupled first order equations to form two uncoupled second order equations. ” 2 2 ,_ 9n + (In; ‘ m )917 = 0 (1.8.0) [I .' In + (711% "N12101:” (1.81) The solutions are simply sums of sines and cosines, whose coefficients are (.letermined by reimposing the first order equations and the boundary conditions. For instz-mce. .if we impose 1.1"? = 0 at both .: = t) and z = u R. we. obtain: fn(z) = an sm —.—, (1.82) 11’ a ,7 n n .: , m: ‘ 917(2) = —— (— cos —- — msm —) , (1.8.3 mn If R, H ) where, 'm-n —. (1.84) and the coefficient an is fixed by the normalizatimi condition '7]? 0 / dzf7';(..) —1 (183) . (’1 The boundary conditions also allow for a zero mode for x: 91)) ~ ——mz . ' : = —— e . 1.150 got > \/1 __H.,m,.1. t > 23 We see that the 5-D mass does not contribute to the mass of the lightest fermion (it. stays n'iassless). This is important - in a realistic theory, we should have. all the SM fermions nearly massless (except the top—quark). N ow that we have discussed gauge theories and fermions in extra dimensions. let. us move on to see if we can develop an understanding of the features of these theories based purely on physics in 4-D. 1 .5 Deconstruction As we have seen. compactified extra dimensions naturally have an associated. le' tower. We would like to see if we can write down a. simple gauge invariant. Lagrangian to describe these KK modes without the full machinery of the extra dimension. This is done using the idea of “Deconstruction” (23, ‘24] -— a manifestly 4-D (.lescription of 5-D physics. Consider a gauge theory living in a slice of extra (jlimension. Now. if we imagine slicing up the. extra dimension into an infinite munber of segments. each plane is described by a 4-D gauge. theory. So a 5-D gauge theory can be thrmght of as in infinite collection. of 4-D gauge groups. Let us sum'iosc the gauge group is 813(2). \-\"e have to let the gauge symmetries be broken so the KK resonances become massive. We can do this by having a Higgs field. (NI). that. transforms between two adjacent gauge groups 311(2) groups as (2. '2). When the Higgs field (.levclops a diagonal vev. c 0 i.e., ((1)) = , it breaks SI.."(2) >< SU(2) down to the diagonal sub group. and t) 1.! thus we will have 1‘1‘1assive W and Z bosons plus their KK pz'u‘tners. But since our only goal is to break the gauge symmetry, we can do away with the scalar Higgs (‘legrer-i of freedom and employ a. non-linear sigma model, in the spirit of the Higgsless Standard Model. Thus. the picture that emerges is one that is exactly like in Figure (1.3). 24 Figure 1.5: A deconstructed picture of the extra dimension. The 4—D gauge groups are connected by means of non-linear sigma model fields. albeit with replicated gauge groups. This picture is called a “;\'Ioose" or “Quiver“ diagram [‘25]. We sl‘iow a general deconstructed model in F igure 1.5. The non—linear sigma fields transform as fundamentals and anti-fumlamentals un- der the adjacent gauge groups. Zirt+1 —+ L.7z-712i..,-+1Uj +1. The action for this theory can be written down by simply extending Eqn. (1.29) to include multiple sigma fields and gauge groups: ' 1 “3 1 . . ' 4 “3 f2 9 k_ ’,. . firu;e ,, ..;L , s_' “ -7 s _ /.z izjginp I 1,”) +/d .12: 4 T1 Iran“) (125.) ' 2'21 4 2:1 where the covariant derivative is given by: “113mm = ”#3144 - ”Magma +"':'l‘i+1;1£i.~i+l' (L88) To check that this picture is indeed right, let. us look at the continuum limit, i.e.. in the limit N ——> 00. In this limit, we should recover the full 5-D gauge theory. To do this, let us start by relabeling the couplings and the decay constants. g.) = V 1V + 2 H7: (1.89) fz- = x/N + 1 hi- (1.90) with the constraints N+1 . N+1 l. l 1 s 1 .___ = . . — = 1. 1.S)1 fV + 2 _ 52 1V +1 2‘ I}? I ( ) 1:0 7 1:0 I which come from 1 1V+l 1 (,2 Z (1.2 l l ‘ F20 '2 Air-+1 1 1 . . F = Z -—2 (1.9.3) i=0 1?. Let us now define the dimensionless coordinate: ~. = , (1.91) ~i '_’ . 1 AZ: —— 12.. N+1—H ' 1 N+1 iv +1 :0 d/(iz. 1;: A/jhr) —’ le-(J'. z). M (rm-1) [ii/.’_+_51(r-) — Aide] —, ( ”4 J J ('9: 51,1“ —’ r;(:)./z(:), N+1 . . 1 l l — —> 12— =1. N ‘ Z 2 /( {2 5 +2 {:0 ml . I (~) N—l—l 1 . 1 1 —~ -—‘> (13%— Z l. N +1 7;” I)? / [13(3) Using the above rel)lacemcnts and working in the unitary gauge ()3; = l ), the action, (Eqn. (1.87:). becomes: 9 /15 1 T (r rim) + f2h'2(:)T (r r112) (10') r ya )' i Z ( IL‘ —_.-—.——— I' ‘ / 4 ' —— 1’ ,‘ -- ' ' . A .u) gauge . 292H2(:) in 4 [n . 26 where F”; = 82.4“. (1.91)) since in the unitary gauge. .4;- = 0. The above equation represents the action for a 5- D gaugc in a “xx-'arped" or “curved" background. i.c., a geometry that. is not flat“ (Note that if we had started with all the gi’s and ffs the same. we would have ended up with the action for a gauge theory in a flat extra dimension, Eqn. (l.37)). Relabcling g ——> gr; and f ——> f h. we see that this is reflected in the fact that the gauge couplings and f constants depend on the extra dimensional co—ordinate 2. Thus, the process of deconstruction allows us to recover the complete 5-D theory in the continuum limit. 1.6 Deconstructed Higgsless Model \vVe. will develop a dcconstructed Higgsless model [26, 27, 28 ‘29] derived from a flat extra dimension. i.e.. we will choose all the bulk gauge crmplings and the. decay constants, f, to be the same through the moose. When we integrate out all the heavy le' states, we. should be left with an SL/'(‘2) >< U(1) Higgsless standard model. We will thus choose the coupling at the first site to be 9. which is almost the same as the SM 811(2) coupling and. at the last. site, we will gauge the U ('1) part. of the 812(2) and give it. a SM-like hypercharge coupling 9' - the couplings of the first and last site gauge groups being different from the rest of the moose is indicative of the boundary ctmditions imposed at the two ends of the. extra dimension in the underlying 5-D .1, - ,. .. - N—lwv: .. ‘-., theory. Thus. the gauge group of this IV site moose 1s “1.20 51 (2),- x I (1) (P igure 1.6). The sigma fields connecting two adjacent gauge groups transform, as before. as $21241 —~.» I F2712 m- +1132- +1. \Vhen the sigma fields develop a vev (f) and break every adjacent. Sl.-'(‘2) x SU ( 2) groups down to the diagonal SU("2). the lowest lying modes (which we identify as the SM H" and Z bosons) and the le’ tower get masses. 27 Figure 1.6: A deconstructed Higgsless model deriver'l from a flat extra (limensitm. All the bulk gauge couplings are the same and so are the decay constants. The left. and right handed fermions have Yukawa couplings to the sigma fields and also hz-rve a bulk Dirac mass term. 1.6.1 Gauge sector The gauge sector Lagrangian reads: 1 ,am/ viaw/ 1 ~p1/ . f2 "1 2 -. 15-32117. u: -113 Emma—Z‘npzz-Hll . (1.9;) i i where the covariant derivative is given by: . _, , .- 7 1.- , 7,711-1 . ”Him“ = ()pEm'H +1.9”;1L'i,i+l - ’QLLHIHH (1-98) and f; is the gauge. coupling of the internal (or the bulk) SU ( 2) groups. The mass terms for the gauge bosons can be derived from the last. term of Eqn. (1.97) by working in the unitary gauge (2 = l). The charged gauge boson mass matrix is given by: (2:3 —:c 0 0 0 0) 2 2 —.7: 2 —1 0 (l (l ‘ {,7 ' . - .1157, = —4j— o —.r 2 —1 0 0 (1.99) \0 0 0 0 —1 2} 28 where :r = 9/3) is a small parameter. The matrix can be diagonalized pertiurln-uivcly in .7: to yield the mass eigenvalues of the standard W boson and the l\' K resonances. ‘2 '2 r r . (I / N(21~ 'l‘ 1)l ‘_) J 11/2,; = —,——— —— —— ‘ + - (1.101) ” Inw+1)l 6(N +1) ) 112 —— = "fl' 2 2112 - ' ’1” 2 1 101 " ”'I H .9 2.2f {3111 m + 1 ll" L015 m . ( . - ) In the continuum limit. (N —-+ 00), we see that the second term inside the parenthesis goes to zero. Thus. to recover the correct formula for the mass of the W gauge boson. f should scale like 1.1/\/ N +1. Similarly. the neutral gauge. boson mass matrix is given Irv: (x3 —w t) t) () -~ 0) —r 2 -4. 0 0 --- 0 ~22 0 —$ 2 —1 e --- 0 .2 III . AI = run Z 4 I I I I I ( ) 0 0 0 0 -~ 2 y (t) 0 0 0 --- —y y?) where y = {1’ / f]. The light and heavy eigenvalues are given by: 2 ’2 2 T t’ .7 I I 2 2 .2 o +9)! -NQW+4) 2 :2 Amy . A/ =-—~—¢—s—-1-—7e————»-+~ +—~w—-».~ 11am Z 4(.»\ + 1) [ em +1) ( y ) (3:2 +7172 ( 2 2 [271' 2 2 ”71' 2 ’llZ,‘ —- g 2f (sin m) + ZAIZ (COS m) . (1.104) 1 .6.2 Fermion sector To construct a realistic theory. we have to put. in the fermions. In Figure (1.6). we show left (right) handed fermions as top (bottom) lines attached to each gauge 29 group. How do we, write down mass terms for these fermions in the (jleconstriieted language? In the spirit of Eqn.(l.68), we can write down a "bulk” mass term of the form M it). But the gauge symmetries also allow a term that couples fermions in adjacent sites - this “hopping” term ties left and right. handed fermions through a Yukawa coupling to the sigma field and takes the. form 'QEL.,jZ.i.,Z-+112R.”1. When the. sigma field develops a vev. this becomes a mass term for the fermions. It. was shown in [26. 27] that a Higgsless model with the light SM fermions localized on the branes (in the deconstructed picture, deriving the 311(2) charge only from the first. site.) does not satisfy precision electrowealt measurements. Thus, one has to allow the fermions to “(,lelocalize". i.e.. derive their SU(2) charge from more than one. site. In the 5—D picture. this corretmnds to the wavefunction 0f the fermion “leaking" into the hulk of the extra dimensitm. We write. down the fermion Lagrangian below: ['fermicms : A‘IDEL (3510201 ’f’Rl + ‘MD 2 (fl/”'Ll'i'i’Ri (L105) 'i .. 3 u r) + 11/1) “(’LNENJ’Va—l ”R R“ + Ire. Ede dRQ To get. fermion flavor mixing. we could add generatimial indices to all the fermion fields. and choosing EL and M D to be generation-(1iagonal, embed all the min-trivial flavor structure in the Yukawa matrix in the last term of Eqn. (1.106). Here, 5L and sz an be understood as the degree of delocalization of the left and right-handed fermions respectively. We will show how to determine the values of these parameters in the context of a model with only one extra 813(2) group (a “three site model”) in the next chapter. Diagonalizing the fermion mass matrix. we obtain the light and heavy eigenvalues: l _ ,7 72 _ - g2 A. 73 __ 7 72 yr 7 :11 'f '— , 9 _ Ngz IE:- v R - R 48 1+ fif— H + (1.106) 7711.72.71“). (1.107) Thus. a simple (ileconstructed model can be constructed without. a physical Higgs boson in the spectrum. We have. only sketched the general outline of such a theory in this section. We will construct simpler models in the next chapters and investigate their pheuommxology in detail. This thesis is organized as follows: in the next. chapter. we will construct a simple model with only one site in the “bulk” ( a three. site model). In Chapter 3. we will look at. a simple extension of the three site model by appealing to the idea of “top-color” that will enable us to have KK fermions in the spectrum that are light enough to be discrwered at the LHC - we will investigate the phenomenology of these heavy fermions in detail and show that they are discoverable at the LHC for a. wide range of masses. In Chapter 4. we will address the issue of unitarity in the process If ——-> W; W; in a family of deconstructed Higgsless models and show how the Appelquist-Chanowit7. bound can be. substantially weakened for a proper choice of the heavy fermion mass. Finally, in Chapter 5. we offer our conclusions. Chapter 2 A Three Site Higgsless Model 2.1 A minimal model H iggsless models, as we have seen. break the elect-roweak symmetry without requiring a fundamental scalar in. the spectrum and the W L WI. scattering amplitude in these theories is unitari'zed by a tower of heavy gauge bosons. analagous to the SM W and Z bosons. Typically. these gauge l.)()S(_)IlS get. progressively heavier and one can only see a. few of the lowest lying resonances at the CERN LHC. Thus, it is phenomenologically useful to have an effective theory that retains only a few of the extra gauge bosons and yet captures all the phenomenologically interesting features of Higgsless models. In this chapter, we will present the Three Site Model [30], the simplest possible example of (let-(mstructed Higgsless models of the kind described in the introduction. This chapter is based on work published in [30]. The model has the same color group as in the Standard ;\‘Iodel and an extended 3151(2) x SIM?) X (7(1) electroweak gauge group. Accordingly, there is one set of extra H" and Z ’ bosons that. are heavy compared to their S;\l counterparts. This theory is in the same class as models of extended electroweak gauge symmetries [31, 3‘2] motivated by i‘nodels of hidden local symmetry [33, 34. 35, 36, 37]. “’0 will 32 Figure 2.1: The three site model analyzed in this paper. go and g are the gauge couplings of the S U(2) groups, while the coupling of the U (1) is represented by g]. The left-handed fermions are denoted by the lower vertical lines (located at sites 0 and 1), and the right—handed fermions are denoted by the upper vertical lines (at sites 1 and 2). The dashed lines correspond to Yukawa couplings, as described in the text. As discussed below, we will take. (201) = (212) = $2 11. also introduce a heavy fermionic partner for every SM fermion and these, along with the heavy gauge bosons, complete the extra particles in the spectrum. The scale of unitarity violation in the W LW L scattering amplitude is delayed by the exchange of the H”, as opposed to a tower of gauge bosons [43, 44. 45, 46. 47]. In Figure 2.1, we illustrate the model using “Moose notation” [25]. The. model incorporates an SU(2) X SU(2) x U (1) gauge group, and 2 nonlin- ear (SU(2) x SU(2))/SU(‘2) sigma models in which the global symmetry groups in adjacent sigma models are identified with the corresponding factors of the gauge group. The symmetry breaking between the middle S U(‘2) and the U(1) follows an SU(2)L x SU(2)R/SI.7(2)V symmetry breaking pattern with the U(1) embedded as the T3—gene.rator of SU(2)R. The left—handed fermions are 511(2) doublets coupling to the groups at the first two sites, and which we will correspondingly label a“ L 0 and ”I’LL The right-handed fermions are an 807(2) doublet at site 1, 117131, and two singlet. fermions. denoted in figure 2.1 as “residing" at site 2. u R2 and d R2- The fermions 71/10, y”: L17 and L R 1 33 have U(1) charges typical of the left-handed doublets in the stzn‘idard model, +1/6 for quarks and —1 / 2 for leptons. Similarly, the fermion 11.er has U(1) charges typical for the right.