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Phenomenology of Universal Extra Dimensions

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PAGE 1

PHENOMENOLOGY OF UNIVERSAL EXTRA DIMENSIONS By KYOUNGCHUL K ONG A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2006

PAGE 2

Cop yrigh t 2006 b y Ky oungc h ul Kong

PAGE 3

T o m y family

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A CKNO WLEDGMENTS First and foremost, I am deeply indebted to m y advisor, Prof. Konstan tin Matc hev, for his time, patience, encouragemen t, m uc h stim ulating advice and nancial supp ort for the researc h and man y academic tra v els. My collab orator, Prof. Asesh K. Datta, deserv es sp ecial thanks for useful discussions, and his honest advice in ph ysics and life. I w an t to thank Dr. Andreas Birk edal, Dr. Lisa Ev erett and Dr. Hy e-Sung Lee for their useful discussions, imp ortan t commen ts and suggestions on m y talks and researc hes. I w ould lik e to thank Prof. Pierre Ramond for constan t supp ort, encouragemen t and teac hing necessary particle ph ysics courses, and Prof. Ric hard W o o dard for b eing a constan t source of inspiration and sharing his en th usiasm for ph ysics with graduate studen ts. I also w an t to thank Prof. Andrey Koryto v, Prof. Ric k Field and Prof. Da vid Groisser for reading m y thesis and questions. In m y hometo wn univ ersit y I w an t to thank Prof. Chang Gil Han, Prof. Deog Ki Hong and Prof. Hyun-Ch ul Kim for teac hing and their supp ort when I applied for graduate program in US. I am grateful for Prof. Mic hael P eskin, Prof. Jonathan F eng, Prof. HsinChia Cheng, Prof. Bogdan Dobrescu and Prof. Tim T ait who ga v e me useful commen ts and advice, among man y other ph ysicists that I ha v e met at w orkshops and seminar visits. Man y thanks should go to Bobb y Scurlo c k and Craig Group for getting me closer to exp erimen ts and computer languages, Karthik Shank ar and Sudarshan Anan th for commen ts and gossip, and m y ocemate, Sung-So o Kim and T aku W atanab e, for stim ulating me with their diligence and commen ts. iv

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I can not forget m y old friends Dr. Suc kjo on Jun in AMOLF and By oung-Ch ul Kim in BNL, Junghan Lee in Mainz and Dr. By oung-Ik Hur in cancer treatmen t cen ter and Seung-Hw a Sheen in nance. I feel a deep sense of gratitude for m y paren ts and brothers who b eliev ed in me although they do not understand what I am doing. But I will nev er b e able to thank m y wife enough, for all her lo v e, supp ort, and friendship and m y little son, Casey for his gen uine smiles that mak e me the happiest man in the w orld. I ha v e b een really blessed to ha v e her and Casey b eside me. v

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T ABLE OF CONTENTS page A CKNO WLEDGMENTS . . . . . . . . . . . . . . iv LIST OF T ABLES . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . ix KEY TO ABBREVIA TIONS . . . . . . . . . . . . . xii ABSTRA CT . . . . . . . . . . . . . . . . . . xiii CHAPTER1 INTR ODUCTION . . . . . . . . . . . . . . . 1 2 UNIVERSAL EXTRA DIMENSIONS . . . . . . . . . 9 2.1 Massiv e Scalar Field in Fiv e Dimensions . . . . . . . 9 2.2 Univ ersal Extra Dimensions . . . . . . . . . . . 10 2.3 Comparison b et w een UED and Sup ersymmetry . . . . . 18 3 COLLIDER PHENOMENOLOGY . . . . . . . . . . 22 3.1 Searc h for Lev el 2 KK P articles at the LHC . . . . . . 22 3.1.1 Phenomenology of Lev el 2 F ermions . . . . . . . 23 3.1.2 Phenomenology of Lev el 2 Gauge Bosons . . . . . 29 3.1.3 Analysis of the LHC Reac h for Z 2 and r 2 . . . . . 37 3.2 Spin Determination at the LHC . . . . . . . . . . 42 3.2.1 Dilepton In v arian t Mass Distributions . . . . . . 44 3.2.2 Lepton-Jet In v arian t Mass Charge Asymmetry . . . 47 3.3 UED and SUSY at Linear Colliders . . . . . . . . . 53 3.3.1 Ev en t Sim ulation and Data Analysis . . . . . . . 54 3.3.2 Comparison of UED and Sup ersymmetry in + / E T . . 58 3.3.2.1 Angular Distributions and Spin Measuremen ts . 58 3.3.2.2 Threshold Scans . . . . . . . . . 60 3.3.2.3 Pro duction Cross-Section Determination . . . 61 3.3.2.4 Muon Energy Sp ectrum and Mass Measuremen ts 62 3.3.2.5 Photon Energy Sp ectrum and Radiativ e Return to the Z 2 . . . . . . . . . . . . 63 3.3.3 Prosp ects for Disco v ery and Discrimination in Other Final States . . . . . . . . . . . . . . . 65 vi

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3.3.3.1 Kaluza-Klein Leptons . . . . . . . . 65 3.3.3.2 Kaluza-Klein Quarks . . . . . . . . 67 3.3.3.3 Kaluza-Klein Gauge Bosons . . . . . . 69 4 COSMOLOGICAL IMPLICA TIONS . . . . . . . . . . 73 4.1 Dark Matter Abundance . . . . . . . . . . . . 73 4.2 The Basic Calculation of the Relic Densit y . . . . . . . 76 4.2.1 The Standard Case . . . . . . . . . . . 76 4.2.2 The Case with Coannihilations . . . . . . . . 78 4.2.3 Thermal Av erage and Nonrelativistic V elo cit y Expansion . 79 4.3 Relic Densit y in Minimal UED . . . . . . . . . . 81 4.4 Relativ e W eigh t of Dieren t Coannihilation Pro cesses . . . . 88 4.4.1 Coannihilations with KK Leptons . . . . . . . 88 4.4.2 Coannihilations with KK Quarks and KK Gluons . . . 94 4.4.3 Coannihilations with Electro w eak KK Bosons . . . . 95 4.5 Other Dark Matter Candidates and Direct Detection . . . . 96 5 CONCLUSIONS . . . . . . . . . . . . . . . 101 APPENDIX . . . . . . . . . . . . . . . . . . 103 A ST AND ARD MODEL IN 5D . . . . . . . . . . . . 103 A.1 Lagrangian of the Standard Mo del in 5D . . . . . . . 103 A.2 The Kaluza-Klein F ermions and Gauge b osons . . . . . . 109 A.3 The Deca y Widths of KK P articles . . . . . . . . . 111 A.4 Running Coupling Constan ts in Extra Dimensions . . . . . 114 B ANNIHILA TION CR OSS-SECTIONS . . . . . . . . . 117 B.1 Leptons . . . . . . . . . . . . . . . . 118 B.2 Gauge Bosons . . . . . . . . . . . . . . . 122 B.3 F ermions and Gauge Bosons . . . . . . . . . . . 124 B.4 Quarks . . . . . . . . . . . . . . . . . 126 B.5 Quarks and Leptons . . . . . . . . . . . . . 128 B.6 Higgs Bosons . . . . . . . . . . . . . . . 129 B.7 Higgs Bosons and Gauge Bosons . . . . . . . . . . 133 B.8 Higgs Bosons and F ermions . . . . . . . . . . . 136 REFERENCES . . . . . . . . . . . . . . . . . 139 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 149 vii

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LIST OF T ABLES T able page 3{1 Masses of the KK excitations for R 1 = 500 GeV and R = 20 . . 56 3{2 MSSM parameters for a SUSY study p oin t . . . . . . . . 58 A{1 F ermion con ten t of the Standard Mo del and the corresp onding KaluzaKlein fermions . . . . . . . . . . . . . . . . 110 A{2 Quan tum n um b ers of KK fermions . . . . . . . . . . 110 A{3 F ermions and gauge b osons in the Standard Mo del . . . . . . 110 B{1 A guide to the form ulas in the App endix B . . . . . . . . 118 viii

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LIST OF FIGURES Figure page 1{1 de Broglie's particle-w a v e dualit y . . . . . . . . . . . 4 1{2 An illustration of bulk and brane . . . . . . . . . . . 5 1{3 S 1 = Z 2 orbifold . . . . . . . . . . . . . . . . 6 2{1 KK states after a compactication on the orbifold . . . . . . 12 2{2 KK n um b er conserv ation and KK parit y . . . . . . . . . 14 2{3 The sp ectrum of the rst KK lev el at (a) tree lev el and (b) one-lo op . 16 2{4 Qualitativ e sk etc h of the lev el 1 KK sp ectroscop y . . . . . . 17 2{5 A disco v ery reac h for MUEDs at the T ev atron (blue) and the LHC (red) in the 4 ` + / E T c hannel . . . . . . . . . . . . . . 18 3{1 One lo op corrected mass sp ectrum of the n = 1 and n = 2 KK lev els . 23 3{2 Cross-sections of n = 2 KK particles at the LHC . . . . . . . 24 3{3 Branc hing fractions of the lev el 2 \up" quarks v ersus R 1 . . . . 26 3{4 Branc hing fractions of the lev el 2 KK electrons v ersus R 1 . . . . 28 3{5 Masses and widths of lev el 2 KK gauge b osons . . . . . . . 30 3{6 Cross-sections for single pro duction of lev el 2 KK gauge b osons . . 33 3{7 Branc hing fractions of the n = 2 KK gauge b osons v ersus R 1 . . . 35 3{8 5 Disco v ery reac h for (a) r 2 and (b) Z 2 . . . . . . . . . 38 3{9 The r 2 Z 2 diresonance structure in UED with R 1 = 500 GeV . . 40 3{10 Twin diagrams in SUSY and UED . . . . . . . . . . 43 3{11 Comparison of dilepton in v arian t mass distributions . . . . . . 44 3{12 A closer lo ok in to dilepton in v arian t mass distributions . . . . . 46 3{13 Jet-lepton in v arian t mass distributions . . . . . . . . . 50 3{14 Asymmetries for UED and SUSY . . . . . . . . . . . 51 ix

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3{15 Asymmetries with relaxed conditions . . . . . . . . . . 53 3{16 The dominan t F eynman diagrams for KK m uon pro duction . . . . 55 3{17 The dominan t F eynman diagrams for sm uon pro duction . . . . 55 3{18 Dieren tial cross-section d =d cos . . . . . . . . . . 59 3{19 The total cross-section in pb as a function of the cen ter-of-mass energy p s near threshold . . . . . . . . . . . . . . 60 3{20 The m uon energy sp ectrum resulting from KK m uon pro duction ( 3{14 ) in UED (blue, top curv e) and sm uon pro duction ( 3{15 ) in sup ersymmetry (red, b ottom curv e) . . . . . . . . . . . . . . 62 3{21 Photon energy sp ectrum in e + e +1 1 r . . . . . . . . 64 3{22 The dominat F eynman diagrams for KK electron pro duction . . . 66 3{23 ISR-corrected pro duction cross-sections of lev el 1 KK leptons . . . 67 3{24 Dieren tial cross-section d =d cos e for UED and sup ersymmetry . . 68 3{25 ISR-corrected pro duction cross-sections of lev el 1 KK quarks . . . 69 3{26 ISR-corrected pro duction cross-sections of lev el 1 KK gauge b osons . 71 3{27 ISR-corrected pro duction cross-sections of lev el 2 KK gauge b osons . 72 4{1 The a -term of the annihilation cross-sections for (a) r 1 r 1 e + e and (b) r 1 r 1 . . . . . . . . . . . . . . . . 84 4{2 The n um b er of eectiv ely massless degrees of freedom and freeze-out temp erature . . . . . . . . . . . . . . . . . . 86 4{3 Relic densit y of the LKP as a function of R 1 in the minimal UED mo del 87 4{4 Coannihilation eects of (a) 1 generation or (b) 3 generations of singlet KK leptons . . . . . . . . . . . . . . . . . 89 4{5 Plots of v arious quan tities en tering the LKP relic densit y computation 91 4{6 The eects of v arying the S U (2) W -doublet KK electron mass . . . 93 4{7 The eects of v arying KK quarks masses . . . . . . . . . 94 4{8 The eects of v arying KK gluon mass . . . . . . . . . . 95 4{9 The eects of v arying EW b osons . . . . . . . . . . . 96 4{10 The c hange in the cosmologically preferred v alue for R 1 as a result of v arying the dieren t KK masses a w a y from their nominal MUED v alues 98 x

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4{11 The spin-indep enden t direct detection limit from CDMS exp erimen t . 99 A{1 Dep endence of the \W ein b erg" angle n for the rst few KK lev els ( n = 1 ; 2 ; ; 5) on R 1 for xed R = 20 . . . . . . . . . . 112 A{2 Running coupling constan ts in SM (a) and UED (b) . . . . . . 114 xi

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KEY TO ABBREVIA TIONS CDMS: Cry ogenic Dark Matter Searc h CLIC: Compact Linear Collider EW: Electro w eak EWSB: Electro w eak Symmetry Breaking ISR: Initial State Radiation KK: Kaluza-Klein LHC: Large Hadron Collider LKP: Ligh test KK particle LSP: Ligh test Sup ersymmetric particle MSSM: Minimal Sup ersymmetric Standard Mo del mSugra: Minimal Sup ergra vit y MUED: Minimal Univ ersal Extra Dimensions SM: Standard Mo del SPS: Sno wmass P oin ts and Slop es: Benc hmarks for SUSY searc hes SUSY: Sup ersymmetry UED: Univ ersal Extra Dimensions WIMP: W eakly In teracting Massiv e P article WMAP: Wilkinson Micro w a v e Anisotrop y Prob e xii

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y PHENOMENOLOGY OF UNIVERSAL EXTRA DIMENSIONS By Ky oungc h ul Kong August 2006 Chair: Konstan tin T. Matc hev Ma jor Departmen t: Ph ysics A ma jor motiv ation for studying new ph ysics b ey ond the Standard Mo del is the dark matter puzzle whic h nds no explanation within the Standard Mo del. Mo dels with extra dimensions ma y naturally pro vide p ossible dark matter candidates if the theory is compactied at the T eV scale. In this dissertation, the phenomenology of Univ ersal Extra Dimensions (UED), in whic h all the Standard Mo del elds propagate, is explored. W e fo cus on mo dels with one univ ersal extra dimension, compactied on an S 1 = Z 2 orbifold. W e in v estigate the collider reac hes for new particles and the cosmological implications of this mo del. Mo dels with Univ ersal Extra Dimensions ma y pro vide excellen t coun ter examples for t ypical sup ersymmetric theories with dark matter candidates. Therefore w e con trast the exp erimen tal signatures of lo w energy sup ersymmetry and mo dels with Univ ersal Extra Dimensions and discuss v arious metho ds for their discriminations at colliders. W e rst study the disco v ery reac h of the T ev atron and the LHC for lev el 2 Kaluza-Klein mo des, whic h w ould indicate the presence of extra dimensions, since suc h particles are guaran teed b y extra dimensions but not sup ersymmetry W e also in v estigate the p ossibilit y to dieren tiate the spins of the sup erpartners xiii

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and KK mo des b y means of a dilepton mass metho d and the asymmetry metho d in the squark cascade deca y to electro w eak (EW) particles. W e then study the pro cesses of Kaluza-Klein m uon pair pro duction in univ ersal extra dimensions in parallel to sm uon pair pro duction in sup ersymmetry at a linear collider. W e nd that the angular distributions of the nal state m uons, the energy sp ectrum of the radiativ e return photon and the total cross-section measuremen t are p o w erful discriminators b et w een the t w o mo dels. W e also calculate the pro duction rates of v arious Kaluza-Klein particles and discuss the asso ciated signatures. A prediction of the mo dels with Univ ersal Extra Dimensions with conserv ed KK-parit y is the existence of dark matter. W e calculate the relic densit y of the ligh test Kaluza-Klein particle. W e include coannihilation pro cesses with al l lev el one KK particles. In our computation w e consider a most general KK particle sp ectrum, without an y simplifying assumptions. W e rst calculate the KaluzaKlein relic densit y in the minimal UED mo del, turning on coannihilations with all lev el one KK particles. W e then go b ey ond the minimal mo del and discuss the size of the coannihilation eects separately for eac h class of lev el 1 KK particles. Our results pro vide the basis for consisten t relic densit y computations in arbitrarily general mo dels with Univ ersal Extra Dimensions. All these studies not only bring us to deep er understanding of new p ossibilities b ey ond the Standard Mo del but also pro vide strong phenomenological bac kgrounds and to ols to iden tify the nature of new ph ysics. xiv

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CHAPTER 1 INTR ODUCTION The Standard Mo del of particle ph ysics is a theory whic h describ es the strong, w eak, and electromagnetic fundamen tal forces. This theory has b een astonishingly successful in explaining m uc h of the presen tly a v ailable exp erimen tal data. Ho w ev er, the Standard Mo del still lea v es op en a n um b er of outstanding fundamen tal questions whose answ ers are exp ected to emerge in a more general theoretical framew ork. One of the ma jor motiv ations for pursuing new ph ysics b ey ond the Standard Mo del is the dark matter problem whic h nds no explanation within the Standard Mo del. F rom the accum ulated astroph ysical data, w e no w kno w that ordinary matter comprises only ab out 4% (n B ) of the Univ erse. The remaining 96% are divided b et w een a m ysterious form of matter called \dark matter" (22%, n C D M ) and an ev en more p erplexing en tit y called \dark energy" (74%, n ). F rom the inrationary big bang mo del, 1 = n = n + n C D M + n B ; (1{1) is exp ected where n B is the fractional energy densit y in bary ons, n C D M the fractional energy densit y in dark matter, and n the fractional energy densit y in dark energy (The precise measured v alues are n C D M = 0 : 22 +0 : 01 0 : 02 n = 0 : 74 0 : 02, n B = 0 : 044 +0 : 002 0 : 003 and n = 1 : 02 0 : 02 [1].) The microscopic nature of the dark matter is at presen t unkno wn. P erhaps the most attractiv e explanation is pro vided b y the WIMP (w eakly in teracting massiv e particle) h yp othesis: dark matter is assumed to consist of h yp othetical stable particles with masses around the scale of electro w eak symmetry breaking, whose 1

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2 in teractions with other elemen tary particles are of the strength and range similar to the familiar w eak in teractions of the Standard Mo del. Suc h WIMPs naturally ha v e a relic abundance of the correct order of magnitude to accoun t for the observ ed dark matter, making them app ealing from a theoretical p oin t of view. The relic densit y n W I M P of the WIMP dark matter is roughly estimated b y n W I M P 1 10 2 2 m W I M P 1 T eV 2 (1{2) O (1) ; (1{3) where m W I M P is the mass of the WIMP dark matter candidate and the magnitude of electro w eak in teraction, is exp ected to b e of order 0 : 01. Therefore the relic densit y of WIMP dark matter is exp ected to b e of order 1 if the mass scale is O (1) T eV. The precise relic densit y including the correct co ecien ts in the ab o v e equation needs to b e calculated using the Boltzmann equation and the result dep ends on the particular mo del. The ab o v e estimation tells us that the WIMP h yp othesis can naturally explain all or part of the dark matter. Moreo v er, man y extensions of the Standard Mo del con tain particles whic h can b e iden tied as WIMP dark matter candidates. Examples include sup ersymmetric mo dels, mo dels with Univ ersal Extra Dimensions, little Higgs theories, etc. An excellen t candidate for suc h thermal WIMP arises in the R-parit y conserving sup ersymmetric theories. New particles, called sup erpartners, predicted b y the sup ersymmetry are c harged under this R-parit y while the Standard Mo del particles are neutral under the symmetry So the ligh test sup ersymmetric particle (LSP) is stable and can b e a dark matter candidate. The sup ersymmetric mo dels ha v e other side b enets: 1. R-parit y also implies that sup erpartners in teract only pairwise with SM particles, whic h guaran tees that the sup ersymmetric con tributions to lo w energy precision data only app ear at the lo op lev el and are small.

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3 2. If the sup erpartners are indeed within the T eV range, the problematic quadratic div ergences in the radiativ e corrections to the Higgs mass are absen t, b eing canceled b y lo ops with sup erpartners. The cancellations are enforced b y the symmetry and the Higgs mass is therefore naturally related to the mass scale of the sup erpartner. 3. The sup erpartners w ould mo dify the running of the gauge couplings at higher scales, and gauge coupling unication tak es places with astonishing precision. Therefore sup ersymmetric extensions of the SM b ecame the primary candidates for new ph ysics at the T eV scale. Not surprisingly therefore, the signatures of sup ersymmetry at the T ev atron and the LHC ha v e b een extensiv ely discussed in the literature. Ho w ev er, sup ersymmetry is not the only mo del whic h has WIMP candidates. Recen t dev elopmen ts in string theory ha v e spurred a reviv al of in terest in the phenomenology of theories with extra spatial dimensions. Some or ev en all of the Standard Mo del particles could also propagate in the extra dimensions and it is suggested that a stable particle in the extra dimensional mo dels ma y b e able to accoun t for the observ ed dark matter. The immediate result from the h yp othesis of extra dimensions is the existence of extra particle states. This can b e understo o d easily in the follo wing w a y In 4 dimensions, w e ha v e the follo wing energy-momen tum relation, E 2 = p 2x 1 + p 2x 2 + p 2x 3 + m 2 ; (1{4) where x 1 x 2 x 3 are the co ordinates of the usual 3 dimensions, E is an energy of a particle and m is a mass of a particle. Supp ose there w as an extra dimension with a co ordinate y ; then this relation b ecomes E 2 = p 2x 1 + p 2x 2 + p 2x 3 + p 2y + m 2 : (1{5)

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4 2 R = 2 2 R = 5 2 R = 6 Figure 1{1: de Broglie's particle-w a v e dualit y As w e go around the circle, w e m ust t an in teger m ultiple of 's in its circumference. No w recall the particle-w a v e dualit y p y = 2 : (1{6) If the extra dimension is compact, e.g., a circle, then as w e go around the circle, w e m ust t an in teger m ultiple of 's in its circumference as sho wn in g. 1{1 Therefore p erio dicit y implies a quan tization of momen tum along the extra dimension, = 2 R n ) p y = 2 n 2 R = n R : (1{7) Substitute eqn. 1{7 in to eqn. 1{5 then the energy-momen tum relation b ecomes E 2 = p 2x 1 + p 2x 2 + p 2x 3 + n 2 R 2 + m 2 ~ p 2 + M 2 n ; (1{8) where M n = q n 2 R 2 + m 2 is the eectiv e mass of the particle mo ving in the extra dimensions. This translates in to a ric h and exciting phenomenology at the LHC, since quan tization of the particle momen tum along the extra dimension necessarily implies the existence of whole to w er of massiv e particles, called Kaluza-Klein (KK) mo des or partners. The KK particles within eac h to w er are nothing but hea vier v ersions of their Standard Mo del coun terpart. A disco v ery of a compact extra dimension at a collider can only b e made through the disco v ery of the KK particles and measuremen t of their prop erties. In g. 1{2 our 4 dimensional spacetime is one of the t w o branes, and the space b et w een the t w o branes is usually referred to as \the bulk." SM particles can either freely propagate in to the bulk or remain on the brane. The mass sp ectrum of the KK partners ev en enco des information ab out

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5 bulk brane (4 dimensional spacetime) Figure 1{2: An illustration of bulk and brane. 4 dimensional spacetime is sho wn as a brane and the space b et w een t w o brane is called a bulk, where extra dimensions existthe space-time geometry: if the extra dimension is rat, the KK masses are roughly equally spaced [2, 3], and if the extra dimension is w arp ed, the KK mass sp ectrum follo ws a non-trivial pattern [4, 5]. No w consider, for example, the most \demo cratic" scenario (whic h has b ecome kno wn as Univ ersal Extra Dimensions) in whic h all Standard Mo del particles propagate in the bulk. Its simplest incarnation has a single extra dimension of size R whic h is compactied on an S 1 = Z 2 orbifold [6]. In g. 1{3 w e sho w S 1 = Z 2 orbifold where the extra dimension is sho wn as a line in this geometry In terestingly the dark matter puzzle can b e resolv ed in a comp elling fashion in mo dels with Univ ersal Extra Dimensions. A p eculiar feature of UED is the conserv ation of Kaluza-Klein n um b er at tree lev el, whic h is a simple consequence of momen tum conserv ation along the extra dimension. Ho w ev er, bulk and brane radiativ e eects break KK n um b er do wn to a discrete conserv ed quan tit y called KK-parit y The KK-parit y adorns the UED scenario with man y of the virtues t ypically asso ciated with sup ersymmetry:

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6 brane brane bulk S 1 Z 2 S 1 = Z 2 Z 2 Figure 1{3: S 1 = Z 2 orbifold. A half of the circle ( S 1 ) is iden tied with the other half with a Z 2 symmetry The geometry b ecomes a line with t w o xed p oin ts. The line b et w een t w o xed p oin ts represen ts the bulk. 1. The ligh test KK-partners (those at lev el 1) m ust alw a ys b e pair-pro duced in collider exp erimen ts, whic h leads to relativ ely w eak b ounds from direct searc hes. 2. The KK-parit y conserv ation implies that the con tributions to v arious precisely measured lo w-energy observ ables only arise at the lo op lev el and are small. 3. Finally the KK-parit y guaran tees that the ligh test KK partner is stable, and th us can b e a cold dark matter candidate. As w e will see in the next c hapters, the phenomenology of this scenario clearly resem bles that of sup ersymmetry In this sense, man y of the SUSY studies in the literature apply and it is p erhaps more imp ortan t to nd metho ds to distinguish b et w een the t w o mo dels. Recen tly other mo dels suc h as little Higgs theory with T-parit y ha v e b een prop osed as new ph ysics b ey ond the Standard Mo del. Our studies can also apply in the case of little Higgs mo dels since the rst lev el of the UED mo del lo oks lik e the little Higgs particle sp ectrum. Except for its abundance, no other prop erties of dark matter candidates are kno wn at presen t. Therefore it is imp ortan t to study the prop erties of new t yp es of dark matter candidates in the extra dimensional mo dels and compare them with those in sup ersymmetry Then a n um b er of questions can arise: What are the

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7 prop erties of dark matter candidates in the extra dimensional mo dels? Ho w dieren t are they from ones in sup ersymmetry? Can w e see an y evidences for the extra dimensions in dark matter or collider exp erimen ts? etc. In this dissertation, w e w an t to answ er at least some of these questions. Hence w e in v estigate the collider phenomenology and astroph ysical implications of Univ ersal Extra Dimensions. In c hapter 2, w e rst sho w a simple example of a Lagrangian in extra dimensions and later in tro duce the complete mo del with Univ ersal Extra Dimensions. W e review the basic phenomenology of the UED mo del, con trasting it with a generic sup ersymmetric mo del as describ ed ab o v e. The detailed prop erties of UED mo dels are summarized in app endix A. In c hapter 3, w e iden tify t w o basic discriminators b et w een UED and SUSY, and pro ceed to consider eac h one in turn in the follo wing sections. One of the c haracteristic features of extra dimensional mo dels is the presence of a whole to w er of Kaluza-Klein (KK) partners, lab elled b y their KK lev el n In con trast, N = 1 sup ersymmetry predicts a single sup erpartner for eac h SM particle. One migh t therefore hop e to disco v er the higher KK mo des of UED and th us pro v e the existence of extra dimensions. In section 3.1, w e study the disco v ery reac h for lev el 2 KK gauge b oson particles and the resolving p o w er of the LHC to see them as separate resonances. This study w as done b y our group for the rst time [7{11]. The other fundamen tal dierence b et w een SUSY and UED is the spin of the new particles (sup erpartners or KK partners). Therefore in section 3.2, w e in v estigate ho w w ell the t w o mo dels can b e distinguished at the LHC based on spin correlations in the cascade deca ys of the new particles. In particular, w e use the asymmetry v ariable recen tly adv ertised b y Barr [12], as w ell as dilepton mass distributions. Un til recen tly there w ere no kno wn metho ds for measuring the spins of new particles at the LHC but no w the spin determination at the LHC has b ecome a hot topic in collider ph ysics [13{ 18]. In section 3.3, w e con trast the

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8 exp erimen tal signatures of lo w energy sup ersymmetry and the mo del of Univ ersal Extra Dimensions, and this time at a linear collider, discuss v arious metho ds for their discrimination. This w as also the rst study to con trast SUSY and UED at a linear collider [10, 19]. In c hapter 4, w e consider the astroph ysical implications of the UED. W e calculate the relic densit y of the KK dark matter and sho w new results on direct detection limits [20]. The rst calculation of KK dark matter [21] w as done in the past but under the assumption that all KK particles ha v e the same masses. In addition, only a subset of the relev an t coannihilation pro cesses w as included. Therefore in our new calculation [20], w e include all p ossible coannihilation pro cesses without assuming KK mass degeneracy A similar calculation ab out one particular t yp e of dark matter w as done b y a group at Princeton [22] indep enden tly and our results are in agreemen t. W e then go b ey ond the minimal mo del and discuss the size of coannihilation eects separately for eac h class of lev el 1 KK particles. This calculation with dieren t t yp es of KK dark matter in nonminimal UED mo dels w as p erformed b y our group only The annihilation cross-sections for the dark matter calculation are listed in app endix B. In c hapter 5, w e conclude.