-hande(.1 tip—quarks (+2/3), and dim has the U (1) charge associated with the right-handed dram-quarks (—1/3) or the leptons (—1). In the analysis of a general linear moose model in Ref. [48], it was shown that a Higgsless model with localized fermions does not satisfy precision electroweak mea- surements. Thus. for these models to be viable, the fermions have to be “delocalized” — in the (":ontext of the three site model, this means that the fermions derive their 81.5(2) charges from site 0 and site 1. (In an extra dimensional scenario, this is analagous to the “leale-rge” of the fermion wavefunction into the bulk). We will denote the amount of delocalization of the left(right) handed fermions by €L(.€R). \v-Vith the arrangement of fermions in Figure 2.1. we can write down Ynkawa cou- plings linking adjacent left and right handed fermion fields via the non linear sigma model of the form tam-31?. Thus, the fermion mass terms read: u. “R? Cf = Afl 'JLUEIERI + fixv'lf’filfi’Ll + f2 1.71122 + /I.(‘. (2.1) I Ad (1R2 We will set the vev’s of the sigma fields the same - f1 2 f2 :2 fit! (The reason for the. V? was explained in the introduction - for a Higgsless model derived from a flat extra dimension, the f constant should scale like \/ N + 1 to recover the right. continuum limit. and in the three site model, N =2 .1). - r , “ ,— 5 R “R? 3f ZN!) €LL"I.()ZH"Ri+wn174-5’Li+‘b”"L1$2 u +7”?- 5d]? (’32 (2.2,) “’0 have set. x/ZRI :: M D and set A/;\ = a L and /\’/5t 2: 5R. It is now straightforward to incorporate quark flavor and mixing in a minimal way. To get the SM quark flavor 34 mixing, we could add generational indices to each of the fermion fields. and, choosing s L and the mass term M D to be generation-(1iagonal, embed all of the nontrivial flavor structure in the Yukawa matrix in the last term in Eqn. (2.2) -- preciser as in the standard model; the only mixing parameters that appear are. the ordini-iry Cabiblm—Kohayashi—l\=1askawa (CK \l) angles and phase. 2.2 Masses and Eigenstates This section reviews the mass eigenvalues and the waveflinctions of the gauge bosons and fermions of the three-site model. The gauge sector is the same as that ol' the. BESS model [31]. Ref. [43] has also previously discussed the gauge boson eigenfunctions. but wrote them in terms of the. parameters 0. zl/IW, MZ. :l'lwl. and ill-12,. 2.2.1 Gauge bosons The gauge boson 111185808 arise from the kinetic: energy terms for the. sigma fields: where the covariant derivatives are: l#15301 = “#201 + WWSZOI -- 'i.?1$01ll",l (2.4) 0,,201 = 0,312 + wit/1212 — ig’zlgufi (2.5) In the unitary gauge (with 2301 : 212 = 1), Eqn. (2.3) gives terms quadratic in the gauge fields. for example. .2 ‘2 .t j I l" (2.6) TTrlD/‘Z'gunxlzml -+ :4— ~—2‘.gw/9 H.011"; 35 from which we can read off the mass matrix for the gauge bosons. \-\'e will work in the. limit :1: = 9/5) <<1 , y = gI/f} <<1 , (2.7) in which case we expect a. massless photon, light W and Z bosons. and a. heavy set of hosons H" and Z '. Numerically. then, 9 and g’ are approximately equal to the standard model SU(2)W and U(1)y winnings. Vt’e also define an angle 6 such that 9’ sin 6 (22s 9 cos 6 Ill film The charged gauge-boson mass-squared matrix may be written in terms of the small parz-rineter .7: as J) 2 ,.2 __ U .l- L ‘72 (21» —.’1: 2 Diagonalizing this matrix perturbatively in {17. we find the light eigenvalue 2 2 .2 .6 - q 1: :1. IL , .112,.v==——1——+—+... . 2.10 it 4 4 64 ‘ ( l and the. corresponding t-éigenstate [L _ ,0 , 7/1 ,‘1 [ll—11 ,.2 r, .4 3 r ,.5 .r 0.1. , t ;r. I 3.1, , /1 == —-——-— -+... 1tt +— <—+—-+-— -+ It . 211 s 128 0 2 16 256 1 l l where l-lv’o‘l are the gauge bosons associz-lted with the SU(2) groups at sites 0 and 1. Note that the light W is primarily located at site 0. The heavy eigenstat e has an eigenvector orthogonal to that in Eqn. (2.11) and a. mass '2 4 ‘ ~‘ ‘ ‘1‘ :1‘ . ‘ , Alli” =3 {[2122 1 + T "'l’ T6 + . . . , (2.12) 36 Comparing Eqns. (2.10) and (2.12). we find ‘1”2 1‘2 1'4 (176 . . Mil; =—4——§+6—+... . (2.13) ‘“ W’ or, equivalently. 2 1le I'll/,2 ,' 2 A, 2 r 3 (13.) 2.9:. .___;V +8 _rv- +28 _;L +.... (2.1.4) 9 1‘] VI All...” A] 7, w W W which confirms that the W’ boson is heavy in the limit of small 1'. The neutral bosons” mass—squared matrix is r2 —:r 0 22 2 g u ——1: 2 —.rt (212) 2 0 ~11 .7212 where t E tan/7’ = 3/('. This matrix has a zero eigenvalue. corresponding to the massless photon. with an eigenstate which may be written f (‘ (’ , , A" = Lu}; + :l/Vf" + —,B/" . (2.16) g .0 .9 where ”11.1 are the gauge bosons associated with the 51.7(2) groups at sites () and 1, the B is the gauge boson associated with the U(1) group at site 2. and the electric charge c satislies 1 'l '1 c + :7 ' —-—. . (2.17) 92 92 9/2 The light. neutral gauge boson, which we associate with the 7., has a mass ' Z M 402 4 (:2 G4 (‘6 ’ _. i 37 with a. corresponding eigenvector fluugmfiwéwf+fiefl (2w) w%=0_x%%1+33—mh Om» z W2- e + < > + . .. (2.21) 1.22 = —s — 32862” —8212 — [1) + . .. . (2.22) The heavy neutral boson has a mass 2 .1 .2 2 .2 ~2 2 it 1' (1 — t ) r .' . AI, = '2 1+——.—+—-——————+... , 2.2.5 [I .(1 f 462 16 i l with the corresponding eigenveetor VII-1v _ 40 2“ ,1 ”7” 1,2 [l .1 .4 ~— 12,110 +1..Z,ll1 +1'Z’B (2.24) 1‘0 _ _£ _ (1T3(1— 3(2) (2 )5) Z" 2 16 '— K) . 2 (‘1 ___.’II‘(l+/) ‘0‘ . 3. - 2 .- l. : l .3 —— I , _ #,=—i E—LT—ls”. (2%) Z 2 lb 2.2.2 Fermions Consider the fermion mass matrix ‘ 5L (l m 0 ' ‘Uu. (I = ill/D E (2.28) . - g I l "URJiR MD mud The notation introduced at the far right is used to emphasize that in the limit. M D >> m. m’. the above matrix displays a “see-saw" form. In what. follows. we will largely 38 be interested in the top- and bottom-quarks. and therefore in it]? and EbR. Diartmalizin ‘ the. to )-( uark seesaw—style mass matrix )erturbativelv in s . we _ l \l L [I find the light eigenvalue 2 M 3 a 5 . - mt = M 1-- +— + . .. . (2.29)) r ":2 2 l+€tZR 2(“fR+1) This is precisely the same. form as found in a. continuum model (51]. For the bottmn- quark. we find the same expression with 511? —~+ EbRi and therefore (neglecting higher order terms in £2 ) ' ”M? H 5 . Lfig,ifi 1+€32 (2am mt 51R ’ The heavy eigenstate (T) corresptmding to the top-quark has a mass r— . - ,2 ' ; r 11sz MD 1+: 1+——-.————:—+... . (2.51) *1? 2 (53 +1)2 (I? _ and similarly for the heavy eigenstate corresponding to the bottom-quark (B) with 5”? -—> 57],]? (or. equivalently, m; —’ mg). The left- and right-handed light mass eigenstates of the top quark are 0,t 1a 1L=¢Ltmettrwri -2 .-2 1 _ —1+ cL wfiR‘3kll ’ _ M1+“ ) M3 +1” [0 CH? CH? 3 ( 3-2 “1)5'5 . 39 and 1 It 2 9 CIR 3$35.11 = — + t 9 2 5 2 R1 1+3)?” (1+5le/ 1 32 :2 + ’R L IR; (2.34) 2 (1+ p2 )r/Z ‘— 1+C R ctlf and similarly for the left- and right-handed b—quarks with it]? ——+ 51,5» Here we denote * ‘ " ‘v‘ '3 '\ it -’-)t r‘ ‘-‘i the upper components of the fall (2) douliltt fit lds as 1;: L0, L1, R1’ (1ch the smaller the value of E L (311?): the more strongly the left-handed (right-handed) eigenstate will be concentrated at site. 0 (site 2). The left- and right-handed heavy fermion nurss eigenstates are the orthogonal eornlnnations b‘fl 'l 0,5]. , l ,-'t I ‘7‘ ' ,2 3 _ _ 3L _ (2CtR_1)‘L 1 " 2 2 3 +~- Lm 52 (852 — 3)a4 . + (—1 + {)(1 L + if? (L -l- . . .) @5le (2.30) TR = Th4}, + 7%! n2 , (2.37) x _ 1 _ 51.21? 5% + . .. U1 1??? <1 + sew/2 “1 g :2 + 5H? + "R“L (R2 , (2.38) :2 5’2 1+ 52?]? (1 VB) / and similarly for the left- and right-handed heavy B (marks with 5t I? —-> 5M?- Analogous results follow for the other ordinary fermions and their heavy partners with the appropriate 5 f1; substituted for 5,3 in the expressions above. As mentioned 40 before, 5: L is flavor universal whose Value is dictated by ideal fermion dtrltiealization. which we will explain in the next section. We will choose SR for each fermion in accordance w1th Illf a: A’IDEI’JEfR. In the hunt mf ——> 0. 5f I? is very small. 2.3 Couplings 2.3.1 Ideal fermion delocalization We mentioned before that for higgsless models to satisfy precision electroweak data, the fermions have to be delocalized. Most tree-level corrections to precision observ- ables eome necessarily from the coupling of SM fermions to heavy gauge bosons, and this suggests that a phcnomenologically efficient means of deloealization is one that will render this coupling zero. The coupling of the heavy W’ to SM fermions is of ”19 form 2.010;” fifth/V" Thus choosing the. light fermion profile such that (t-’:/~I,)2 2' . “i is proportional to Law. would make this coupling automatically vanish because the z _. heavy and light It" fields are orthogonal to one. another [49]. Thus we retpiire: ' 4 gift-9,1 )2 = giv'z’iv (239,) “-"e will refer the above as Ideal Fermion Delocalization (IF D) [49]. In the three- site model, if we. write the wavefunction of a deloealized left-handed fermion (L = f flllf’ft) + f 1111/: [1 then ideal delocalization imposes the following condition (having taken the ratio of the separate constraints for 2' = 0 and i = 1): i 9(f2)‘2 _ LlL - - . -— . (2.40) My 1er Based on our general expressions for fermion mass eigenstates (Eqns. (“3.32) and (2.34)) and the. IV mass eigenstate (Eqr1.(f2.11)), it is clear that Eqn.(‘2.4()) relates 41 the flavor-independent. quantities 2' and a L to the flav<_)r-sper::ilir 5 f R‘ Hence, if we construe this as an equation for E‘ L and solve perturbatively in the small quantity 2:. we find 1'3 1 ‘2“? 5 Pin "6 -2 -2 2 ;_ _ _ , ,.4 “ .' . , Note that, as we will see. ff]? is only substantiz—rl for the top-quark - and so ideal de- localization for the light fermions corresponds to the case 5f}; 2: t). Regardless of the precise value of 5 f1? involved, it is immediately clear that ideal delocalization implies 5 L = 0(1). Since. 1' << 1. this justifies the expansions used above in diagonalizing the fermion mass matrix. We will now derive the fermion couplings to light. and heavy gauge. boson imposing this condition. 2.3.2 Charged currents We will start by computing the left. handed coupling of the W to the fermions. Throughmit this section and the next. we will be writing down the couplings specif- ically for the top—bottom doublet. The couplings for other fermions can simply read off by the replacen'ient 5,}? —+ E f R‘ Also, we will work in the limit Eb]? ——> t), which simply means that. the couplings are. computed in the mb —> 0 limit. IV”) _ 0 b0 ‘0 ~. 1 b1 1 r) if) 9L “WL L'H’+gtL LUW f—-‘ -l which can be evaluated to be: . -4 .2 .- .- :8 .56 . .4 .2 . L ‘ 8(sz + 1)2 128(512R4r1)4 42 The (‘-('.)1‘res1i)omling equation for the coupling of standard model fermions other than the top—quark to the W may be obtained by taking SIR —-v 0 in the equation above. yielding 3 15 g}? =g(1——:L'2+ , .4 ‘ . .: . 2.44 8 1281 + ) l ) Combining this with eqns. (2.8). (2.10). (2.17). and (2.18) we find [1 + 0(32 251)) . (2.45) which shows that the l‘I"-feai'rrii()ri couplings (for fermions other than top) are of very nearly standard model form, as consistent with ideal delocalization. Eqn. (2.44) corresponds to a value of (3' F W 2 2 7 4 G»-_(9Ll -1 _-‘Z- i; '9 ~ v-6). _ 411131,. -12 (1 2 + 4 +... . (-.40) The W also couples to the heavy partners of the ordinary fermions. Here. we quote the results for the ’1‘ and B fermions; analogous results follow for other generations when 5t]? is replaced by the appropriate EQR. There. is a (iliagonal lit-"TB coupling of the f( )rm g? *1 B = ngBgear + Qi“1f13]1}=,1,,. , (2.47) c4 _ ~2 _ = ‘1 (1— "R 6c”? 5 .253 + ) (2 4s) ‘ - a2 ‘2 u . o .... k W -.1-2 = fl 1+ "1? + be”? + 4r? + (2 4.9) 2 , ,2 2 . ... , . , 0.1 . . . . 1. where T? 1 and I3? 1 are the heavy—fermion analogs of the. components (ll 1 and b] . I There are also smaller off-diagonal couplings involving one heavy and one ordinary 43 fermion 77)) Tb -— “171.0”1’ 91' + gT)b)/.,s)V , (2.51)) 9(1 -— E . 7 1:21?) (.17 + (9(zrr'3)) , (2.51) =2fi(:? +1) and 1;)3’3 91-11132?) + g1)113)1;,1V , (2.5-2) g(1+2s (R) )($ +023 . 2.3’3 21/2 (321ml 1 >) (,) Because 1 1? is a doublet under 511(2)1. the threc-site model i1111ud1s right handtd couplings of the W £1171? x IV);f [F] 117171.11:er ’7'"f “Rlll + h.(':. . (2.54) Note that the right—handed fermions exist only on sites 1 and 2 while the W is limited to sites (I and 1: hence. the right—handed coupling comes entirely from the overlap at site 1. For the tb doublet we find )3 ”9 — 31)?!) u3V (2.5.3) 53R ab]? (1 + C(12)) (2.56) :2 \/1+:1R\/1+51711 N91!» 511?2 2 mt 1 +c L 2]? where reaching the last line requires use of Eqn.(2.3()). We thus see. that. the. right handed coupling for all fermions vanishes in the limit "If ——-> 0. The H" also has 44 right-handed couplings to T and I3, for which we compute the diagonal coupling A .2 C + ()1 Al— 1 1 I . =__JL__ 1+t32 1R..fl+.n new and the off-diagonal coupling W318 __~l 14,1 1 .1 - A (g _. Own (V+vn+2wn 3g+_n) . 1‘) 2 2J1+§R MqR+1) ”(in As in the case of .1} I? . the right—handed coupling 1;)3'7717’ turns out to be proportional men to 311R? and is therefore very small. ()ther right-handed W f f ’ couplings involfing the light. standard fermions are straightforward to deduce from eqn. (2.56) and clearly I .. . . l'II’F suppressed by the small values of E f [1" Similarly. the off-diagonal g)? f are propor— tion-11 to small “ Tl ‘ l" 0‘ ‘l WFF, ‘ 1 -' "l I" ' ' f t ‘2 ’0) . 1. . . .. CH). 1e11agona 1])? are anaogous 111 orm o( 1 . 2.3.3 Neutral Currents \Ve will now compute the Z coupling to fermions. Like the W". the Z may couple to a pair of ordinary or heavy-partner fermions, or to a mixed pair with one ordinary and one heavy-partner fermion. The left-handed coupling of the light. Z-hoson to quark fields may be written 3 () I f. T 1 "V “ 1 ‘7 T 1 r £21xswzemogfiflmw4rrmzemi2¢®1n41 .‘l, 1,2 7, )1 . . T. «)1, (‘7 (51) ‘6 kg (1411,07 lrt‘Lt) + Li’Ll é’ LI’LI) ‘ “' “) where the first two terms give rise to the left-handed “T3" coupling and the last. term (1‘111'1p11rtional to 9') gives rise to the left-handed hypereharge coupling. The expression for leptons would be similar, replacing hypercharge 17/6 with —1/2. Similarly. the right-handed coupling of the Z to quark fields is 7.3 q! - ~ 2 [ZR 9‘ -‘I’17;("°1?1'2_7/ "’R1)Z/1-+ %'Z(UR1 ”1111)th '2 1 .. 9,1322 (317]?27/JUR2 — ng‘) ”(udRQ) Z), , (2.0.5) where the last three terms arise from the hypereharge. For leptons, 1/6 —+ —1/2 in the second term. ‘2/3 ——1 0 in the third term (for neutrinos), and —1/3 —+ —1 in the last term for the charged leptons. The left handed T3 coupling of the light fermion field to the Z is given by: Z .. ¢ ’ s n 931’ f = g1f£>21%+ 911112112 12.14) T202 (3 + 612 — (‘1) The coupling of left—handed light fermions to hypercl‘iarge arises from the overlap be- tween the fraction of the Z wavefunction arising from site 2 (the locus of hypercharge) and the left-handed fermion wavefunetions which are limited to sites 0 and 1: qf)’ =11"; [<12>+ +1.1 >2]= 1% (“Z-“61 7,2,2.-__+.2__4 = -g’.s (1+ I (' (‘5 8% f ') +...). (2.67) 46 The left-handed coupling of the top—quark to T3 is .1132]? =1}(f%)27‘% +1}(f}1)21‘lz (2.08) =72 2 + 572 r i :1?” 1+ ’3‘ ’R). :12 + . .. 12.1111) 13L ) 9 ) , ‘ 4 all + SIR)“ and the corresponding right. handed coupling is: th_~,1 I2 12 2'70 113;; -(.9"Z‘97'Zl(tfil t .. ) ”‘2 C = %——fl—22——(1 + C(12)). ((3-71) -‘ ’ 1 + TTR The left and right handed couplings to hypercharge can be. evaluated: J I ) | '6 Z ~; 917% = 9122 [(1)12 + (1)412] =11y‘f’ (2 12) t I 2 ‘ Z ( , _. ”21$: 91% [(#2) + (1%?)2] = 9Y3?! . ' (2.1.5) The T3 couplings of the Z to a pair of heavy-partner fermions or an off-diagonal pair can all be similarly computed. We give the result. for the top-bottom pair and their heavy partners below: (The. couplings for the other generatirms can be read off by replacing ‘tR —+ ‘le- 47 1 . 1 .1 1 ()5 2,(2+32 1).. 11.57,“ : ('11——_—1:3g(3+6f‘2—f1).1'2+ H H 8 4c(l+€tR) 5‘2 « ”‘2 2 2:2 ,2 _ I _ 11-2 _ ‘2 _ 2 .2 ”‘2 - ‘2 ' 21cm+n 81 (emu) . . . 2 . 3 cg (4 (1,3 + 1) — (:3 (sz + 1) (13 — 1) ) ZH' 1 02_O+_ 0 93L :- —I3(l(/ 2 .r ‘ 16(3 ER+1) (132151“ : 7—79—— f1R+ll . 13 ..., 3 " 211’ 9 9 = I ‘L 2V§thrtll -2 2 + 1(1:,1,+ 1)2 +‘(112+011? 3—41-41: 31.4.11) 3 1 :1; ‘ 16\/§c3(€ ;R+1)3 (ZtT __ 95H? .13]? “‘ ""— 2(‘(522 +1) 1‘]? + 11:? .. .7 . J' ‘1? 16c315fiR-+-115 The hyperclmrge couplings of the Z to a pair of left—handed or right-handed heavy— partner fermions follow the. pattern of the ordinary fermions: ZFF 1;) 7FF . gYR =gtz=91L , (- 1v 5] ,4— v and the hypercharge coupling of the Z to an off-diagonal (fla.\'or-c()11ser\-'i11g) fl" pair a 1 ways vanishes ZfF_ ZfF__0 (9,.) '(IYL —- gyrR -—-' q .1... (1,) 48 because the F and f wavefunctions are orthogonal. Weak mixing angle: Before closing this section, let us calculate the “Z stan- dard" weak mixing angle. Using Eqns. (2.17) and (2.18) and the relation (ll- )2 \/— ., 9L 1 11:2 :1,"1 _, 20F: g. =— 1— +—+... , (2.16) 4.115;, 1,2 we can calculate: ‘) P“ 8212 — _-——.—2 41/21,, F1112 - + 1 , , , . = 521:2 + 52(1'2 — 52) (c2 — 1).!‘2 + 0(1‘1). (2.77) l where 5 Z E sin HWI Z and c Z E cos HWI Z- The relationship between the weak mixing angle (In) 2 and the angle 0 defined in Eq. (2.8) is expressed as follows: 822 = .52 + A, (:22 = (:2 ~ A. (2.78) r 1 r , A a .92 (c2 — :1) 13 + 011:4). 12.79) In other words. 5“ and $22 differ by corrections of order :rZ. 2.4 Phenomenological bounds The three site model has the following parameters in addition to the SM: 5 [1.5 f R and MD. W'e are interested in finding bounds on I)ll__\;"Sl(_:Zl.l 1.):11'21111eters. in particular. Alum the mass of the gauge boson and MD. the mass scale of the heavy fermions. 49 2-4-1 9711/11" and lull" Exl.)1—)r'i111ental constraints on the Z ll'l'V vertex in the three—site model turn out. to provide useful bounds on the fermion delocalizat ion parameter EL. To leading order. in the absence of CID-violation, the triple gauge boson vertices may be written in the. Hagiwara-Peccei-Zeppenfeld—Hikasa triple—gauge-vertex nota~ tion [58] ['TGV I “’"(fZ‘ l1 + AHzl W; W17 Z’w * 1'1: [1+ Am] 1411'1T11’,7111“’ _ “CZ [1+ A1112] (”Pi-#171»): _ Wr-N-L/WJ)ZV (2,80) .52 — azoi'fwwlj — 11"‘Wu’pay . where the two-index tensors denote. the Lorentz field-strength tensor of the corre- sponding field. In the standard model. AHZ : A») : 13ng E 0. As noted in ref. [59]. in any vector-resonance model, such as the Higgsless mod— els considered here, the interactions (2.80) come from re-ex1:)ressirig the nonalwlian couplings in the kinetic energy terms in the original Lagrangian in terms of the mass- eignestate fields. In this case one obtains equal contributions to the deviations of the first. and third terms, and the second and fourth terms in Eqn. (2.80). In addition the coeffirient of the fourth term is fixed by electron)agneti1" gallge-invariance. and therefore in these models we find Asia]. E 0 AHZ E A1112 . (2.81) Computing the Z W W coupling explicitly in the three-site model yields ‘ 1 ‘ . fizww=9 Write) v%+.w‘z( iflzrz (2.82) [-1-7 1 212-14 , =()c(1:— - ‘1 ' ( +4 H...) (ass) fo ° thus. the deviation of the coupling from the SM value is given by: 2 .72 Ang = fl (2.85,) a;- The 95% CL. upper limit. from LEP-Il is A912 < 0.028 [52]. Approximating (:2 3 cos2 (2W x 0.77, we find the bound on :1: <0. 42 A92 (’2 86) 0 028 "‘ and hence. from eqn. (2.13), (l. (l .28 - n H . A91 This lower bound on :1le translates into an upper bound on 5L through the IF D condition, Eqn. (2.41). Finally, we recall that, in the. absence of a Higgs boson, ll’Lll’L spin-t) isospin=0 scattering would violate unitarity at a scale of \/8—mr and that: exchange. of the heavy electroweak bosons is what unitarizes lrll’lrl" scattering in Higgsless models. Hence, i‘lzlwl g 1.2 TeV in the. three-site model. This, along with Eq. (2.87) constrains a L to lie in the range 0.09:3 3 3L 3 0.30 . (2.88) 2.4.2 13/) and MD The isospin violating p parameter is defined as the ratio of isotriplet neutral current and charged current interactions at zero momentum. Neglecting the exchange of heavy gauge bosons. as apprtmriate in the case of ideal fermion ('l<.:loca.lization. p can also be expressed in terms of the masses of the H" and Z bosons as follows: flit-‘1; = ————. (2.89 ,0 Mg cos2 6 ) At tree level in the SM. /) is one - the reason for this is an (I.(I(."t(1(’71.f'(Lf syntmetrjt/ in the. Higgs sector of the SM. To see. what this symmetry is. let us start by writing the components of this Higgs field as [:33]: <15 = (2.90) Then, i026?“ is also an 843(2) L doublet with components: . 950* 2'0ng)" = . (2.91) _¢— This lets us define the Higgs matrix field: (I 1 (420* 62+ (2 ( )) > = — ‘ .J‘. \/§ _0 — 990 N ow we can rewrite the Higgs Lagrangian as: cHiggS = —;12Tr (Ducal (We) — we). (2.03) r 02 where the potential is given by: , ,, ‘2 ‘ ’ v («11) = — )1,“T1-c1)l(1‘» + XII-(NMZ. and the (:-(,)variant derivative is: I ) . (2.94) (2.95) Now, in the limit. 9’ —-» 0. the Higgs Lagrangian has an SU(2) L x 812(2)}? global synnnetry. Under this syrmnetry. matrix field transforms as: (1» —-. teal. \Vhen the Higgs field develops a vev, (2.96) (2.97) it. breaks both 812(2) L and SU(2)L. leaving only the diagonal subgroup .8”! 1(2) 11+]? = SU(2)V unbroken. Thus. there are three massless Goldstone bosons generated that. are eaten by the W and the Z to make them massive. It. is this accidental St} ( 2) global synnnetry. called the custodial symmetry. that. guarantees the relation between the W and the Z masses (Eqn. (2.89)) in the SM. In the fermionic sector, the S'U( 2) L x 811(2) R symmetry guarantees that the masses of the up and down components of a. fermion doublet. have. equal masses. Significant fermionic one loop corrections to the p parameter arise from the breaking of this custodial isospin symmetry. thus making the up and down type fermions non—degenerate. The largest. correction wines from the top-bottom doublet and is given by [55]: . 2. 2 ‘2 30F 2 2 Ht, mb mt ) = 1+ -—.—-—- m + m — 2—.——————ln—— . (2.98) I 87tZ\/2 t b m; — mg mg For the other fermion doublets. the up and down type components have. almost the same mass and hence the correction vanishes. In the three site model, in addition to the SM 1. and b quark contribution. the existence of the T and B quarks gives rise to new contrilmtions to Ap. We will evaluate this correction and use this and mt to constrain MD. Since 6? L is flavor independent. it cannot contribute to custodial synnnetry vi- olation and hence we will work in the limit. 5 L —> 0 to extract. only the leading contrilmtion in 5t R- The corrections due to the heavy top-bottom doublet arises out. of vacuum. polarization diagrams (symbol 11(0), where the 0 indicates that. these are. evalua ted at zero momentum) shown in Fig. (2.2). In particular. the formula for Ap is given by: 4 Ap = (7 (1111(0) - 1133(0)]. (2.99) Note that the subscripts 11 and 33 in this formula refer to the Ill-'1 and H73 bosons that couple to the fermions. The WI and 1V3 are not mass eigenstates. Subsequently. we have defined (mantities like 11 L L~ and the subscripts here should be understood as the currents to which the bosons couple. for example, 11 L L refers to the coupling to two left handed currents. We will call the vacuum polarizatitm diagrams with left handed currents H L L' Similarly. we will also define HR}? (only right handed currents) and 11 l. R (with both left and right. hantiled currents). At zero momentum 54 nu. 1Tue Figure 2.2: One-loop contributions to Ap arise from vacuum polarization (gliagrams involving two left handed fermionic currents (left) and mixed left and right handed currents (right). The RR piece is the same as the LL piece. The X and Y indicate the type of fermions in the loop. \Ve compute the leading contribution in the limit: 5L —» 0 and g, —> 0. these functions are [54] . 1 . . 1 HLL(O) —-: (57r2 (mi, + ”lg/)1? — 2(nzib1(mX,'/ny;0) + m?b1(my,m_\v;0))Jl (2.100) 1 . I1L1?(()l : 1(‘—‘2 [*2771X'myE + 2meyb0(-mJ\n 'my; 0)], (2.101) m where .- '1 :1‘1n2+(1——3:)rr12:,:1—1—3(2 meX, my; (fl) = / dr log ( X 2} )1 (2.102 . 0 . '1 331122 +1—3:7112,—3:(1— 3.)q2 121(mx\-.my;q‘2) = / (13' 3'log( Y ( I) 2} (2.105) . 0 p. .s,y's 'U“ t") ) "we. ‘0" ' is" 2. why. . Here F 1 the dmrgtnt p nt (f the l( op di igr 1111 ft in (llIIltIl 1011 11 rtgnluuition [:7 = — 11 + log(47r) —— log( [12) (5 = 4 — d), and [I is the renormalization mass scale. MIN; The RR piece has the same form as the. LL piece. We will treat. the LL, RR and the LR pieces separately and show explicitly that. the divergences cancel and compute the finite part. For sinuilicity. we will rclabel b(ml, mg, 0) as simply 0(1. 2,0). We will be ct)ncentrating on the top-bottom doublet and their heavy partners, as they C"! C)" codify maximum flavor violation. LL diagrams: The vacuum polarization amplitude with left handed external currents at zero momentum is given by: 4 -~1 ‘ .— 1 ( . «- Hill”) = —T4_"')—2— [—4— (mf + 71%)]? + 5 (I125h1(120)+mfbl(210))] (2.101) There are contril'mtions due to (tb). (1.13), (T, b) and (T B) fermions in the loop. The divergent part is thus given by: - "3 . L 2 2 9 . . ‘ 2 2 Hllfoldiv = W (2(5),!) (””t + mg) + 201,113)2 (7171 +1113) +2fQ-510)? (1111721 +1113) + 2(9TB)2 ("311+ n33» ' (2.105,) Plugging in the couplings, we find the the divergent part of 1111(0) is: .2 E 171’. 1 ‘2 ‘2 ‘ . “llfoldiv : W [-7 + ,— ("114- 7223)] . (2.106) which cancels the divergent contribution from the 1133(0) piece: ( I: t ‘ “ ‘) ‘ ~‘.) ‘1 ”33(0)div = (—1_7r)—2 [2(y,{2)2,11? + AUTTVWT + 2f.‘lf§Bl“"’i3 , ‘7 ‘ r) +2(gtLT)‘ (m? + 777%) + 2(515’B)2112.B] 2 E m, 1 9 .2 ’ Z .. —+— ("'7'+ ’ ) 2.107 (4”,)2 [ 2 8 1 ”B :l f. ) To evaluate the finite part, we will consider two cases. "’1 = 711-2 and m1 # III-2. _2 . .. —-—):2—m log 7772 (3 (2.108) ’ 71" m1 = m2 : “LLfUlfinite = 1 2(47r).2 (ml +1173 ml 75 mg : I‘llLLfolfinite = ) (mf— m2— 2m, log m1 +2mg log mg] (2.100) .7,.‘,. ,. ‘. -3 .,_ '..3.,- ,, _,. ‘ .. __ 3,2 \M. recall that to thc ()I‘(l(..1‘ M are interested 111, MB — .1 ID and MT — IUD ‘ / 1 + 23,3. Evaluating the finite contributions due to [111(0) and 1133(0), we get: ‘) , . 4 m" . U2 +107 905.72,, + - ,,3+1210gu,2, +63%,310g mg) (2.110) and 4 TI?- 2 1 , I) ‘2 1 “33(0)finite = “W ("Fig—103% mt + 15MB 10s MD + 32 ”D IR +1 1 . , . and thus the finite part of the difference in the vacuum polarization amplitude for the LL piece is given by: 4 "'f 4 wDEII? n- '—n«« 0 .» - .=——.— f 11(0) .33( )lfimtt (4%)) 16 +(47t)2192 (2.112) we recognize the first term as the SM contribution to the one loop correction to the. p parameters [5 5] and the second. term is due to the heavy t.op-l_)(_)ttom (ltmblet. RR diagrams: As before. the divergent pieces of 1111 and U33 are equal and 57 ("aneel F 3? Inland-33,. : 11:53“wa = —(_3332 (mgr +1233) (1 — 532]?) (2.113) 003...; ‘ (In;2 + mg?) + Using Eqn. (2.108). we can evaluate the finite (‘(_)Iltl'll)tlt.lt)l)S due to the RR enr- I'PIllSI 4 ‘_’D 2 L’D 1 ., , 3 4 112 11,3 3 3 [133((UfilllTHZ—W [-1—16 l-lDlogfllD'i' 64 DCt)R— 1)8 51R (2.115) Thus. the finite eontrilmtion from the vacuum polarization diagrams involving two right handed currents is given by: (Hum)—I133(()))finm, = (433.3 3383* . (2.111,) LR diagrams: The LR diagrams are evaluated to be (Ref. [54]): 2 11,3];(0) = — (4w)? [1171711313 — ‘1)11'7I72f)()(120)]. (2.117) The divergent part. of 1111 (.‘tlll now be evaluated: 2E ”11(01111v = — 2rnth(_q,1%3)(g3LB) + 2meBthLB)({/¥B)] . (2.118.) In the above formula, we have. omitted terms proportiorml to rub. Plugging in the eouplings. we see that: 1111mm“, :: —W Z‘rl’IT‘I‘H.B 16 1 -— 7 . (2.11)) r 08 It. can be easily shown that the divergent part of' 1133 exactly cancels this. We now proceed to compute the finite contributions. ‘) " . L I? ll11(0)3333330 = )2 2(933)(gtB)‘mt-meM-mt. 172.87 0) (4 :1 ‘) H 471' 2 2 2 . 2 . 2 . . 2 2 M M s M AI. log .\I (471') 8 16 , 48 1(3 )2 207143) (”138 l’""Tme0(mT-. 721.1330) + A‘ Iv (2.120) () 3 . .. r) 3 . 1133(1))3333333. = m [(113131)(ggbnfbmmt, mt. 0) + ((15111)(IQ/£271)Illfiubohllfr. INT. 0) +(91I§B)(.qu)”’%3b()(mB~ NIB. 0) + 2(gf’T)(1);?)7123711'11120(mt. (”"1“ 0)] . , 2 2 -2 , 2 ,. 2 . ,. , 2 (471—)? s ' - ' D 113 H? :32 16 ' (2.121) Thus, we see that the finite contribution due to the LR diagrams is: 4 1‘12 3.4 (1111(0) — H3st0llfinae = D m- (2123) (47.)2 192 The total fermionic contribution to Ap in the three site model is obtained by adding Eons. (2.112), (2.116) and (2.122) and is given by (after subtracting out. the SM contrilmtion and multiplying by a factor of 3 for color): 1 2 -4 . ... y, .P . . 3 3 . _ _ 3 3 . . 