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CHAPTER 2 UNIVERSAL EXTRA DIMENSIONS 2.1 Massiv e Scalar Field in Fiv e Dimensions The full Lagrangian of Univ ersal Extra Dimensions is giv en in app endix A and here w e consider a simple example to illustrate the ph ysics of a theory with extra compact dimensions. As the simplest example of a Lagrangian in higher dimensions, w e consider the action for a massiv e scalar led in 5 dimensions, S = Z d 4 xdy @ M ( x; y ) @ M ( x; y ) m 2 ( x; y )( x; y ) : (2{1) Here M ; N = 0 ; 1 ; 2 ; 3 ; 5 ; 5, 5 dimensional metric is g M N = (+ ), @ M = ( @ ; @ 5 ), and y is the extra dimensional co ordinate (5th comp onen t of a Loren tz index, M ). In the case of a circular extra dimension ( S 1 ), the 5 dimensional scalar eld is expressed in terms of an exp onen tial basis as follo ws: ( x; y ) = 1 p 2 R 1 X n = 1 n ( x ) exp iny R ; (2{2) where R is the radius of the extra dimension. This exp onen tial basis satises the follo wing orthogonalit y relation b et w een dieren t mo des, 2 R n;m = Z 2 R 0 dy exp i ( n m ) y R : (2{3) No w w e in tegrate out the extra dimensional co ordinate y to get a 4 dimensional eectiv e theory Then the action b ecomes S = 1 X n = 1 Z d 4 x @ n ( x ) @ n ( x ) n 2 R 2 + m 2 n ( x ) n ( x ) ; (2{4) 9

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10 Here n is a 4 dimensional scalar eld of mass m n = q n 2 R 2 + m 2 W e started with one massiv e scalar eld in 5D and compactied this theory on S 1 As a result, w e get an innite n um b er of scalar elds (called a KK-to w er) with mass, m n = q n 2 R 2 + m 2 in 4 dimensions. n = 0 (the \zero" mo de) corresp onds to a regular massiv e scalar eld in 4 dimensions with mass, m F or the nonzero mo des (or KK mo des), the mass comes mostly from the deriv ativ e with resp ect to the extra dimension ( @ 5 ). Notice that all KK mo des ha v e the same spin. 2.2 Univ ersal Extra Dimensions The mo dels of Univ ersal Extra Dimensions are similar to this example. In the simplest and most p opular v ersion, there is a single extra dimension of size R compactied on an orbifold ( S 1 = Z 2 ) instead of circle ( S 1 ) [6]. The orbifold can in tro duce c hiral fermions and pro ject out un w an ted 5th comp onen ts of the gauge elds (see app endix A). More complicated 6-dimensional mo dels ha v e also b een built [23{ 25]. The Standard Mo del is written in 5 dimensions as follo ws. L Gaug e = 1 2 Z R R dy 1 4 B M N B M N 1 4 W a M N W aM N 1 4 G AM N G AM N ; L GF = 1 2 Z R R dy 1 2 ( @ B @ 5 B 5 ) 2 1 2 @ W a @ 5 W a 5 2 1 2 @ G A @ 5 G G5 2 ; L Leptons = 1 2 Z R R dy i L ( x; y ) M D M L ( x; y ) + i E ( x; y ) M D M E ( x; y ) ; L Quar k s = 1 2 Z R R dy i Q ( x; y ) M D M Q ( x; y ) + i U ( x; y ) M D M U ( x; y ) + i D ( x; y ) M D M D ( x; y ) ; (2{5) L Y uk aw a = 1 2 Z R R dy u Q ( x; y ) U ( x; y ) i 2 H ( x; y ) + d Q ( x; y ) D ( x; y ) H ( x; y ) + e L ( x; y ) E ( x; y ) H ( x; y ) ; L H ig g s = 1 2 Z R R dy h ( D M H ( x; y )) y D M H ( x; y ) + 2 H y ( x; y ) H ( x; y ) H y ( x; y ) H ( x; y ) 2 i ;

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11 where co v arian t deriv ativ es are dened in the app endix and eac h Standard Mo del eld is expressed in terms of cos and sin mo des on the orbifold, H ( x; y ) = 1 p R ( H ( x ) + p 2 1 X n =1 H n ( x ) cos ( ny R ) ) ; B ( x; y ) = 1 p R ( B 0 ( x ) + p 2 1 X n =1 B n ( x ) cos( ny R ) ) ; B 5 ( x; y ) = r 2 R 1 X n =1 B n 5 ( x ) sin( ny R ) ; W ( x; y ) = 1 p R ( W 0 ( x ) + p 2 1 X n =1 W n ( x ) cos( ny R ) ) ; W 5 ( x; y ) = r 2 R 1 X n =1 W n 5 ( x ) sin ( ny R ) ; G ( x; y ) = 1 p R ( G 0 ( x ) + p 2 1 X n =1 G n ( x ) cos( ny R ) ) ; (2{6) G 5 ( x; y ) = r 2 R 1 X n =1 G n5 ( x ) sin( ny R ) ; Q ( x; y ) = 1 p R ( q L ( x ) + p 2 1 X n =1 h P L Q nL ( x ) cos ( ny R ) + P R Q nR ( x ) sin ( ny R ) i ) ; U ( x; y ) = 1 p R ( u R ( x ) + p 2 1 X n =1 h P R u nR ( x ) cos ( ny R ) + P L u nL ( x ) sin( ny R ) i ) ; D ( x; y ) = 1 p R ( d R ( x ) + p 2 1 X n =1 h P R d nR ( x ) cos( ny R ) + P L d nL ( x ) sin ( ny R ) i ) ; L ( x; y ) = 1 p R ( L 0 ( x ) + p 2 1 X n =1 h P L L nL ( x ) cos( ny R ) + P R L nR ( x ) sin ( ny R ) i ) ; E ( x; y ) = 1 p R ( e R ( x ) + p 2 1 X n =1 h P R e nR ( x ) cos( ny R ) + P L e nL ( x ) sin ( ny R ) i ) ; where H ( x; y ) is the 5D scalar eld and ( B ( x; y ) ; B 5 ( x; y )), ( W ( x; y ) ; W 5 ( x; y )) and ( G ( x; y ) ; G 5 ( x; y )) are the 5D gauge elds for U (1), S U (2) and S U (3) resp ectiv ely Q ( x; y ) and L ( x; y ) are the S U (2) fermion doublets while U ( x; y ), D ( x; y ) and E ( x; y ) are resp ectiv ely the generic singlet elds for the up-t yp e quark, the do wn-t yp e quark and the lepton.

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12 n = 3 R L R L n = 0 n = 1 n = 2 A A 5 Q U A Figure 2{1: KK states after a compactication on the orbifold. Q (U) is a fermion state whic h is a doublet (singlet) under S U (2) and A is a gauge b oson. L/R represen t the c hiralit y of eac h state. There is one corresp onding KK fermion ( n 6 = 0) for eac h c hiralit y of SM fermion ( n = 0). Tw o KK states sharing an arro w mak e one Dirac fermion while a SM Dirac fermion needs one S U (2) W -doublet and one S U (2) W singlet. A 5 is eaten b y A at eac h KK lev el after compactication and KK gauge b osons b ecome massiv e while a SM gauge b oson remains massless. These states are equally spaced since all KK states ha v e the same mass n R b efore electro w eak symmetry breaking. Crossed states do not exist on an orbifold compactication. The 4 dimensional eectiv e Lagrangian is obtained b y in tegrating out the extra dimension using orthogonalit y relations b et w een these trigonometric functions, whic h are giv en in eqns. (A-6). As a result of the compactication, w e nd the follo wing prop erties of the 4 dimensional eectiv e theory and list them b elo w rather than sho wing the actual Lagrangians, whic h are are quite length y 1. Eac h SM particle has an innite n um b er of KK partners. This is illustrated in g. 2{1 where n = 0 corresp onds to a SM particle and non-zero mo des corresp ond to KK states. In fact, R is the n um b er of KK lev els b elo w a cuto scale, since this theory requires a cuto at high energy 2. KK particles ha v e the same spin as SM particles. All KK particles at lev el n ha v e the same mass, n R b efore the Higgs gets a v acuum exp ectation v alue through the EWSB. The EWSB giv es masses to Standard Mo del particles and c hanges KK masses to q m 2 + n 2 R 2 where m is the mass term

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13 from EWSB. A p eculiar feature of UED is that there are t w o KK Dirac fermions for eac h Dirac fermion in the SM. In g. 2{1 L (R) represen ts a left (righ t) handed c hiralit y of the SM fermion or KK fermion. In the SM, a fermion doublet (denoted b y Q) is left handed and a fermion singlet (denoted b y U) is righ t handed. Therefore there is no righ t handed fermion doublet and left handed fermion singlet in the SM. Ho w ev er, due to the orbifold b oundary condition, there are t w o KK states with dieren t c hiralities for b oth Q and U. These t w o KK states mak e one Dirac spinor and therefore there are t w o Dirac fermions for eac h Dirac fermion in SM. In other w ords, the zero mo de (SM particle) is either righ t handed or left handed but the KK mo de (KK particle) comes in c hiral pairs. This c hiral structure is a natural consequence of the orbifold b oundary conditions. The mass from EWSB app ears as an o-diagonal en try in a fermion mass matrix (see eqn. B.96 ). 1 g. 2{1 sho ws that the 5th comp onen ts of 5 dimensional gauge b osons are eaten b y KK gauge b osons and these KK gauge b osons b ecome massiv e while the SM gauge b osons remain massless since there is no 5th comp onen t. The SM gauge b oson can get a mass through EWSB. 2 3. All v ertices at tree lev el satisfy KK n um b er conserv ation. F or eac h term in the Lagrangian, w e ha v e a (see eqn. A-21 ) whic h is a linear com bination of the Kronec k er delta functions. Due to this structure, the 1 Then there could b e, in principle, a mixing b et w een t w o KK Dirac fermions but the mixing angle is small since R 1 is larger than fermion mass in the SM. 2 Similarly to the fermion case, there is a nonzero con tribution to the diagonal part of the gauge b oson mass matrix from EWSB (eqn. A-22 ) and therefore there is a mixing b et w een KK partners of U (1) h yp erc harge gauge b oson ( B n ) and KK partners of neutral S U (2) W gauge b oson ( W 3 n ), as in the SM (see eqn. A-24 ). This mixing angle turns out to b e small and w e ignore it in our analysis.

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14 1 01 (a) 2 0 2 (b) (c) (d) 0 11 2 1 0 Figure 2{2: KK n um b er conserv ation and KK parit y KK parit y is alw a ys conserv ed in all cases. (a) KK n um b er is conserv ed and therefore this v ertex exists at tree lev el. This coupling is the same as a SM coupling. (b) KK n um b er is not conserv ed and it do es not exist at tree lev el. It is generated at 1 lo op. (c) KK n um b er is conserv ed and it exists at tree lev el. This coupling do es not in v olv e an y SM particle and its magnitude is less than SM coupling b y p 2 (d) Either KK n um b er or KK parit y are not conserv ed. It do es not exist at an y lo op. allo w ed v ertices satisfy one of the follo wing conditions, j m n k j = 0 ; (2{7) j m n k l j = 0 : This is the conserv ation of Kaluza-Klein n um b er at tree lev el, whic h is a simple consequence of momen tum conserv ation along the extra dimension. Therefore it is easy to see whic h v ertices are allo w ed or whic h v ertices are not. In g. 2{2 (a) and (c) satisfy KK n um b er conserv ation and those t w o v ertices are allo w ed at tree lev el. (b) and (d) are not allo w ed at tree lev el. 4. KK-parit y is alw a ys conserv ed ev en at higher order. Bulk and brane radiativ e eects [26{28] break KK n um b er do wn to a discrete conserv ed quan tit y the so called KK parit y ( 1) n where n is the KK lev el. KK parit y ensures that the ligh test KK partners (those at lev el one) are alw a ys pair-pro duced in collider exp erimen ts, just lik e in the R -parit y conserving sup ersymmetry mo dels. KK parit y conserv ation also implies that the con tributions to v arious lo w-energy observ ables [29{ 39] only arise at lo op lev el and are small. As a result, the limits on the scale R 1 of the extra

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15 dimension from precision electro w eak data are rather w eak, constraining R 1 to b e larger than appro ximately 250 GeV [33]. Fig. 2{2 (b) can b e generated b y 1 lo op corrections with lev el 1 KK particles, ho w ev er, KK-parit y is not conserv ed in g. 2{2 (d), hence it can nev er b e generated b y higher order corrections. 5. New v ertices are basically the same as SM couplings (up to normalization). V ertices whic h ha v e b oth SM and KK particles are the same as the v ertices in the SM if the KK particles are replaced b y the corresp onding SM particles. V ertices with KK particles only can dier b y a factor suc h as p 2 due to orthogonalit y relations (eqns. A-20 ) and normalization factors (eqn. 2{7 ). Of course, KK-parit y m ust b e alw a ys conserv ed in an y case. This UED framew ork has b een a fruitful pla yground for addressing dieren t puzzles of the Standard Mo del, suc h as electro w eak symmetry breaking and v acuum stabilit y [40{42], neutrino masses [43, 44], proton stabilit y [45] or the n um b er of generations [46]. T o con tin ue the study on the phenomenology of UED mo del, w e need to kno w the mass sp ectrum. It dep ends on the in terpla y b et w een the one-lo op radiativ e corrections to the KK mass sp ectrum and the brane terms generated b y unkno wn ph ysics at high scales [28]. In g. 2{3 the sp ectrum of the rst KK lev el is sho wn at tree lev el (a) and one-lo op (b), for R 1 = 500 GeV, R = 20, and assuming v anishing b oundary terms at the cut-o scale Fig. 2{4 sho ws a qualitativ e sk etc h of the lev el 1 KK sp ectroscop y depicting the dominan t (solid) and rare (dotted) transitions and the resulting deca y pro duct, based on the mass sp ectrum giv en in g. 2{3 As indicated in g. 2{3 in the minimal UED mo del (MUED) dened b elo w, the LKP turns out to b e the KK partner r 1 (or the KK partner B 1 of h yp erc harge gauge b oson since the W ein b erg angle for KK states is small) of the photon [28] and its relic densit y is t ypically in the righ t ballpark:

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16 Figure 2{3: The sp ectrum of the rst KK lev el at (a) tree lev el and (b) one-lo op, for R 1 = 500 GeV, R = 20, m h = 120 GeV, and assuming v anishing b oundary terms at the cut-o scale The gures are tak en from Cheng et al. [28]. in order to explain all of the dark matter, the B 1 mass should b e in the range 500-600 GeV [20{22, 47{ 49]. Kaluza-Klein dark matter oers excellen t prosp ects for direct [50{52] or indirect detection [50, 53{ 61]. Once the radiativ e corrections to the Kaluza-Klein masses are prop erly tak en in to accoun t, the collider phenomenology of the minimal UED mo del exhibits striking similarities to sup ersymmetry [62, 63] and represen ts an in teresting and w ell motiv ated coun terexample whic h can \fak e" sup ersymmetry signals at the LHC. A t hadron colliders, the dominan t pro duction mec hanisms are KK gluon ( g 1 ) or KK quark ( q 1 or Q 1 ) pro ductions. As sho wn in g. 2{4 an S U (2) W -singlet KK quark ( q 1 ) dominan tly deca ys in to a jet and a KK photon ( r 1 ) while an S U (2) W doublet KK quark ( Q 1 ) deca ys in to lev el 1 EW gauge b osons ( Z 1 or W 1 ). Lev el 1 gauge b osons deca y in to a KK lepton pro ducing a SM lepton and later the KK lepton also pro duces a SM lepton. W e can notice that this cascade deca y lo oks lik e a t ypical SUSY cascade. F or the purp oses of our study w e ha v e c hosen to w ork with the minimal UED mo del considered in [62]. In UED the bulk in teractions of the KK mo des are xed b y the SM Lagrangian and con tain no unkno wn parameters other than

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17 Figure 2{4: Qualitativ e sk etc h of the lev el 1 KK sp ectroscop y depicting the dominan t (solid) and rare (dotted) transitions and the resulting deca y pro duct. The gure is tak en from Cheng et al. [62]. the mass, m h of the SM Higgs b oson. In con trast, the b oundary in teractions, whic h are lo calized on the orbifold xed p oin ts, are in principle arbitrary and their co ecien ts represen t new free parameters in the theory Since the b oundary terms are renormalized b y bulk in teractions, they are scale dep enden t [26] and cannot b e completely ignored since they will b e generated b y renormalization eects. Therefore, one needs an ansatz for their v alues at a particular scale. Lik e an y higher dimensional Kaluza-Klein theory the UED mo del should b e treated only as an eectiv e theory v alid up to some high scale at whic h it matc hes to some more fundamen tal theory The minimal UED mo del is then dened so that the co ecien ts of all b oundary in teractions v anish at this matc hing scale but are subsequen tly generated through R GE ev olution to lo w er scales. The minimal UED mo del therefore has only t w o input parameters: the size of the extra dimension, R and the cuto scale, The n um b er of KK lev els presen t in the eectiv e theory is simply R and ma y v ary b et w een a few and 40, where the upp er limit comes from the breakdo wn of p erturbativit y already b elo w the scale Unless sp ecied otherwise, for our n umerical results b elo w, w e shall alw a ys c ho ose the v alue of

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18 Figure 2{5: A disco v ery reac h for MUEDs at the T ev atron (blue) and the LHC (red) in the 4 ` + / E T c hannel. A 5 excess or the observ ation of 5 signal ev en ts is required, and lines sho w the required total in tegrated luminosit y p er exp erimen t (in f b 1 ) as a function of R 1 for R = 20. In either case the t w o exp erimen ts are not com bined. The gure is tak en from Cheng et al. [62]. so that R = 20. Changing the v alue of will ha v e v ery little impact on our results since the dep endence of the KK mass sp ectrum is only logarithmic. F or R 1 > 500 GeV, sin 2 n < 0 : 01 where n is the W ein b erg angle for lev el n. Fig. 2{5 sho ws the disco v ery reac h for MUEDs at the T ev atron (blue) and the LHC (red) in the 4 ` + / E T c hannel. A 5 excess or the observ ation of 5 signal ev en ts is required, and lines sho w the required total in tegrated luminosit y p er exp erimen t (in f b 1 ) as a function of R 1 for R = 20. 2.3 Comparison b et w een UED and Sup ersymmetry W e are no w in a p osition to compare in general terms the phenomenology of UED and sup ersymmetry at colliders. The discussion of Section 2.2 leads to the follo wing generic features of UED: 1. F or eac h particle of the Standard Mo del, UED mo dels predict an innite 3 to w er of new particles (Kaluza-Klein partners). 3 Strictly sp eaking, the n um b er of KK mo des is R see Section 2.2

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19 2. The spins of the SM particles and their KK partners are the same. 3. The couplings of the SM particles and their KK partners are equal. 4. The generic collider signature of UED mo dels with WIMP LKPs is missing energy Notice that the dening features 3 and 4 are common to b oth sup ersymmetry and UED and cannot b e used to distinguish the t w o cases. W e see that while R -parit y conserving SUSY implies a missing energy signal, the rev erse is not true: a missing energy signal w ould app ear in an y mo del with a dark matter candidate, and ev en in mo dels whic h ha v e nothing to do with the dark matter issue, but simply con tain new neutral quasi-stable particles, e.g. gra vitons [2, 64, 65]. Similarly the equalit y of the couplings (feature No. 3) is a celebrated test of SUSY, but from the ab o v e comparison w e see that it is only a necessary but not a sucien t condition in pro ving sup ersymmetry In addition, the measuremen t of sup erpartner couplings in order to test the SUSY relations is a v ery c hallenging task at a hadron collider. F or one, the observ ed pro duction rate in an y giv en c hannel is only sensitiv e to the pro duct of the cross-section times the branc hing fractions, and so an y attempt to measure the couplings from a cross-section w ould ha v e to mak e certain assumptions ab out the branc hing fractions. An additional complication arises from the fact that at hadron colliders all kinematically a v ailable states can b e pro duced sim ultaneously and the pro duction of a particular sp ecies in an exclusiv e c hannel is rather dicult to isolate. The couplings could also in principle b e measured from the branc hing fractions, but that also requires a measuremen t of the total width, whic h is imp ossible in our case, since the Breit-Wigner resonance cannot b e reconstructed, due to the unkno wn momen tum of the missing LSP (LKP). W e are therefore forced to concen trate on the rst t w o iden tifying features as the only promising discriminating criteria. Let us b egin with feature 1: the n um b er of new particles. The KK particles at n = 1 are analogous to sup erpartners in

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20 sup ersymmetry Ho w ev er, the particles at the higher KK lev els ha v e no analogues in N = 1 sup ersymmetric mo dels. Disco v ering the n 2 lev els of the KK to w er w ould therefore indicate the presence of extra dimensions rather than SUSY. In this study w e shall concen trate on the n = 2 lev el and in Section 3.1 w e in v estigate the disco v ery opp ortunities at the LHC and the T ev atron (for linear collider studies of n = 2 KK gauge b osons, see [10, 19, 66, 67]). Notice that the masses of the KK mo des are giv en roughly b y m n n=R where n is the KK lev el n um b er, so that the particles at lev els 3 and higher are rather hea vy and their pro duction is sev erely suppressed. The second iden tifying feature { the spins of the new particles { also pro vides a to ol for discrimination b et w een SUSY and UED: the KK partners ha v e iden tical spin quan tum n um b ers as their SM coun terparts, while the spins of the sup erpartners dier b y 1 = 2 unit. Ho w ev er, spin determinations are kno wn to b e dicult at the LHC (or at hadron colliders in general), where the parton-lev el cen ter of mass energy E C M in eac h ev en t is unkno wn. In addition, the momen ta of the t w o dark matter candidates in the ev en t are also unkno wn. This prev en ts the reconstruction of an y rest frame angular deca y distributions, or the directions of the t w o particles at the top of the deca y c hains. The v ariable E C M also rules out the p ossibilit y of a threshold scan, whic h is one of the main to ols for determining particle spins at lepton colliders. W e are therefore forced to lo ok for new metho ds for spin determinations, or at least for nding spin correlations. Recen tly it has b een suggested that a c harge asymmetry in the lepton-jet in v arian t mass distributions from a particular cascade, can b e used to discriminate SUSY from the case of pure phase space deca ys [12]. The p ossibilit y of discriminating SUSY and UED b y this metho d will b e the sub ject of Section 3.2 (see also [7{10] and [13]). F or the purp oses of our study w e ha v e implemen ted the relev an t features of the minimal UED mo del in the CompHEP ev en t generator [68]. The minimal

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21 Sup ersymmetric Standard Mo del (MSSM) is already a v ailable in CompHEP since v ersion 41.10. W e incorp orated all n = 1 and n = 2 KK mo des as new particles, with the prop er in teractions, widths, and one-lo op corrected masses [28]. Similar to the SM case, the neutral gauge b osons at lev el 1, Z 1 and r 1 are mixtures of the KK mo des of the h yp erc harge gauge b oson and the neutral S U (2) W gauge b oson. Ho w ev er, as sho wn in [28], the radiativ ely corrected W ein b erg angle at lev el 1 and higher is v ery small. F or example, r 1 whic h is the LKP in the minimal UED mo del, is mostly the KK mo de of the h yp erc harge gauge b oson. F or simplicit y in the co de w e neglected neutral gauge b oson mixing for n 1.

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CHAPTER 3 COLLIDER PHENOMENOLOGY In this c hapter, w e consider collider implications of Univ ersal Extra Dimensions at the LHC and a future linear collider. Since the disco v ery of the rst KK lev el is discussed in [62], w e rst fo cus on the disco v ery of lev el 2 KK particles at the LHC and the T ev atron. W e then consider discrimination b et w een sup ersymmetry and Univ ersal Extra Dimensions with sev eral dieren t metho ds at the LHC and a linear collider. 3.1 Searc h for Lev el 2 KK P articles at the LHC In this section w e shall consider the prosp ects for disco v ery of lev el 2 KaluzaKlein particles in UED. Our notation and con v en tions follo w those of Ref. [62]. F or example, S U (2) W -doublet ( S U (2) W -singlet) KK fermions are denoted b y capital (lo w ercase) letters. The KK lev el n is denoted b y a subscript. In g. 3{1 w e sho w the mass sp ectrum of the n = 1 and n = 2 KK lev els in minimal UED, for R 1 = 500 GeV, R = 20 and SM Higgs b oson mass m h = 120 GeV. W e include the full one-lo op corrections from Cheng et al. [28]. W e ha v e used R GE impro v ed couplings to compute the radiativ e corrections to the KK masses (see app endix A.4 ). It is w ell kno wn that in UED the KK mo des mo dify the running of the coupling constan ts at higher scales. W e extrap olate the gauge coupling constan ts to the scale of the n = 1 and n = 2 KK mo des, using the appropriate functions dictated b y the particle sp ectrum [69{ 71]. As a result the sp ectrum sho wn in g. 3{1 diers sligh tly from the one in [28]. Most notably the colored KK particles are somewhat ligh ter, due to a reduced v alue of the strong coupling constan t, and o v erall the KK sp ectrum at eac h lev el is more degenerate. 22

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23 Figure 3{1: One-lo op corrected mass sp ectrum of the n = 1 and n = 2 KK lev els in minimal UED including the eect of running coupling constan ts in the presence of extra dimensions, for R 1 = 500 GeV, R = 20 and m h = 120 GeV. W e sho w the KK mo des of gauge b osons, Higgs and Goldstone b osons and rst generation fermions.3.1.1 Phenomenology of Lev el 2 F ermions W e b egin our discussion with the n = 2 KK fermions. Since the KK mass sp ectrum is prett y degenerate, the pro duction cross-sections at the LHC are mostly determined b y the strength of the KK particle in teractions with the proton constituen ts. As KK quarks carry color, w e exp ect their pro duction rates to b e m uc h higher than those of KK leptons. W e shall therefore concen trate on the case of KK quarks only In principle, there are t w o mec hanisms for pro ducing n = 2 KK quarks at the LHC: through KK-n um b er conserving in teractions, or through KK-n um b er violating (but KK-parit y conserving) in teractions. The KK n um b er conserving QCD in teractions allo w pro duction of KK quarks either in pairs or singly (in asso ciation with the n = 2 KK mo de of a gauge b oson). The corresp onding pro duction cross-sections are sho wn in g. 3{2 (the cross-sections for pro ducing n = 1 KK quarks ha v e b een calculated in [13, 72, 73]). In g. 3{2 a w e sho w the cross-sections (in pb) for n = 2 KK-quark pair pro duction, while in g. 3{2 b w e sho w the results for n = 2 KK-quark/KK-gluon asso ciated pro duction and

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24 Figure 3{2: Cross-sections of n = 2 KK particles at the LHC for (a) KK-quark pair pro duction (b) KK-quark/KK-gluon asso ciated pro duction and KK-gluon pair pro duction. The cross-sections ha v e b een summed o v er all quark ra v ors and also include c harge-conjugated con tributions suc h as Q 2 q 2 Q 2 q 2 g 2 Q 2 etc. for n = 2 KK-gluon pair pro duction. W e plot the results v ersus R 1 and one should k eep in mind that the masses of the n = 2 particles are roughly 2 =R In calculating the cross-sections of g. 3{2 w e consider 5 partonic quark ra v ors in the proton along with the gluon. W e sum o v er the nal state quark ra v ors and include c harge-conjugated con tributions. W e used CTEQ5L parton distributions [74] and c ho ose the scale of the strong coupling constan t s to b e equal to the parton lev el cen ter of mass energy All calculations are done with CompHEP [68] with our implemen tation of the minimal UED mo del. Sev eral commen ts are in order. First, g. 3{2 displa ys a sev ere kinematic suppression of the cross-sections at large KK masses. This is familiar from the case of SUSY, where the ultimate LHC reac h for colored sup erpartners extends only to ab out 3 T eV. Notice the dieren t mass dep endence of the cross-sections for the three t yp es of nal states with n = 2 particles: quark-quark, quark-gluon, and gluon-gluon. This can b e easily understo o d in terms of the structure functions of the quarks and gluon inside the proton. W e also observ e minor dierences in the cross-sections for pair pro duction of KK quarks with dieren t S U (2) W quan tum n um b ers. This is partially due to the dieren t masses for S U (2) W -doublet and

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25 S U (2) W -singlet quarks (see g. 3{1 ), and the remaining dierence is due to the con tributions from diagrams with electro w eak gauge b osons. Notice that since the cross-sections in g. 3{2 a are summed o v er c harge conjugated nal states, the mixed case of Q 2 q 2 con tains t wice as man y quark-an tiquark con tributions (compare Q 2 q 2 + Q 2 q 2 to q 2 q 2 or Q 2 Q 2 alone). If w e compare the cross-sections for n = 2 KK quark pro duction to the cross-sections for pro ducing squarks of similar masses in SUSY, w e realize that the pro duction rates are higher in UED. This is due to sev eral reasons. Consider, for example, s -c hannel pro cesses. W ell ab o v e threshold, the UED cross-sections are larger b y a factor of 4 [19]. One factor of 2 is due to the fact that in UED the particle con ten t at n 1 is duplicated { for example, there are b oth lefthanded and righ t-handed S U (2) W -doublet KK fermions, while in SUSY there are only \left-handed" S U (2) W -doublet squarks. Another factor of 2 comes from the dieren t angular distribution for fermions, 1 + cos 2 v ersus scalars, 1 cos 2 When in tegrated o v er all angles, this accoun ts for the second factor of 2 dierence. F urthermore, at the LHC new hea vy particles are pro duced close to threshold, due to the steeply falling parton luminosities. In SUSY, the new particles (squarks) are scalars, and the threshold suppression of the cross-sections is 3 while in UED the KK-quarks are fermions, and the threshold suppression of the cross-section is only This distinct threshold b eha vior of the pro duction cross-sections further enhances the dierence b et w een SUSY and UED. F or example, w e nd that for R 1 = 500 GeV the pair pro duction cross-section for c harm KK-quarks is ab out 6 times larger than the cross-section for c harm squarks. F or pro cesses in v olving rst generation KK-quarks, where t -c hannel diagrams con tribute signican tly the eect can b e ev en bigger. F or example, up KK-quark pro duction and up squark pro duction dier b y ab out factor of 8.