3 3 3 - The. H” and 1’ contributions to Ap are discussed in [56]. The phenontenological bounds on the value of Ap depend (since they include the one-loop standard model 59 corrections) on the refereniee. Higgs mass chosen. \N’e are interested in the bounds on Ap corresponding to Higgs masses between about 380 GeV (from Eqn.(2.87)) and the unitarity bound 1.2 TeV [56]. Current. bounds (see for example, Langacker and Erler in (57]) yield (approximately) Ap g 2.5 X 10"3, at 90% C.L.. assuming the existence of a moderately heavy (340 GeV) Higgs boson. while it is relaxed to approximately Ap S 5 x 10‘3 in the ease of a heavy (1000 GeV) Higgs boson. \\-"e therefore expect. that the upper bound on Ap in the three site model varies from a})proxiinz-rtely 2.5 x 10_3 to 5 X 10'3. For (IT = 5 X 10‘3, we find the upper bound 3,, 1/2 533<0.94 (U) . (2121) Our upper limit. on a, R and our knowledge of the top quark mass allow us to derive a. lower bound on I'll . Our expression (2.29) for Tllf reminds 11s that. 5 5 M 3,3, : _I___t£_3_ (2.125) 1 53,? For a given value of MD. the existence of an upper bound on 53 I? implies that there is a smallest allowed value of EL. which we denote 5 o 1 1 r ,3. . 2.5 X10”3 / In, (17" 1/3 1,- ,, . a ,= 1.211 —,—— 1+0.6.5 ——+ —— . (2.1211) . of 3/qu 2.5 x 111-3 M :1: Since eqn. (2.88) requires 5L < 0.30, for (IT = 2.5 X 10"“3 we find that MD must :1: be greater than 2.3 TeV. and for (1T = 5 x 10—3 we. find that M D must be greater than 1.8 TeV. The joint range of M D and 11-13333; is summarised in Fig. (“2.3), for both values of QT. Using 11/ > 1.8 TeV and the bound in Eqn. (2.1.24), we see that. 53113 < 0.35. The right handed Wtb coupling contrilimtes to the process I) ——> .97, and this gives an upper bound on 53.1; of 0.67. which is superceded by this limit... 60 25000 20000 15000 10000 5000 600 800 1000 1200 MW, Figure 2.3: I’henon1enologically acceptable values of 51/ D and Il'IW/ in GeV for (17' = 2.5 x 10“3 (solid curve) and 5 x 10‘3 (dashed curve). The region bounded by the lines 380GeV < MW, < 1200 GeV and above the appropriate curve, is allowed. For a given M D and M334. the. value of 5112 is determined by Eqn. (2.125). 2 .5 Remarks In this chapter, we have described in detail a. minimal deconstructed Higgsless model that. is simple, in the sense that there is only one extra set of vector bosons in- stead of‘ the infinite tower of vector bosons present. in the continuum limit. Likewise. there need be only a single heavy fermion partner for each of the standard. model fermions, instead of a tower of such states. The three site model serves as a conve- nient framework to understand many important ideas in Higgsless models. like the concept. of ideal fermion delocalization. After deriving the mass eigenstates and cou— plings, we investigated the phenomenological bounds on the mass scales of the gauge and fermionic sector by appealing to precision low energy measurements. We found that the lower bound of 1113,33 is around 380 GeV. which makes its discovery at the CERN LHC a realistic possibility. However. the s "ale that sets the mass of fermions. 11:! D has a. lmver bound exceeding a TeV, because. of the twin r(&*(',p.1ireinents of getting 61 the correct value of the top quark mass and having a phent)menologieally acceptable value for Ap. This renders the discovery of the heavy fermirms rather difficult. It. is interesting to explore avenues to relax these constraints, so we could have extra fermions that are light. enough to have a strong discovery potential. This. however, will involve extending the three site. model in some specific way so as to free MD from the constraints of mt and Ap and this will be the subject of the next chapter. 62 Chapter 3 Triangle Moose Model In Chapter 2. we presented the details of the three site model [30], a maximally deconstriictcd version of a higgsless extra. dimensional model, with only one extra. SU( 2) gauge group, as compared to the SM. Thus, there are three extra gauge bosons. which contribute to unitarizing the W L W L scattering in place of a higgs. (The LHC phcnonrenology of these extra vector bosons can be found in [60]). Also incorporated in the three site model is a heavy Dirac partner for every SM fermion. The presence of these new fermions, in particular, the heavy top and bottom quarks. gives rise to new one, loop contributions to Ap. Low energy precision measurements require Ap to be < O(.1.()*3) and so, the combination of parameters 51 R and M D have to be tuned to both make Ap small and obtain the large top quark mass. These twin constraints push the heavy quark mass into the multi TeV range, too high to be seen at the LHC. Our goal in this chapter is to construct a model that retains the. features of the higgsless mechanism, but allows for Dirac fermions that are lighter. To achieve this. we. separate top quark mass generation from the rest of electroweak synn’netry breaking. by analogy with top color models. 63 F ignre 3.1: The gauge structure of the model in Moose notation [25] g and g’ are approximately the SM 811(2) and hypercharge gauge couplings while 9 represents the “bulk" gauge coupling. The left (right) handed light fermions are mostly localized at site 0 (2) while their heavy counterparts are mostly at site 1. The links connecting sites 0 and 1 and sites 1 and 2 are non linear sigma model fields while the one connecting sites 0 and 2 is the top Higgs field. 3.1 The Model The clectroxwak gauge structure of the model is the same as that in the three. site model and is 811(2) >< SU(2) >< U(1) (shown using the “Moose Notation" in Figure (3.1)). with the SM fermions deriving their 81.}(2) charges mostly from site 0 (which is most. closely associated with the SM S U (2)”,1) and the bulk fern‘iions mostly from site 1. The extended electroweak gauge structure of the theory is the same as that of the BESS models (31. 3’2], motivated by models of hidden local syn‘rmetry (with a 74- 1) [33. 34, 35. 36. 37]. The non linear sigma field 201 is responsible for breaking the. SU( 2) X SU (2) gauge symmetry down to SU(2). The left. handed fermions are 57%?) doublets residing at sites 0 (#10) and 1 (t'2L-1). while the right handed fermions are a 64 doublet under 8 I (2)1011) 1) and two 311(2) singlet, fermions at site ‘2 (1le and (1132). The fermions 't-Z'LO. t": L1~ and a”: R1 have. 811(2) charges typical of the left—handed SU(2) doublets in the. SM, +1/6 for quarks and —1/2 for leptons. Similarly. the fermion 7.1.32 has U(1) charge typical for the right-handed tip—quarks ( +2 / 3) and (132 has the U(1) charge typical for the right-handed down—quarks (~1/ 3). Our goal is to separate top quark mass generation from the rest of electroweak syimuetry breaking. We do this by introducing a “top Higgs" field (I), motivated by top-color models [38. 39[, and let. the top quark couple preferentially to the top Higgs via the Lagrangian: £t0p: —/\tt;'LU SIR. 3.3.2 Ideal fermion delocalization The leat‘ling tree level contributions to precision measurements in Higgsless models come necessarily from the coupling of standard model fermions to the heavy gauge bosons. It was shown in [49] that it is possible to delocalize the light fermions in such a way that they do not couple to these heavy fields and thus minimize the deviations in precision electroweak parameters. The coupling of the heavy it” to SM fermitms is 72 *ll of the form 2 92(1’9fl- )ZU’W.” Thus choosing the light fermion profile such that. (ti-f. )2 I I is proportional to t’l'Wz. would make this coupling automatically vanish because the heavy and light W fields are orthogonal to one another. This procedure (called Ideal fermion delocalizatitm) also equates the coupling of the W to two light fermions to the SM value. Thus. an equivalent way to impose ideal fermion delocalization (IFD) is to demand that the tree level 9W“, coupling (say) equal the SM value. we will use the latter procedure to implement IFD. The deviation of the ow“, coupling from the SM value can be parametrized in terms of the Peskin-Takeuchi parameters [42] S. T and U parameters as [50]: . 0 . ‘ ‘ . (LS (:“(i'F ((32 — .sz)r.tfj 432— 23-2 — 832 gll-"eI/ = E 1 + (3.30) where c = cos 0w = ..r'lVIW/i’llz and s =2 sin 0w = V1 — c2 are the “mass defined" angles. It was shown in [48] that at tree level, in models of this kind. the parameters ‘T and U have negligible values that are (9(14), and so we can impose ideal. fermion delocalization by requiring S to vanish at tree level (which would make own, the SM value. from Eqn. (3.36)). In computing the couplings, we will use the mass defined angles. (W'e will indicate this by a suffix w in all the couplings). From Eqns.(3.12) and (3.19). we can see that 811103,: is related to sin!) defined implicitly in the couplings in Eqn. (3.10) by: , .2 sum)”: l—iT sint). (3.37) L Using the W and the fermion wave functions. we can calculate the (itnlrflillg .‘ltt’m/ “5 ,-2 e :1: c, L + 1 — 3.33 8111010 4 8 ( ) gll’cu : Thus. we find the ideal fermion delocalization condition in the model to be: 73 . . 1‘} t‘l) (5a.: {\3 (3.39) Note that. this relation is the same as the one obtained in the three site model. 3.4 Light Fermion couplings to the gauge bosons 3.4.1 Charged Currents Now that we have the wave functions of the vector bosons and the fermions, we can compute the couplings between these states. Since all the light fermions are approximately massless. we set sz for all the light fermions to zero in this section. We will calculate all couplings to C(12). We begin with the left handed ll"ud coupling. (7? 0}} ud = goritugdg + (Hamid) = (3.40) sin Hw ‘ This result follows from the fact that we have implemented ideal fermion delocaliza— tion in the model. All other char ‘ed current cou )line's both left. and ri‘rht handed can be similarlv b t.» .. computed. The couplings in this model are only very slightly different from the ones in the three site model. The difference is attributable to the fact that the expansion parameter we have chosen. .1'. is sine), as opposed to tang“) in the three site model. \Ve 74 now summarize the results: 113331“: 1101 L3, 11([) 119 + {/1 LL.111,11L= sniff/11,1 (3.411) g‘L""'f‘r"(=11‘L‘""'U) 2 11012331333113 + 13113113113 = ———9 fit“ 011- (3.42) 1133”” = 90133.1.12113+131},,11L11L—2L—‘i'l’m77 (1 + 21-2) (3.43) 1133 “-—" _1,1-,1,,1133113L = o (3.44) 33 U"(= 133 “0): 1, 1,1,.1'1 1,133 :0 (3.45) ”3:21.113 : 9.3.1.1313 = m (1 - $173) (3.46) 9333"”! = 1)(‘)'1'0;,11(l)-L(I(L)L + .6113111311'3’ '2 t) (3.47) Llp”1.111(:Lap/"1111) = 1,0113, ,1 L11 L + 1,113,03113 = —m (3.48) 1133'“) =11010,1 (Ll/1L +1111 13113 = m (1 —- 143-19) (3.49) 1133 "d‘!“l"--’.1,,1“i1di1 = 0 (3.50) 1133 ’f "(= 1133"”) = 1111:3311331133 = 0 (3.51) 1133;,UD = QNIILJXRDI = 1sii10w (l — £12) (33.52) Two comments are in order. The right. handed ll’lz’d. W’ud couplings are zero in the limits in which we are working (EUR = 5d}? = (l). The right handed coupling of H' with two heavy fields arises. in this limit, solely from site 1 and is not zero. The left. and right-handed H", coupling to two heavv fermions is enhantitet‘l by a factor 1/:1: 1e. latiVe to qLfl’ with .1 being the small expansion paramettr Thus for Very small values of .1'. this coupling becomes large and consequently, F (1114”) [WWI >> 1. We. therefore exclude the region 1111”,, 2 2M D of the M D — MW; parameter space in our phent)menological study of the heavy quark production in Section 3.6. *1 Cf! 3.4.2 Neutral Currents We can now calculate the coupling of the fermions to the neutral bosons. All the charged fermions couple to the photon with their standard electric charges. 0 511.7111 = 9113121. =51tc/glft’1. We will. be calculating the couplings in the “T3 -— Q” basis. To do this we use the standard relation between the three quantum numbers: Q = F; + Y. Since the fermions derive their SU(2) charge from more. than one site, we will calculate, for ( example. the T3 coupling of two light fields to the Z as 293(11332113. The left. 7. handed Z coupling to SM fermions is calculated to be: ) 2 ~ 1 . 'l 2 I 2 ' O ' l 2 gLZuu. : (90712019.) + gl’ZOl'L-l ) T3 + 9 "Z (MIL).z + (UL) ) (Q '— T-ll e - , 2 . = —————————— F — . sm (7.1.) 354 sinHw cos ”11.: ( '3 Q u ( ') All the other couplings can be similarly computed and we summarize the. results 76 fl '- 1 below: Z1111 (" (- 1 - ‘2 ) . rr' 1 = —— ]« —- ( sin (I ,- 3.. IL Sill 6'1" COS 6'“) .3 2 U ( ) )) 2111/" PT . _.. . (I = — T (3.00) L 2 \/§ Sill H'IL! ('05 ”(U . 2 ZL/YL'T (1' 1 II" 2 r 1 ' 2 r -w 1 =——.———-—— — l+— 4—sec 0 ,1 /. —Qs1n ()1 13.111) 'IL s1nfiu1cos6m 2 8 ( 11) ‘5 11 i ’ 1]?” = (’(Q —— 733mm)“; (358) 9121"U = 0 (3.59) 7.1111 ‘1 1(1 1'2” Q - 21,1 ('3 ("(1) ( " =—— — —— -— .‘m 1 ..1 JR Sill H1“, C05 6“) 2 8 .3 S U , 1, 1 11121 11.11. = —§ user-19,1,- tant‘tm (T3 — Q) (3.01) Z'UU ‘0' 1 $21 211 T (‘5 f") f] ' = ——'—— — — $111 1 .1‘ ' Ll . )— L VT 81110“, 8 11 ‘5 _ 1 2 .- ‘2 Z/UU c: 15.7? . 2 0 m 1 5111 H?!" . ~- 1 = ——_———— 1— — 2.‘ec, (l ,1 —— 3tan‘t) 1 I + — c:'————( 13.03) IL 131110”: 8 ( S U U ) 3 2 l (7050”: 2 ( I "I. FIE ,, . . 11g 111 = E—secflwtanflw(Q — ‘13) (3.51) r I , 1114‘. “U = 0 (3.65) ZIUU C 1 I132 (0 0 f) r' t 2 f) ) ,1, + l SlIl2 H4112 () (3 (if) ‘ = —— - -.-— ..sec“ ~1— 1.) an 1 1 — e .1: —— . .1) 9R :1 sin 011,- 8 11 U’ ‘5 ‘2 cos (1w ' ' While ideal fermion tlelt)calization makes gW’ud zero. in the case of the neutral cur- rents. the corrcsponding (1)7121“ is seen to have a small hypercharge coupling. (The 'T3 portion is. of course, zero). Also, 9qu is seen to have only a T3 coupling because the term multiplying Q — T3 (hypercharge) vanishes due to the orthogonality of the fermion wave functions. In the limit cost)“, ——> 1, 1317:1131 to the off diagonal coupling of the H", g33'UD. As is seen to correspond exactly in the case of charged currents. the coupling of two heavy quarks to the Z ’ is enhanced by a factor 1/:1:. This makes l‘(Z')/.lIZ/ >> 1 for small values of .‘I‘, and hence we will restrict. ourself to the region 77 M7; 3 2ND in our phenornenogical analysis in Section 3.6. J 3.5 The Top quark The top quark in the model has different properties than the light. quarks since its mass is generated by the top Higgs. This section reviews the masses and eigenstates of the top quark and proceeds to analyze the delocalization pattern of the top and bottom quarks. 