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26 Figure 3{3: Branc hing fractions of the lev el 2 \up" quarks v ersus R 1 for (a) the S U (2) W -doublet quark U 2 and (b) the S U (2) W -singlet quark u 2 In g. 3{2 w e ha v e only considered pro duction due to KK n um b er conserving bulk in teractions. The main adv an tage of those pro cesses is that the corresp onding couplings are unsuppressed. Ho w ev er, the disadv an tage is that w e need to pro duce two hea vy particles, eac h of mass 2 =R whic h leads to a kinematic suppression. In order to o v ercome this problem, one could in principle consider the single pro duction of n = 2 KK quarks through KK n um b er violating, but KK parit y conserving in teractions, for example Q 2 r T a P L Q 0 A a0 ; (3{1) where A a is a SM gauge eld and T a is the corresp onding group generator. Ho w ev er, ( 3{1 ) is forbidden b y gauge in v ariance, and the lo w est order coupling of an n = 2 KK quark to t w o SM particles has the form [28] Q 2 T a P L Q 0 F a 0 : (3{2) Suc h op erators ma y in principle b e presen t, as they ma y b e generated at the scale b y the unkno wn ph ysics at higher scales. Ho w ev er, b eing higher dimensional, w e exp ect them to b e suppressed at least b y 1 = hence in our subsequen t analysis w e shall neglect them. Ha ving determined the pro duction rates of lev el 2 KK quarks,

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27 w e no w turn to the discussion of their exp erimen tal signatures. T o this end w e need to determine the p ossible deca y mo des of Q 2 and q 2 A t eac h lev el n the KK quarks are among the hea viest states in the KK sp ectrum and can deca y promptly to ligh ter KK mo des (this is true for the top KK mo des [75, 76] as w ell). As can b e seen from g. 3{1 the KK gluon is alw a ys hea vier than the KK quarks, so the t w o b o dy deca ys of KK quarks to KK gluons are closed. Instead, n = 2 KK quarks will deca y to the KK mo des of the electro w eak gauge b osons whic h are ligh ter. The branc hing fractions for n = 2 \up"-t yp e KK quarks are sho wn in g. 3{3 Fig. 3{3 a (g. 3{3 b) is for the case of the S U (2) W -doublet quark U 2 (the S U (2) W -singlet quark u 2 ). The results for the \do wn"-t yp e KK quarks are similar. W e observ e in g. 3{3 that the branc hing fractions are almost indep enden t of R 1 unless one is close to threshold. This feature will p ersist for all branc hing ratios of KK particles whic h will b e sho wn later. Once w e ignore the KK n um b er violating coupling ( 3{2 ), only deca ys whic h conserv e the total KK n um b er n are allo w ed. The case of the S U (2) W -singlet quarks suc h as u 2 is simpler, since they only couple to the h yp erc harge gauge b osons. Recall that at n 1 the h yp erc harge comp onen t is almost en tirely con tained in the r KK mo de [28]. W e therefore exp ect a singlet KK quark q 2 to deca y to either q 1 r 1 or q 0 r 2 as seen in g. 3{3 b. The case of an S U (2) W -doublet quark Q 2 is m uc h more complicated, since Q 2 couples to the (KK mo des of the) w eak gauge b osons as w ell, and man y more t w o-b o dy nal states are p ossible. Since the w eak coupling is larger than the h yp erc harge coupling, the deca ys to W and Z KK mo des dominate, with B R ( Q 2 Q 00 W 2 ) =B R ( Q 2 Q 0 Z 2 ) = 2 and B R ( Q 2 Q 01 W 1 ) =B R ( Q 2 Q 1 Z 1 ) = 2, as evidenced in g. 3{3 a. The branc hing fractions to the r KK mo des are only on the order of a few p ercen t. The threshold b eha vior seen in g. 3{3 a near R 1 = 400 GeV is due to the nite masses for the SM W and Z b osons, whic h en ter the tree-lev el masses of W 1 and Z 1 Since

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28 Figure 3{4: Branc hing fractions of the lev el 2 KK electrons v ersus R 1 The same as g. 3{3 but for the lev el 2 KK electrons: (a) the S U (2) W -doublet E 2 and (b) the S U (2) W -singlet e 2 the mass splitting of the KK mo des is due to the radiativ e corrections, whic h are prop ortional to R 1 the c hannels with W 1 and Z 1 op en up only for sucien tly large R 1 W e are no w in a p osition to discuss the exp erimen tal signatures of n = 2 KK quarks. The deca ys to lev el 2 gauge b osons will simply con tribute to the inclusiv e pro duction of r 2 Z 2 and W 2 whic h will b e discussed at length later in Section 3.1.2 On the other hand, the deca ys to t w o n = 1 KK mo des will con tribute to the inclusiv e pro duction of n = 1 KK particles whic h w as discussed in [62]. Naturally the direct pair pro duction of the ligh ter n = 1 KK mo des has a m uc h larger cross-section. Therefore, the indirect pro duction of n = 1 KK mo des from the deca ys of n = 2 particles can b e easily sw amp ed b y the direct n = 1 signals and the SM bac kgrounds. F or example, the exp erimen tal signature for an n = 2 KK quark deca ying as Q 2 Q 1 r 1 ( q 2 q 1 r 1 ) is indistinguishable from a single Q 1 ( q 1 ). This is b ecause r 1 do es not in teract within the detector, and there are at least t w o additional r 1 particles in eac h ev en t, so that w e cannot determine ho w man y r 1 particles caused the measured amoun t of missing energy The deca ys to W 1 and Z 1 ma y ho w ev er, lead to nal states with up to four n = 1 particles,

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29 eac h with a leptonic deca y mo de. The resulting m ultilepton signatures N ` + / E T with N 5 are therefore v ery clean and p oten tially observ able. Distinguishing those ev en ts from direct n = 1 pair pro duction w ould b e an imp ortan t step in establishing the presence of the n = 2 lev el of the quark KK to w er. Unfortunately the n = 2 sample is statistically v ery limited and this analysis app ears v ery c hallenging. W e p ostp one it for future w ork [77]. Muc h of the previous discussion applies directly to the lev el 2 KK leptons. Assuming the absence of the KK n um b er violating coupling analogous to ( 3{2 ), the branc hing fractions of the n = 2 KK electrons are sho wn in g. 3{4 A t eac h KK lev el, the KK mo des of the w eak gauge b osons are hea vier than the KK leptons, therefore the only allo w ed deca ys are to r 2 and r 1 Just lik e KK quarks, KK leptons can b e pro duced directly through KK n um b er conserving couplings, or indirectly in W 2 and Z 2 deca ys. In either case, the resulting cross-sections are to o small to b e of in terest at the LHC. 3.1.2 Phenomenology of Lev el 2 Gauge Bosons W e no w discuss the collider phenomenology of the n = 2 gauge b osons V 2 As w e shall see, the KK gauge b osons oer the b est prosp ects for detecting the n = 2 structure, since they ha v e direct (but not tree lev el) couplings to SM particles, and can b e disco v ered as resonances, e.g. in the dijet or dilepton c hannels. This is in con trast to the case of n = 2 KK fermions, whic h, under the assumptions of Sec. 3.1.1 do not ha v e fully visible deca y mo des. Bump h un ting will also help discriminate b et w een n = 2 and n = 1 KK particles, since the latter are KK-parit y o dd, and necessarily deca y to the in visible r 1 There are four n = 2 KK gauge b osons: the KK \photon" r 2 the KK \ Z -b oson" Z 2 the KK \ W -b oson" W 2 and the KK gluon g 2 Recall that the W ein b erg angle at n = 2 is v ery small, so that r 2 is mostly the KK mo de of the h yp erc harge gauge b oson and Z 2 is mostly the KK mo de of the neutral W -b oson of the SM. An imp ortan t consequence of the extra dimensional nature of the mo del

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30 Figure 3{5: Masses and widths of lev el 2 KK gauge b osons. (a) Masses of the four n = 2 KK gauge b osons as a function of R 1 (b) T otal widths of the n = 2 KK gauge b osons as a function of the corresp onding mass. W e also sho w the width of a generic Z 0 whose couplings to the SM particles are the same as those of the Z -b oson. is that all four of the n = 2 KK gauge b osons are relativ ely degenerate, as sho wn in g. 3{5 a. The masses are all roughly equal to 2 =R The mass splitting b et w een the KK gauge b osons is almost en tirely due to radiativ e corrections, whic h in the minimal UED mo del yield the mass hierarc h y m g 2 > m W 2 m Z 2 > m r 2 The KK gluon receiv es the largest corrections and is the hea viest particle in the KK sp ectrum at eac h lev el n The W 2 and Z 2 particles are degenerate to a v ery high degree, due to S U (2) W symmetry The KK n um b er conserving in teractions allo w an n = 2 KK gauge b oson V 2 to deca y to t w o n = 1 particles, or to one n = 2 KK particle and one n = 0 (i.e., Standard Mo del) particle, pro vided that the deca ys are allo w ed b y phase space. F or example, the partial widths to fermion nal states are giv en b y ( V 2 f 2 f 0 ) = c 2 g 2 48 m 3V 2 m 2V 2 m 2f 2 m 2f 0 + m 4V 2 ( m 2f 2 m 2f 0 ) 2 m 2V 2 q m 2V 2 ( m f 2 m f 0 ) 2 m 2V 2 ( m f 2 + m f 0 ) 2 (3{3) c 2 g 2 48 m 3V 2 m 2V 2 m 2f 2 2 1 + m 2V 2 + m 2f 2 m 2V 2 c 2 g 2 m V 2 4 ^ m V 2 m 2 ^ m f 2 m 2 2 ;

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31 ( V 2 f 1 f 1 ) = c 2 g 2 24 m 2V 2 m 2V 2 4 m 2f 1 3 2 (3{4) c 2 g 2 m V 2 6 p 2 ^ m V 2 m 2 ^ m f 1 m 1 3 2 m 2 m V 2 3 c 2 g 2 m V 2 6 p 2 ^ m V 2 m 2 ^ m f 1 m 1 3 2 ; where c Y N f c = 2 for V 2 r 2 c N f c = 2 for V 2 Z 2 c = V C K M N f c = p 2 for V 2 = W 2 and c = 1 = p 2 for V 2 = g 2 with Y b eing the fermion h yp erc harge in the normalization Q = T 3 + Y = 2, V C K M is the CKM mixing matrix, and N f c = 3 for f = q and N f c = 1 for f = ` Here ^ m stands for the total radiativ e correction to a KK mass m including b oth bulk and b oundary con tributions [28], m 2 2 =R and g is the corresp onding gauge coupling. The rst lines in ( 3{3 ) and ( 3{4 ) giv e the exact result, while the last lines are the appro ximate form ulas deriv ed in [62] as leading order expansions in ^ m=m The second line in ( 3{3 ) is an appro ximation neglecting the SM fermion mass m f 0 The second line in ( 3{4 ) is an alternativ e appro ximation whic h incorp orates subleading but n umerically imp ortan t terms. In our co de w e ha v e programmed the exact expressions and quote the appro ximations here only for completeness. Note that the KK n um b er conserving deca ys of the n = 2 KK gauge b osons are suppressed b y phase space. This is eviden t from the appro ximate expressions in eqs. ( 3{3 ) and ( 3{4 ). The partial widths are prop ortional to the one-lo op corrections, whic h op en up the a v ailable phase space and allo w the corresp onding deca y mo de to tak e place. Ho w ev er, not all of the fermionic nal states are a v ailable, for example, Z 2 and W 2 ha v e no hadronic deca y mo des to lev el 1 or 2, while r 2 has no KK n um b er conserving deca y mo des at all. The n = 2 KK gauge b osons also ha v e KK n um b er violating couplings whic h can b e generated either radiativ ely from bulk in teractions, or directly at the scale

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32 [28]. F or example, the op erator f 0 r T a P L f 0 A a2 (3{5) couples V 2 directly to SM fermions f 0 and leads to the the follo wing V 2 partial width ( V 2 f 0 f 0 ) = c 2 g 2 m V 2 12 m V 2 m 2 m f 2 m 2 2 1 m 2f 0 m 2V 2 s 1 4 m 2f 0 m 2V 2 (3{6) c 2 g 2 m V 2 12 m V 2 m 2 m f 2 m 2 2 ; where m stands for a mass correction due to b oundary terms only [28]. In the second line w e ha v e neglected the SM fermion mass m f 0 reco v ering the result from Cheng et al. [62]. As w e see from ( 3{6 ), the KK n um b er violating deca y is also suppressed, this time b y a lo op factor, and is prop ortional to the size of the radiativ e corrections to the corresp onding KK masses. In spite of this suppression, the V 2 f 0 f 0 deca ys is most promising for exp erimen tal disco v ery As long as the nal state fermions can b e reconstructed, the V 2 particle can b e lo ok ed for as a bump in the in v arian t mass distribution of its deca y pro ducts. In this sense, the searc h is v ery similar to Z 0 searc hes, with one ma jor dierence. Since al l partial widths ( 3{3 3{6 ) are suppressed, the total width of V 2 is m uc h smaller than the width of a t ypical Z 0 This is illustrated in g. 3{5 b, where w e plot the widths of the KK particles r 2 W 2 Z 2 and g 2 in UED, as a function of the corresp onding particle mass, and con trast to the width of a Z 0 with SM-lik e couplings. W e see that the widths of the KK gauge b osons are extremely small. This has imp ortan t ramications for the exp erimen tal searc h, since the width of the resonance will then b e determined b y the exp erimen tal resolution, rather than the in trinsic particle width. In this sense the width m ust b e included in the set of basic parameters of a Z 0 searc h [78]. Before w e elab orate on the exp erimen tal signatures of the n = 2 KK gauge b osons,

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33 Figure 3{6: Cross-sections for single pro duction of lev el 2 KK gauge b osons through the KK n um b er violating couplings ( 3{5 ). let us briery discuss their pro duction. There are three basic mec hanisms: 1. Single pro duction through the KK n um b er violating op erator. The corresp onding cross-sections are sho wn in g. 3{6 as a function of R 1 One migh t exp ect that these pro cesses will b e imp ortan t, esp ecially at large masses, since w e need to mak e only a single hea vy n = 2 particle, alleviating the kinematic suppression. If w e compare the mass dep endence of the Drell-Y an cross-sections in g. 3{6 to the mass dep endence of the n = 2 pair pro duction cross-sections from g. 3{2 indeed w e see that the former drop less steeply with R 1 and b ecome dominan t at large R 1 On the other hand, the Drell-Y an pro cesses of g. 3{6 are mediated b y a KK n um b er violating op erator ( 3{5 ) and the coupling of a V 2 to SM particles is radiativ ely suppressed. This is another crucial dierence with the case of a generic Z 0 whose couplings t ypically ha v e the size of a normal gauge coupling and are unsuppressed [78]. Notice the roughly similar size of the four cross-sections sho wn in g. 3{6 This is somewhat surprising, since the cross-sections scale as the corresp onding gauge coupling squared, and one w ould ha v e exp ected a wider spread in the v alues of the four cross-sections. This is due to a couple of things. First, for a

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34 giv en R 1 the masses of the four n = 2 KK gauge b osons are dieren t, with m g 2 > m W 2 m Z 2 > m r 2 Therefore, for a giv en R 1 the hea vier particles suer a suppression. This explains to an exten t wh y the cross-section for r 2 is not the smallest of the four, and wh y the cross-section for g 2 is not as large as one w ould exp ect. There is, ho w ev er, a second eect, whic h go es in the same direction. The coupling ( 3{5 ) is also prop ortional to the mass corrections of the corresp onding particles: m V 2 m V 2 m f 2 m f 2 : (3{7) Since the QCD corrections are the largest, for V 2 = f r 2 ; Z 2 ; W 2 g the second term dominates. Ho w ev er, for V 2 = g 2 the rst term is actually larger, and there is a cancellation, whic h further reduces the direct KK gluon couplings to quarks. 2. Indirect pro duction. The electro w eak KK mo des r 2 Z 2 and W 2 can b e pro duced in the deca ys of hea vier n = 2 particles suc h as the KK quarks and/or KK gluon. This is w ell kno wn from the case of SUSY, where the dominan t pro duction of electro w eak sup erpartners is often indirect { from squark and gluino deca y c hains. The indirect pro duction rates of r 2 Z 2 and W 2 due to QCD pro cesses can b e readily estimated from gs. 3{2 and 3{3 Notice that B R ( Q 2 W 2 ), B R ( Q 2 Z 2 ) and B R ( q 2 r 2 ) are among the largest branc hing fractions of the n = 2 KK quarks, and w e exp ect indirect pro duction from QCD to b e a signican t source of electro w eak n = 2 KK mo des. 3. Direct pair pro duction. The n = 2 KK mo des can also b e pro duced directly in pairs, through KK n um b er conserving in teractions. These pro cesses, ho w ev er, are kinematically suppressed, since w e ha v e to mak e two hea vy particles in the nal state. One w ould therefore exp ect that they will b e the least relev an t source of n = 2 KK gauge b osons. The only exception is KK gluon pair pro duction whic h is imp ortan t and is sho wn in g. 3{2 b. W e see that it is comparable in size to KK quark pair pro duction and q 2 g 2 / Q 2 g 2 asso ciated pro duction. W e ha v e

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35 Figure 3{7: Branc hing fractions of the n = 2 KK gauge b osons v ersus R 1 : (a) g 2 (b) Z 2 (c) W 2 and (d) r 2 also calculated the pair pro duction cross-sections for the electro w eak n = 2 KK gauge b osons and conrmed that they are v ery small, hence w e shall neglect them in our analysis b elo w. In conclusion of this section, w e discuss the exp erimen tal signatures of n = 2 KK gauge b osons. T o this end, w e need to consider their p ossible deca y mo des. Ha ving previously discussed the dieren t partial widths, it is straigh tforw ard to compute the V 2 branc hing fractions. Those are sho wn in g. 3{7 (a-d). Again w e observ e that the branc hing fractions are v ery w eakly sensitiv e to R 1 just as the case of gs. 3{3 and 3{4 This can b e understo o d as follo ws. The partial widths ( 3{3 ) and ( 3{4 ) for the KK n um b er conserving deca ys are prop ortional to the a v ailable phase space, while the partial width ( 3{6 ) for the KK n um b er violating deca y is prop ortional to the mass corrections (see eq. ( 3{7 )).

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36 Both the phase space and the mass corrections are prop ortional to R 1 whic h then cancels out in the branc hing fraction. Similarly to the case of n = 2 KK quarks discussed in Sec. 3.1.1 KK n um b er conserving deca ys are not v ery distinctiv e, since they simply con tribute to the inclusiv e n = 1 sample whic h is dominated b y direct n = 1 pro duction. The deca ys of n = 1 particles will then giv e relativ ely soft ob jects, and most of the energy will b e lost in the LKP mass. In short, n = 2 signatures based on purely KK n um b er conserving deca ys are not v ery promising exp erimen tally | one has to pa y a big price in the cross-section in order to pro duce the hea vy n = 2 particles, but do es not get the b enet of the large mass, since most of the energy is carried a w a y b y the in visible LKP W e therefore concen trate on the KK n um b er violating c hannels, in whic h the V 2 deca ys are fully visible. Fig. 3{7 a sho ws the branc hing fractions of the KK gluon g 2 Since it is the hea viest particle at lev el 2, all of its deca y mo des are op en, and ha v e comparable branc hing fractions. The KK n um b er conserving deca ys dominate, since the KK n um b er violating coupling is sligh tly suppressed due to the cancellation in ( 3{7 ). In principle, g 2 can b e lo ok ed for as a resonance in the dijet [79] or t t in v arian t mass sp ectrum, but one w ould exp ect large bac kgrounds from QCD and DrellY an. Notice that there is no indirect pro duction of g 2 and its single pro duction cross-section is not that m uc h dieren t from the cross-sections for r 2 Z 2 and W 2 (see g. 3{6 ). Therefore, the inclusiv e g 2 pro duction is comparable to the inclusiv e r 2 and Z 2 pro duction, and then w e an ticipate that the searc hes for the n = 2 electro w eak gauge b osons in leptonic c hannels will b e more promising. Figs. 3{7 b and 3{7 c giv e the branc hing fractions of Z 2 and W 2 corresp ondingly W e see that the deca ys to KK quarks ha v e b een closed due to the large QCD radiativ e corrections to the KK quark masses. Among the p ossible KK n um b er conserving deca ys of Z 2 and W 2 only the leptonic mo des surviv e, and they will

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37 b e con tributing to the leptonic disco v ery signals of UED [62]. Recall that the KK n um b er conserving deca ys are phase space suppressed, while the KK n um b er violating deca ys are lo op suppressed, and prop ortional to the mass corrections as in ( 3{7 ). The precise calculation sho ws that the dominan t deca y mo des are Z 2 q q and W 2 q q 0 This can b e understo o d in terms of the large m q 2 correction app earing in ( 3{7 ). The resulting branc hing ratios are more than 50% and in principle allo w for a Z 2 =W 2 searc h in the dijet c hannel, just lik e the case of g 2 Ho w ev er, w e shall concen trate on the leptonic deca y mo des, whic h ha v e m uc h smaller branc hing fractions, but are m uc h cleaner exp erimen tally Finally g. 3{7 d sho ws the branc hing fractions of r 2 This time all KK n um b er conserving deca ys are closed, and r 2 is forced to deca y through the KK n um b er violating in teraction ( 3{5 ). Again, the jett y mo des dominate, and the leptonic mo des (summed o v er lepton ra v ors) ha v e rather small branc hing fractions, on the order of 2%, whic h could b e a p oten tial problem for the searc h. In the follo wing section w e shall concen trate on the Z 2 ` + ` and r 2 ` + ` signatures and analyze their disco v ery prosp ects in a Z 0 -lik e searc h [80, 81]. 3.1.3 Analysis of the LHC Reac h for Z 2 and r 2 W e are no w in a p osition to discuss the disco v ery reac h of the n = 2 KK gauge b osons at the LHC and the T ev atron. W e will consider the inclusiv e production of Z 2 and r 2 and lo ok for a dilepton resonance in b oth the e + e and + c hannels. An imp ortan t parameter of the searc h is the width of the reconstructed resonance, whic h in turn determines the size of the in v arian t mass windo w selected b y the cuts. Since the in trinsic width of the Z 2 and r 2 resonances is so small (see g. 3{5 b), the mass windo w is en tirely determined b y the mass resolution in the dim uon and dielectron c hannels. F or electrons, the resolution in CMS is appro ximately constan t, on the order of m ee =m ee 1% in the region of in terest [82]. On

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38 Figure 3{8: 5 disco v ery reac h for (a) r 2 and (b) Z 2 W e plot the total in tegrated luminosit y L (in fb 1 ) required for a 5 excess of signal o v er bac kground in the dielectron (red, dotted) or dim uon (blue, dashed) c hannel, as a function of R 1 In eac h plot, the upp er set of lines lab elled \D Y" mak es use of the single V 2 pro duction of g. 3{6 only while the lo w er set of lines (lab elled \All pro cesses") includes indirect r 2 and Z 2 pro duction from n = 2 KK quark deca ys. The red dotted line mark ed \FNAL" in the upp er left corner of (a) rerects the exp ectations for a r 2 e + e disco v ery at the T ev atron in Run I I. The shaded area b elo w R 1 = 250 GeV indicates the region disfa v ored b y precision electro w eak data [33]. the other hand, the dim uon mass resolution is energy dep enden t, and in preliminary studies based on a full sim ulation of the CMS detector has b een parametrized as [83] m m = 0 : 0215 + 0 : 0128 m 1 T eV : Therefore in our analysis w e imp ose the follo wing cuts 1. Lo w er cuts on the lepton transv erse momen ta p T ( ` ) > 20 GeV. 2. Cen tral rapidit y cut on the leptons j ( ` ) j < 2 : 4. 3. Dilepton in v arian t mass cut for electrons m V 2 2 m ee < m ee < m V 2 + 2 m ee and m uons m V 2 2 m < m < m V 2 + 2 m With these cuts the signal eciency v aries from 65% at R 1 = 250 GeV to 91% at R 1 = 1 T eV. The main SM bac kground to our signal is Drell-Y an, whic h w e ha v e calculated with the PYTHIA ev en t generator [84]. With the cuts listed ab o v e, w e compute the disco v ery reac h of the LHC and the T ev atron for the r 2

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39 and Z 2 resonances. Our results are sho wn in g. 3{8 W e plot the total in tegrated luminosit y L (in fb 1 ) required for a 5 excess of signal o v er bac kground in the dielectron (red, dotted) or dim uon (blue, dashed) c hannel, as a function of R 1 In eac h panel in g. 3{8 the upp er set of lines lab elled \D Y" only utilizes the single V 2 pro duction cross-sections from g. 3{6 The lo w er set of lines (lab elled \All pro cesses") includes in addition indirect r 2 and Z 2 pro duction from the deca ys of n = 2 KK quarks to r 2 and Z 2 (w e ignore secondary r 2 pro duction from Q 2 Z 2 ` 2 r 2 ). The shaded area b elo w R 1 = 250 GeV indicates the region disfa v ored b y precision electro w eak data [33]. Using the same cuts also for the case of the T ev atron, w e nd the T ev atron reac h in r 2 e + e sho wn in g. 3{8 a and lab elled \FNAL." F or the T ev atron w e use electron energy resolution E =E = 0 : 01 0 : 16 = p E [85]. The T ev atron reac h in dim uons is w orse due to the p o orer resolution, while the reac h for Z 2 is also w orse since m Z 2 > m r 2 for a xed R 1 Fig. 3{8 rev eals that there are go o d prosp ects for disco v ering lev el 2 gauge b oson resonances at the LHC. Already within one y ear of running at lo w luminosit y (L = 10 fb 1 ), the LHC will ha v e sucien t statistics in order to prob e the region up to R 1 750 GeV. Notice that in the minimal UED mo del, the \go o d dark matter" region, where the LKP relic densit y accoun ts for all of the dark matter comp onen t of the Univ erse, is at R 1 500 600 GeV [20{22]. This region is w ell within the disco v ery reac h of the LHC for b oth n = 1 KK mo des [62] and n = 2 KK gauge b osons (g. 3{8 ). If the LKP accoun ts for only a fr action of the dark matter, the preferred range of R 1 is ev en lo w er and the disco v ery at the LHC is easier. F rom g. 3{8 w e also see that the ultimate reac h of the LHC for b oth r 2 and Z 2 after sev eral y ears of running at high luminosit y (L 300 fb 1 ), extends up to just b ey ond R 1 = 1 T eV. One should k eep in mind that the actual KK masses