3.5.1 Masses and wave functions The top quark mass n‘iat'rix may be read from Eqns. (3.1) and (3.5) and is given by: MDELt Atesinw (3.67) 1WD AIDE”? Let us define the parameter ,\ esinw _, a = -—’-,——. (3.68) in terms of which the aliiove matrix can be written as: a a. M) = MD I” . (3.69) 1 51R Note that we have introduced the left. handed delocalization parameter EL). that is technically distinct from the one for the light. fermions. We will see in the next. subsection that “1’7Lt = L is the preferred value. i.e.. the top quark is delocalized in exactly the same way as the light. quarks. 7'8 - ”Ir Diagonalizing the top quark mass matrix perturbatively in SLt and 511;, we can find the light. and heavy eigenvalues: 2 2 2- A C: + 5 + ‘1‘ 5 mt = At‘t’SlIlw‘ [1+ Lt O(t1§;+(12)Lt ”2] . (3.70) _ — a -2 2 . ,- C: +5 +2tlc 5- M, = MD 1— L’ ’1? 2“ ’R (3.71) 2(—1+a ) Thus. we see (hat. mt depends mainly on 1' and only slightly on 5H?» in contrast to the light fermion mass, Eqn. (3.30), where the dominant term is s [1? dependent. The wave functions of the left and right. handed top quark are: _ .0 t 1..-.t ,.2 2~2 ,. é: ‘ N = 1 —— : [if ‘f (1. Cf]? ii— 2“; Lt‘fR Lt + (W) Li.” ((5.72) 2(-1+a2)2 "‘0 —1 +3" U __ 1 ,.t 2 ’1? - ’1?‘='m + ’R’RQ 2,2 .2 __ c . _ ” CI.r+~'1R+2“'Lt'tR t “CL! +=tR _ i. .. . 1 2.2 9131+ 2 (HZ- (3(5) 2(—1+a ) —1 +11 The. left. and right handed heavy top wave functions are the orthogonal combinations: . u .r . 1 ,t TI = TLC'LOJVTLL’II J 2 2-2 ,_ 5‘ +0.6 . ”5 +‘1 1 +2“ 5 . . - = ( Li, (1?) 9,1“) + (_1 + Lt (1? LI 11%))..21 (3.14) ~1+r12 2(—l+r12)2 , 1 ...; 2 TR = Tat-”R1 “t Tie/R2 2-2 2 , - asLt+5tR it ‘1 CLt+5m+2a=chrn , —.—_ -'= ——-—- w + —1+ 1, , .5. .3) ( _..__1_+_.(,2 ) R1 ( 2(_1+"2)2 132 t I ) 3.5.2 Z bf) and choice of em Since the (left-hai'ided) bottom is the SU(2) partner of the (left-handed) top. it is delocalized in exactly the same way as t L' Thus. we can compute the tree level value of the ZbLchouplmg and use it to constrain 5L1: This coupling is given by: ( ‘ '4 . 1 T ‘ ‘ i 2 r 1 ”Lbe = (Ilut'ozf’ff +(I1'1'lszil2) 73 + 9'27; (“9.12 + (bi) ) (Q - l3) 0 1 + .172 5%f )T Q . 2 6 ('3 ”(3) = _— ———- -—-."i )- sin Hw cos 9w ( ' 4 2 3 S n a. I ) Now this exactly corresponds to the tree-level SM value provided that SLt satisfies We see that. this matches the delocalization condition for the. light quarks, Eqn. (3.39). T hus, we see that the left—handed top quark is to be delocalized in exactly the. san’ie way as the light. fermions if we are to avoid significant tree-level corrections to the SM Z I) Lf’ L value. Henceforth. we shall be choosing this value for 6L). 3.5.3 Ap and All) The (,‘mitril'nititm of the heavy top-bottom doublet. to Ap can be evaluated in this model in the same way as in the three site model discussed in Chapter 2 and the result is the same as in Eqn. (2.123). We give it. below: A/) = 412-113. (3.78) The import ant. (..liffcrence now is that, since the top quark mass is (‘lominated by the vev of the top Higgs instead of MI) (see Eqn. (3.70)), it]? could be as low as the. ER 80 of any light fermion. Thus, there is no constraint between the twin goals of getting a large. top quark mass and having an experimentally admissible value. of Ap. This enables us to have. heavy fermions in this model that are light. enough to be seen at the LHC. We explore this in detail in the next section. 3.6 Heavy fermion phenomenology at hadron col- liders We are now prepared to investigate the collider phenomenology of this model. As we have. just seen. there is no tension between getting the correct values of the top quark mass and the p parameter in this model. Thus, the mass of the heavy quark does not necessarily lie in the TeV range as in the three site nmdel discussed in Chapter 2 [3(ll. This enables us to investigate the phenomenology of these heavy quarks for MD of the order of hundreds of GeV. The current CDF lower bounds on heavy up (decaying via charged currents) and down type quarks (decaying via neutral currents) are 284 GeV and ‘270 GeV respectively at. 95% CL. [61}. Thus, in our phenomenologk'al analysis. we will be concentrating on quarks whose masses are lmtween 300 GeV and 1 TeV. Let us recall that. the diagonal coupling of the heavy ll" or Z I with two heavy fermimis is enhanced by a factor 1 /;r. , where .I.‘ = sin (I) is our small expansion parame- ter. Thus. if the masses are such that the heavy gauge bosons can decay to two heavy fermions. then we are in a situation where Pug/MW; > 1. rendering perturbative analysis invalid. Thus. for perturbative consistency, we will always stay in the region of the sly/W, — ii I D paran‘ieter space where M W’. Z’ g ‘21‘l I D- We. will study both pair and single production channels. 81 3.6.1 Heavy fermion decay The heavy fermions in the model decay to a. vector boson and a light fermion. If the heavy fermion is massive enouglh the vector boson could even be the W’ or Z ’ in the. theory (Fig. 3.2). (The Situation changes slightly for the heavy top quark for which decay into top pions is allowed). V f Figure 3.2: The decay modes of the heavy quarks in the theory. The decay rate is controlled by the off-diagonal left handed coupling of the vector boson to a heavy fermion and the corresponding light fermion (the cm‘responding right: handed coupling vanishes in the limit of massless light fermions). In the limit. that the mass of the light fermion is zero, the rate of decay to charged gauge bosons is given by: 2 3 0 2 2 {I ,r M m“ ’m x ': £217}? 2D _ l: 1+ 2____l;)_ . (3.79) ‘ 7T ”IL; j‘ID A [D In the hunt that. the Dirac mass 18 much higher than the H and ll boson masses, the terms in the parentheses can be approximated by 1. Thus, in this limit, the dmm.’ width is ('lcterminctl by the factor ”3’1" [/2116 This can be evaluated for the. H7 and 82 the l’V’ couplings to be: 2 , 2 gW’Qq/mW’ 2 2 ~ -(]1v'Qq/7Hlv' N (6212/8 sin2 6“,) (e2212/4 sin2 6“,) 1:2 —— (3.80) 1'2 . (6.2/2 sin2 6w) (c2v2/4 sin2 6.11.172) 2 {1.7 —. 3.81 [,2 < > Thus, we see that in this limit, the decay of the heavy fermion into 14/ and W ' become equally important because gfvoq/miv a: 9124" Qq / ma”. This is further illustrated in Figure (3.3). BR ”L-JIIE-iunr. l ' I I I I x I , I r 0.6“ II. U_)W,d H : x -— U->Z,u I 0.5_ ‘. _ _ U—>W',d 4 . ‘ d : ‘ — U—>Z',u 1 0.4- N. a Q a Q . .. 'I-I-I—I- .. a 0'3:— ~ \ ...-.--.----------IIII{ : ~\ ’ ' — - — _ - — — ‘: ” ~ " _ 0.2 ~ ’~ ~. - : ----—-—-_-_-_-_-_-: 0.1_— : 0.0 :1 L l 1 1 l 1 1 1 l 1 1 1 l 1 1: 400 600 800 1000 1200 MD (GeV) Figure 3.3: The plot of the branching ratio of the heavy quark into the charged and neutral gauge bosons. The masses of the W, and Z I gauge bosons were taken to be 500 GeV each. 83 3.6.2 Heavy quarks at the LHC Our goal in this section is to analyze the possible discovery modes of the heavy quarks at the LHC. We will show that it is possible to discover them at 50 level for a large range in the NW; — M D parameter space. We will consider both the (QCD dominated) pair production and the (electroweak) single prtxluction of the heavy quarks. Each produced quark inunediately decays to either a SM gauge boson plus a light. quark (for Al D < MW’) or a heavy gauge boson plus a light quark ( for ill D > A IWI). W'e will (,‘onsider the first possil,>lity in the pair production scenario (subscct ion 1) and the second in the single production analysis (subsection ‘2) and show that these cover much of the M D — MW; parameter space. For our phenomenological analysis, we used the CachEP package [62]. Pair production: pp —+ (X) —, ”"qu —+ (”l/(ii We first consider the process pp —> QQ at the LHC. Pair production of heavy quarks occurs via gluon fusion and quark annihilation processes (Figure 34) In Figure 3.5. we present the production cross section as a function of Dirac mass for a single flavor. we see that the cross-section for the gluon fusion process is higher than its ctmnterpart for low values of MD. However. as M D increases, the qr’i channel starts to dominate. This is because the parton distribution function (pdf) of the gluon falls rapidly with increasing parton momentum fraction, 1'. Each heavy quark decays to a vector boson and a light fermion. For M D < iii/”42;. the decay is purely to the standard model gaulge bosons. The decay to heavy gauge bosons opens up for I” D > 1")le Z” and we will analyze this channel while discussing single production of heavy fermions in the next subsection. Here. we look at the signal in the case where one of the heavy quarks decays to a Z and the other decays to a W. with the gauge bosons subsequtmtly decaying leptonical1y. Thus. the final state is pp -—-> QC? -+ [Ill/jj. 84 QI 0| 9 9 Q Figure 3.4: (a). Pair production of the heavy quarks occurs through (jq annihilation and gluon fusion. I I I F I I 1 I Y I r I I I If r I I 10 E— u... ggqUU _i i 1 :~ _ _ : _ I-I qanU l 1 f' 1 A I .D . 8‘ b 0-1 r ‘2 0.01 5* ~. ‘g . ~.~ 1 ~§ L ~~ ~~ Q t ..~ 0.00] _1 1 l 1 4 L l 1 1 1 l L 4 4 l L 1 1 l 1. 400 600 800 1000 1200 MD (GCV) Figure 3.5: The cross section for pair production (for one flavor) as a function of the Dirac mass. As can be seen from the figure, for low values of AID, the cross section for the gluon fusion channel is higher than the quark annihilation process. As MD increases, the quark annihilation process becomes equally important because the pdf of the gluon falls rapidly with increasing parton momentum fraction, :r. To enhance the signal to backgrotmd ratio, we have imposed a variety of cuts. We note that the. the two jets in the signal should have a high PT (~ AID/2), since they each come from the 2-body decay of a very heavy fermion. Thus, imposing strong pT cuts on the outgoing jets can eliminate much of the SM background without affecting the signal too much. We also expect the 7) distribution of the jets to be largely central (see Figure 3.6), which suggests an 1) cut: (”I S 2.5. Lastly, we. will unpose a separation cut on the jets to avoid IR divergences in our computations. 1200 1000 800 600 400 200 O I —( q q q .— .. q .( u ... q I- q q q _. Nevents li11l1111111linLL111Ime I T I T r I I I l I I I I I I I I I I I I I I I p— i. h h .— p L P h P h p— b P h h .h —4 —2 O 2 77th Figure 3.6: The 1} distribution of the outgoing hard jets for the process pp —’ QQ —+ W qu —+ lllujj, corresponding to M D = 700 GeV and MW, 2 500 GeV for a luminosity of 100 f b“1. One can see that the events are in the central region: —2.5 < 7) < 2.5. The slight asymmetry in the shape of the curve is because we add the distributions corresponding to the jets from both the Q and the Q decays. We also impose PT cuts on the leptons and missing energy (Table 3.3). In re- constructing the heavy fermion nuiss, we have a choice between reconstructing the Q and the Q. We let one of them decay to VII", j and the other to Z, j with the ”7's and 86 Kinematic variable Cuts PT( jets) >100 GeV pT(leptons) >15 GeV |AR|(jets) _>_0.4 Missing ET >15 GeV njetsl ——<— 2-5 Ill“ 89 GBV< All) < 93 GeV T able 3.1: The complete set of cuts employed to enhance the signal to background ratio in the process pp —-v QQ ——+ Wqu —-> lllz/jj. Z ’ 3 further decaying to leptons. Since the leptonic decay of W involves neutrinos, it is more conveniant to use the Z. 3' combination (to avoid the two fold ambiguity in determing momenta when one uses neutrinos). One could simply identify the leptons that came out of the Z and construct the invariant mass of the lepton pair with the outgoing jet. Thus. we will impose the cut (M Z -— ‘2)GeV < M“ < (M Z + ‘3)GeV. \Ve present the complete set of cuts in Table 3.3 \Vhen generating the signal events. we, included 4 flavors of heavy quarks. We do not. consider the heavy top and bottom in this analysis. Including them would further enhance the signal, but since the top quark couples to the uneaten top pions. the branching ratios to gauge bosons would be different from that of the heavy partners of the first two generations. In F ignre 3.6, we present the invariant mass distribution (of jet. 41 2 leptons) for two dil‘fercnt values of M D (with Milt”: 500 GeV) with the cuts (Table 3.3) imposed. Since one cannot distinguish between the jets from the Q and Q decays. we. added the distrilfmtions corresponding to each jet. i.e.. the invariant. mass distribution was constructed by identifying the leptons from the decay of the Z and cornlnning that. with both jets separately and adding the two distributions. This enhances the number of signal events, but also creates the small off-peak events in the distril'mtions (Figure 3.6). We checked that. for the M D values we are interested in. these ell-peak events are not. big enough to compete with the signal. This can be directly seen from F igure 3.6 - the fluctuation around the signal peaks is just too 87 small to overwhelm the signal. 6000_-..,............ _ 5000:- — _ l . g : l @J I i > 3000: 1 O - . O : 1 g 2000i i z 1000} ; OtJ‘l—r. . l l—1.l'_rL'Ill—I. . . n . . . 1 _ _ . 200 400 600 800 1000 1200 All-"(Gem 100 ' ' ' r ' ' ' l ' r . I v r v I - "l 1 L T T 80: 4 <3 . j 8 - . v—1 60— .1 © t . % C . CD 40c ~ 8 i ‘ 2 got _‘ 0- . .— .M‘Tr‘l—I‘lu—‘I—rl—lll—l—H r-u. . . 200 400 600 800 1000 1200 MjutceV) Figure 3.7: Predicted signal invariant mass distributions Mll’ for MD 2 300 GeV and MD 2 700 GeV for a fixed MW’ 2 500 GeV. The sma off peak events arise because we added the distributions corresponding to the jets from both Q and Q decays. 88 In each of the plots, the signal distribution is clearly seen to peak at the value of MD. We estimate. the size of the peak by counting the signal events in the invariant mass window: (MD —- 10)GeV < Mfl, < (MD +10)ch. (3.82) To analyze the SM background, we fully calculated. the irreducible pp —+ Zl't'jj process and subsequently decayed the W and Z leptonitally. Once we imposed all the cuts discussed above on the final state [I l u j j, we find that the cuts entirely eliminate the background for the range of MD values of interest. to us. The most effective cut- for reducing the SM background is the strong PT cut imposed on both the jets. “7e find there is an appreciable number of signal events in the region of parameter space where Q —+ Vq decays are allowed but Q ——> V, q decays are. kinematically forliiidden. The precise number is controlled by the l_)ranching ratio of the heavy fermion into the standard model vector bosons. In Fig 3.8. we present. a contour plot. of the number of expected events in the MD —— MW; plane for a fixed luminosity of 100 fb—l. Since the SM background is negligible. we can take 10 events to represent a 50 discovery (this is the minimum number of events required to report discovery). Thus. we see that this process spans almost. the entire parameter space. However. as may be seen from Figure 3.8, the region where M D _>_ 900 GeV and “it" _<_ M D will not. yield enough signal events for the discovery of the heavy quark. In order to explore this region, we will now investigate the single production channel where the heavy quark decays to a heavy gauge boson. Single production: pp ——> Qq ——) lit/”qq’ —+ Wqu’ The. single production channel of heavy fermions is electroweak in nature. in contrast to pair production where gluon fusion is important. But. the smaller cross sections 89 l 200 .l .4 1000 50 10 l- 9 . D g 800 _ E 2 600 - 400. ..1...