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40 Figure 3{9: The r 2 Z 2 diresonance structure in UED with R 1 = 500 GeV, for (a) the dim uon and (b) the dielectron c hannel at the LHC with L = 100 fb 1 The SM bac kground is sho wn with the (red) con tin uous underlying histogram. are at least t wice as large: m V 2 m 2 = 2 =R so that the KK resonances can b e disco v ered for masses up to 2 T eV. While the n = 2 KK gauge b osons are a salien t feature of the UED scenario, an y suc h resonance b y itself is not a sucien t discriminator, since it resem bles an ordinary Z 0 gauge b oson. If UED is disco v ered, one could then still mak e the argumen t that it is in fact some sort of non-minimal sup ersymmetric mo del with an additional gauge structure con taining neutral gauge b osons. An imp ortan t corrob orating evidence in fa v or of UED w ould b e the sim ultaneous disco v ery of sev eral, rather degenerate, KK gauge b oson resonances. While SUSY also can accommo date m ultiple Z 0 gauge b osons, there w ould b e no go o d motiv ation b ehind their mass degeneracy A crucial question therefore arises: can w e separately disco v er the n = 2 KK gauge b osons as individual resonances? F or this purp ose, one w ould need to see a double p eak structure in the in v arian t mass distributions. Clearly this is rather c hallenging in the dijet c hannel, due to the relativ ely p o or jet energy resolution. W e shall therefore consider only the dilepton c hannels, and in v estigate ho w w ell w e can separate r 2 from Z 2 Our results are sho wn in g. 3{9 where w e sho w the in v arian t mass distribution in UED with R 1 = 500 GeV, for (a) the dim uon and (b) the dielectron

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41 c hannel at the LHC with L = 100 fb 1 W e see that the diresonance structure is easier to detect in the dielectron c hannel, due to the b etter mass resolution. In dim uons, with L = 100 fb 1 the structure is also b eginning to emerge. W e should note that initially the t w o resonances will not b e separately distinguishable, and eac h will in principle con tribute to the disco v ery of a bump, although with a larger mass windo w. In our reac h plots in g. 3{8 w e ha v e conserv ativ ely c hosen not to com bine the t w o signals from Z 2 and r 2 but sho w the reac h for eac h one separately In this section w e ha v e discussed the dierences and similarities in the hadron collider phenomenology of mo dels with Univ ersal Extra Dimensions and sup ersymmetry W e iden tied the higher lev el KK mo des of UED and as a reliable discriminator b et w een the t w o scenarios. W e then pro ceeded to study the disco v ery reac h for lev el 2 KK mo des in UED at hadron colliders. W e sho w ed that the n = 2 KK gauge b osons oer the b est prosp ects for detection, in particular the r 2 and Z 2 resonances can b e sep ar ately disco v ered at the LHC. Is this a pro of of UED? Not quite { these resonances could still b e in terpreted as Z 0 gauge b osons, but their close degeneracy is a smoking gun for UED. F urthermore, although w e did not sho w an y results to this eect, it is clear that the W 2 KK mo de can also b e lo ok ed for and disco v ered in its deca y to SM leptons. One can then measure m W 2 and sho w that it is v ery close to m Z 2 and m r 2 whic h w ould further strengthen the case for UED. Here w e only concen trated on the minimal UED mo del, it should b e k ept in mind that there are man y in teresting p ossibilities for extending the analysis to a more general setup. F or example, non-v anishing b oundary terms at the scale can distort the minimal UED sp ectrum b ey ond recognition. A priori, in suc h a relaxed framew ork the UED-SUSY confusion can b e \complete" in the con text of a hadron collider and a preliminary study is under w a y to address this issue [14, 15].

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42 The UED collider phenomenology is also v ery dieren t in the case of a \fat" brane [86, 87], c harged LKPs [88] or KK gra viton sup erwimps [89, 90]. Notice that Little Higgs mo dels with T -parit y [16, 91{ 94] are v ery similar to UED, and can also b e confused with sup ersymmetry 3.2 Spin Determination at the LHC The fundamen tal dierence b et w een SUSY and UED is rst the n um b er of new particles and second, the spins of new particles. The KK particles at n = 1 are analogous to sup erpartners in sup ersymmetry Ho w ev er, the particles at the higher KK lev els ha v e no analogues in N = 1 sup ersymmetric mo dels. Disco v ering the n 2 lev els of the KK to w er w ould therefore indicate the presence of extra dimensions rather than SUSY. Ho w ev er these KK particles can b e to o hea vy to b e observ ed. Ev en if they can b e observ ed at the LHC, they can b e confused with other new particles [10, 11] suc h as Z 0 or dieren t t yp es of resonances from extra dimensions [25]. The second feature { the spins of the new particles { also pro vides a to ol for discrimination b et w een SUSY and UED: the KK partners ha v e iden tical spin quan tum n um b ers as their SM coun terparts, while the spins of the sup erpartners dier b y 1 = 2 unit. Ho w ev er, spin determinations are kno wn to b e dicult at the LHC (or at hadron colliders in general), where the parton-lev el cen ter of mass energy E C M in eac h ev en t is unkno wn. In addition, the momen ta of the t w o dark matter candidates in the ev en t are also unkno wn. This prev en ts the reconstruction of an y rest frame angular deca y distributions, or the directions of the t w o particles at the top of the deca y c hains. The v ariable E C M also rules out the p ossibilit y of a threshold scan, whic h is one of the main to ols for determining particle spins at lepton colliders. W e are therefore forced to lo ok for new metho ds for spin

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43 SUSY: ~ q ~ 02 ~ ` L ~ 01 UED: Q 1 Z 1 ` 1 r 1 q ` (near) ` (far) Figure 3{10: Twin diagrams in SUSY and UED. The upp er (red) line corresp onds to the cascade deca y ~ q q ~ 02 q ` ~ ` L q ` + ` ~ 01 in SUSY. The lo w er (blue) line corresp onds to the cascade deca y Q 1 q Z 1 q ` ` 1 q ` + ` r 1 in UED. In either case the observ able nal state is the same: q ` + ` / E T determinations, or at least for nding spin correlations 1 The purp ose of this section is to in v estigate the prosp ects for establishing sup ersymmetry at the LHC b y discriminating it from its lo ok-alik e scenario of Univ ersal Extra Dimensions b y measuring spins of new particles in t w o mo dels 2 As discussed b efore, the second fundamen tal distinction b et w een UED and sup ersymmetry is rerected in the prop erties of the individual particles. Recen tly it has b een suggested that a c harge asymmetry in the lepton-jet in v arian t mass distributions from a particular cascade (see g. 3{10 ), can b e used to discriminate SUSY from the case of pure phase space deca ys [12] and is an indirect indication of the sup erparticle spins (A study of measuring sleptons spins at the LHC can b e found in [17]). It is 1 Notice that in simple pro cesses with t w o-b o dy deca ys lik e slepton pro duction e + e ~ + ~ + ~ 01 ~ 01 the rat energy distribution of the observ able nal state particles (m uons in this case) is often regarded as a smoking gun for the scalar nature of the in termediate particles (the sm uons). Indeed, the sm uons are spin zero particles and deca y isotropically in their rest frame, whic h results in a rat distribution in the lab frame. Ho w ev er, the rat distribution is a necessary but not sucien t condition for a scalar particle, and UED pro vides a coun terexample with the analogous pro cess of KK m uon pro duction [19], where a rat distribution also app ears, but as a result of equal con tributions from left-handed and righ t-handed KK fermions. 2 The same idea can apply in the case of little Higgs mo dels since the rst lev el of the UED mo del lo oks lik e the new particles in little Higgs mo dels [91{94].

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44 Figure 3{11: Comparison of dilepton in v arian t mass distributions in the case of (a) UED mass sp ectrum with R 1 = 500 GeV (b) mass sp ectrum from SPS1a. In b oth cases, UED (SUSY) distributions are sho wn in blue (red). All distributions are normalized to L = 10 fb 1 and the error bars represen t statistical uncertain t y therefore natural to ask whether this metho d can b e extended to the case of SUSY v ersus UED discrimination. F ollo wing [12], w e concen trate on the cascade deca y ~ q q ~ 02 q ` ~ ` L q ` + ` ~ 01 in SUSY and the analogous deca y c hain Q 1 q Z 1 q ` ` 1 q ` + ` r 1 in UED. Both of these pro cesses are illustrated in g. 3{10 Blue lines represen t the deca y c hain in UED and red lines the deca y c hain in SUSY. Green lines are SM particles. 3.2.1 Dilepton In v arian t Mass Distributions First w e will lo ok for spin correlations b et w een the t w o SM leptons in the nal state. In sup ersymmetry the slepton is a scalar particle and therefore there is no spin correlation b et w een the t w o SM leptons. Ho w ev er in UED, the slepton is replaced b y a KK lepton and is a fermion. W e migh t therefore exp ect a dieren t shap e in the dilepton in v arian t mass distribution. T o in v estigate this, w e rst c ho ose a study p oin t in UED (SPS1a in mSugra) with R 1 = 500 GeV tak en from Cheng et al. [28, 62] and then w e adjust the relev an t MSSM parameters (UED parameters) un til w e get a matc hing sp ectrum. So the masses are exactly the same and they can not b e used for discrimination and the only dierence is the spin. In

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45 g. 3{11 w e sho w in v arian t mass distributions in UED and SUSY for t w o dieren t t yp es of mass sp ectrum. In g. 3{11 (a), all UED masses are adjusted to b e the same as the SUSY masses in SPS1a ( m 0 = 100 GeV, m 1 = 2 = 250 GeV, A 0 = 100, tan = 10 and > 0) while in g. 3{11 (b) the SUSY masses are replaced b y KK masses for R 1 = 500. In b oth cases, UED (SUSY) distributions are sho wn in blue (red). Squark/KK quark pair-pro duction cross-sections are tak en from Smillie et al. [13] and the relev an t branc hing fractions are obtained from Cheng et al. [62] for UED and [95] for SUSY. All distributions are normalized to L = 10 fb 1 and the error bars represen t statistical uncertain t y In sup ersymmetry the distribution is the same as the one in the case of pure phase space deca y since the slepton has no spin. As w e can notice, the t w o distributions are iden tical for b oth UED and SUSY mass sp ectrum ev en if the in termediate particles in UED and SUSY ha v e dieren t spins. The minor dierences in the plot will completely disapp ear once the bac kground, radiativ e corrections and detector sim ulation are included. The in v arian t mass distributions for UED and SUSY/Phase space can b e written as [13, 96] Phase Space : dN d ^ m = 2 ^ m SUSY : dN d ^ m = 2 ^ m (3{8) UED : dN d ^ m = 4( y + 4 z ) (1 + 2 z )(2 + y ) ^ m + r ^ m 3 where the co ecien t r in the second term of the UED distribution is dened as r = (2 y )(1 2 z ) y + 4 z ; (3{9) ^ m = m `` m max`` is the rescaled in v arian t mass, y = m ~ ` m ~ 02 2 and z = m ~ 01 m ~ ` 2 are the ratios of the masses in v olv ed in the deca y y and z are less than 1 in the case of on-shell deca y F rom eqn. 3{8 there are t w o terms in UED. The rst term is a

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46 Figure 3{12: A closer lo ok in to dilepton in v arian t mass distributions. (a) Con tour dotted lines represen t the size of the co ecien t r in eqn. 3{9 The minimal UED is a blue dot in the upp er-righ t corner since y and z are almost 1 due to the mass degeneracy The red dots represen t sev eral sno wmass p oin ts: SPS1a, SPS1b, SPS5 and SPS3 from left to righ t. The green line represen ts gaugino unication so all SUSY b enc hmark p oin ts are close to this green line. (b) The dashed line represen ts the dilepton distribution in SUSY or pure phase space. The solid cy an (magen ta) line represen ts the dilepton distribution in UED for r = 0 : 3 ( r = 0 : 7). linear in ^ m lik e phase space and the second term is prop ortional to ^ m 3 So w e see that whether or not the UED distribution is the same as the SUSY distribution dep ends on the size of the co ecien t r in the second term of the UED distribution. The UED distribution b ecomes exactly the same as the SUSY distribution if r = 0 : 5. Therefore w e scan the ( y ; z ) parameter space, calculate the co ecien t r and sho w our result in g. 3{12 (a). In g. 3{12 (a), the con tour dotted lines represen t the size of the co ecien t r in eqn. 3{9 The minimal UED is blue dot in upp er-righ t corner since y and z are almost 1 due to the degeneracy in the masses while red dots represen t sev eral sno wmass p oin ts [97]: SPS1a, SPS1b, SPS5 and SPS3 from left to righ t. The green line represen ts gaugino unication so all SUSY b enc hmark p oin ts are close to this green line. As w e see r is small for b oth MUED and sno wmass p oin ts and this is wh y w e did not see an y dierence in the distributions from g. 3{11 If the mass sp ectrum is either narro w (MUED mass sp ectrum) or generic mSugra t yp e, the dilepton distributions are v ery similar

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47 and w e can not tell an y spin information from this distribution. Ho w ev er a w a y from the the mSugra mo del or MUED, w e can easily nd the regions where this co ecien t r is large and the spin correlation is big enough so that w e can see a dierence in shap e. W e sho w t w o p oin ts (denoted b y `Go o d' and `Better') from g. 3{12 (a) and sho w the corresp onding dilepton distributions in g. 3{12 (b). F or the `Go o d' p oin t, the mass ratio is m ~ 01 : m ~ ` : m ~ 02 = 9 : 10 : 20 and for the `Better' p oin t, m ~ 01 : m ~ ` : m ~ 02 = 1 : 2 : 4. In g. 3{12 (b), the dashed line represen ts dilepton distribution in SUSY or pure phase space and the solid cy an (magen ta) line represen ts the dilepton distribution in UED for r = 0 : 3 ( r = 0 : 7). Indeed for larger r the distributions lo ok dieren t but bac kground and detector sim ulation need to b e included. Notice that in the mSugra mo del, the maxim um of the co ecien t r is 0.4. 3.2.2 Lepton-Jet In v arian t Mass Charge Asymmetry No w w e lo ok at spin correlations b et w een q and ` in g. 3{10 In this case, there are sev eral complications. First of all, w e don't kno w whic h lepton w e need to c ho ose. There are t w o leptons in the nal state. One lepton, called `near', comes from the deca y of ~ 02 in SUSY or Z 1 in UED, while the other lepton, called `far', comes from the deca y of ~ ` in SUSY or ` 1 in UED. One can form the lepton-quark in v arian t mass distributions m `q The spin of the in termediate particle ( Z 1 in UED or ~ 02 in SUSY) go v erns the shap e of the distributions for the near lepton. Ho w ev er, in practice w e cannot distinguish the near and far lepton, and one has to include the in v arian t mass com binations with b oth leptons (it is imp ossible to tell near and far leptons ev en t b y ev en t but there can b e an impro v emen t on their selection [96].). Second, w e do not measure c harge of jets (or quarks). Therefore w e do not kno w whether a particular jet (or quark) came from the deca y of squark or an ti-squark. This doubles the n um b er of diagrams that w e need to consider. These complications tend to w ash out the spin correlations, but a residual eect remains,

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48 whic h is due to the dieren t n um b er of quarks and an ti-quarks in the proton, whic h in turn leads to a dierence in the pro duction cross-sections for squarks and an ti-squarks [12]. Most imp ortan tly w e do not kno w whic h jet is actually the correct jet in this cascade deca y c hain. W e pair-pro duce t w o squarks (or KK quarks) particles and eac h of them pro duces one jet. Once ISR is included, there are man y jets in the nal state. F or no w, as in [13], w e will assume that w e kno w whic h jet is the correct one w e need to c ho ose. One nev er kno ws for sure whic h is the correct jet although there can b e clev er cuts to increase the probabilit y that w e pic k ed the righ t one [96]. There are t w o p ossible in v arian t distributions in this case: d dm q ` + with a p ositiv ely c harge lepton and d dm q ` with a negativ ely c harged lepton. In principle, there are 8 diagrams that need to b e included (a factor of 2 from quark/an ti-quark com bination, another factor of 2 from the t w o dieren t leptons with dieren t c hiralities, a factor of 2 from the am biguit y b et w een near and far leptons). F or this study as in the dilepton case, w e rst start from a UED mass sp ectrum and adjust the MSSM parameters un til w e get a p erfect matc h in the sp ectrum. In this case, Z 1 do es not deca y in to righ t handed leptons. There are 4 con tributions and they all con tribute to b oth d dm q ` + and d dm q ` distributions whic h are in g. 3{13 d dm q ` + = f q dP 2 dm n + dP 1 dm f + f q dP 1 dm n + dP 2 dm f d dm q ` = f q dP 1 dm n + dP 2 dm f + f q dP 2 dm n + dP 1 dm f ; (3{10) where P 1 ( P 2 ) represen ts distribtuion for a deca y from a squark or KK quark (an ti-squark or an ti-KK quark) and f q ( f q ) is the fraction of squarks or KK quarks (an ti-squarks or an ti-KK quarks) and b y denition, f q + f q = 1. This quan tit y f q tells us ho w m uc h squarks or KK quarks are pro duced compared to their an ti-particles. F or a UED mass sp ectrum and SPS1a, f q 0 : 7 [13]. These t w o

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49 distributions in UED (SUSY) are sho wn in g. 3{13 (a) (g. 3{13 (b)) in dieren t colors. The distributions are normalized to L = 10 f b 1 and the v ery sharp edge near m q ` 60 GeV ( m q ` 75 GeV) is due to the near (far) letp on. Ho w ev er, once bac kground and detector resolutions are included, the clear edges are smo othed out. No w with these t w o distributions, a con vien t quan tit y `asymmetry' [12] is dened b elo w A + = d dm q ` + d dm q ` d dm q ` + + d dm q ` : (3{11) Notice that if f q = f q = 0 : 5, d dm q ` + = d dm q ` + and A + b ecomes zero. This is the case for pure phase space deca y So zero asymmetry means w e don't obtain an y spin information from this deca y c hain, i.e., if w e measure non-zero a ymmetry it means that the in termediate particle ( ~ 02 or Z 1 ) has non-zero spin. So for this metho d to w ork, f q m ust b e dieren t from f q So this metho d do es not apply at p p collider suc h as the T ev atron since a p p collider pro duces the same amoun t of quarks and an ti-quarks. The spin correlations are enco ded in the c harge asymmetry [12]. Ho w ev er, ev en in a pp collider suc h as the LHC, whether or not w e measure non-zero asymmetry dep ends on parameter space, e.g.., in the fo cus p oin t region, gluino pro ductioin dominates and gluino pro duces equal amoun ts of squarks and an ti-squarks. Therefore w e exp ect f q f q 0 : 5 and the asymmetry will b e w ashed out. Our comparison b et w een A + in the case of UED and SUSY for UED mass sp ectrum is sho wn in g. 3{14 (a). W e see that although there is some minor dierence in the shap e of the asymmetry curv es, o v erall the t w o cases app ear to b e v ery dicult to discriminate unam biguously esp ecially since the regions near the t w o ends of the plot, where the deviation is the largest, also happ en to suer from p o orest statistics. Notice that w e ha v e not included detector eects

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50 Figure 3{13: Jet-lepton in v arian t mass distributions. dN dm q ` + (blue) and dN dm q ` + (red) in the case of (a) UED and (b) SUSY for UED mass sp ectrum with R 1 = 500 GeV. q stands for b oth a quark and an an tiquark, and N ( q ` + ) ( N ( q ` )) is the n um b er of en tries with p ositiv ely (negativ ely) c harged lepton. The distributions are normalized to L = 10 f b 1 A v ery sharp edge near m q ` 60 GeV ( m q ` 75 GeV) is due to near (far) letp on. Once bac kground and detector resolutions are included, the clear edges are smo othed out. or bac kgrounds. Finally and p erhaps most imp ortan tly this analysis ignores the com binatorial bac kground from the other jets in the ev en t, whic h could b e misin terpreted as the starting p oin t of the cascade depicted in g. 3{10 Ov erall, g. 3{14 sho ws that although the asymmetry (eqn. 3{11 ) do es enco de some spin correlations, distinguishing b et w een the sp ecic cases of UED and SUSY app ears c hallenging. Similarly in g. 3{14 (b), w e sho w the asymmetry in UED and SUSY for a mass sp ectrum from the SPS1a p oin t in the mSugra mo del. In this case, the mass sp ectrum is broad compared to the UED sp ectrum and ~ 02 in SUSY ( Z 1 in UED) do es not deca y in to left handed sleptons ( S U (2) W KK letp ons). Unlik e the narro w mass sp ectrum, in this study p oin t with larger mass splittings, as exp ected in t ypical SUSY mo dels, the asymmetry distributions app ear to b e more distinct than the case sho wn in g. 3{14 (a), whic h is a source of optimism. These results ha v e b een recen tly conrmed in [13]. It remains to b e seen whether this

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51 Figure 3{14: Asymmetries for UED and SUSY are sho wn in blue and red, resp ectiv ely in the case of (a) UED mass sp ectrum with R 1 = 500 GeV and (b) SPS1a mass sp ectrum. The horizon tal dotted line represen ts pure phase space. The error bars represen t statistical uncertain t y with L = 10 fb 1 conclusion p ersists in a general setting, and once the com binatorial bac kgrounds are included [96]. Notice that comparing (a) and (b) in g. 3{11 the signs of the t w o asymmetries ha v e c hanged. The dierence is the c hiralit y of sleptons or KK leptons. In g. 3{11 (a) (g. 3{11 (a)), left handed sleptons or S U (2) W doublet KK leptons (righ t handed sleptons or S U (2) W singlet KK leptons) are onshell and the asymmetry starts out p ositiv e (negativ e) and ends negativ e (p ositiv e). By lo oking at the sign of the asymmetry w e can see whic h c hiralit y w as onshell. What w e did so far w as, rst w e c ho ose a study p oin t in one mo del and fak e parameters in other mo dels un til w e see p erfect matc h in the mass sp ectrum. Ho w ev er not all masses are observ able and sometimes w e get less constranin ts than the n um b er of masses in v olv ed in the deca y So what w e need to do is to matc h endp oin ts in the distributions instead of matc hing mass sp ectrum and ask whether there is an y p oin t in parameter space whic h is consisten t with the exp erimen tal data. In other w ords, w e ha v e to ask whic h mo del ts the data b etter. W e consider three kinematic endp oin ts: m q `` m q ` and m `` (see g. 3{10 ). In principle, w e can nd more kinematic endp oin ts suc h as a lo w er edge, here w e are b eing conserv ativ e

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52 and tak e upp er edges only [98{ 100]. In case of an onshell deca y of 02 and ~ ` these three kinematic endp oin ts are written in terms of masses m q `` = m ~ q p (1 x )(1 y z ) m q ` = m ~ q p (1 x )(1 z ) (3{12) m `` = m ~ q p x (1 y )(1 z ) where m ~ q is squark mass or KK quark mass and x = m ~ 02 m ~ q 2 y = m ~ ` m ~ 02 2 and z = m ~ 01 m ~ ` 2 are the ratios b et w een masses in the cascade deca y c hain. By denition, x y and z are less than 1. W e are no w left with 2 free parameters: f q and x and solv e for y z and m ~ q in terms of t w o free parameters. W e minimize 2 2 = n X i =1 ( x i i ) 2 2 i ; (3{13) b et w een the t w o asymmetries in the ( x f q ) parameter space to see whether w e can fak e a SUSY asymmetry in the UED mo del. x i is the theory prediction and i is the exp erimen tal v alue with uncertain t y i 2dof = 2 =n is the `reduced' 2 or 2 for n degrees of freedom. Our result is sho wn in g. 3{15 (a). W e found the minim um 2 is around 3 in the region where all KK masses are the same as the SUSY masses in the deca y and f q is large. This means that 2 is minimized when w e ha v e p erfect matc h in mass sp ectrum. The red circle is the SPS1a p oin t. No w since w e don't ha v e exp erimen tal data y et, w e generated data samples from SPS1a assuming 10 f b 1 and constructed the asymmetries in SUSY and UED in g. 3{15 (b). W e included 10% jet energy resolution. Red dots represen t data p oin ts and the red line is the SUSY t to the data p oin ts and the blue lines are the UED ts to the data p oin ts for t w o dieren t f q 's. F or SUSY, 2 is around 1 as w e exp ect. W e can get b etter 2 for UED from 9.1 to 4.5 b y increasing f q

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53 Figure 3{15: Asymmetries with relaxed conditions. (a) The con tour lines sho w 2 in the ( x f q ) parameter space and the red dot represen ts the SPS1a p oin t. 2 is minimized when f q 1 and x is the same as for SPS1a. (b) Red dots represen t the data p oin ts with statistical error bars generated from SPS1a with L = 10 fb 1 including 10% jet energy resolution. 2 -minimized UED (SUSY) ts to data are sho wn in blue (red). Since data w as generated from SUSY, small 2 in the SUSY t is exp ected. 2 in the UED ts is 9.1 (blue sold) and 4.5 (blue dotted) for f q = 0 : 7 and f q =1, resp ectiv ely It is still to o big to t the exp erimen tal data. So our conclusion for this study is that a particular p oin t lik e SPS1a can not b e fak ed through the en tire parameter space of UED. Ho w ev er w e need to c hec k whether this conclusion will remain the same when w e include the wrong jets whic h ha v e nothing to do with this deca y c hain [96]. Notice that the clear edge at m q ` 300 GeV in g. 3{14 (b) disapp eared in g. 3{15 (b) after including jet energy resolution. F rom g. 3{14 w e see that SUSY has a larger asymmetry 3.3 UED and SUSY at Linear Colliders Univ ersal Extra Dimensions and sup ersymmetry ha v e rather similar exp erimen tal signatures at hadron colliders. The prop er in terpretation of an LHC disco v ery in either case ma y therefore require further data from a lepton collider. In this section w e iden tify metho ds for discriminating b et w een the t w o scenarios at the linear collider. W e will consider 3 T eV Compact Linear Collider (CLIC). W e study the pro cesses of Kaluza-Klein m uon pair pro duction in univ ersal extra

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54 dimensions in parallel to sm uon pair pro duction in sup ersymmetry accoun ting for the eects of detector resolution, b eam-b eam in teractions and accelerator induced bac kgrounds. W e nd that the angular distributions of the nal state m uons, the energy sp ectrum of the radiativ e return photon and the total cross-section measuremen t are p o w erful discriminators b et w een the t w o mo dels. Accurate determination of the particle masses can b e obtained b oth b y a study of the momen tum sp ectrum of the nal state leptons and b y a scan of the particle pair pro duction thresholds. W e also calculate the pro duction rates of v arious Kaluza-Klein particles and discuss the asso ciated signatures. 3.3.1 Ev en t Sim ulation and Data Analysis In order to study the discrimination of UED signals from sup ersymmetry w e ha v e implemen ted the relev an t features of the minimal UED mo del in the CompHEP ev en t generator [68]. The MSSM is already a v ailable in CompHEP since v ersion 41.10. All n = 1 KK mo des are incorp orated as new particles, with the prop er in teractions and one-lo op corrected masses [28]. The widths can then b e readily calculated with CompHEP on a case b y case basis and added to the particle table. Similar to the SM case, the neutral gauge b osons at lev el 1, Z 1 and r 1 are mixtures of the KK mo des of the h yp erc harge gauge b oson and the neutral S U (2) W gauge b oson. Ho w ev er, it w as sho wn in [62] that the radiativ ely corrected W ein b erg angle at lev el 1 and higher is v ery small. F or example, r 1 whic h is the LKP in the minimal UED mo del, is mostly the KK mo de of the h yp erc harge gauge b oson. F or simplicit y in the co de w e neglect neutral gauge b oson mixing for n 1. In the next section w e concen trate on the pair pro duction of lev el 1 KK m uons e + e +1 1 and compare it to the analogous pro cess of sm uon pair pro duction in sup ersymmetry: e + e ~ + ~ In UED there are t w o n = 1 KK m uon Dirac fermions: an S U (2) W doublet D1 and an S U (2) W singlet S1 b oth of whic h con tribute in eqn. ( 3{14 ) b elo w (see also g. 3{16 ). In complete

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55 e + e r ; Z +1 1 (a) e + e r 2 ; Z 2 +1 1 (b) Figure 3{16: The dominan t F eynman diagrams for KK m uon pro duction e + e +1 1 in Univ ersal Extra Dimensions. The blac k dot represen ts a KKn um b er violating b oundary in teraction [28]. e + e r ; Z ~ + ~ Figure 3{17: The dominan t F eynman diagrams for sm uon pro duction e + e ~ + ~ in sup ersymmetry analogy in sup ersymmetry there are t w o sm uon eigenstates, ~ L and ~ R b oth of whic h con tribute in eqn. ( 3{15 ). The dominan t diagrams in that case are sho wn in g. 3{17 In principle, there are also diagrams mediated b y r n ; Z n for n = 4 ; 6 ; ::: but they are doubly suppressed b y the KK-n um b er violating in teraction at b oth v ertices and the KK mass in the propagator and here can b e safely neglected. Ho w ev er, r 2 and Z 2 exc hange (g. 3{16 b) ma y lead to resonan t pro duction and signican t enhancemen t of the cross-section, as w ell as in teresting phenomenology as discussed b elo w in Section 3.3.2.5 W e ha v e implemen ted the lev el 2 neutral gauge b osons r 2 ; Z 2 with their widths, including b oth KK-n um b er preserving and the KK-n um b er violating deca ys as in Ref. [62]. W e consider the nal state consisting of t w o opp osite sign m uons and missing energy It ma y arise either from KK m uon pro duction in UED e + e +1 1 + r 1 r 1 ; (3{14)

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56 T able 3{1: Masses of the KK excitations for R 1 = 500 GeV and R = 20 used in the analysis. P article Mass D1 515.0 GeV S1 505.4 GeV r 1 500.9 GeV with r 1 b eing the LKP or from sm uon pair pro duction in sup ersymmetry: e + e ~ + ~ + ~ 01 ~ 01 ; (3{15) where ~ 01 is the ligh test sup ersymmetric particle. W e reconstruct the m uon energy sp ectrum and the m uon pro duction p olar angle, aiming at small bac kground from SM pro cesses with minimal biases due to detector eects and selection criteria. The goal is to disen tangle KK particle pro duction ( 3{14 ) in UED from sm uon pair pro duction ( 3{15 ) in sup ersymmetry W e also determine the masses of the pro duced particles and test the mo del predictions for the pro duction cross-sections in eac h case. W e rst x the UED parameters to R 1 = 500 GeV, R = 20, leading to the sp ectrum giv en in T able 3{1 The ISR-corrected signal cross-section in UED for the selected nal state + r 1 r 1 is 14.4 fb at p s = 3 T eV. Ev en ts ha v e b een generated with CompHEP and then reconstructed using a fast sim ulation based on parametrized resp onse for a realistic detector at CLIC. In particular, the lepton iden tication eciency momen tum resolution and p olar angle co v erage are of sp ecial relev ance to this analysis. W e assume that particle trac ks will b e reconstructed through a discrete cen tral trac king system, consisting of concen tric la y ers of Si detectors placed in a 4 T solenoidal eld. This ensures a momen tum resolution p=p 2 = 4.5 10 5 GeV 1 A forw ard trac king system should pro vide trac k reconstruction do wn to 10 W e also accoun t for initial state radiation (ISR) and for b eamstrahlung eects on the cen ter-of-mass energy W e assume that

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57 m uons are iden tied b y their p enetration in the instrumen ted iron return y ok e of the cen tral coil. A 4 T magnetic eld sets an energy cuto of 5 GeV for m uon tagging. The ev en ts from the CompHEP generation ha v e b een treated with the Pythia 6.210 parton sho w er [101] and reconstructed with a mo died v ersion of the SimDet 4.0 program [102]. Beamstrahlung has b een added to the CompHEP generation. The luminosit y sp ectrum, obtained b y the GuineaPig b eam sim ulation for the standard CLIC b eam parameters at 3 T eV, has b een parametrised using a mo died Y ok o y a-Chen appro ximation [103, 104]: This analysis has bac kgrounds coming from SM + nal states, whic h are mostly due to gauge b oson pair pro duction W + W + Z 0 Z 0 + and from e + e W + W e e e + e Z 0 Z 0 e e follo w ed b y m uonic deca ys. The bac kground total cross-section is 20 fb at p s = 3 T eV. In addition to its comp etitiv e cross-section, this bac kground has leptons pro duced preferen tially at small p olar angles, therefore biasing the angular distribution. In order to reduce this bac kground, a suitable ev en t selection has b een applied. Ev en ts ha v e b een required to ha v e t w o m uons, missing energy in excess to 2.5 T eV, transv erse energy b elo w 150 GeV and ev en t sphericit y larger than 0.05. In order to reject the Z 0 Z 0 bac kground, ev en ts with di-lepton in v arian t mass compatible with M Z 0 ha v e also b een discarded. The underlying r r collisions also pro duces a p oten tial bac kground to this analysis in the form of r r + This bac kground has b een sim ulated using the CLIC b eam sim ulation and Pythia Despite its large cross-section, it can b e completely suppressed b y a cut on the missing transv erse energy E missing T > 50 GeV. Finally in order to remo v e ev en ts with large b eamstrahlung, the ev en t sphericit y had to b e smaller than 0.35 and the acolinearit y smaller than 0.8. These criteria pro vide a factor 30 bac kground suppression, in the kinematical region of in terest, while not signican tly biasing the lepton momen tum distribution.

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58 T able 3{2: MSSM parameters for the SUSY study p oin t used in the analysis. This c hoice of soft SUSY parameters in CompHEP leads to an exact matc h b et w een the corresp onding UED and SUSY mass sp ectra. MSSM P arameter V alue 1000 GeV M 1 502.65 GeV M 2 1005.0 GeV M ~ L 512.83 GeV M ~ R 503.63 GeV tan 10 3.3.2 Comparison of UED and Sup ersymmetry in + / E T In order to p erform the comparison of UED and MSSM, w e adjusted the MSSM parameters to get the t w o sm uon masses M ~ L and M ~ R and the ligh test neutralino mass M ~ 01 matc hing exactly those of the t w o Kaluza-Klein m uons M D1 and M S1 and of the KK photon M r 1 for the c hosen UED parameters. It m ust b e stressed that suc h small mass splitting b et w een the t w o m uon partners is t ypically rather acciden tal in sup ersymmetric scenarios. The sup ersymmetric parameters used are giv en in T able 3{2 W e then sim ulate b oth reactions ( 3{14 ) and ( 3{15 ) with CompHEP and pass the resulting ev en ts through the same sim ulation and reconstruction. The ISR-corrected signal cross-section in SUSY for the selected nal state + ~ 01 ~ 01 is 2.76 fb at p s = 3 T eV, whic h is ab out 5 times smaller than in the UED case. 3.3.2.1 Angular Distributions and Spin Measuremen ts In the case of UED, the KK m uons are fermions and their angular distribution is giv en b y d d cos U E D 1 + E 2 1 M 2 1 E 2 1 + M 2 1 cos 2 : (3{16) Assuming that at CLIC the KK pro duction tak es place w ell ab o v e threshold, the form ula simplies to: d d cos U E D 1 + cos 2 : (3{17)

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59 0 100 200 300 400 500 600 700 800 -1 -0.5 0 0.5 1 Entries / 0.1 ab-1cos qm 0 25 50 75 100 125 150 175 200 -1 -0.5 0 0.5 1 Figure 3{18: Dieren tial cross-section d =d cos for UED (blue, top) and sup ersymmetry (red, b ottom) as a function of the m uon scattering angle The gure on the left sho ws the ISR-corrected theoretical prediction. The t w o gures on the righ t in addition include the eects of ev en t selection, b eamstrahlung and detector resolution and acceptance. The left (righ t) panel is for the case of UED (sup ersymmetry). The data p oin ts are the com bined signal and bac kground ev en ts, while the y ello w-shaded histogram is the signal only As the sup ersymmetric m uon partners are scalars, the corresp onding angular distribution is d d cos S U S Y 1 cos 2 : (3{18) Distributions ( 3{17 ) and ( 3{18 ) are sucien tly distinct to discriminate the t w o cases. Ho w ev er, the p olar angles of the original KK-m uons and sm uons are not directly observ able and the pro duction p olar angles of the nal state m uons are measured instead. But as long as the mass dierences M 1 M r 1 and M ~ M ~ 01 resp ectiv ely remain small, the m uon directions are w ell correlated with those of their paren ts (see gure 3{18 a). In g. 3{18 b w e sho w the same comparison after detector sim ulation and including the SM bac kground. The angular distributions are w ell distinguishable also when accoun ting for these eects. By p erforming a 2 t to the normalised p olar angle distribution, the UED scenario considered here

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60 R s (GeV)s (pb) 10-3 10-2 10-1 1010 1015 1020 1025 1030 1035 1040 1045 1050 E (GeV)s (pb) 10 -3 10 -2 1010 1015 1020 1025 1030 1035 1040 1045 Figure 3{19: The total cross-section in pb as a function of the cen ter-of-mass energy p s near threshold for e + e +1 1 + r 1 r 1 Left: the threshold onset with (line, blue) and without (dots) b eamstrahlung eects. Righ t: a threshold scan at selected p oin ts. The green curv e refers to the reference UED parameters while for the red (blue) curv e the mass of S1 ( D1 ) has b een lo w ered b y 2.5 GeV. The p oin ts indicate the exp ected statistical accuracy for the cross section determination at the p oin ts of maxim um mass sensitivit y Eects of the CLIC luminosit y sp ectrum are included. could b e distinguished from the MSSM, on the sole basis of the distribution shap e, with 350 fb 1 of data at p s = 3 T eV. 3.3.2.2 Threshold Scans A t the e + e linear collider, the m uon excitation masses can b e accurately determined through an energy scan of the onset of the pair pro duction threshold. This study not only determines the masses, but also conrms the particle nature. In fact the cross-sections for the UED pro cesses rise at threshold / while in sup ersymmetry their threshold onset is / 3 where is the particle v elo cit y Since the collision energy can b e tuned at prop erly c hosen v alues, the p o w er rise of the cross-section can b e tested and the masses of the particles in v olv ed measured. W e ha v e studied suc h threshold scan for the e + e +1 1 + r 1 r 1 pro cess at p s = 1 T eV, for the same parameters as in T able 3{1 W e accoun t for the an ticipated CLIC cen tre-of-mass energy spread induced b oth b y the energy spread in the CLIC

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61 linac and b y b eam-b eam eects during collisions. This b een obtained from the detailed GuineaPig b eam sim ulation and parametrised using the mo died Y ok o y aChen mo del [103, 105]. An optimal scan of a particle pair pro duction threshold consists of just t w o energy p oin ts, sharing the total in tegrated luminosit y in equal fractions and c hosen at energies maximising the sensitivit y to the particle widths and masses [106]. F or the UED mo del scan w e ha v e tak en three p oin ts, one for normalisation and t w o at the maxima of the mass sensitivit y (see gure 3{19 ). Inclusion of b eamstrahlung eects induces a shift of the p ositions of these maxima to w ards higher nominal p s v alues [107]. F rom the estimated sensitivit y d =dM and the cross-section accuracy the masses of the t w o UED m uon excitations can b e determined to 0 : 11 GeV and 0 : 23 GeV for the singlet and the doublet states resp ectiv ely with a total luminosit y of 1 ab 1 shared in three p oin ts, when the particle widths can b e disregarded. 3.3.2.3 Pro duction Cross-Section Determination The same analysis can b e used to determine the cross-section for the pro cess e + e + / E T The SM con tribution can b e determined indep enden tly using an ti-tag cuts, and subtracted. Since the cross-section for the UED pro cess at 3 T eV is ab out v e times larger compared to sm uon pro duction in sup ersymmetry this measuremen t w ould reinforce the mo del iden tication obtained b y the spin determination. This can b e quan tied b y p erforming the same 2 t to the m uon p olar pro duction angle discussed ab o v e, but no w including also the total n um b er of selected ev en ts. Since the cross-section dep ends on the mass of the pair pro duced particles, w e include a systematic uncertain t y on the prediction corresp onding to a 0 : 05 % mass uncertain t y whic h is consisten t with the results discussed b elo w. A t CLIC the absolute luminosit y should b e measurable to O (0 : 1 %) and the a v erage eectiv e collision energy to O (0 : 01 %).

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62 0 100 200 300 400 500 600 20 40 60 80 100 Entries / 0.1 ab-1pm (GeV) 0 20 40 60 80 100 120 140 20 40 60 80 100 Figure 3{20: The m uon energy sp ectrum resulting from KK m uon pro duction ( 3{14 ) in UED (blue, top curv e) and sm uon pro duction ( 3{15 ) in sup ersymmetry (red, b ottom curv e). The UED and SUSY parameters are c hosen as in g. 3{18 The plot on the left sho ws the ISR-corrected distribution, while that on the righ t includes in addition the eects of ev en t selection, b eamstrahlung and detector resolution and acceptance. The data p oin ts are the com bined signal and bac kground ev en ts, while the y ello w-shaded histogram is the signal only 3.3.2.4 Muon Energy Sp ectrum and Mass Measuremen ts The c haracteristic end-p oin ts of the m uon energy sp ectrum are completely determined b y the kinematics of the t w o-b o dy deca y and hence they don't dep end on the underlying framew ork (SUSY or UED) as long as the masses in v olv ed are tuned to b e iden tical. W e sho w the ISR-corrected exp ected distributions for the m uon energy sp ectra at the generator lev el in g. 3{20 a, using the same parameters as in g. 3{18 As exp ected, the shap e of the E distribution in the case of UED coincides with that for MSSM. The lo w er, E min and upp er, E max endp oin ts of the m uon energy sp ectrum are related to the masses of the particles in v olv ed in the deca y according to the relation: E max=min = 1 2 M ~ 1 M 2 ~ 01 M 2 ~ r (1 ) (3{19)

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63 where M ~ and M ~ 01 are the sm uon and LSP masses and r = 1 = (1 2 ) 1 = 2 with = q 1 M 2 ~ =E 2 beam is the ~ b o ost. In the case of the UED the form ula is completely analogous with M 1 replacing M ~ and M r 1 replacing M ~ 01 Due to the splitting b et w een the ~ L and ~ R masses in MSSM and that b et w een the D1 and S1 masses in UED, in g. 3{20 a w e see the sup erp osition of t w o b o x distributions. The left, narro w er distribution is due to S1 pair pro duction in UED ( ~ R pair pro duction in sup ersymmetry). The underlying, m uc h wider b o x distribution is due to D1 pair pro duction in UED ( ~ L pair pro duction in sup ersymmetry). The upp er edges are w ell dened, with smearing due to b eamstrahlung and, but less imp ortan tly to momen tum resolution. The lo w er end of the sp ectrum has the o v erlap of the t w o con tributions and with the underlying bac kground. F urthermore, since the splitting b et w een the masses of the D1 S1 and that of r 1 is small, the lo w er end of the momen tum distribution can b e as lo w as O (1 GeV) where the lepton iden tication eciency is cut-o b y the solenoidal eld b ending the lepton b efore it reac hes the electro-magnetic or the hadron calorimeter [109]. Nev ertheless, there is sucien t information in this distribution to extract the mass of the r 1 particle, using the prior information on the D1 and S1 masses, obtained b y the threshold scan. In g. 3{20 b w e sho w the m uon energy distribution after detector sim ulation. A one parameter t giv es an uncertain t y on the r 1 mass of 0.19 (stat.) 0.21 (syst) GeV, where the statistical uncertain t y is giv en for 1 ab 1 of data and the systematics rerects the eect of the uncertain t y on the 1 masses. The b eamstrahlung in troduces an additional systematics, whic h dep ends on the con trol of the details of the luminosit y sp ectrum. 3.3.2.5 Photon Energy Sp ectrum and Radiativ e Return to the Z 2 With the e + e colliding at a xed cen ter-of-mass energy ab o v e the pair pro duction threshold a signican t fraction of the KK m uon pro duction will pro ceed

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64 Ephoton (GeV)Events/500 fb-1 0 5 10 15 20 25 30 100 120 140 160 180 200 220 240Figure 3{21: Photon energy sp ectrum in e + e +1 1 r for R 1 = 1350 GeV, R = 20 and E C M = 3 T eV b efore (left) and after (righ t) detector sim ulation. The acceptance cuts are E r > 10 GeV and 1 < r < 179 The mass of the Z 2 resonance is 2825 GeV. through radiativ e return. Since this is mediated b y s -c hannel narro w resonances, a sharp p eak in the photon energy sp ectrum app ears whenev er one of the mediating s -c hannel particles is on-shell. In case of sup ersymmetry only Z and r particles can mediate sm uon pair pro duction and neither of them can b e close to b eing on-shell. On the con trary an in teresting feature of the UED scenario is that 1 pro duction can b e mediated b y Z n and r n KK excitations (for n ev en) as sho wn in g. 3{16 b. Among these additional con tributions, the Z 2 and r 2 exc hange diagrams are the most imp ortan t. Since the deca y Z 2 1 1 is allo w ed b y phase space, there will b e a sharp p eak in the photon sp ectrum, due to a radiativ e return to the Z 2 The photon p eak is at E r = 1 2 E C M 1 M 2 Z 2 E 2 C M : (3{20) On the other hand, M r 2 < 2 M 1 so that the deca y r 2 1 1 is closed, and therefore there is no radiativ e return to r 2 Notice that the lev el 2 W ein b erg angle is v ery small [28] and therefore Z 2 is mostly W 0 2 -lik e and couples predominan tly to

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65 D1 and not S1 The photon energy sp ectrum in e + e +1 1 r for R 1 = 1350 GeV, R = 20 and E C M = 3 T eV is sho wn in g. 3{21 On the left w e sho w the ISR-corrected theoretical prediction from CompHEP while the result on the righ t in addition includes detector and b eam eects. It is clear that the p eak cannot b e missed.3.3.3 Prosp ects for Disco v ery and Discrimination in Other Final States Previously in section 3.3.2 w e considered the + / E T nal state resulting from the pair pro duction of lev el 1 KK m uons. Ho w ev er, this is not the only signal whic h could b e exp ected in the case of UED. Due to the relativ e degeneracy of the KK particles at eac h lev el, the remaining n = 1 KK mo des will b e pro duced as w ell, and will yield observ able signatures. In those cases, the discrimination tec hniques whic h w e discussed earlier can still b e applied, pro viding further evidence in fa v or of one mo del o v er the other. In this section w e compute the cross-sections for some of the other main pro cesses of in terest, and discuss ho w they could b e analyzed. 3.3.3.1 Kaluza-Klein Leptons W e rst turn to the discussion of the other KK lepton ra v ors. The KK leptons, 1 are also pro duced in s -c hannel diagrams only as in g. 3{16 hence the + 1 1 pro duction cross-sections are v ery similar to the +1 1 case. The nal state will b e + / E T and it can b e observ ed in sev eral mo des, corresp onding to the dieren t options for the deca ys. Ho w ev er, due to the lo w er statistics and the inferior jet energy resolution, none of the resulting c hannels can comp ete with the discriminating p o w er of the +1 1 / E T nal state discussed in the previous section. The case of KK electrons is more in teresting, as it con tains a new t wist. The pro duction of KK electrons can also pro ceed through the t -c hannel diagram sho wn in g. 3{22 c. As a result, the pro duction cross-sections for KK electrons can b e m uc h higher than for KK m uons. W e illustrate this in g. 3{23 where w e

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66 e + e r ; Z e 1 e +1 (a) e + e r 2 ; Z 2 e 1 e +1 (b) e + e r 1 ; Z 1 e 1 e +1 (c) Figure 3{22: The dominat F eynman diagrams for KK electron pro duction, e + e e +1 e 1 the same as g. 3{16 sho w separately the cross-sections for S U (2) W doublets (solid lines) and S U (2) W singlets (dotted lines), as a function of R 1 (F or the n umerical results throughout section 3.3.3 w e alw a ys x R = 20.) A t lo w masses (i.e. lo w R 1 ) the e +1 e 1 cross-sections can b e up to t w o orders of magnitude larger, compared to the case of +1 1 Another in teresting feature is the resonan t enhancemen t of the cross-section for R 1 1450 GeV, whic h is presen t in either case ( e or ) for the S U (2) W doublets (solid lines), but not the S U (2) W singlets (dotted lines). The feature is due to the on-shell pro duction of the lev el 2 Z 2 KK gauge b oson, whic h can then deca y in to a pair of lev el 1 KK leptons (see diagram (b) in gs. 3{16 and 3{22 ). Since the W ein b erg angle at the higher ( n > 0) KK lev els is tin y [28], Z 2 is predominan tly an S U (2) W gauge b oson and hence do es not couple to the S U (2) W singlet fermions, whic h explains the absence of a similar p eak in the e S1 and S1 cross-sections 3 Because of the higher pro duction rates, the e + e / E T ev en t sample will b e m uc h larger and ha v e b etter statistics than + / E T The e + e / E T nal state has b een recen tly adv ertised as a discriminator b et w een UED and sup ersymmetry in [108]. Ho w ev er, the additional t -c hannel diagram (g. 3{22 c) has the eect of not only enhancing the o v erall cross-section, but also distorting 3 One migh t ha v e exp ected a second p eak closeb y due to r 2 resonan t pro duction, but in the minimal UED mo del the sp ectrum is suc h that the deca ys of r 2 to lev el 1 fermions are all closed.

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67 Figure 3{23: ISR-corrected pro duction cross-sections of lev el 1 KK leptons ( e 1 in red, 1 in blue) at CLIC, as a function of R 1 Solid (dotted) lines corresp ond to S U (2) W doublets (singlets). the dieren tial angular distributions discussed previously in Section 3.3.2.1 and creating a forw ard p eak, whic h causes the cases of UED and sup ersymmetry to lo ok v ery m uc h alik e. W e sho w the resulting angular distributions of the nal state electrons in g. 3{24 F or prop er comparison, w e follo w the same pro cedure as b efore: w e c ho ose the UED sp ectrum for R 1 = 500 GeV, whic h yields KK electron masses as in T able 3{1 W e then c ho ose a sup ersymmetric sp ectrum with selectron mass parameters as in T able 3{2 This guaran tees matc hing mass sp ectra in the t w o cases (UED and sup ersymmetry) so that an y dierences in the angular distributions should b e attributed to the dieren t spins. Unlik e g. 3{18 where the underlying shap es of the angular distributions w ere v ery distinctiv e (see eqs. ( 3{17 ) and ( 3{18 )), the main eect in g. 3{24 is the uniform enhancemen t of the forw ard scattering cross-section, whic h tends to w ash out the spin correlations exhibited in g. 3{18 3.3.3.2 Kaluza-Klein Quarks Lev el 1 KK quarks will b e pro duced in s -c hannel via diagrams similar to those exhibited in g. 3{16 The corresp onding pro duction cross-sections are sho wn in g. 3{25 as a function of R 1 W e sho w separately the cases of the S U (2) W

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68 Figure 3{24: Dieren tial cross-section d =d cos e for UED and sup ersymmetry The same as g. 3{18 (left panel), but for KK electron pro duction e + e e +1 e 1 with e b eing the electron scattering angle. doublets u D1 and d D1 and the S U (2) W singlets u S1 and d S1 In the minimal UED mo del, the KK fermion doublets are somewhat hea vier than the KK fermion singlets [28], so naturally the pro duction cross-sections for u D1 and d D1 cut o at a smaller v alue of R 1 Since singlet pro duction is only mediated b y U (1) h yp erc harge in teractions, the singlet pro duction cross-sections tend to b e smaller. W e notice that u S1 u S1 is larger b y a factor of 2 2 compared to d S1 d S1 in accordance with the usual quark h yp erc harge assignmen ts. The observ able signals will b e dieren t in the case of S U (2) W doublets and S U (2) W singlets. The singlets, u S1 and d S1 deca y directly to the LKP r 1 and the corresp onding signature will b e 2 jets and missing energy The jet angular distribution will again b e indicativ e of the KK quark spin, and can b e used to discriminate against (righ t-handed) squark pro duction in sup ersymmetry follo wing the pro cedure outlined in section 3.3.2.1 The jet energy distribution will again exhibit endp oin ts, whic h will in principle allo w for the mass measuremen ts discussed in section 3.3.2.4 A threshold scan of the cross-section will pro vide further evidence of the particle spins (see section 3.3.2.2 ). The only ma jor dierence with resp ect to the + / E T nal state discussed in section 3.3.2 is the absence of the

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69 Figure 3{25: ISR-corrected pro duction cross-sections of lev el 1 KK quarks at CLIC, as a function of R 1 mono c hromatic photon signal from section 3.3.2.5 since Z 2 is to o ligh t to deca y to KK quarks. In spite of the man y similarities to the dim uon nal state considered in section 3.3.2 notice that jet angular and energy measuremen ts are not as clean and therefore the lepton (m uon or electron) nal states w ould still pro vide the most con vincing evidence for discrimination. The signatures of the S U (2) W doublet quarks are ric her { b oth u D1 and d D1 predominan tly deca y to Z 1 and W 1 whic h in turn deca y to leptons and the LKP [62]. The analogous pro cess in sup ersymmetry w ould b e left-handed squark pro duction with subsequen t deca ys to ~ 02 or ~ 1 whic h in turn deca y to ~ ` L and ~ 01 In principle, the spin information will still b e enco ded in the angular distributions of the nal state particles. Ho w ev er, the analysis is m uc h more in v olv ed, due to the complexit y of the signature, and p ossibly the additional missing energy from an y neutrinos. 3.3.3.3 Kaluza-Klein Gauge Bosons The ISR-corrected pro duction cross-sections for lev el 1 electro w eak 4 KK gauge b osons ( W 1 Z 1 and r 1 ) at a 3 T eV e + e collider are sho wn in g. 3{26 as 4 The lev el 1 KK gluon, of course, has no tree-lev el couplings to e + e

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70 a function of R 1 The three relev an t pro cesses are W + 1 W 1 Z 1 Z 1 and Z 1 r 1 ( r 1 r 1 is unobserv able). In eac h case, the pro duction can b e mediated b y a t -c hannel exc hange of a lev el 1 KK lepton, while for W + 1 W 1 there are additional s -c hannel diagrams with r Z r 2 and Z 2 Z 1 and W 1 are almost degenerate [28], th us their cross-sections cut o at around the same p oin t. The analogous pro cesses in sup ersymmetry w ould b e the pair pro duction of gaugino-lik e c harginos and neutralinos. The nal states will alw a ys in v olv e leptons and missing energy since W 1 and Z 1 do not deca y to KK quarks. In conclusion of this section, for completeness w e also discuss the p ossibilit y of observing the higher lev el KK particles and in particular those at lev el 2. F or small enough R 1 lev el 2 KK mo des are kinematically accessible at CLIC. Once pro duced, they will in general deca y to lev el 1 particles and th us con tribute to the inclusiv e pro duction of lev el 1 KK mo des. Unco v ering the presence of the lev el 2 signal in that case seems c hallenging, but not imp ossible. W e c ho ose to concen trate on the case of the lev el 2 KK gauge b osons ( V 2 ), whic h are somewhat sp ecial in the sense that they can deca y directly to SM fermions through KK n um b er violating in teractions. Th us they can b e easily observ ed as dijet or dilepton resonances. In principle, there are t w o t yp es of pro duction mec hanisms for lev el 2 gauge b osons. The rst is single pro duction e + e V 2 whic h can only pro ceed through KK n um b er violating (lo op suppressed) couplings. The second mec hanism is e + e V 2 V 2 pair pro duction whic h is predominan tly due to KK n um b er conserving (tree-lev el) couplings. In g. 3{27 w e sho w the corresp onding cross-sections for the case of the neutral lev el 2 gauge b osons, as a function of R 1 F or lo w v alues of R 1 pair pro duction dominates, but as the lev el 2 gauge b oson masses increase and approac h E C M single pro duction b ecomes resonan tly enhanced. Th us the rst indication of the presence of the lev el 2 particles ma y come from pair pro duction ev en ts, but once the mass of the dijet

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71 Figure 3{26: ISR-corrected pro duction cross-sections of lev el 1 KK gauge b osons at CLIC, as a function of R 1 or dilepton resonance is kno wn, the collider energy can b e tuned to enhance the cross-section and study the V 2 resonance prop erties in great detail. Sup ersymmetry and Univ ersal Extra Dimensions are t w o app ealing examples of new ph ysics at the T eV scale, as they address some of the theoretical puzzles of the SM. They also pro vide a dark matter candidate whic h, for prop erly c hosen theory parameters, is consisten t with presen t cosmology data. Both theories predict a host of new particles, partners of the kno wn SM particles. If either one is realized in nature, the LHC is exp ected to observ e signals of these new particles. Ho w ev er, in order to clearly iden tify the nature of the new ph ysics, one ma y need to con trast the UED and sup ersymmetric h yp otheses at a m ulti-T eV e + e linear collider suc h as CLIC 5 W e studied in detail the pro cess of pair pro duction of m uon partners in the t w o theories, KK-m uons and sm uons resp ectiv ely W e used the p olar pro duction angle to distinguish the nature of the particle partners, based on their spin. The same 5 Similar studies can also b e done at the ILC pro vided the lev el 1 KK particles are within its kinematic reac h. Since precision data tends to indicate the b ound R 1 250 GeV for the case of 1 extra dimension, one w ould need an ILC cen ter-ofmass energy ab o v e 500 GeV in order to pair-pro duce the lo w est lying KK states of the minimal UED mo del.