|...1.‘ 800 1000 1200 MD(GCV) Figure 3.8: Contour plot of number of events in the pair production case for a fixed integrated luminosity of 100 fb‘l. The shaded region corresponds to MW’ > 2M0 and is non perturbative and is excluded from our analysis, as discussed in the begin- ning of this section. can be compensated if we exploit the fact that the u and d are valence quarks, and hence their parton distribution functions do not fall as sharply as the gluon’s for large I (parton momentum fraction). Also. there is less phase space suppression in the single production channel than in the pair production case. Thus, we analyze the processes [11. u —> 11., U l, [d, d —> d, D] and [u., (l —> u, 1)]. These occur through a t channel exchange of Z and 2' (Figure 3.8). In Figure 3.9, we show the cross section for the single production of one flavor of the heavy quark as a function of the Dirac mass. Since we want to look at the region of parameter space where JWWI is smaller 90 than xiv/D. we let. the heavy quark decay to a W’. (One can also consider decays to Z’. The only (small) difference would be that. the Z’ does not. decay to a pair of it"s 100% of the time - the ideal fermion delocalization condition only makes the. T3 crmpling of the Z to SM fermions zero, but there is a small non zero hyrmrrflmrgc coiniling proportitmal to .r). The W, decays 100% of the time to a W and Z , because its coupling to two SM fermions is zero in the limit of ideal fermion deloealization (see Eqn.(3.39)). \Ve constrain both the Z and W to decay leptonically so the final state is lll_j_jET. u,d ‘ U,D Z,Z’ u,d , u,d Figure. 3.9: Feynman diagram for the, I channel single production of the heavy fermion via the exchange of the Z and the Z I bosons. As in the ease of pair production. we expect the jet from the. decay of the heavy quark to have a large. pp and hence we will impose a strong T’T cut on the hard jet. Also. as before, this jet is going to be largely in the central direction and hence one can impose the same 7] cut (Table 3.3) on the hard jet. W'e also impose the same p71 cuts on the leptons and missing energy as in Table 3.3. Also, we expect the 1] distribution of the soft jet to be in the forward region, 2 < Inl <’ -’1. And finally, we. impose a separatitm cut, AR, on the outgoing jets. \Ve presmit the. complete set. of cuts in Table 3.4. The. lel'itonie W decay introduces the. usual two fold ambiguity in determining the neutrino IllOll‘lelltllIIl and hence, we did a transverse mass analysis of the process, 91 1.00 r' (l70~ (150- l U(pb) (130 l 0.20 _ 0.15 i T rnJng L l 1 1 L A 1 l 1 l A l l l l l l 4 400 600 800 1000 1200 AhJGSCVW Figure 3.10: Cross section for the I. channel single production of the heavy fermion as a function of the Dirac mass M D- It is seen to fall more gradually as compared to that of the pair production case. defining the transverse mass variable of interest as: 2 211% = (\/M2(1uj) + pQTnzzj) + lpT(mi88)|) —- Wain) + ?T(miss)|2 (3.83) The signal to background ratio increases appreciable after applying the transverse mass cut. We expect the distribution to fall sharply at M D in the narrow Width armreximation, and indeed we find that there are typically few or no events beyond M D + 20 GeV in the (,listributitms. Thus, we have chosen the following cut on the tI'EIIlSVOI‘SO IIIHSS variable: (MD — 200)ch < MT < (MD + 20)GeV. (3.84) In Figure 3.9, we show a few exanmle transverse mass distrilmtions of the signal (with the cuts (Table 3.4) imposed. The distributions can be seen to fall off sharply 9‘2 Kinematic variable Cuts pT( hard jet.) >100 GeV PT( soft. jet) >15 GeV pT(lcpto1'is) >15 GeV |AH|(jets) _>_0.4 Missing ET >15 GeV lflhardjet $2.5 lnsoftjetl 2< I’ll <4 Table 3.2: The complete set. of cuts employed to enhance the signal to bz-ickground ratio in the process pp —+ Qq —+ W ’ q, q —> WZq'q ——> 1111/ j j . Ell. AID. We show a. contour plot of the number of signal events for an intergrated luminosity of 100 ftfl in Figure (3.12). It is seen that there are no events in the M D < MW, region because we allow the heavy quarks to decay to W, and hence are cmrsidering only the region M D > MW" Also, in the region of interest. one an see that there is an appreciable number of events. The SM background for this process, pp ——«: Wij —+ J'jl 11/1 . was (ZHltf‘lllttlt-‘étl surn- niing over the u. d, c, s and gluon jets and the first two families of leptons. Since we apply a. stong prr cut on only one of the jets (unlike in the pair prrxluction case), there is a non zero SM bt-ickground. We show the SM transverse mass distribution in Figure 3.13. The luminosity necessary for a 5 (7 (‘liscovery can be calculi-lied by requiring ("IV-signal / t / NBC) 2 5, as per a Gaussian approximatitm to a Poisson distribution. It is instructive to look at the results of this analysis by combining it with the previous pair prmluction case. as the two cover the M H” < MD and 111W; > M D regitms of the .ill-I‘WI -— M D parameter space respectively. Thus, we present a combined plot: of the required luminosity for a 5 0 discovery of these heavy vector quarks at the. LHC in Figure (3.14). One can see that almost the. entire parameter space is covered. with the pair and 93 E .4 i a 12 - T 10 r j :3 ' 1 8 . .. 8 s a @ C 3 31>, - . o 6; i o r -« Q . ~ 2 4 :- j l‘ 2 L 5 . C y b 0 1 1 1 L 1 7 1 1 1 1 1 1 1 4 1 1 1 1 400 600 800 1000 1200 MT(GCV) 14 _ r T ' . I I T r Y I ' ' I I T T v r .. 12 L T 10 r I '53 I i 8 t . 1—1 8 t‘ 4 © : I a r- .l o 6T i O r a Q . . Z 4 f j r . k -4 2 f 3 [ i 0 1 l 1 1 1 l 1 1 1 1 L 1 1 1 1 l 1 400 600 800 1000 1200 MT(GCV) Figure 3.11: The. transverse mass distribution for the single production of a heavy quark in the model, for M D 2 800 GeV and 1 TeV, for a fixed A ,W’ = 500 GeV. It is seen that the signal falls sharply at M D- 94 1200 1’ 1* T iii ff 1 I v I 1000 ‘ 1 q 9 i 0 g 800 10 § E 25 600 ‘ 50 100 400—111...I../.'T'$.I..111'1 400 600 800 1000 1200 MD(G6V) Figure 3.12: Contour plot of the number of signal events for the single production channel for an integrated luminosity of 100 fb‘ . The shaded region is where MW’ > 2M D and is non perturbative. One can see there is a considerable number of events in the low MW’ region of the parameter space. single production channels nicely complementing each other. Before we conclude, however, we would like to comment briefly on how our analysis compares with other models with vector quarks. 3.7 Related Vector Quark Models There are other BSM theories that feature heavy quarks with vector-like. couplings, as in the present model. In this section, we would like to briefly explain how our phe- 95 35 I r I—I I fifi I I ‘r I I I l I I I I f E i 30: 1 Le 25? —i s 2 2 '51 2°: 3 E 15;— -; o ; 2 § 10; j 5 3 0 ' m L 1 1 1 1 1 1 1 1 1 1 L L 1 1 1 1 1 400 600 800 1000 1200 MT(GCV) Figure 3.13: The SM background for the single production channel, pp —+ ”'ij —+ jjlull, calculated by summing over the u, d, c, s and gluon jets and the first two families of leptons, and with the cuts in Table 3.4 imposed. The bin size is 20 GeV. nomenological analysis con’ipares with these. One import-(mt feature of deconstructed Higgsless models of the kind discussed in this paper is ideal fermion (.lelocalization, which does not allow the heavy gauge bosons in the theory to couple to two standard model fermions. This constrains the W, to decay only to W and Z, thus providing a tool to distinguish this class of models from others. There are. however, certain features of this model that are generic, like the vector nature of the heavy quark couplings. In the context of Little Higgs Models [63], there have been studies of the LHC 1'.)henomenology of the T-odd heavy quarks [64]. The cross sections for the production of heavy T -quark pairs are comparable to the ones in our study. However. in those models, the heavy T-quark necessarily decays to a heavy photon (due to constraints of conserving T parity). Also, in [65], the authors study the pair production of 96 1200 1000 800 M W (GeV) 600 L 1 l 1 1 l 400 600 800 1000 1200 MD (GeV) Figure 3.14: Luminosity required for a 5 a discovery of the heavy vector fermions at the. LHC in the single (blue) and pair (red) production channels. The shaded portion is non perturbative and not included in the study. It is seen that the two channels are c()niplementary to one another and allow almost the entire region to be covered 111 300 f1; heavy partners of the 1st. and 2nd generation quarks in the context of the Littlest Higgs Models (66, 67, 68, 69, '70]. They consider decays exclusively to the heavy gauge bosons in the theory, which then decay to the standard model gauge bosons plus a heavy photon. Thus, the final state. though still ”(ij T, is kinemat.ically different. In particular, strong cuts on the missing energy are. now an important. part of the analysis, because part. of E T is due to the heavy photons. Kelli?” presents a cmnprehensive study of the. production and decay of heavy quarks by separating out. the partners of the 3rd generation from the others and analyzing them separately. The authors let the heavy quark decay to a SM W boson and a light quark, but in their analysis, they neglect the mass of the W boson compared to its momentum (since. it is highly boosted). Thus, when the W decays to a 11/ pair. the directitm of the neutrino momentum can be approximated to be parallel to that of the charged lepton, which enables them to recontruct the full neutrino momentum and create a invariant mass peak. for the heavy quark (as opposed to a. transverse. mass analysis). In the context of the three site model, the authors of [72] consider the single production of the heavy top quark. As mentioned before, the heavy top in this model is necessarily around a few TeV’s and the paper concludes that the most viable scenario at the LHC is the subprocess qb —+ q'T ——+ q’ll'b with the H" de 'aying leptonically. Ref. [73] presents a model independent analysis of the discovery prospeds of heavy quarks at the Tevatron. The authors write down generic charged and neutral current interactions mixing the heavy and the light fermions and proceed to analyze both the pair and single production of these heavy quarks, with decays to the SM gauge bosons. Understandably, the Tevatron reach is much lower than that of the. LHC. 98 3.8 Remarks In this chapter. we presented a. minimal extension of the three site model to lift the constraint that exists between getting the top quark mass right. and having the p parameter under experimental bounds. This we did by separating the third generation (marks from the light ones, and having the top quark mass arise from the vev of a "top Higgs’. This enabled us to have additional vector-like quarks in the model that are light enough to be discovered at the LHC, without afi'ecting the tree level couplings of the three site model too much. We encoded the model in CachEP and analyzed the pherunnenology ol' the heavy quarks. We first considered pair product ion (pp -~+ (2Q ~> Wij -—~ lllujj) of these heavy fermions. We found that the 50 reach of the pair prt'.)duction channel was a 1 TeV, but for high Dirac masses, this channel. is viable only if the W’ mass is in the TeV region also. This is because. in the A] D > MW; region, there is the possibility of heavy quark decay into W’ and the signal is don’iinated by the branching ratio of the heavy quark into a W and light quark. The single production channel (pp —> Qj -+ l"l»"jj —+ ”’ij ——* lllz/jj) complements this nicely because. we choose to decay the heavy quark to a W, and hence are. necessarily in the region AIWI < A! D' By combining both these analyses, we were able to cover most of the M D — MW; parameter space between MD z 300 GeV and M D m 1200 GeV. We conclude that the reach at the LHC for the vector quarks in this theory can be 2 1.3 TeV (for a 50 disctwery) for an appropriate choice. of til/”,4. In doing the pheru)menology, we have implicitly assumed that the value of the fermion n'iass M D could be anything higher than the experimental lower bounds. But clearly, in low energy effective theories like the one. discussed in this chapter, the .Iuasses of particles cannot be. arbitrarily high, since the theory itself breaks down at some scz-rle. Thus, it. is interesting to see whether we can determine an upper bound 99 E t. for the fermion mass scale. We will address this question in the next chapter. 1 ()0 Chapter 4 Unitarity and Bounds on the Scale of Fermion Mass Generation The mechanism of electroweak symmetry breaking must give mass to two very dif— ferent classes of particles: the electroweak gauge bosons and the fermions. In the standard model. the scalar Higgs doublet couples directly to both classes of particles. l\-loreover, the gauge and Yukawa couplings through which the Higgs interacts, respec- tively, with gauge bosons and fermions are proportional to the masses generated for those states when the scalar doublet. acquires a vacuum expectation value. Nonethe- less, in considering physics beyond the standard model. the possibility remains that the gauge boson and fermion masses are generated through different mechanisms. In particular. it is possible that electroweak symmetry breaking is transmitted to the fermions via some intermediary physics specifically associated with fermion mass generation. This Chapter is based on work published in [74]. Appelquist and Chanowitz [75] have shown (see also [76. 77]) that. the tree—level, spin-(l scattering amplitude for fermion-anti—fermion pairs to scatter into longitudiimlly- polarized electroweak gauge. bosons grows linearly with energy below the scale of the. phxg'sics responsible for transmitting electroweak symmetry breaking to the fermions. 101 As the amplitude must. be unitary, one can derive an upper bound on the scale of fermion mass generation by finding the energy at which the amplitude would grow to be of order 1/2 . The rate of energy growth is proportional to the mass of the feri’nions involved. The most stringent bound, therefore. arises from top—quark annihilation. and the bound on the scale of top—quark mass generation is found to be of order a few TeV.For light fermions, the scattering of fermions into many garuge-liiosons yields a stronger result than the Appelquist-Chanowitz bound [78, 79]. For the top-quark, howwcr, two-body final states yield the strongest bound. As enmhasized by Golden [77]. the interpretation of the Appelquist-Chanowit7. (AC) bound on the scale of top-quark mass generation can be problematic: longitudi- nal electroweak gauge-boson elastic scattering itself grows quadratically with energy [80. 81, 3, 4. 82] below the scale of the physics responsible for electroweak gauge-bosrm mass generation. As the scale of the physics responsible for electroweak symmetry breaking is also bounded by of order a 'l‘eV. it can be difficult to be sure that the violation of unitarity in fermion annihilation is truly independent of the violation of unitarity in the gauge-boson sector. The standard model illustrates this difficulty. as in that case the Higgs boson is responsible for restoring unitarity in both the fermion annihilatitm and gauge-boson scattering processes. In this chapter, we will discuss unitarity violation and the resulting bounds on the scale of top-quark mass generation in the context of deconstructed Higgsless models. It is straigl‘itforward to generalize the three site model to an arlfltrary number of sites [83]. In the continuum limit (the limit in which the number of sites goes to infinity). this model reproduces the fix-"i-dimensional model introduced in [51]. A fermion field in a general compactificd fiVC-(lllllt¥IlSl0Hfll theory gives rise to a tower of Kaluza-Klein (KK) modes, the lightest of which can (under chiral boundary conditions) be massless in the absence of electroweak symmetry breaking. The lightest. states can therefore be identified with the ordinary fermions. The massive Kaluza— 10‘2 Klein fermion modes are. however. massive Dirac fermions from the four-dimensit)nal point of view. Correspondingly. the fermions in a deconstructed Higgsless model include both chiral and vector-like electroweak states [30, 83]. and generation of the masses of the ordinary fermions in these models involves the mixing of the chiral and vector states [84, 50]. As we will demonstrate, the scale of top-quark mass generation in these models depends on the masses of the vector-like fermions (the “KK” modes). as well as on the number of sites in the deconstructed lattice. What is particularly interesting about deconstruct-ed Higgsless models, in this ccmtext. is that one can. distinguish between the unitarity—derived bounds on the scales of gauge-lumen and top-quark mass generation. We will demonstrate that. for an appropriate number of decrmstructed lattice sites, spin—t) top-quark annihilation to longittudinally-polarized gauge-bosons remains unitary at tree-level up to energies much higher than the naive AC bound if the vector-like fern'iions are light. However the AC bound is reproduced as the mass of the vector-like fermion is increased. 'l‘herefore. for fixed top—quark and gauge-boson masses. the bound on the scale of fermion mass genmation interpolates smoothly between the AC bound and one that can, potentially, be much higher as the mass of the vector—like fermion varies. The unitarity bounds on elastic scattering of longitudinal electroweak gauge bosons in H iggsless models [85]. however. depend only on the masses of the ga.uge-boson KK modes. In this sense. the bound on the scale of fermion mass generation is in(leperi.t1erz.t. of the bound on the scale of gauge-boson mass generation. While our discussion is restricted to deconstructed Higgsless models. many mod- els of dynamical electroweak symmetry breaking incorporate the mixing of chiral and vector fermions to accommodate top-quark mass generation. Examples include. the top-quark seesaw model [86, 87, 88] and models in which the top mixes with composite fermions arising from a dynamical electroweak symmetry breaking sector [89. 9t), 91]. Indeed. the fermion delocalization retutired to construct a realistic Higgs- 103 less model is naturally interpreted, in the context of AdS / CF T duality [19. 2t). 21. ‘22], as mixing between fundamental and composite fermions [92]. As chiral—vector fermion mixing is the basic. feature required for our results, we expect similar effects in these other models. In the next section, to set notation and make contact. with the literature, we reproduce [T7] the Appelquist—Chanowitz bound in the electroweak chiral Lagrangian [93. 94, 95, 96, 97] - which may be interpreted as a “two—site" Higgsless model. In section three, we introduce the n(+2) site Higgsless models that we will use for our calculations. Section four contains our calculations and primary results. The last section sunnnarims our findings. 4.1 The Appelquist-Chanowitz Bound In the standard model (SML the helicity non-conserving process I +1- + ——> H"; WI rot-renters contrilmtions at. tree level from the diagrams in Figure 4.1. We are interested in the behavior of the amplitude for large center of mass energy. x/Z >> MW-mt- This allows us to expand the amplitude in the small parameters ”121/" and mfg/5. Practically, this means that we use the following leading order am)r<;)xi1nat,ions. For the longitudinal pt.)larizat.ion of the H" gauge bosom we use k“ W" :II' N L 4 1 v ll L _ 4‘ Ill' 3 ( I ) I. . v . . . where A'lVI is the tour-moment,um of the corresponding boson. For the spinor chain in the .5 channel. we use m (n - #2) out + 91?, Pu) n+ z nu fcosfi (u, +91?) (4.2) TL. (#1 — 5(2) (gLPL + 9RPR) u,“ 2 —mt\/;cos9 (9L + .01?) , (4.3) 1 ()4 7 + w; t+ WE“ b Figure 4.1: The diagrams that contribute to the process t+T+ —* WE WE in the Higgsless SM. There are analogous diagrams for the process LL, ——+ WEI/I71": . Each diagram has an amplitude that grows linearly with \/3 for all energies. However. most (but not all) of this linear J3 growth cancels when the diagrams are summed. The remaining piece that grows linearly with J; comes from the t channel diagram, and it. eventually surpasses the unitarity bound. In the SM, this unitarity violation is eliminated by the contribution of the Higgs in the 3 channel. where It}: and HZ” are the momenta of the outgoing bosons, and for the spinor chain in the t channel we find I? ’ t. s . fill—5i (1 + cos 0) 51L (4.4) nun/E Tabb/1 '" 651) [fill/L I’L"+ 'fi—lf‘zll/l‘lelflLPLU— 2 — 2 "(1+Ct‘eHML (4.5) where ) 1 11. = 30—2/5) (40) l p. Pu = 5cm) en) are chirz-rlity projection operators, and {IL and 91?, are chiral electroweak coupling constants. Since the It ——> W+W_ amplitude is the same for each color and only differs 105 by a sign for the opposite helicity, we get the largest. amplitude by considering the incoming state , 1 - _ - _ It") = %( |t1+t1+) + l”2+’2+> + "3+’3+> (*1-8) — |{1—"1—>- If2—t2—>— It’s—en) ) where the numerical subscripts (1,2, and 3) label the three. different; colors. The state we consider here. differs from that chosen by [7.3 as we include both comifinations of incoming helicities. This state allows us to derive a slightly stronger bound, c. f. Eqn (4.23). Putting the pieces together gives the scatterii’ig amplitude c c n -' \, 1m“ .sros ‘ _ 2 M = 2 “2 X (29217.93. w W + thzzgzwu' + .qthyZL-VW — gum!!!) ‘ W V/t3 inn/s 3 4t “fr—Hum" ~ ( "3) zilw for \/§ >> MW, mt, where the electroweak crmplings are given by: gtte, = 3‘9 . (4.10) yawn? = f , (4311) QLUZ = sin flit/(cos ”W (i _ 38in? 6”,.) i (4.12) HRH Z = sin (in/“cos 6W (—§s11120w) , (4.13) .(Izww = 2%}???— . (4.14) .‘JL-th'V = W - (4.15) Our exln‘ession here differs in the sign of the term proportional to gifbtt” from that given in {75]. and is correct for the top-quark which is the T33 = +1 f2 member of an electroxwak (,loublet. The (_~.or1‘(~*spo11ding expression in [75], which is from [98. 99), is 106 Figure 4.2: The diagram that contributes linear growth in M3 to the process 1+ t+ ——> + - ~ - 1. . - . 7r ' '7: in the Higgsless SM, where we have used the eqtuvalence theorem to replace the longitudinally polarized gauge—boson by the corresponding “eaten“ Goldstzone Bosons. There is an analogous diagram for the process t_t__ —-> n+7?" correct. for the lower member of an electroweak doublet with T3 = ~1/2. With these couplings. we find the identity 0. a 29m!1»,ww + {ILIIZgZWW + 912/ l.Z.(lZl'VIV — gmw = 0 - (4-10) The Hunt-lining armilitude is, therefore, ‘ . . 2 ”Lth’ - ' Zulu, which grows linearly with J: for \/s >> 1“"1‘47.nlf. We note that since (Jump 2 g/ \/2 and It I”: = gr / 2. where g is the weak coupling and e 2 246 GeV is the weak scale. our expression for M simplifies to [7.3] ff- . Mzw. (4.18) 1! We. can check this result using the (xiiiivalence theorem [3. 100:}. where one re- places the longitudinal gauge—bosons by the corresponding “eaten“ Nambu-Gohlstone Bosons. In this limit... the only diagram that contributes to the .l 2 0 amplitude is shown in Figure 4.2. The leading order approximations {411+ 2 VG emu... 2 —\/§, (4.19) 107 coinlnned with the four point coupling mt ,, .‘Imfirn— — Tr? (4.20) yield the same amplitude as in Eqn. (4.18) M = M . (4.21) .Iffi Note that the potential s-channel contribution, illustrated in Figure 4.3, does not contribute in the J = 0 channel. 1+ 7? {I}. \ 77— Figure 4.3: This diagram, corresponding to s—ehannel Z—boson exchz‘tnge in the eqilivalence-theorem limit, does not contribute to the J = 0 partial wave. scatter- ing amplitude for the process l+lt+ —> n+7r'“ in the Higgsless SM. The J 2 0 partial wave is extracted from Eqn. (4.18.) as (4.22) 1 1 win/Gs (l = —'— (ICOS 0 1M :2 —— 0 3271 /—1 167m2 To satisfy partial wave unitarity. this tree-level amplitude must be less than 1/2, the maximum value for the real part of any amplitude lying in the Argand circle. This produces the bound 2 8 t‘,’ W “V 3.5 TeV . (4.23) 'Int\/6 N Our result differs numerically from that given in [75]. as we include both helicity x/ES channels in Eq. 4.8. and bound the. amplitude by 1/2 rather than 1. One may obtain a slight ly stronger upper bound by considering an isosinglet, spin—(l. final state 108 Figure 4.4: Moose [25] diagram of the n(+2) site model. Each solid (dashed) circle i'e[.')resents an SU(2) (U(1)) gauge group. Each horizontal line is a non-linear sigma model. Vertical lines are fermions. and diagonal lines represent. Yukawa couplings. '9 (I = .1 = 0) of ga.uge-l_)osons [16). This amounts to a reduction in the value of the upper bound in Eqn. (4.23) by a factor of \/’2/1 x 0.8. 4.2 The -n(+'2) Site Deconstructed Higgsless Model We will be studying the Higgsless model introduced in [83]. denoted the. n(+2) site model. As we will discuss in subsection 4.2.1, the gauge sector is an HI ;'(2)"+1 x (1(1) extended electroweak group; the label. 11 thus denotes how many extra 811(2) groups the model contains relative to the Standard Model. The electroweak chiral lagrangian [93. 94. 95. 90. 97] can be obtained by setting n = 0 while the Higgsless Three Site Model [30]. which has one extra SI,.-”'(2) group. can be obtained by setting n = 1. This model may be schematically represented by a “.\’loose“ diagram [25] as shown in Figure 4.4. After discussing the gauge. sector, we examine the fermion sector (subsection 4.2.2). the “eaten N ambu-Goldstone bosons” (subsection 4.2.3) and then the couplings that. are relevant. to our calculation of If —» ll"+W_. 4.2.1 Gauge Boson Sector The gauge group of the n(+2) site model. as illustrated in Figure 4.4. is n (z = 311(2)U x H 31/12),- x U(1)...+1 (4.24) i=1 109 where Sl-"(2)0 is represented by the leftmost circle and has coupling g; the gauge groups 1875(2) ' 7 are represented consecutively by the internal circles and have a com— 1non coupling g. Connnon couplings for the “internal” SU (2) groups corresponds to a continuum model with spatially independent. gauge-coupling [51]. Qualitatively. our results do not depend on this assumption and should apply in any case in which the mass of the lt-boson is much less than that of the first gauge-boson KK mode. The U(1)” +1 is represented by the dashed circle at the far right and has coupling g’. The coupling {1 is taken to be much larger than g, so we expand in the small quantity ...? ll GHQ A 1;. to 2.1 v We also find it convenient. to define the parameters (4.26) 2 9 . . . . where .s“ + c = 1. In the continuum limit. It ——> 00, this model reduces to the. one (jlcscribed in [-51]. The horimntal bars in Figure 4.4 represent nonlinear sigma models 27- which break the gauge symmetry down to electromagnetism 7 —+ U(1)}.jg” (4.27) giving mass to the other 3(rt + 1) gauge bosons. To leading order. the effective Lagrangian for these fields is 2 em = [Li—Tr Z (r),,.xj)l1)“z:j (4.28) 3' 110 where “#2,; = 19,127- +ig‘jl1/j‘flgj - ”Zia—133‘ 1134,14,, (4.29) with go = g. 93- = f] and gn+1 = g’. The nonlinear sigma model fields may be written 5271'; . 27' = c! [/f , (41.30) in terms of the Goldstone bosons (7rj) which become the longitudinal con‘iponents of the massive gauge bosons. The 7rj and W]- are written in matrix form and are 1 0 1 + 571'; —7r - ”J, 2 ~ J fl J (4.31) _1/__7r— -17.?) v2 1 9 J 11101112+ W111- : i ’1: Vi 3" (4.32) 7‘11): 2”}1: 1 0 w 0 W'nJrLu : 2 PH“ (41-33) ' 0 4,110 The. mass matrices of the gauge bosons can be obtained by going to unitary gauge (SJ —+ 1). For the neutral gauge bosons, we find ( x2 —.1: 0 11 - 0 (,1 ) -—:Ir 2 —1 0 - 0 0 .112 _ 914]“? 0 —1 2 —1 - 0 t) (4.34) —1 0 0 0 0 - —1 2 —:rt ( 0 0 (1 0 41 212 ) while the matiLx Vt 101 the (harged gauge bosons is 111,7 with the last 10“ and 111 column removed. The photon is massless and given by the wavefunction Qll’b A H'H ,..; h—J HIH 1" v ,r—x p... 1.40 g,‘ v where. 