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72 Figure 3{27: ISR-corrected pro duction cross-sections of lev el 2 KK gauge b osons at CLIC, as a function of R 1 analysis could b e applied for the case of other KK fermions, as discussed in section 3.3.3 W e ha v e also studied the accuracy of CLIC in determining the masses of the new particles in v olv ed b oth through the study of the energy distribution of nal state m uons and threshold scans. An accuracy of b etter than 0.1% can b e obtained with 1 ab 1 of in tegrated luminosit y Once the masses of the partners are kno wn, the measuremen t of the total cross-section serv es as an additional cross-c hec k on the h yp othesized spin and couplings of the new particles. A p eculiar feature of UED, whic h is not presen t in sup ersymmetry is the sharp p eak in the ISR photon energy sp ectrum due to a radiativ e return to the KK partner of the Z The clean nal states and the con trol o v er the cen ter-of-mass energy at the CLIC m ulti-T eV collider allo ws one to unam biguously iden tify the nature of the new ph ysics signals whic h migh t b e emerging at the LHC already b y the end of this decade.

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CHAPTER 4 COSMOLOGICAL IMPLICA TIONS 4.1 Dark Matter Abundance In this c hapter, w e no w fo cus on the cosmological implications of Univ ersal Extra Dimensions. W e revisit the calculation of the relic densit y of the ligh test Kaluza-Klein particle (LKP) in the mo del of Univ ersal Extra Dimensions. The rst and only comprehensiv e calculation of the UED relic densit y to date w as p erformed in [21]. The authors considered t w o cases of LKP: the KK h yp erc harge gauge b oson B 1 and the KK neutrino 1 The case of B 1 LKP is naturally obtained in MUED, where the radiativ e corrections to B 1 are the smallest in size, since they are only due to h yp erc harge in teractions. The authors of [21] also realized the imp ortance of coannihilation pro cesses and included in their analysis coannihilations with the S U (2) W -singlet KK leptons, whic h in MUED are the ligh test among the remaining n = 1 KK particles. It w as therefore exp ected that their coannihilations will b e most imp ortan t. Subsequen tly Refs. [48, 49] analyzed the resonan t enhancemen t of the n = 1 (co)annihilation cross-sections due to n = 2 KK particles. Our goal in this c hapter will b e to complete the LKP relic densit y calculation of Ref. [21]. W e will attempt to impro v e in three dieren t asp ects: W e will include coannihilation eects with al l n = 1 KK particles. The motiv ation for suc h a tour de force is t w ofold. First, recall that the imp ortance of coannihilations is mostly determined b y the degeneracy of the corresp onding particle with the dark matter candidate. In the minimal UED mo del, the KK mass splittings are due almost en tirely to radiativ e corrections. In MUED, therefore, one migh t exp ect that, since the corrections to KK particles other 73

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74 than the KK leptons are relativ ely large, their coannihilations can b e safely neglected. Ho w ev er, the minimal UED mo del mak es an ansatz [28] ab out the cut-o scale v alues of the so called b oundary terms, whic h are not xed b y kno wn SM ph ysics, and are in principle arbitrary In this sense, the UED scenario should b e considered as a lo w energy eectiv e theory with a m ultitude of parameters, just lik e the MSSM, and the MUED mo del should b e treated as nothing more than a simple to y mo del with a limited n um b er of parameters, just lik e the \minimal sup ergra vit y" v ersion of sup ersymmetry for example. If one mak es a dieren t assumption ab out the inputs at the cut-o scale, b oth the KK sp ectrum and its phenomenology can b e mo died signican tly In particular, one could then easily nd regions of this more general parameter space where other coannihilation pro cesses b ecome activ e. On the other hand, ev en if w e c ho ose to restrict ourselv es to MUED, there is still a go o d reason to consider the coannihilation pro cesses whic h w ere omitted in the analysis of [21]. While it is true that those coannihilations are more Boltzmann suppressed, their cross-sections will b e larger, since they are mediated b y w eak and/or strong in teractions. Without an explicit calculation, it is imp ossible to estimate the size of the net eect, and whether it is indeed negligible compared to the purely h yp erc harge-mediated pro cesses whic h ha v e already b een considered. W e will k eep the exact v alue of eac h KK mass in our form ulas for all annihilation cross-sections. This will render our analysis self-consisten t. All calculations of the LKP relic densit y a v ailable so far [21, 48, 49], ha v e computed the annihilation cross-sections in the limit when all lev el 1 KK masses are the same. This appro ximation is somewhat con tradictory in the sense that al l KK masses at lev el one are tak en to b e degenerate with LKP y et only a limite d numb er of coannihilation pro cesses w ere considered. In realit y

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75 a completely degenerate sp ectrum w ould require the inclusion of all p ossible coannihilations. Con v ersely if some coannihilation pro cesses are b eing neglected, this is presumably b ecause the masses of the corresp onding KK particles are not degenerate with the LKP and are Boltzmann suppressed. Ho w ev er, the masses of these particles ma y still en ter the form ulas for the relev an t coannihilation cross-sections, and using appro ximate v alues for those masses w ould lead to a certain error in the nal answ er. Since w e are k eeping the exact mass dep endence in the form ulas, within our approac h hea vy particles naturally decouple, coannihilations are prop erly w eigh ted, and all relev an t coannihilation cross-sections b eha v e prop erly Notice that the assumption of exact mass degeneracy o v erestimates the corresp onding cross-sections and therefore underestimates the relic densit y This exp ectation will b e conrmed in our n umerical analysis in Section 4.3 W e will try to impro v e the n umerical accuracy of the analysis b y taking in to accoun t some minor corrections whic h w ere neglected or appro ximated in [21]. F or example, w e will use a temp erature-dep enden t g (the total n um b er of eectiv ely massless degrees of freedom, giv en b y eq. ( 4{6 ) b elo w) and include subleading corrections ( 4{19 ) in the v elo cit y expansion of the annihilation cross-sections. The a v ailabilit y of the calculation of the remaining coannihilation pro cesses is imp ortan t also for the follo wing reason. Coannihilations with S U (2) W -singlet KK leptons w ere found to reduce the eectiv e annihilation cross-section, and therefore increase the LKP relic densit y This has the eect of lo w ering the range of cosmologically preferred v alues of the LKP mass, or equiv alen tly the scale of the extra dimension. Ho w ev er, one could exp ect that coannihilations with the other n = 1 KK particles w ould ha v e the opp osite eect, since they ha v e stronger in teractions compared to the S U (2) W -singlet KK leptons and the B 1 LKP As a

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76 result, the preferred LKP mass range could b e pushed bac k up. F or b oth collider and astroparticle searc hes for dark matter, a crucial question is whether there is an upp er limit on the WIMP mass whic h could guaran tee disco v ery and if so, what is its precise n umerical v alue. T o this end, one needs to consider the eect of all coannihilation pro cesses whic h ha v e the p oten tial to enhance the LKP annihilations. W e will see that the lo w ering of the preferred LKP mass range in the case of coannihilations with S U (2) W -singlet KK leptons is more of an exception rather than the rule, and the inclusion of al l remaining pro cesses is needed in order to deriv e an absolute upp er b ound on the LKP mass. 4.2 The Basic Calculation of the Relic Densit y 4.2.1 The Standard Case W e rst summarize the standard calculation for the relic abundance of a particle sp ecies whic h w as in thermal equilibrium in the early univ erse and decoupled when it b ecame nonrelativistic [21, 110, 111]. The relic abundance is found b y solving the Boltzmann equation for the ev olution of the n um b er densit y n dn dt = 3 H n h v i ( n 2 n 2eq ) ; (4{1) where H is the Hubble parameter, v is the relativ e v elo cit y b et w een t w o 's, h v i is the thermally a v eraged total annihilation cross-section times relativ e v elo cit y and n eq is the equilibrium n um b er densit y A t high temp erature ( T m ), n eq T 3 (there are roughly as man y particles as photons). A t lo w temp erature ( T m ), in the nonrelativistic appro ximation, n eq can b e written as n eq = g mT 2 3 2 e m=T ; (4{2) where m is the mass of the relic T is the temp erature and g is the n um b er of in ternal degrees of freedom of suc h as spin, color and so on. W e see from eq. ( 4{2 ) that the densit y n eq is Boltzmann-suppressed. A t high temp erature,

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77 particles are abundan t and rapidly con v ert to ligh ter particles and vice v ersa. But shortly after the temp erature T drops b elo w m the n um b er densit y decreases exp onen tially and the annihilation rate = h v i n drops b elo w the expansion rate H A t this p oin t, 's stop annihilating and escap e out of the equilibrium and b ecome thermal relics. h v i is often appro ximated b y the nonrelativistic expansion 1 h v i = a + b h v 2 i + O ( h v 4 i ) a + 6 b=x + O 1 x 2 ; (4{3) where x = m T : (4{4) By solving the Boltzmann equation analytically with appropriate appro ximations [21, 110, 111], the abundance of is giv en b y n h 2 1 : 04 10 9 M P l x F p g ( x F ) 1 a + 3 b=x F ; (4{5) where the Planc k mass M P l = 1 : 22 10 19 GeV and g is the total n um b er of eectiv ely massless degrees of freedom, g ( T ) = X i = bosons g i + 7 8 X i = f er mions g i : (4{6) The freeze-out temp erature, x F is found iterativ ely from x F = ln c ( c + 2) r 45 8 g 2 3 mM P l ( a + 6 b=x F ) p g ( x F ) x F ; (4{7) where the constan t c is determined empirically b y comparing to n umerical solutions of the Boltzmann equation and here w e tak e c = 1 2 as usual. The co ecien t 7 8 in 1 Note, ho w ev er, that the metho d fails near s -c hannel resonances and thresholds for new nal states [112]. In the in teresting parameter region of UED, w e are alw a ys sucien tly far from thresholds, while for the treatmen t of resonances, see [48, 49].

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78 the righ t hand side of ( 4{6 ) accoun ts for the dierence in F ermi and Bose statistics. Notice that g is a function of the temp erature T as the thermal bath quic kly gets depleted of the hea vy sp ecies with masses larger than T 4.2.2 The Case with Coannihilations When the relic particle is nearly degenerate with other particles in the sp ectrum, its relic abundance is determined not only b y its o wn self-annihilation cross-section, but also b y annihilation pro cesses in v olving the hea vier particles. The previous calculation can b e generalized to this \coannihilation" case in a straigh tforw ard w a y [21, 111, 112]. Assume that the particles i are lab eled according to their masses, so that m i < m j when i < j The n um b er densities n i of the v arious sp ecies i ob ey a set of Boltzmann equations. It can b e sho wn that under reasonable assumptions [112], the ultimate relic densit y n of the ligh test sp ecies 1 (after all hea vier particles i ha v e deca y ed in to it) ob eys the follo wing simple Boltzmann equation dn dt = 3 H n h ef f v i ( n 2 n 2eq ) ; (4{8) where ef f ( x ) = N X ij ij g i g j g 2 ef f (1 + i ) 3 = 2 (1 + j ) 3 = 2 exp ( x ( i + j )) ; (4{9) g ef f ( x ) = N X i =1 g i (1 + i ) 3 = 2 exp ( x i ) ; (4{10) i = m i m 1 m 1 : (4{11) Here ij ( i j S M ), g i is the n um b er of in ternal degrees of freedom of particle i and n = P Ni =1 n i is the densit y of 1 w e w an t to calculate. This Boltzmann equation can b e solv ed in a similar w a y [21, 112], resulting in n h 2 1 : 04 10 9 M P l x F p g ( x F ) 1 I a + 3 I b =x F ; (4{12)

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79 with I a = x F Z 1 x F a ef f ( x ) x 2 dx ; (4{13) I b = 2 x 2F Z 1 x F b ef f ( x ) x 3 dx : (4{14) The corresp onding form ula for x F b ecomes x F = ln c ( c + 2) r 45 8 g ef f ( x F ) 2 3 mM P l ( a ef f ( x F ) + 6 b ef f ( x F ) =x F ) p g ( x F ) x F : (4{15) Here a ef f and b ef f are the rst t w o terms in the v elo cit y expansion of ef f ef f ( x ) v = a ef f ( x ) + b ef f ( x ) v 2 + O ( v 4 ) : (4{16) Comparing eqs. ( 4{9 ) and ( 4{16 ), one gets a ef f ( x ) = N X ij a ij g i g j g 2 ef f (1 + i ) 3 = 2 (1 + j ) 3 = 2 exp ( x ( i + j )) ; (4{17) b ef f ( x ) = N X ij b ij g i g j g 2 ef f (1 + i ) 3 = 2 (1 + j ) 3 = 2 exp ( x ( i + j )) ; (4{18) where a ij and b ij are obtained from ij v = a ij + b ij v 2 + O ( v 4 ). Considering relativistic corrections [110, 113, 114] to the ab o v e treatmen t results in an additional subleading term whic h can b e accoun ted for b y the simple replacemen t b b 1 4 a (4{19) in the ab o v e form ulas, whic h will b e explained in detail in next section. 4.2.3 Thermal Av erage and Nonrelativistic V elo cit y Expansion The thermally a v eraged cross-section times relativ e v elo cit y is dened as [110], h v r el i = R d 3 p 1 d 3 p 2 e E 1 =T e E 2 =T v r el R d 3 p 1 d 3 p 2 e E 1 =T e E 2 =T ; (4{20)

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80 where v r el is the relativ e v elo cit y b et w een t w o incoming particles. Since the relic particle decouples from the equilibrium when the particle is nonrelativistic, w e can use nonrelativistic energy-momen tum relation, p = mv + O ( v 2 ) and E = m + 1 2 mv 2 + O ( v 4 ). In the CM frame, ab o v e equation b ecomes h v r el i = R d 3 v 1 d 3 v 2 e E 1 =T e E 2 =T v r el R d 3 v 1 d 3 v 2 e E 1 =T e E 2 =T = R d 3 V d 3 v r el e m 4 T ( V 2 + v 2 r el ) v r el R d 3 V d 3 v r el e m 4 T ( V 2 + v 2 r el ) (4{21) = R dv r el e 1 4 xv 2 r el v r el R dv r el e 1 4 xv 2 r el ; where ~ V = ~ v 1 + ~ v 2 ~ v r el = ~ v 1 ~ v 2 and E 1 + E 2 = 2 m + 1 2 m 2 ( v 2 1 + v 2 2 ) = 2 m + m 4 ( V 2 + v 2 r el ) are used and x is x = m T No w all w e need to do is to expand v r el in terms of v r el and in tegrate o v er it. The cross-section is giv en b y = 1 2 E 1 2 E 2 v r el Z (2 ) 4 4 p 1 + p 2 X j p j Y i d 3 p i (2 ) 3 2 p 0i : (4{22) No w w e dene function w ( s ) using ab o v e equation, w ( s ) = 1 2 E 1 2 E 2 Z (2 ) 4 4 p 1 + p 2 X j p j Y i d 3 p i (2 ) 3 2 p 0i : (4{23) No w in CM frame w e expand 1 E 2 w ( s ) in terms of s around 4 m 2 with s = 4 m 2 + m 2 v 2 r el + O ( v 4 r el ) and E = m + 1 2 mv 2 C M + O ( v 4 r el ) (2 V C M = v r el ), v r el = 1 E 1 E 2 w ( s ) (4{24) = 1 E 2 w ( s ) = w (4 m 2 ) + dw ds ( s 4 m 2 ) + O ( v 4 r el ) m + 1 2 mv 2 C M + O ( v 4 r el ) 2 = w 0 + w 0 0 v 2 r el 4 + O ( v 4 r el ) m 2 1 + v 2 r el 4 + O ( v 4 r el ) (4{25) = 1 m 2 w 0 + ( w 0 0 w 0 ) 4 v 2 r el + O ( v 4 r el ) ;

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81 where w 0 = w (4 m 2 ) w 0 0 = 4 m 2 dw ds s =4 m 2 : (4{26) Let us dene t w o co ecien ts in the v elo cit y expansion as a and b a = w 0 m 2 ; b = w 0 0 w 0 4 m 2 : (4{27) Therefore the thermally a v eraged cross-section is h v r el i = a + b h v 2 r el i + hO ( v 4 r el ) i = a + 6 bx + O ( x 2 ) : (4{28) Ho w ev er the full relativistic calculation giv es us [110] h v r el i = w 0 m 2 + 3 2 m 2 ( w 0 0 2 w 0 ) + O ( x 2 ) = a + 6( b 1 4 a ) x + O ( x 2 ) : (4{29) So w e expand v r el in terms of relativ e v elo cit y in the nonrelativistic limit to get t w o co ecien t a and b and w e substitute b b y b 1 4 a to reco v er relativistic correction [113, 114]. 4.3 Relic Densit y in Minimal UED F or the purp oses of our study w e ha v e implemen ted the relev an t features of the minimal UED mo del in the CompHEP ev en t generator [68]. W e incorp orated all n = 1 and n = 2 KK mo des as new particles, with the prop er in teractions and one-lo op corrected masses [28]. Similar to the SM case, the neutral gauge b osons at lev el 1, Z 1 and r 1 are mixtures of the KK mo des of the h yp erc harge gauge b oson and the neutral S U (2) W gauge b oson. Ho w ev er, as sho wn in [28],

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82 the radiativ ely corrected W ein b erg angle at lev el 1 and higher is v ery small. F or example, r 1 whic h is the LKP in the minimal UED mo del, is mostly the KK mo de of the h yp erc harge gauge b oson. Therefore, for simplicit y in the co de w e neglected neutral gauge b oson mixing for n = 1. W e then use our UED implemen tation in CompHEP to deriv e analytic expressions for the (co)annihilation cross-sections b et w een an y pair of n = 1 KK particles. Our co de has b een sub jected to n umerous tests and cross-c hec ks. F or example, w e repro duced all results from Serv an t et al. [21]. W e ha v e also used the same co de for indep enden t studies of the collider and astroparticle signatures of UED [10, 19, 61, 115] and th us ha v e tested it from a dieren t angle as w ell. The mass sp ectrum of the n = 1 KK partners in minimal UED can b e found, for example, in g. 1 of [62]. In MUED the next-to-ligh test KK particles are the singlet KK leptons and their fractional mass dierence from the LKP is 2 ` R 1 m ` R 1 m r 1 m r 1 0 : 01 : (4{30) Notice that the Boltzmann suppression e ` R 1 x F e 0 : 01 25 = e 0 : 25 is not v ery eectiv e and coannihilation pro cesses with ` R 1 are denitely imp ortan t, hence they w ere considered in [21]. What ab out the other, hea vier particles in the n = 1 KK sp ectrum in MUED? Since their mass splittings from the LKP i m i m r 1 m r 1 (4{31) 2 In this c hapter w e follo w the notation of [21] where the t w o t yp es of n = 1 Dirac fermions are distinguished b y an index corresp onding to the c hiralit y of their zero mo de partner. F or example, ` R 1 stands for an S U (2) W -singlet Dirac fermion, whic h has in principle b oth a left-handed and a righ t-handed comp onen t.

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83 are larger, their annihilations suer from a larger Boltzmann suppression. Ho w ev er, the couplings of all n = 1 KK partners other than ` R 1 are larger compared to those of r 1 and ` R 1 F or example, S U (2) W -doublet KK leptons ` L 1 couple w eakly and the KK quarks q 1 and KK gluon g 1 ha v e strong couplings. Therefore, their corresp onding annihilation cross-sections are exp ected to b e larger than the crosssection of the main r 1 r 1 c hannel. W e see that for the other KK particles, there is a comp etition b et w een the increased cross-sections and the larger Boltzmann suppression. An explicit calculation is therefore needed in order to ev aluate the net eect of these t w o factors, and judge the imp ortance of the coannihilation pro cesses whic h ha v e b een neglected so far. One migh t exp ect that coannihilations with S U (2) W -doublet KK leptons migh t b e n umerically signican t, since their mass splitting in MUED is 3% and the corresp onding Boltzmann suppression factor is only e 0 : 03 25 e 0 : 75 In our co de w e k eep all KK masses dieren t while w e neglect all the masses of the Standard Mo del particles. As an illustration, let us sho w the a and b terms for r 1 r 1 annihilation only F or fermion nal states w e nd the a -term and b -term of ( r 1 r 1 f f ) v as follo ws a = X f 32 2 1 N c m 2r 1 9 Y 4 f L ( m 2r 1 + m 2f L 1 ) 2 + Y 4 f R ( m 2r 1 + m 2f R 1 ) 2 (4{32) X f 8 2 1 9 m 2r 1 N c Y 4 f L + Y 4 f R = 8 2 1 9 m 2r 1 95 18 ; (4{33) b = X f 4 2 1 N c m 2r 1 27 Y 4 f L 11 m 4r 1 + 14 m 2 m 2f L 1 13 m 4f L 1 ( m 2r 1 + m 2f R 1 ) 4 + Y 4 f R 11 m 4r 1 + 14 m 2 m 2f L 1 13 m 4f L 1 ( m 2r 1 + m 2f R 1 ) 4 (4{34) X f 2 1 9 m 2r 1 N c Y 4 f L + Y 4 f R = 2 1 9 m 2r 1 95 18 ; (4{35)

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84 Figure 4{1: The a -term of the annihilation cross-section for (a) r 1 r 1 e + e and (b) r 1 r 1 as a function of the mass of the t -c hannel particle(s). W e x the LKP mass at m r 1 = 500 GeV and v ary (a) the KK lepton mass m e R 1 = m e L 1 or (b) the KK Higgs b oson mass m 1 The blue solid lines are the exact results ( 4{32 ) and ( 4{36 ), while the red dotted lines corresp ond to the appro ximations ( 4{33 ) and ( 4{37 ). where g 1 is the gauge coupling of the h yp erc harge U (1) Y gauge group, 1 = g 2 1 4 and N c = 3 for f = q and N c = 1 for f = ` Y f is the h yp erc harge of the fermion f F or the Higgs b oson nal states w e get a = X i 2 2 1 Y 4 i 9 m 2r 1 11 m 4r 1 2 m 2r 1 m 2 i + 3 m 4 i ( m 2r 1 + m 2 i ) 2 (4{36) X i 2 2 1 Y 4 i 3 m 2r 1 = 4 2 1 Y 4 3 m 2r 1 ; (4{37) b = X i 2 1 Y 4 i 108 m 2r 1 121 m 8r 1 + 140 m 6r 1 m 2 i 162 m 4r 1 m 4 i + 60 m 2r 1 m 6 i 15 m 8 i ( m 2r 1 + m 2 i ) 4 (4{38) X i 2 1 Y 2 i 12 m 2r 1 = 2 1 Y 2 6 m 2r 1 : (4{39) In the limit where all KK masses are the same (the second line in eac h form ula ab o v e), w e reco v er the result of [21]. Notice the tremendous simplication whic h

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85 arises as a result of the mass degeneracy assumption. In g. 4{1 w e sho w the a terms of the annihilation cross-section for t w o pro cesses: (a) r 1 r 1 e + e and (b) r 1 r 1 as a function of the mass of the t -c hannel particle(s). W e x the LKP mass at m r 1 = 500 GeV and v ary (a) the KK lepton mass m e R 1 = m e L 1 or (b) the KK Higgs b oson mass m 1 The blue solid lines are the exact results ( 4{32 ) and ( 4{36 ), while the red dotted lines corresp ond to the appro ximations ( 4{33 ) and ( 4{37 ) in whic h the mass dierence b et w een the t -c hannel particles and the LKP has b een neglected. W e see that the appro ximations ( 4{33 ) and ( 4{37 ) can result in a relativ ely large error, whose size dep ends on the actual mass splitting of the KK particles. This is wh y in our co de w e k eep all individual mass dep endencies. Another dierence b et w een our analysis and that of Ref. [21] is that here w e shall use a temp erature-dep enden t g function as dened in ( 4{6 ). The relev an t v alue of g whic h en ters the answ er for the LKP relic densit y ( 4{12 ) is g ( T F ), where T F = m r 1 =x F is the freeze-out temp erature. In g. 4{2 a w e sho w a plot of g ( T F ) as a function of R 1 in MUED, while in g. 4{2 b w e sho w the corresp onding v alues of x F In g. 4{2 a one can clearly see the jumps in g when crossing the b b W + W Z Z and hh thresholds (from left to righ t). The t t threshold is further to the righ t, outside the plotted range. As w e shall see b elo w, cosmologically in teresting v alues of n h 2 are obtained for R 1 b elo w 1 T eV, where g ( T F ) = 86 : 25, since w e are b elo w the W + W threshold. The analysis of Ref. [21] assumed a constan t v alue of g = 92, whic h is only v alid b et w een the W + W and Z Z thresholds. The exp ert reader has probably noticed from g. 4{2 b that the v alues of x F whic h w e obtain in MUED are somewhat larger than the x F v alues one w ould ha v e in t ypical SUSY mo dels. This is due to the eect of coannihilations, whic h increase g ef f (see g. 4{5 c b elo w) and therefore x F in accordance with ( 4{15 ). W e are no w in a p osition to discuss our main result in MUED. In g. 4{3 w e sho w the LKP relic densit y as a function of R 1 in the minimal UED mo del. W e sho w

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86 Figure 4{2: The n um b er of eectiv ely massless degrees of freedom and freeze-out temp erature. (a) g ( T F ) and (b) x F as a function of R 1 in MUED. the results from sev eral analyses, eac h under dieren t assumptions, in order to illustrate the eect of eac h assumption. W e rst sho w sev eral calculations for the academic case of no coannihilations. The three solid lines in g. 4{3 accoun t only for the r 1 r 1 pro cess. The (red) line mark ed \a" recreates the analysis of Ref. [21], assuming a degenerate KK mass sp ectrum. The (blue) line mark ed \b" rep eats the same analysis, but uses T -dep enden t g according to ( 4{6 ) and includes the relativistic correction to the b -term ( 4{19 ). The (blac k) line mark ed \c" further relaxes the assumption of KK mass degeneracy and uses the actual MUED mass sp ectrum. Comparing lines \a" and \b," w e see that, as already an ticipated from g. 4{2 a, accoun ting for the T dep endence in g has the eect of lo w ering g ( x F ), ef f ( x F ), and corresp ondingly increasing the prediction for n h 2 This, in turns, lo w ers the preferred mass range for r 1 Next, comparing lines \b" and \c," w e see that dropping the mass degeneracy assumption has a similar eect on ef f ( x F ) (see g. 4{1 ), and further increases the calculated n h 2 This can b e easily understo o d from the t -c hannel mass dep endence exhibited in ( 4{32 ) and ( 4{36 ). The t -c hannel