1 1 n 1 . _=_,-+—‘+—T-, 4.30 .112 .512 .11” ( ) After diagonalizing the gauge boson mass matrices. we find that the other masses and wavefunctions are given, at leading order in 1', by the following expressions. The mass and mwefunction of the. light W boson are (7 1'4!“ ._ 11.1 .. : _-—-_ 4.31 11 0 2 (n, + 1) ( ) ((11.0 —_— 1 (1.38) . 'j _ Ir) — j + 1 ,.. u L"!‘,’() — T? + 1 J" (40'39) where the superscript 0 refers to the left-most 91(2) group on the moose while the superscript j = [1...11] refers to the SU(2) gauge groups on the interior of the moose. The masses and wavefunctions of the charged KK modes are (7f 1371' , 111,,' = 44—— 1— ' .‘ 4.40 11 k [2 ‘0“ [12+ 1] t ) 0 —.'1,‘ 1177? ‘1: 7 ' 1.: —— ('01, — 4511 ‘1' t 21n+11 be + 111 1 1 1' '111 ’1'] . = . 2 sin J T . (4.42) l’l’k 11 + 1 n + l Likewise. the mass and wavefunetion of the light Z boson are in M = —-————— (4.43 20 2e (71. + 1) ) '1 1%0 = e ( 4.411) ”j _ (:(n + 1) — j/e , :- L-ZO —— n + 1 .I‘ (4.49) 1 7}?)- 1 = —— s, (4.46) v I where, superscript T) + 1 refers to the 1,-(1) group. The masses and wzjm3funet,ions of the neutral KK modes are. ” k1 A1211? = 15% 1. —— COS [n11] = Alurk (4.47) 0 —1“ kw ’t’r , = ——-——————..__ cot, —— 14-48 1’” \/2(n + 1) [2("+1)] ' ) . r) "k 7 a . J 77 , = q 4.49 721“ n+1s1n [n+1] ( ) . 2 (—1)k :1: _ ,II.+ _ , . r '21- —— 11 +1 1‘. [(11+l)al+ b1] (4.o()) PUk 2 km , = —— ': —— 4.51 ‘” «n+n‘“ 2m+i) (’1 . k ‘ 11“ 1 \< (—1)" sin 7r —12sin ”T 11+ 1 n + 1 —1 1'71 1 ‘2 (0 [201 +1)] ( ) ) We note that the W gauge boson mass is given by 1111“," = 1111170 E g] —' 9—0 (4.53) mM+1“2‘ and. hence, we have the I‘t‘l'dl ion f = \/n +1 11 . (454) 113 The ratio of the W and Z mass is MZ .1120 c. identifying c with cos 9W at. leading order in '1'. The ratio of NW to the mass of the first KK mode Mil-'1 is 111 r ;. Vt ____ T (4, 111 ,7 ”1 \/2(n. + 1) (1 — cos fig-Ii) which relates :1.’ to the mass ratio MW / MW1 for a given 11. at leading order. From 0) CI! this we see that. expansion in '1‘ is justified as long as Mwl >> MW. 4.2.2 Fermion Sector The vertical lines in Figure 4.4 represent the fermionic fields in the theory. The vertical lines below the circles represent the left. chiral fermions while the. vertical lines above the circles are the right. chiral fermions. Each fermion is in a fundamental representation of the gauge. group to which it is attached and a. singlet. under all the other gauge groups except 1.1(1)n+1. The charges under U(1).n+1 are as follows: If the fermion is attached to an 811(2) then its charge is 1 / 3 for quarks and —-1 for leptons. If the fermion is attached to U(1)n+1 its charge is twice its electromagnetic charge: 0 for neutrinos. —2 for charged leptons. 4 / 3 for up type quarks and —2/ 3 for down type quarks. The fermion mass terms can be written down by extending the one for the three site model. Eqn. 2.2, to include 12 bulk terms. 114 52.523 = =11}? [ELIELozti'UC’Ri ~ZLTILJ'U’RJ 14-57) J j where the value of e L is the same for all fermions. while 5R is a diagomil matrix which r l (..listinguishes flavors 130. 83]. For example for the top and bottom quark we have 5 . 0 , ER: Rt (4.58) 0 5R1) The. fermion mass matrix can be (‘liagonalized by performing unitary t ransforma— tions on the left- and right-handed fermions separately. To leading order in e L. I? we find the following masses and wavefunctions for the lightest fermion. F0. in a given tower (which we associate with an ordinary standard model fermion) 1111.10 2 MFELERf (4.59) 1.1%,?) = 1 (4.611) .1in = 5L (4.151) tam) = st (4.62) 2117?;3 = 1 (1.11:1) while the expressions for the heavier states, Pk» are - —- k 1 ill [17k = QAIF ('08 [Kim—iii] ' ‘II 0 EL (n —- k +1)7T >— .I‘ _, —- _._—..____ t‘ —-—————-—-—— 4.0:. (ka V2” +1 an[ Zn +1 ( )) - 2-13' 2'a—k+1 ,. 1’11: = ———,____( ) sin [ _7(n{ )W] (4.66) k \/2'n. +1 211 +1 7' (-1)”+"+j+12 . 2(n—j+1)(n— k+1)7r 't“ = ,_____ sm RF}: V212 + 1 2n. + 1 (4.67) k . (-1) 31: .. —k 1 ”(Jig-Pl : .___—f tan Lil (468) I.- fiFl—l 2n. +1 For small EL. we see that. the left-handed component of the lightest. fermion in each tower is primarily located at site 0 - and the flavor-universal factor 5 L controls the. amtmnt of fermion “delocalization” along the moose. Likewise, the right—handed component. is primarily located at. site 71+ 1, and the flavor-deptindent quantities a R f control the degree of (‘lelocalization Since the amplitude for If ——+ W+W— scattering will depend on the values of 5 L and E R t’ we need to evaluate these quantities: we will start with E L and then use it to constrain ER]. Precision electroweak (;-.()rrect.ions provide a useful source of constraints on the parameters of Higgsless models. While custodial symmetry generally keeps the t ree— level value of Ap = oT sufficiently small, satisfying the bounds on S at". tree level requires some degree of fermion delocalization [101, 102. 84. 50. 51. ‘26, 103, 49]. We will implement ideal fermion delocalization exactly as in chapter ‘2, i.e.. by demanding that the tree—level value of gww equal the SM value. An explicit. calculation of my”, in this model. which requires expanding the wavel‘unctions, masses. and couplings to order 5% and order 1'2, yields 116 'n(-n.+2) 2 1i. 2 , .v, = ,.,r ,, _——— — —€ . 41. 9” “’n 9” WSM (1+ 6(n+1)‘r 2 L) f (’9) Tl‘ierefore. the condition 9 n + 2 2 E“- = ——-——-—.’L‘ 470 L 3(n + 1) ( l ) causes 3 to vanish at tree-level. Using Eqn. (4.56) this is equivalent. to ' ‘ A12 I 2 2 ‘ _ _ 7r l'l' _. €'=—(n+2 (l—c0s[ ]) .. , 4.11) L 3 l n +1 Mfi-l ( in terms of physical masses. Here again, note that E L is small so long as M W << .91 I W1- Finally. the parameter s R f can be determined by taking the ratio of the masses of the light fermion and the first KK mode. "UFO _ :1st Mn 2 cos [Q—Tg: ] Since we know 5L, this gives a prediction for 5 R f in terms of physical masses x/Gcos [sf—17:7] MFO MWI , —,. R _ = . (4.1.3) ' ..f . M ill ,. ,.r \/(n + 2))(1— cos lfiID F1 H For all flavors except the top quark, this parameter is tiny; at. leading order. we (I) therefore set. 5 = t) for all the light fermions. The size of ER, affects Ap at one ”1‘ loop; comparison of the exI‘)erimental bounds on Ap with the value calculatml in Higgsless models [30, 83] shows that aRt must also be relatively small. In what. follows. we therefore keep only the leading terms in ERt. 117 4.2.3 Goldstone Boson Sector We will perform the computation of the process t+f+ ——’ WE WI: in the. n(+~2) site model using the equivalence theorem. We must. therefore. determine the wzwefunc- tion of the Goldstone bosons associated with (eaten by) the. massive gauge bosmis. This is determined by the mixing between the two given in Eqn. (4.28). To find the mixing, we expand the nonlinear sigma-model field Ej and keep the terms linear in both the gauge bosons (llj) and the Goldstone bosons (79-). After these manipula- tions. Eqn. (4.28) becomes cm- = 49;; {8“qu ., 7- W6‘ — wf’} (4.7.4) "—1 . . ru _ . x!- + 21:1 {WI = ”‘2' Wm} . 1. . 1 +{al’7r‘n * ”ft — It ill/2+1} from which we may read off the wavefunctions for the charged Goldstone bosons as [01 1 ( 0 1 > —- ‘t’ = 1‘?! .r —7-‘ , 4. t ”it: Na + u A. u k f t ’l ' k ,- 1 - .- , .,.l./:lt = _’V (1%1 — (a); I) (4.76) Wk 1 Wi k A.” k lnl 1 n , —. .1, __L = Y “H" . (4.71 ) Wk IV”? 1.. where the N}.— k are. normalization factors. Note that. Namlm—Goldstone boson compo- nents are associated with the links rather than the gauge grtmps: the superscript [0] refers to the left-most link, the superscript [n] refers to the right—most link, and the siuierscripts [j] range from 1 through n—1 and denote the interior links of the ;\'Ioose. 1'18 The wavefimetions for the neutral Goldstone bosons are similar .135 ._. r17, (. 9k _ 117%) (4.78) ‘1: (Egg) 2 N:0 ("é/r —1'%:1) (4.79) ' ‘k Flt—Iii : it; (sz -— art 192:1) . (4.80) ' k but note that 12:}; includes a contribution from the Z k wavefunction on the U (1) site. k These wavefunctions are particularly simple for the. lightest modes, the W and Z: they are flat 2 =1, )) (4.s1) with the same value on all links [I = 0...n] of the .\rloose. 4.2.4 Couplings To obtain the couplings of the Goldstone bosons to the fermions, we start from Eqn. (4.38). expand the nonlinear sigma-model fields. and plug in the eigenmode wavcfimetions we. have just. derived. Doing this, we find viz-Mr 0 H1 [2 .‘ILtFkvr I —"—— f [L’LWREk 7: +2: Lt k‘t‘w ,n+1 [n] +vallfIIfIk’n k Ill/QMFEL tan [—(n _ k + 1W] (4.82) 1) = —1 ( )\/‘2n+1(n+' 2I2+1 119 .x/ZM (i 1 III-1171 f/RtFkIr = *7- f SU’Lle Ref? +24" IPA RI ’Ir . . v+1 11‘ “if [”12sz I)?“ lift {l = IfiA'IFER tan [(n (— f.‘ +.1)7r:| (4.83) \/2H+1(n + 1)c 2n +1 LIF f0] )2 +1 ’l '2 ‘qtffi’ffl— 2 f2 >3[ L7 Lt Rtva + Elwin (7’71" ) z - . nl .- +Enfzxztcgtf1(1rir)2] In, . = ——\—— . 4.84) (II + 1)'1I2 ( ’ Here we have denoted the lightest fermions (previously denoted FU) by t. and b. as apIiropriate. while leaving the corresponding KK modes as F k (which. t.o leading order in 5 L. R. have the same properties for all quarks). Note that. the four point vertex has an extremely simple form. and vanishes in the limit n —> oo. 4.3 Unitarity Bounds on if. ——> W’LWL The diagrams that contribute at. tree level to t+f+ —> lift/171: are shown in Figure 4.5. \Ve are again interested in the behavior at large energies. so we expand in the small parameters M 121 /s and th/s; we also include all colors and both helicity polarizations in a coupled channel analysis (Eqn. (4.8)). The calculation is most easily performed using the equivalence theorem [3, 100]. Again, as in the SM (see Figure 4.3) the potential 3- -ehannel diagrams do not c.ontribute to the l:- 0 amplitude and the only diagrams that contribute are shown in Figure 4.6. 120 I+ w Figure 4.5: the process {4.1-5+ -—+ 11’3”"; in the n(+‘2) site Higgsless model. There are. analogous diagrams for the process Lt: —> WEWE . As in the SM. most of the. linear growth in \/§ will cancel. All the persisting linear growth in v37 comes from the f chainiel diagrams. [.+ , 77+ / / / \ \ _ \ (.1. \ 71'— L+ , 7T+ / / Fl: \ _ \ __ t+ \ 7T Figure 4.6: Diagrams contributing to unitarity violation at high energies in the process for far -—> 7r+7r_. There are analogous diagrams for the process LIT- —» 7T+7T_. The top diagram grows linearly with \/§ for all energies, whereas the bottom diagrams only grow with \/s up to Mpk , after which they fall off as 1/ \/§. 121 The scattering amplitude arising from the diagrams in Figure 6 is F— MP ‘lLtF’ ‘lRtF _ a . k‘. kn. 1‘ A77]- , 3 M — V6.5 .qmirfi- —— E f_ 112 (4-65) It ' ‘ Fl: where the couplings are given in Eqns. (4.82) ~~~ (4.84). The ._I = 0 partial wave can be extracted as 1 1 _ (20 = m ‘1dcosfiA/l (4.86) \/6 k \/E M Fr V W] re re 1 . g(.I:) = $1110 + 11:2) (4.87) This partial wave must be less than 1/2 to maintain unitarity. giving a bound on J3 and/or 1111.11. We have plotted this bound in Figures 4.7 and 4.8 for n = 0. 1. 2. ~ - . . 1t). 20. 30 and 00. The n = O bound corresponds to the original AC bound of Eqn. (4.23). We see from these figures that there are two important domains corresponding to dill‘erent ranges of values for M F1. In the first. domain. where M F1 S 4.5 TeV, we. find that unitarity can be satisfied up to very large energies. In this limit. we find that the. I. channel diagram becomes irrelevant and the process is controlled by the four point vertex (Figure 4.6). For the lmvest fermion masses, M F1 << 4.5TeV. we lind (10 2 wfiit 1) S % (4-88) which gives the bound \/§ S (n + 1) 3.5 TeV (4.89) 50 . w 40* ‘20 30 n=oo f; 30_ 10 G) E", I; 20 10' I 0 4|; 1 2 4 6 8 10 MF1(TeV) Figure 4.7: The scale where unitarity breaks down in the helicity nonconserving channel in the n(+2) site model. Unitarity is valid in the region below and to the left of a given curve. The bottom—most curve is for n = 0 and is the AC bound. The line directly above the bottom one is for n = 1 and corresponds to the Three Site Model. The line directly above that is for n. = 2 and so on until 11 = 10. The line above that is for n = 20, the line to the right of that is for n = 30 and the line to the right of that is the continuum limit (71 —> 00). We find that unitarity breaks down if either «.3 is large or [111.11 is large. If M F1 is large, then unitarity breaks down for J; very close to the AC bound. On the other hand, if M F1 S 4.5 TeV, unitarity can be valid in this process to very high energies, with the precise value depending on the number of sites In. 123 25 . . . + 10 20 30 11:00 20* 9 15» (D 5'1 1% 10 5* ‘ , ,,,,, 7 0 2 3 4 5 6 MF1(TeV) Figure 4.8: Expanded View of low (fl region of Figure 4.7. In this “low” KK fermion-mass region, unitarity is valid to approximately (11+ 1) times the AC bound. In the second domain, where M F1 > 4.5 TeV. we find that, for all n. unitarity breaks down at a value of \/§ given approximately by the AC bound (Eqn. (4.23)) In Figures 4.7 and 4.8, we see that at M F1 ~ 4.5 TeV, the curves corresponding to small 71 approach the n : 0 curve, while the curves for large 71 turn back on themselves, defining a wedge-shaped area in which unitarity is always violated starting at \/§ of order a few TeV. To understand why M F1 2 4.5 TeV is the fermion mass value at which the theory crosses from the first to the second domain. we consider what happens as n —I 00. In this limit, the four point vertex disappears and we are left with the partial wave 124 amplitude 2\/(id/F1 "’7‘ (_1)k+l \/§ «47,2 gals—1V!) (:2k-—1).i/F1 (10 = lim (4.90) -n—+oc This sum is dominated by the first. KK mode (A? = 1). Thus. to locate the left most edge of the wedge—shaped in the (J? M p1) plane where unitarity is violated, we need only keep the first KK fermion mode. ‘2 x/b M F1 m t fl 9 . ‘ AIFI lim ”U(k = 1) z n —4 «x. it'd-U2 (4.91) The function g(\/§ /.=‘l l F1) determines the shape of this bound. It is maximized for \/E = 2:)! F1 and gives the upper limit of M F1, -41 2 A l}, 5 __‘_"___ ~ 4.25 TeV , (41-92) 1 ‘2 0m,1n(5) if we want this amplitude to be unitary up to very high scales. Including the higher fermion KK modes changes this upper bound only slightly, to ~ 4.5TeV. Note that, in the continuum limit, the scattering amplitude does not. grow at. asymptotically high energies — a property ensured by various sum-rules satisfied by the couplings [104. 105]. Nonetheless, as illustrated in Figures 4.7 and 4.8, the properly normalized spin—0 couple