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87 Figure 4{3: Relic densit y of the LKP as a function of R 1 in the minimal UED mo del. The (red) line mark ed \a" is the result from considering r 1 r 1 annihilation only follo wing the analysis of Ref. [21], assuming a degenerate KK mass sp ectrum. The (blue) line mark ed \b" rep eats the same analysis, but uses T -dep enden t g according to ( 4{6 ) and includes the relativistic correction to the b -term ( 4{19 ). The (blac k) line mark ed \c" relaxes the assumption of KK mass degeneracy and uses the actual MUED mass sp ectrum. The dotted line is the result from the full calculation in MUED, including all coannihilation pro cesses, with the prop er c hoice of masses. The green horizon tal band denotes the preferred Wilkinson Micro w a v e Anisotrop y Prob e (WMAP) region for the relic densit y 0 : 094 < n C D M h 2 < 0 : 129. The cy an v ertical band delineates v alues of R 1 disfa v ored b y precision data. masses app ear in the denominator, and they are b y denition larger than the LKP mass. Therefore, using their actual v alues can only decrease ef f and increase n h 2 The dotted line in g. 4{3 is the result from the full calculation in MUED, including all coannihilation pro cesses, with the prop er c hoice of masses. The green horizon tal band denotes the preferred WMAP region for the relic densit y 0 : 094 < n C D M h 2 < 0 : 129. The cy an v ertical band delineates v alues of R 1 disfa v ored b y precision data [33]. W e see that according to the full calculation, the cosmologically ideal mass range is m r 1 500 600 GeV, when r 1 accoun ts for al l of the dark matter in the Univ erse. This range is somewhat lo w er than earlier studies ha v e indicated, mostly due to the eects discussed ab o v e. Since the MUED mo del

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88 will b e our reference p oin t for the in v estigations in Section 4.4 the dotted line from g. 4{3 will b e app earing in all subsequen t plots in Section 4.4 b elo w. 4.4 Relativ e W eigh t of Dieren t Coannihilation Pro cesses As w e already explained in 4.1 the assumptions b ehind the MUED mo del can b e easily relaxed b y allo wing non-v anishing b oundary terms at the scale This w ould mo dify the KK sp ectrum and corresp ondingly c hange our prediction for the KK relic densit y from the previous section [116{ 120]. Our co de is able to handle suc h more general cases with ease, since w e use as inputs the ph ysical KK masses. In order to gain some insigh t in to the cosmology of suc h non-minimal scenarios, w e ha v e studied the eects of v arying the n = 1 KK masses one at a time. The c hange in an y giv en KK mass will not only enhance or suppress the related coannihilation pro cesses, but also impact an y other cross-sections whic h happ en to ha v e a dep endence on the mass parameter b eing v aried. Th us the results in this section ma y allo w one to judge the imp ortance of eac h individual coannihilation pro cess, and an ticipate the answ er for n h 2 in non-minimal mo dels. W e ha v e classied the discussion in this section b y particle t yp es. Section 4.4.1 con tains our results for the annihilation pro cesses with KK leptons. Man y of our results ha v e already app eared in Ref. [21]. The new elemen t here is the discussion of ` L 1 coannihilations. The results presen ted in Sections 4.4.2 and 4.4.3 are completely new { there w e in v estigate the coannihilation eects with strongly in teracting KK mo des and electro w eak gauge b osons and/or Higgs b osons, resp ectiv ely 4.4.1 Coannihilations with KK Leptons W e b egin with a discussion of r 1 coannihilations with the n = 1 S U (2) W singlet leptons ` R 1 and the n = 1 S U (2) W -doublet leptons ` L 1 One migh t exp ect that those pro cesses will b e imp ortan t, since the KK leptons receiv e relativ ely small one-lo op mass corrections. F or example, in the minimal UED mo del ` R 1 1%

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89 Figure 4{4: Coannihilation eects of (a) 1 generation or (b) 3 generations of singlet KK leptons. The lines sho w the LKP relic densit y as a function of R 1 for sev eral c hoices of the mass splitting ` R 1 b et w een the LKP and the S U (2) W -singlet KK fermions ` R 1 In eac h case w e use the MUED sp ectrum to x the masses of the remaining particles, and then v ary the ` R 1 mass b y hand. The solid lines from top to b ottom in b oth (a) and (b) corresp ond to e R 1 = 0 ; 0 : 3 ; 0 : 1. The dotted line is the nominal UED case from g. 4{3 and ` L 1 3%. It is natural to exp ect that this degeneracy migh t p ersist in non-minimal mo dels as w ell. Our approac h is as follo ws. Since w e k eep separate v alues for the KK masses, when w e start v arying an y one of them, w e ha v e to someho w x the remainder of the KK mass sp ectrum. W e c ho ose to use MUED as our reference mo del, hence the masses whic h are not b eing v aried, will b e xed according to their MUED v alues. W e shall still sho w results for n h 2 as a function of R 1 but for v arious xed v alues of the corresp onding mass splitting i dened in eq. ( 4{31 ). W e shall also alw a ys displa y the reference MUED mo del line, for whic h, of course, i tak es its MUED v alue. Our rst example is sho wn in g. 4{4 where w e illustrate the size of the coannihilation eects for (a) 1 generation or (b) 3 generations of degenerate singlet KK leptons ` R 1 The lines sho w the LKP relic densit y as a function of R 1 for sev eral c hoices of the mass splitting ` R 1 b et w een the LKP and the S U (2) W -singlet KK fermions ` R 1 The solid lines from top to b ottom in

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90 b oth (a) and (b) corresp ond to e R 1 = 0 ; 0 : 3 ; 0 : 1, and the dotted line is the nominal UED case from g. 4{3 for whic h ` R 1 = 0 : 01. As exp ected, all lines follo w the general trend of g. 4{3 In accord with the observ ations of Ref. [21], w e see that ` R 1 coannihilations incr e ase the prediction for n h 2 Suc h a b eha vior ma y seem p eculiar at rst sigh t, since in sup ersymmetry one nds the opp osite phenomenon | coannihilations with sleptons tend to r e duc e the SUSY WIMP relic densit y The dierence b et w een the t w o cases can b e in tuitiv ely understo o d as follo ws. In SUSY, the cross-section for the main annihilation c hannel ( ~ 01 ~ 01 f f ) is helicit y suppressed, but the coannihilation pro cesses are not. Adding coannihilations therefore can only increase the eectiv e cross-section ( 4{9 ) and corresp ondingly decrease n h 2 In con trast, in UED the main annihilation c hannel ( r 1 r 1 f f ) is already of normal strength. The eect of coannihilations can b e easily guessed only if the additional pro cesses ha v e either m uc h w eak er or m uc h stronger in teractions. In the case of ` R 1 ho w ev er, the additional pro cesses are of the same order (b oth r 1 and ` R 1 ha v e h yp erc harge in teractions only) and the sign of the coannihilation eect dep ends on the detailed balance of n umerical factors, whic h will b e illustrated in g. 4{5 and discussed in more detail b elo w. The spread in the lines in g. 4{4 is indicativ e of the imp ortance of the coannihilations. Comparing g. 4{4 a and g. 4{4 b, w e see that in the case of three generations, the eects are magnied corresp ondingly A similar conclusion w as reac hed in Ref. [21]. Notice the p eculiar ordering of the lines corresp onding to dieren t ` R 1 With resp ect to v ariations of ` R 1 the maxim um p ossible v alue of n h 2 is obtained for ` R 1 0, where the eect of coannihilations is maximal. Then, as w e increase the mass splitting b et w een ` R 1 and r 1 at rst n h 2 decreases (see the sequence of ` R 1 = 0, ` R 1 = 0 : 01 and ` R 1 = 0 : 1) but then starts increasing again and the n h 2 v alues that w e get for ` R 1 = 0 : 3 are sligh tly larger than those for ` R 1 = 0 : 1. This

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91 Figure 4{5: Plots of v arious quan tities en tering the LKP relic densit y computation, as a function of the mass splitting e R 1 b et w een the LKP and the S U (2) W -singlet KK electron, for R 1 = 500 GeV in MUED. (a) Relic densit y (b) a ef f ( x F ), (c) g ef f ( x F ) and (d) a ef f ( x F ) g 2 ef f ( x F ). b eha vior can b e seen more clearly from g. 4{5 a, where w e v ary the mass of the S U (2) W -singlet KK electron e R 1 and plot n h 2 v ersus e R 1 for a xed R 1 = 500 GeV. The in teresting b eha vior of n h 2 exhibited in g. 4{5 a can b e understo o d in terms of the m ` R 1 dep endence of the eectiv e annihilation cross-section ( 4{9 ) whic h is dominated b y its a -term ( 4{17 ). Both ef f and a ef f are functions of x but for the purp oses of our discussion here it is sucien t to concen trate on the xed v alue x = x F whic h dominates the in tegrals ( 4{13 ) and ( 4{14 ). W e plot a ef f ( x F ) as a function of e R 1 in g. 4{5 b. W e see that a ef f ( x F ) exhibits exactly the

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92 opp osite dep endence to n h 2 and in particular, has an analogous lo cal extrem um at e R 1 0 : 1. Therefore, in order to understand qualitativ ely the b eha vior of n h 2 w e only need to concen trate on a ef f ( x F ). Let us start with the large e R 1 region in g. 4{5 b. The S U (2) W -singlet KK electron e R 1 is then to o hea vy to participate in an y relev an t coannihilation pro cesses. The eectiv e cross-section ( 4{9 ) then receiv es no con tributions from pro cesses with e R 1 Nev ertheless, the mass of e R 1 en ters ef f through the crosssection for the pro cess r 1 r 1 e + e (see eqs. ( 4{32 ) and ( 4{34 )). Then as w e lo w er m e R 1 ( r 1 r 1 e + e ) is increased and this leads to a corresp onding increase in a ef f as seen in g. 4{5 b. This trend con tin ues do wn to e R 1 0 : 1, where coannihilations with e R 1 start b ecoming relev an t. This can b e seen in g. 4{5 c, where w e plot g ef f ( x F ) as a function of e R 1 F rom its dening equation ( 4{10 ) w e see that g ef f ( x ) starts to deviate from a constan t only when the exp onen tial terms (whic h signal the turning on of coannihilations) b ecome non-negligible. The exp onen tial terms are all p ositiv e and increase g ef f A t the same time, there are new cross-section terms en tering the sum for ef f so w e exp ect the n umerator in ( 4{9 ) to increase as w ell. This is conrmed in g. 4{5 d, where w e plot the n umerator of ( 4{9 ) simply as a ef f ( x F ) g 2 ef f ( x F ). F rom gs. 4{5 c and 4{5 d w e see that b oth the n umerator and the denominator of ( 4{17 ) increase at lo w e R 1 and so it is a priori unclear ho w their ratio will b eha v e with e R 1 In this particular case, g ef f wins, and a ef f ( x F ) is eectiv ely decreased as a result of turning on the coannihilations with e R 1 This feature w as also observ ed in Ref. [21]. W e are no w in p osition to rep eat the same analysis, but for the case of the S U (2) W -doublet KK leptons ` L 1 In g. 4{6 in complete analogy to g. 4{4 w e illustrate the eects on the relic densit y from v arying the S U (2) W -doublet KK electron mass. F rom top to b ottom, the solid lines sho w n h 2 as a function of R 1 for e L 1 = 0 : 01 ; 0 : 001 ; 0. The dotted line is again the MUED reference mo del. W e see that the case of

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93 Figure 4{6: The eects of v arying the S U (2) W -doublet KK electron mass. The same as g. 4{4 but illustrating the eects of v arying the S U (2) W -doublet KK electron mass. F rom top to b ottom, the solid lines sho w n h 2 as a function of R 1 for e L 1 = 0 : 01 ; 0 : 001 ; 0. The dotted line is the nominal UED case from g. 4{3 S U (2) W -doublet KK leptons is dieren t. Unlik e ` R 1 they ha v e w eak in teractions, and the extra terms whic h they bring in to the sum ( 4{9 ) are larger than the main annihilation c hannel. The increase in g ef f is similar as b efore. As a result, this time the increase in the n umerator of ( 4{9 ) wins, and the net eect is to increase the eectiv e annihilation cross-section. This leads to a reduction in the predicted v alue for the relic densit y as evidenced from g. 4{6 Notice ho w the decrease in n h 2 is monotonic with e L 1 Another dierence b et w een ` R 1 and ` L 1 coannihilations is rev ealed b y comparing the case of 1 generation (panels (a) in gs. 4{4 and 4{6 ) and 3 generations (panels (b) in gs. 4{4 and 4{6 ). W e see that for S U (2) W singlets, the coannihilations are more prominen t for the case of 3 generations, while for S U (2) W doublets, it is the opp osite. This is due to the dieren t n um b er of degrees of freedom con tributed to g ef f in eac h case, whic h shifts the delicate balance b et w een the n umerator and denominator of ( 4{9 ), as discussed ab o v e.

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94 Figure 4{7: The eects of v arying KK quarks masses. The same as gs. 4{4 (b) and 4{6 (b), but for the case of KK quarks. Eac h solid line is lab eled b y the v alue of (a) q R 1 or (b) q L 1 used. The dotted line is the nominal UED case from g. 4{3 4.4.2 Coannihilations with KK Quarks and KK Gluons W e will no w consider coannihilation eects with colored KK particles (KK quarks and KK gluons). Since they couple strongly w e exp ect on general grounds that the eectiv e annihilation cross-sections will b e enhanced, and the preferred range of the LKP mass will corresp ondingly b e shifted higher. These exp ectations are conrmed b y our explicit calculation whose results are sho wn in gs. 4{7 and 4{8 In g. 4{7 w e sho w the eects on the relic densit y from v arying the masses of all three generations of (a) S U (2) W -singlet KK quarks and (b) S U (2) W -doublet KK quarks. The solid lines sho w n h 2 as a function of R 1 and are lab eled b y the corresp onding v alue of q R 1 or q L 1 used. As b efore, the dotted line is the MUED reference mo del. Comparing the results in gs. 4{7 a and 4{7 b, w e nd that the coannihilations with q R 1 and q L 1 ha v e v ery similar eects, as they are b oth dominated b y the strong in teractions, whic h are the same for q R 1 and q L 1 Fig. 4{8 sho ws the analogous result for the case of v arying the KK gluon mass, where the lab els no w sho w the v alues of g 1 There is a noticeable distortion of the lines around R 1 2300 GeV, whic h is due to the c hange in g (see g. 4{2 a).

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95 Figure 4{8: The eects of v arying KK gluon mass. The same as g. 4{7 but for the case of v arying the KK gluon mass. The lines are lab eled b y the v alue of g 1 The dotted line is the nominal UED case from g. 4{3 F rom gs. 4{7 and 4{8 w e see that in non-minimal UED mo dels where the colored KK mo des happ en to exhibit some sort of degeneracy with the LKP m ulti-T eV v alues for m r 1 are in principle p ossible. F rom that p oin t of view, unfortunately there is no \no-lose" theorem for the LHC or ILC regarding a p oten tial absolute upp er b ound on the LKP mass. 4.4.3 Coannihilations with Electro w eak KK Bosons W e nally sho w our coannihilation results for the case of electro w eak KK gauge b osons ( W 0 1 and W 1 ) and KK Higgs b osons ( H 0 1 G 01 and G 1 ). The results are displa y ed in gs. 4{9 a and 4{9 b, corresp ondingly Due to the S U (2) W symmetry all three n = 1 KK W -b osons are v ery degenerate, and w e ha v e assumed a common parameter W 1 for all three. Similarly the masses of the n = 1 KK Higgs b osons dier only b y electro w eak symmetry breaking eects, whic h w e neglect throughout the calculation. W e ha v e therefore assumed a common parameter H for them as w ell. Since b oth the electro w eak KK gauge b osons and the KK Higgs b osons ha v e w eak in teractions, w e exp ect the results to b e similar to the case of

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96 Figure 4{9: The eects of v arying EW b osons. The same as g. 4{8 but illustrating the eect of v arying sim ultaneously the masses of all (a) S U (2) KK gauge b osons and (b) KK Higgs b osons. In (a) the lines are lab eled b y the v alue of W 1 while in (b) the v alues of H are (from top to b ottom) H = 0 : 05 ; 0 : 01 ; 0 : 001 ; 0. The dotted line is the nominal UED case from g. 4{3 S U (2) W -doublet leptons in the sense that coannihilations w ould lo w er the predictions for n h 2 This is conrmed b y g. 4{9 W e observ e that the eects from the KK W -b osons are actually quite signican t, and can push the preferred LKP mass as high as 1.4 T eV. 4.5 Other Dark Matter Candidates and Direct Detection In previous sections w e revisited the calculation of the LKP relic densit y in the scenario of Univ ersal Extra Dimensions. W e extended the analysis of Ref. [21] to include al l coannihilation pro cesses in v olving n = 1 KK partners. This allo w ed us to predict reliably the preferred mass range for the KK dark matter particle in the minimal UED mo del. W e found that in order to accoun t for all of the dark matter in the univ erse, the mass of r 1 should b e within 500 600 GeV, whic h is somewhat lo w er than the range found in [21]. This is due to a com bination of sev eral factors. Among the eects whic h caused our prediction for n h 2 to go up are the follo wing: w e used a lo w er v alue of g w e k ept the individual KK masses in our form ulas, and w e accoun ted for the relativistic correction ( 4{19 ). On the other hand, as w e sa w

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97 in Section 4.4 including the eect of coannihilations with KK particles other than S U (2) W -singlet KK leptons, alw a ys has the eect of lo w ering the predicted n h 2 Finally the cosmologically preferred range for n h 2 itself has shifted lo w er since the publication of [21]. The lo w er range of preferred v alues for R 1 is go o d news for collider and astroparticle searc hes for KK dark matter. It should b e k ept in mind that it is quite plausible, and in fact v ery lik ely that the dark matter is made up of not one but sev eral dieren t comp onen ts, in whic h case the LKP could b e ev en ligh ter. W e should men tion that sev eral collider studies [10, 13, 19, 62] ha v e already used an MUED b enc hmark p oin t with R 1 = 500 GeV, a c hoice whic h w e no w see also happ ens to b e relev an t for cosmology In Section 4.4 w e also in v estigated ho w eac h class of n = 1 KK partners impacts the KK relic densit y W e summarize the observ ed trends in g. 4{10 where w e x n h 2 = 0 : 1 and then sho w the required R 1 for an y giv en i for eac h class of KK particles. W e sho w v ariations of the masses of one (red dotted) or three (red solid) generations of S U (2) W -singlet KK leptons; three generations of S U (2) W doublet leptons (magen ta); three generations of S U (2) W -singlet quarks (blue) (the result for three generations of S U (2) W -doublet quarks is almost iden tical); KK gluons (cy an) and electro w eak KK gauge b osons (green). The circle on eac h line denotes the MUED v alues of and R 1 Fig. 4{10 summarizes our results from Section 4.4 It also pro vides a quic k reference guide for the exp ected v ariations in the predicted v alue of n h 2 as w e mo v e a w a y from the minimal UED mo del. F or example, it is clear that unlik e the case of coannihilations with ` R 1 whic h w as considered in [21], coannihilations with all other KK particles will lo w er the prediction for n h 2 and corresp ondingly increase the preferred range of R 1 This is due to the larger couplings of those particles. Fig. 4{10 can also b e used to

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98 Figure 4{10: The c hange in the cosmologically preferred v alue for R 1 as a result of v arying the dieren t KK masses a w a y from their nominal MUED v alues. Along eac h line, the LKP relic densit y is n h 2 = 0 : 1. T o dra w the lines, w e rst x the MUED sp ectrum, and then v ary the corresp onding KK mass and plot the v alue of R 1 whic h is required to giv e n h 2 = 0 : 1. W e sho w v ariations of the masses of one (red dotted) or three (red solid) generations of S U (2) W -singlet KK leptons; three generations of S U (2) W -doublet leptons (magen ta); three generations of S U (2) W singlet quarks (blue) (the result for three generations of S U (2) W -doublet quarks is almost iden tical); KK gluons (cy an) and electro w eak KK gauge b osons (green). The circle on eac h line denotes the MUED v alues of and R 1 quan titativ ely estimate the v ariations in the preferred v alue of R 1 in non-minimal mo dels. On a nal note, in the non-minimal UED mo del, other neutral KK particles suc h as Z 1 can also b e dark matter candidates. On dimensional grounds, the relic densit y is in v ersely prop ortional to the square of the LKP mass, n h 2 g 4 1 m 2r 1 ; (4{40) n h 2 g 4 2 m 2Z 1 : (4{41) Due to the larger coupling g 2 of the S U (2) W gauge in teractions, w e exp ect the upp er b ound on m Z 1 consisten t with WMAP to b e larger than the b ound on m r 1 roughly b y a factor of g 2 2 =g 2 1 3. Ho w ev er, in the Z 1 LKP case, S U (2) W symmetry implies that the c harged W 1 states are almost degenerate with Z 1 and therefore

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99 Figure 4{11: The spin-indep enden t direct detection limit from CDMS exp erimen t. W e sho w the relic densit y and spin-indep enden t direct detection limit from CDMS exp erimen t in the plane of mass splitting Q 1 = q 1 = m Q 1 m r 1 m r 1 and LKP mass for (a) r 1 LKP (b) Z 1 LKP The red line accoun ts for all of the dark matter (100%) and the t w o red dotted lines sho w 10% and 1%, resp ectiv ely The blue (green) line sho ws the curren t CDMS limit with Ge-detector (Si-detector) and the three cy an lines represen t pro jected Sup erCDMS limits for eac h phase: A (25 kg), B (150 kg) and C (1 ton) resp ectiv ely In the case of r 1 LKP Sup erCDMS rules out most of parameter space while there is little parameter space left in the case of Z 1 LKP The y ello w region in the case of r 1 LKP sho ws parameter space that could b e co v ered b y the collider searc h in 4 ` + / E T c hannel at the LHC. coannihilations with W 1 will b e v ery imp ortan t and will need to b e considered. The analysis of the cases of Z 1 and H 1 LKP and their direct and indirect detection prosp ects is curren tly in progress [121]. In g. 4{11 w e sho w the relic densit y and spin-indep enden t direct detection limit [50] from CDMS exp erimen t [122, 123] in the plane of mass splitting, Q 1 = q 1 = m Q 1 m r 1 m r 1 and LKP mass for (a) r 1 LKP (b) Z 1 LKP The red line accoun ts for all of the dark matter (100%) and the t w o red dotted lines sho w 10% and 1%, resp ectiv ely The blue (green) line sho ws the curren t CDMS limit with Ge-detector (Si-detector) and the three cy an lines represen t pro jected Sup erCDMS limits for eac h phase: A (25 kg), B (150 kg) and C (1 ton) resp ectiv ely In the case of r 1 LKP Sup erCDMS rules out most of parameter space while there is little parameter space left in the case of Z 1 LKP

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100 The y ello w region in the case of r 1 LKP sho ws parameter space that could b e co v ered b y the collider searc h in 4 ` + / E T c hannel at the LHC [62]. The results presen ted here are also relev an t for the case of KK gra viton sup erwimps [89, 124, 125], whose relic densit y is still determined b y the freeze-out of the next-to-ligh test KK particle. In conclusion, dark matter candidates from theories with extra dimensions should b e considered on an equal fo oting with more con v en tional candidates suc h as SUSY dark matter or axions. The framew ork of Univ ersal Extra Dimensions pro vides a useful pla yground for gaining some exp erience ab out the signals one could exp ect from extra dimensional dark matter. If extra dimensions ha v e indeed something to do with the dark matter problem, the explicit realization of that idea ma y lo ok quite dieren tly (see for example [88, 126, 127]), esp ecially if one w an ts to resolv e the radion stabilization problem [128, 129]. Nev ertheless, w e b eliev e that the metho ds and insigh t w e dev elop ed in this c hapter will pro v e useful in more general con texts.

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CHAPTER 5 CONCLUSIONS A ma jor motiv ation for studying new ph ysics b ey ond the Standard Mo del is the dark matter puzzle whic h nds no explanation within the Standard Mo del. The mo dels with Univ ersal Extra Dimensions naturally pro vide p ossible candidates if the theory is compactied at the T eV scale and KK-parit y is conserv ed.The mo dels are within the reac h of curren t or future collider and astroparticle exp erimen ts. F or the collider studies, w e ha v e written a co de to sim ulate new ph ysics signals in UED and it is already b eing used b y exp erimen talists. F or the dark matter study w e wrote a co de whic h calculates the relic abundance of the KK dark matter. In this co de, w e do not assume an y sp ecic dark matter candidate and therefore it can b e used for an y other neutral dark matter candidates within the mo del. Imp ortan tly an y coannihilation pro cesses can b e included since our co de w as written with a general mass sp ectrum of the UED mo del. In this dissertation, w e concen trated on the collider phenomenology and astroph ysical implications of Univ ersal Extra Dimensions. In c hapter 3, w e studied the disco v ery reac h of the T ev atron and the LHC for lev el 2 Kaluza-Klein mo des, whic h w ould indicate the presence of extra dimensions. W e found that with 100 fb 1 of data the LHC will b e able to disco v er the r 2 and Z 2 KK mo des as separate resonances if their masses are b elo w 2 T eV. W e also in v estigated the p ossibilit y to dieren tiate the spins of the sup erpartners and KK mo des b y means of the dilepton mass metho d and the asymmetry metho d in a squark cascade deca y Ho w ev er this metho d ma y not b e generalized for all p oin ts in parameter space. A t a e + e linear collider, w e found that in the + / E T c hannel the angular distributions of the 101

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102 nal state m uons, the energy sp ectrum of the radiativ e return photon and the total cross-section measuremen t are p o w erful discriminators b et w een the t w o mo dels. In c hapter 4, w e revisited the calculation of the relic densit y of the ligh test Kaluza-Klein particle. W e included coannihilation pro cesses with al l lev el one KK particles. In our computation w e considered a most general KK particle sp ectrum, without an y simplifying assumptions. In particular, w e did not assume a completely degenerate KK sp ectrum and instead retain the dep endence on eac h individual KK mass. W e found that in order to accoun t for all of the dark matter in the univ erse, the mass of r 1 should b e within 500 600 GeV, whic h is somewhat lo w er than the range found in previous calculation. In nonminimal UED mo dels, an y neutral KK particle, in principle, could b e a dark matter candidate and for these candidates more w ork need to b e done. W e ha v e also discussed curren t limits from direct detections. T o conclude, mo dels with Univ ersal Extra Dimensions are in teresting prop osals for ph ysics b ey ond the standard mo del. The attractiv e merit of analogy to sup ersymmetry particularly mak es this mo del more in teresting. Studies of their phenomenology sho w that the idea of Univ ersal Extra Dimensions is viable at presen t. Our studies on their phenomenology will pro vide to ols to understand the nature of new ph ysics b ey ond the Standard Mo del.

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APPENDIX A ST AND ARD MODEL IN 5D A.1 Lagrangian of the Standard Mo del in 5D Here w e consider the minimal mo del of univ ersal extra dimensions. The mo del is dened in v e dimensions with one extra dimension compactied. In 5D there are t w o problems. First w e can not write the usual Dirac mass term b ecause a c hiral represen tation of a fermion do es not exist in 5D. Second w e ha v e the fth comp onen t (scalar in 4D) of the 5D v ector eld after the compactication and therefore w e end up with to o man y zero mo de scalar particles in 4D since there are three gauge groups in the SM. T o in tro duce c hiral fermions and pro ject out un w an ted scalars, a Z 2 symmetry is imp osed on the S 1 There are t w o xed p oin ts in this geometry S 1 = Z 2 called the orbifold (see g. 1{3 ). W e can see that this geometry is still in v arian t under the exc hange of t w o xed p oin ts ( Z 2 ). This symmetry is called KK parit y W e imp ose the follo wing sp ecial b oundary conditions for fermions and v ector elds at the xed p oin ts. W e w an t a eld to b e either ev en or o dd under the transformation P 5 : y y then at y = 0 ; R @ 5 + = 0 for ev en elds ; = 0 for o dd elds ; (A-1) where @ 5 is the deriv ativ e with resp ect to the extra dimensional co ordinate. These are Neumann and Diric hlet b oundary conditions resp ectiv ely at the xed p oin ts. The asso ciated KK expansions are + ( x; y ) = 1 p 2 R +0 + 1 p R 1 X n =1 +n ( x ) cos ny R ; (A-2) ( x; y ) = 1 p R 1 X n =1 +n ( x ) sin ny R ; (A-3) 103

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104 where x is the 4-dimensional spacetime co ordinate x = ( t; x; y ; z ) and y is the extra dimensional co ordinate. R is the size of the extra dimension and n represen ts the KK-lev el. n = 0 is the SM mo de. The orbifold compactication forces the rst four comp onen ts to b e ev en under P 5 while the fth comp onen t is o dd: @ 5 A = 0 A 5 = 0 9>=>; at the xed p oin ts : (A-4) Hence the KK expansion of a v ector eld is A ( x; y ) = 1 p R ( A 0 ( x ) + p 2 1 X n =1 A n ( x ) cos( ny R ) ) ; (A-5) A 5 ( x; y ) = r 2 R 1 X n =1 A n5 ( x ) sin( ny R ) : (A-6) Imp osing @ 5 + R = 0 + L = 0 9>=>; at y = 0 ; R or @ 5 + L = 0 + R = 0 9>=>; at y = 0 ; R ; (A-7) the resp ectiv e KK mo de expansions are + ( x; y ) = 1 p 2 R 0 R ( x ) + 1 p R 1 X n =1 n R ( x ) cos ny R + n L ( x ) sin ny R ; (A-8) ( x; y ) = 1 p 2 R 0 L ( x ) + 1 p R 1 X n =1 n L ( x ) cos ny R + n R ( x ) sin ny R : (A-9) So the zero mo de is either righ t handed or left handed. Ho w ev er KK mo des come in c hiral pairs. This c hiral structure is a natural consequence of the orbifold

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105 b oundary conditions. The 5 dimensional SM elds are dened as follo ws. H ( x; y ) = 1 p R ( H ( x ) + p 2 1 X n =1 H n ( x ) cos ( ny R ) ) ; B ( x; y ) = 1 p R ( B 0 ( x ) + p 2 1 X n =1 B n ( x ) cos( ny R ) ) ; B 5 ( x; y ) = r 2 R 1 X n =1 B n 5 ( x ) sin( ny R ) ; W ( x; y ) = 1 p R ( W 0 ( x ) + p 2 1 X n =1 W n ( x ) cos( ny R ) ) ; W 5 ( x; y ) = r 2 R 1 X n =1 W n 5 ( x ) sin ( ny R ) ; G ( x; y ) = 1 p R ( G 0 ( x ) + p 2 1 X n =1 G n ( x ) cos( ny R ) ) ; (A-10) G 5 ( x; y ) = r 2 R 1 X n =1 G n5 ( x ) sin( ny R ) ; Q ( x; y ) = 1 p R ( q L ( x ) + p 2 1 X n =1 h P L Q nL ( x ) cos ( ny R ) + P R Q nR ( x ) sin ( ny R ) i ) ; U ( x; y ) = 1 p R ( u R ( x ) + p 2 1 X n =1 h P R u nR ( x ) cos ( ny R ) + P L u nL ( x ) sin( ny R ) i ) ; D ( x; y ) = 1 p R ( d R ( x ) + p 2 1 X n =1 h P R d nR ( x ) cos( ny R ) + P L d nL ( x ) sin ( ny R ) i ) ; L ( x; y ) = 1 p R ( L 0 ( x ) + p 2 1 X n =1 h P L L nL ( x ) cos( ny R ) + P R L nR ( x ) sin ( ny R ) i ) ; E ( x; y ) = 1 p R ( e R ( x ) + p 2 1 X n =1 h P R e nR ( x ) cos( ny R ) + P L e nL ( x ) sin ( ny R ) i ) ; where H ( x; y ) is the 5D scalar eld and ( B ( x; y ) ; B 5 ( x; y )), ( W ( x; y ) ; W 5 ( x; y )) and ( G ( x; y ) ; G 5 ( x; y )) are the 5D gauge elds for U (1), S U (2) and S U (3) resp ectiv ely Q ( x; y ) and L ( x; y ) are the S U (2) fermion doublets while U ( x; y ), D ( x; y ) and E ( x; y ) are resp ectiv ely the generic singlet elds for the up-t yp e quark,

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106 the do wn-t yp e quark and the lepton. The S U (2) and S U (3) gauge elds are W M W a M a 2 ; (A-11) G M G AM A 2 ; (A-12) where a 's are the usual P auli's matrices and A 's are the usual Gell-Mann matrices. P L;R = 1 r 5 2 and M = ; 5 and = 0 ; 1 ; 2 ; 3. No w w e write the 5 dimensional Lagrangian in v arian t under S U (3) S U (2) U (1) and compactify o v er the orbifold to get the 4 dimensional eectiv e Lagrangian. First let us set up the con v en tions for the ingredien ts that go in to the Lagrangian. The gamma matrices in 5D M = ( r ; ir 5 ) ; (A-13) satisfy the Dirac-Cliord algebra f M ; N g = 2 g M N ; (A-14) where g M N is the 5D metric g M N = 0B@ g 0 0 1 1CA ; (A-15) and g = (+ ) is the usual 4D metric. The gauge couplings are denoted b y g i = g (5) i p R ; (A-16) where i = 1 ; 2 ; 3 stands for U (1), S U (2), and S U (3), g (5) i 's are the 5-dimensional gauge couplings and g i 's are the 4-dimensional gauge couplings. The co v arian t

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107 deriv ativ e in 5D acts on 5D elds in the follo wing w a y D M L ( x; y ) = @ M + ig (5) 2 W M + i y 1 2 g (5) 1 B M L ( x; y ) ; D M E ( x; y ) = @ M + i y 2 2 g (5) 1 B M E ( x; y ) ; D M Q ( x; y ) = @ M + ig (5) 3 G M + ig (5) 2 W M + i y 3 2 g (5) 1 B M Q ( x; y ) ; (A-17) D M U ( x; y ) = @ M + ig (5) 3 G M + i y 4 2 g (5) 1 B M U ( x; y ) ; D M D ( x; y ) = @ M + ig (5) 3 G M + i y 5 2 g (5) 1 B M D ( x; y ) : The v e dimensional Standard Mo del can b e written as, L Gaug e = 1 2 Z R R dy 1 4 B M N B M N 1 4 W a M N W aM N 1 4 G AM N G AM N ; L GF = 1 2 Z R R dy 1 2 ( @ B @ 5 B 5 ) 2 1 2 @ W a @ 5 W a 5 2 1 2 @ G A @ 5 G G5 2 ; L Leptons = 1 2 Z R R dy i L ( x; y ) M D M L ( x; y ) + i E ( x; y ) M D M E ( x; y ) ; L Quar k s = 1 2 Z R R dy i Q ( x; y ) M D M Q ( x; y ) + i U ( x; y ) M D M U ( x; y ) + i D ( x; y ) M D M D ( x; y ) ; (A-18) L Y uk aw a = 1 2 Z R R dy u Q ( x; y ) U ( x; y ) i 2 H ( x; y ) + d Q ( x; y ) D ( x; y ) H ( x; y ) + e L ( x; y ) E ( x; y ) H ( x; y ) ; L H ig g s = 1 2 Z R R dy h ( D M H ( x; y )) y D M H ( x; y ) + 2 H y ( x; y ) H ( x; y ) H y ( x; y ) H ( x; y ) 2 i : W e express the 5 dimensional elds in terms of the trigonometric functions due to the orbifold structure. So the orthogonalit y relations are v ery imp ortan t when w e

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108 compactify The follo wing are the relations w e need 1 2 Z R R dy cos( my R ) cos ( ny R ) = R 2 m;n ; 1 2 Z R R dy sin ( my R ) sin ( ny R ) = R 2 m;n ; 1 2 Z R R dy cos ( my R ) cos( ny R ) cos ( l y R ) = R 4 1mnl ; 1 2 Z R R dy cos ( my R ) cos( ny R ) cos( l y R ) cos( k y R ) = R 8 2mnl k ; 1 2 Z R R dy sin ( my R ) sin ( ny R ) sin( l y R ) sin ( k y R ) = R 8 3mnl k ; 1 2 Z R R dy sin ( my R ) sin( ny R ) cos ( l y R ) = R 4 4mnl ; (A-19) 1 2 Z R R dy sin( my R ) sin( ny R ) cos( l y R ) cos( k y R ) = R 8 5mnl k ; 1 2 Z R R dy cos ( my R ) sin ( ny R ) = 0 ; 1 2 Z R R dy sin( my R ) sin( ny R ) sin ( l y R ) = 0 ; 1 2 Z R R dy sin ( my R ) cos( ny R ) cos ( l y R ) = 0 ; 1 2 Z R R dy sin ( my R ) cos( ny R ) cos( l y R ) cos( k y R ) = 0 ; 1 2 Z R R dy sin ( my R ) sin( ny R ) sin( l y R ) cos( k y R ) = 0 ; and 's are dened b elo w. 1mnl = l ;m + n + n;l + m + m;l + n ; (A-20) 2mnl k = k ;l + m + n + l ;m + n + k + m;n + k + l + n;k + l + m + k + m;l + n + k + l ;m + n + k + n;l + m ; 3mnl k = k ;l + m + n l ;m + n + k m;n + k + l n;k + l + m + k + l ;m + n + k + m;l + n + k + n;l + m ; 4mnl = l ;m + n + n;l + m + m;l + n ; 5mnl k = k ;l + m + n l ;m + n + k + m;n + k + l + n;k + l + m k + l ;m + n + k + m;l + n + k + n;l + m :

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109 After in tegrating out the extra dimensional co ordinate, w e nd the follo wing prop erties of the 4 dimensional eectiv e theory Eac h SM particle has an innite n um b er of KK partners The KK particles ha v e the same spin as SM particles New v ertices are the same as the SM couplings (up to normalization) All v ertices satisfy KK n um b er conserv ation (see g. 2{2 ) F or eac h term in the Lagrangian w e ha v e a whic h is a linear com bination of the Kronec k er delta functions. Due to this structure, the allo w ed v ertices satisfy one of the follo wing conditions, j m n k j = 0 ; (A-21) j m n k l j = 0 : Therefore it is easy to see whic h v ertices are allo w ed or not. In addition, the coupling is the same as the one in the SM (up to normalization factor p 2). Ho w ev er the KK n um b er conserv ation is brok en to the discrete symmetry called KK-parit y due to the radiativ e corrections. The c haracteristic feature of this theory is that there exist maxim um and minim um b ounds on the size of the extra dimension R The minim um b ound comes from the electro w eak precision data measuremen t whic h tells us R 1 250 GeV [6, 33]. And the maxim um b ound is related to the cosmology According to cosmological observ ations, only 23 % is the matter t yp e and the rest is some sort of energy So if this theory suggests dark matter, the relic densit y for this particle should not b e greater that 23% (n h 2 0 : 13). This giv es us an upp er b ound on R 1 a few T eV [20{ 22]. A.2 The Kaluza-Klein F ermions and Gauge b osons No w let us summarize the fermion con ten t of this theory As w e can see from the table A{1 there are t w o KK particles corresp onding to one SM particles. The

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110 T able A{1: F ermion con ten t of the Standard Mo del and corresp onding KaluzaKlein fermions SU(2) Symmetry SM mo de KK mo de Quark doublet q L ( x ) = U L ( x ) D L ( x ) Q nL ( x ) = U n L ( x ) D n L ( x ) Q nR ( x ) = U n R ( x ) D n R ( x ) Lepton doublet L 0 ( x ) = L ( x ) E L ( x ) L nL ( x ) = n L ( x ) E n L ( x ) L nR ( x ) = n R ( x ) E n R ( x ) Quark Singlet u R ( x ) u nR ( x ), u nL ( x ) Quark Singlet d R ( x ) d nR ( x ), d nL ( x ) Lepton Singlet e R ( x ) e nR ( x ), e nL ( x ) table A{2 summarizes the quan tum n um b ers for the mass eigenstates when w e ignore the mass term from the Y uk a w a coupling. T able A{2: Quan tum n um b ers of KK fermions KK F ermions I 3 Y Q = I 3 + Y 2 Quark U n = U n L ( x ) + U n R ( x ) 1 2 1 3 2 3 Doublet D n = D n L ( x ) + D n R ( x ) 1 2 1 3 1 3 Quark u n = u nL ( x ) + u nR ( x ) 0 4 3 2 3 Singlet d n = d nL ( x ) + d nR ( x ) 0 2 3 1 3 Lepton n = n L ( x ) + n R ( x ) 1 2 1 0 Doublet E n = E n L ( x ) + E n R ( x ) 1 2 1 1 Lepton e n = e nL ( x ) + e nR ( x ) 0 2 1 Singlet no KK singlet n T able A{3: F ermions and gauge b osons in the Standard Mo del SM F ermions SM Gauge Bosons u = U L ( x ) + u R ( x ) W = W 1 iW 2 p 2 d = D L ( x ) + d R ( x ) Z = sin W B + cos W W 3 e = E L ( x ) + e R ( x ) A = cos W B + sin W W 3 L = L ( x ) The particles in the Standard Mo del are summarized in the table A{3 Note that in UED, the Dirac fermions F n ( x ) = F n L ( x ) + F n R ( x ), as sho wn ab o v e, are constructed out of F n L ( x ) and F n R ( x ) whic h ha v e the same S U (2) U (1) quan tum n um b ers. Con trast this with a SM Dirac fermion \ f whic h is constructed out of

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111 f L ( x ) and f R ( x ) whic h ha v e dieren t S U (2) U (1) quan tum n um b ers. This difference sho ws up in the pro cesses in v olving gauge-b osons couplings with fermions. The mass terms for the KK singlets app ear with the wrong sign in the fermionic Lagrangian and this aect sho ws up as a wrong sign in Y uk a w a in teraction through the redinition of fermion eld (see app endix B.6 for detail). The mass eigenstates of KK photons and Z can b e obtained b y diagonalizing the follo wing mass matrix in W 3 n and B n basis 0B@ n 2 R 2 + ^ m 2B n + 1 4 g 2 1 v 2 1 4 g 1 g 2 v 2 1 4 g 1 g 2 v 2 n 2 R 2 + ^ m 2W 3 n + 1 4 g 2 2 v 2 1CA : (A-22) The Lagrangian for the EW gauge b osons can b e obtained using the follo wing denitions W n = W 1 n iW 2 n p 2 ; Z n = sin n B n + cos n W 3 n W 3 n ; (A-23) A n = cos n B n + sin n W 3 n B n : The neutral gauge b oson eigenstates b ecome appro ximately pure B n and W 3 n since the W ein b erg angles for KK states are small as sho wn in A{1 Therefore the tree lev el mass of the n th KK mo de is m 2n = n 2 R 2 + m 2 : (A-24) A.3 The Deca y Widths of KK P articles With the F eynman rules, w e calculate the deca y widths for eac h particle at the rst KK lev el and eac h gauge b oson at the second KK lev el. F or lev el 1 KK

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112 Figure A{1: Dep endence of the \W ein b erg" angle n for the rst few KK lev els ( n = 1 ; 2 ; ; 5) on R 1 for xed R = 20. The gure is tak en from Cheng et al. [28]. particles, the deca y widths are giv en b elo w. F or lev el 1 fermion, ( f 1 f 0 V 1 ) = c 2 g 2 32 m 3f 1 m 2V 1 m 2f 1 + m 2f 0 m 2V 1 + ( m 2f 1 m 2f 0 m 2V 1 )( m 2f 1 m 2f 0 + m 2V 1 ) m 2V 1 q m 2f 1 ( m f 0 m V 1 ) 2 m 2f 1 ( m f 0 + m V 1 ) 2 ; (A-25) c 2 g 2 32 m 3f 1 m 2V 1 1 m 2V 1 m 2f 1 2 1 + 2 m 2V 1 m 2f 1 : (A-26) F or lev el 1 gauge b oson, ( V 1 f 1 f 0 ) = c 2 g 2 48 m 3V 1 m 2V 1 m 2f 1 m 2f 0 + m 4V 1 ( m 2f 1 m 2f 0 ) 2 m 2V 1 q m 2V 1 ( m f 1 m f 0 ) 2 m 2V 1 ( m f 1 + m f 0 ) 2 ; (A-27) c 2 g 2 48 m 4f 1 m 3V 1 1 m 2V 1 m 2f 1 2 2 + m 2f 1 m 2V 1 : (A-28) No w w e giv e the deca y widths for lev el 2 gauge b osons. F or t w o SM fermions in the nal state, ( V 2 f 0 f 0 ) = c 2 g 2 m V 2 12 m V 2 m 2 m f 2 m 2 2 1 m 2f 0 m 2V 2 s 1 4 m 2f 0 m 2V 2 ; (A-29) c 2 g 2 m V 2 12 m V 2 m 2 m f 2 m 2 2 : (A-30)

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113 F or one lev el 2 KK fermion and one SM fermion, ( V 2 f 2 f 0 ) = c 2 g 2 48 m 3V 2 m 2V 2 m 2f 2 m 2f 0 + m 4V 2 ( m 2f 2 m 2f 0 ) 2 m 2V 2 (A-31) q m 2V 2 ( m f 2 m f 0 ) 2 m 2V 2 ( m f 2 + m f 0 ) 2 ; c 2 g 2 48 m 3V 2 m 2V 2 m 2f 2 2 1 + m 2V 2 + m 2f 2 m 2V 2 ; (A-32) c 2 g 2 m V 2 4 ^ m V 2 m 2 ^ m f 2 m 2 2 : (A-33) F or t w o lev el 1 KK fermions, ( V 2 f 1 f 1 ) = c 2 g 2 24 m 2V 2 m 2V 2 4 m 2f 1 3 2 ; (A-34) c 2 g 2 m V 2 6 p 2 ^ m V 2 m 2 ^ m f 1 m 1 3 2 m 2 m V 2 3 ; (A-35) c 2 g 2 m V 2 6 p 2 ^ m V 2 m 2 ^ m f 1 m 1 3 2 : (A-36) F or eac h deca y widths corresp onding to lev el 2 gauge b osons, m 22 = n R 2 + m 22 w as used in the last appro ximation. F or the deca y of lev el 2 KK fermion in to one SM fermion and one lev el 2 gauge b oson, ( f 2 V 2 f 0 ) = g 2 c 2 32 m 3f 2 q m 2f 2 ( m f 0 m V 2 ) 2 m 2f 2 ( m f 0 + m V 2 ) 2 (A-37) m 2f 2 + m 2f 0 m 2V 2 + ( m 2f 2 m 2f 0 m 2V 2 )( m 2f 2 m 2f 0 + m 2V 2 ) m 2V 2 : F or t w o lev el 1 KK particles in the nal state, ( f 2 V 1 f 1 ) = g 2 c 2 32 m 3f 2 q m 2f 2 ( m f 1 m V 1 ) 2 m 2f 2 ( m f 1 + m V 1 ) 2 (A-38) m 2f 2 + m 2f 1 m 2V 1 6 m f 2 m f 1 + ( m 2f 2 m 2f 1 m 2V 1 )( m 2f 2 m 2f 1 + m 2V 1 ) m 2V 2 :

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114 Figure A{2: Running coupling constan ts in SM (a) and UED (b) F or one SM gauge b oson and one lev el 2 KK fermion, ( f 2 W 0 f 0 2 ) = g 2 V 2 C K M 32 m 3f 2 q m 2f 2 ( m f 0 2 m V 0 ) 2 m 2f 2 ( m f 0 2 + m V 0 ) 2 (A-39) m 2f 2 + m 2f 0 2 m 2V 0 6 m f 2 m f 0 2 + ( m 2f 2 m 2f 0 2 m 2V 0 )( m 2f 2 m 2f 0 2 + m 2V 0 ) m 2V 2 ; where c = Y 2 for B n c = 1 2 for Z n c = 1 p 2 for G n and c = V C K M p 2 for W n ^ m represen ts the total mass correction ( m + m ) and m only con tains the mass corrections due to the b oundary terms. (though t ypically ^ m m ). W e can generalize those form ulas as follo ws, ( f 1 f 0 V 1 ) ( f 2 f 0 V 2 ) ( f n f 0 V n ) ( V 1 f 1 f 0 ) ( V 2 f 2 f 0 ) ( V n f n f 0 ) (A-40) ( f 2 f 1 V 1 ) ( f 2 n f n V n ) : A.4 Running Coupling Constan ts in Extra Dimensions The couplings in the Standard Mo del, 1 i ( ) = 1 l ( M Z ) b i 2 ln M Z ; (A-41)

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115 satisfy the follo wing simple dieren tial equation d 1 i dt = 1 2 b i ; (A-42) where M z is the mass of Z and i = g 2 l 4 t = ln and i = 1 ; 2 ; 3, corresp onding to the gauge group U (1) S U (2) S U (3) ( 1 = 5 3 Y and b 1 = 3 5 b Y ). F or an y gauge group, the co ecien ts are giv en b y b = 11 3 C ad j + 2 3 X f C f + 1 6 X h C h ; (A-43) where C ad j is the Dynkin index of the adjoin t represen tation of the gauge group, C f is the Dynkin index of the represen tation of the left-handed W eyl fermions and C h is that of the represen tation of the real Higgs eld. C ad j = n for S U ( n )( n > 2) and C f = 1 2 for the fundamen tal represen tation of S U (2) and C f = 3 5 Y 2 f for U (1). Applying this form ula for one Higgs doublet and n f c hiral families, w e nd b 1 = + 4 3 n f + 1 10 ; b 2 = 22 3 + 4 3 n f + 1 6 ; (A-44) b 3 = 11 + 4 3 n f : In the Standard Mo del, with n f = 3, the n umerical v alues of these co ecien ts are ( b 1 ; b 2 ; b 3 ) = ( 41 10 ; 19 6 ; 7) : (A-45) The couplings in the Univ ersal Extra Dimensions 1 i = 1 i ( M Z ) b i 2 ln M Z + ~ b i 2 ln 0 ~ b i X 2 0 1 # ; (A-46) satisfy d 1 i dt = b i ~ b i 2 ~ b i X 2 0 ; (A-47)

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116 where is the n um b er of extra dimensions and X = 2 = 2 ( = 2) The new b eta function co ecien ts ~ b i corresp ond to the con tributions of the Kaluza-Klein states at eac h massiv e KK excitation lev el. In the minimal univ ersal extra dimension, ( ~ b 1 ; ~ b 2 ; ~ b 3 ) = ( 41 10 ; 17 6 ; 13 2 ) ; (A-48) where w e used b = 11 3 C ad j + 2 3 X f C f + 1 6 X h C h + 1 6 C ad j : (A-49)

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APPENDIX B ANNIHILA TION CR OSS-SECTIONS In this section, w e summarize the annihilation cross sections of an y pair of n = 1 KK particles in to SM elds in the limit of no electro w eak symmetry breaking, as in [21]. In order to render the form ulas manageable, in this App endix w e list our results in the limit of equal KK masses. Ho w ev er, in our n umerical calculation, w e k ept dieren t masses for all KK particles, whic h often leads to enormously complicated analytical expressions. W e also assume all SM particles to b e massless, since w e are w orking in the limit where w e neglect electro w eak symmetry breaking (EWSB) eects of order v R where v is the Higgs v acuum exp ectation v alue in the SM. All cross-sections are calculated at tree lev el. All v ertices satisfy KK-n um b er conserv ation and KK-parit y since KK-n um b er violating in teractions are only induced at the lo op lev el [28]. Some of the cross-sections ha v e already app eared in [21] and w e nd p erfect agreemen t with those results. W e dene a few constan ts b elo w whic h are commonly used in our form ulas for the cross-sections. g 1 = e c w ; (B.1) g 2 = e s w ; (B.2) g Z = e 2 s w c w ; (B.3) = r 1 4 m 2 s ; (B.4) L = log 1 1 + = 2 tanh 1 : (B.5) Here g 1 g 2 and g 3 are the gauge couplings of U (1) Y S U (2) W and S U (3). is the v elo cit y of the incoming KK particle in the annihilation pro cess. Notice that L 117

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118 T able B{1: A guide to the form ulas in the App endix. Eac h b o x in the table corresp onds to a particular t yp e of an initial state. The en try p oin ts to the section in the App endix where the corresp onding annihilation cross-sections can b e found. Here \gauge b osons" include EW KK gauge b osons ( W 1 Z 1 and r 1 ) and the KK gluon ( g 1 ). \Higgses" stands for the KK Higgs ( H 1 ) and KK Goldstone b osons ( G 1 and G 1 ). Leptons con tain b oth S U (2) W -singlet KK leptons ( ` R 1 ) and S U (2) W doublet KK leptons ( ` L 1 and ` 1 ). Quarks include b oth S U (2) W -doublet KK quarks ( q R 1 ) and S U (2) W -singlet KK quarks ( q L 1 ). Gauge b osons Leptons Quarks Higgses Gauge b osons B.2 Leptons B.3 B.1 Quarks B.3 B.5 B.4 Higgses B.7 B.8 B.8 B.6 is negativ e since 0 < < 1. m is the KK mass whic h for the purp oses of this app endix is the same for all KK particles, e is the electric c harge and c w and s w are the cosine and sine of the W ein b erg angle in the SM. T able B{1 pro vides a quic k reference guide for the dieren t pro cess t yp es. B.1 Leptons Coannihilations with S U (2) W -singlet KK leptons ` R 1 are imp ortan t since they are exp ected to b e the next-to-ligh test KK particles in the minimal UED mo del [21, 28]. F or fermion nal states with f 6 = ` the cross-section is ( ` +R 1 ` R 1 f f ) = N c g 4 1 Y 2 ` ( Y 2 f L + Y 2 f R )( s + 2 m 2 ) 24 s 2 ; (B.6) where N c is 3 for quarks and 1 for leptons. F or cases with the same lepton ra v or in the initial and nal state, w e ha v e ( ` +R 1 ` R 1 ` + ` ) = g 4 1 Y 4 ` R 1 (5 s + 2(2 s + 3 m 2 ) L ) 32 2 s 2 + g 4 1 Y 4 ` R 1 ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 m 2 2 s (B.7) + g 4 1 Y 2 ` R 1 ( Y 2 ` R + Y 2 ` L )( s + 2 m 2 ) 24 s 2 ; ( ` R 1 ` R 1 ` ` ) = g 4 1 Y 4 ` ( m 2 (4 s 5 m 2 ) L s (2 s m 2 )) 32 2 s 2 m 2 ; (B.8)

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119 ( ` R 1 ` 0 R 1 ` ` 0 ) = g 4 1 Y 4 ` (4 s 3 m 2 ) 64 sm 2 ; (B.9) ( ` R 1 ` 0 R 1 ` ` 0 ) = g 4 1 Y 4 ` ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 2 sm 2 ; (B.10) where ` and ` 0 are the leptons from dieren t families. F or the remaining nal states w e get ( ` +R 1 ` R 1 ) = g 4 1 Y 2 ` Y 2 ( s + 2 m 2 ) 48 s 2 ; (B.11) ( ` +R 1 ` R 1 B 0 B 0 ) = g 4 1 Y 4 ` (2( s 2 + 4 m 2 s 8 m 4 ) s ( s + 4 m 2 )) 8 2 s 3 : (B.12) Our results, ( B.6 B.12 ), exactly agree with (C.1) (C.8) from Serv an t et al. [21]. The cross-sections among left handed fermions are somewhat complicated since they in v olv e S U (2) W gauge b osons as w ell as U (1) Y gauge b osons. F or KK neutrinos w e nd ( ` 1 ` 1 f f ) = N c g 2 Z ( g 2 L + g 2 R )( s + 2 m 2 ) 24 s 2 ; (B.13) ( ` 1 ` 1 ) = g 2 g 2 Z ( s + 2 m 2 ) 48 s 2 ; (B.14) ( ` 1 ` 1 Z Z ) = g 4 Z ((8 m 4 s 2 4 m 2 s ) L s ( s + 4 m 2 )) 8 2 s 3 ; (B.15) ( ` 1 ` 1 W + W ) = 5 g 4 2 ( s + 2 m 2 ) 96 s 2 + g 4 2 ( s 2 m 2 L ) 32 2 s 2 g 4 2 ( ( s + 4 m 2 ) + ( s + 2 m 2 ) L ) 32 2 s 2 ; (B.16) ( ` 1 ` 1 ` ` ) = g 4 Z ( s (2 s m 2 ) m 2 (4 s 5 m 2 ) L ) 32 2 s 2 m 2 ; (B.17) ( ` 1 ` 0 1 ` ` 0 ) = g 4 Z (4 s 3 m 2 ) 64 sm 2 ; (B.18) ( ` 1 ` 0 1 ` ` 0 ) = g 4 Z ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 2 sm 2 ; (B.19) ( ` 1 ` 0 1 ` ` 0 + ) = g 4 2 ( (4 s + 9 m 2 ) + 8 m 2 L ) 256 2 sm 2 ; (B.20)

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120 ( ` 1 ` 1 ` + ` ) = g Z ^ g 2 L g L (5 s + 2(2 s + 3 m 2 ) L ) 32 2 s 2 + g 2 Z ( g 2 L + g 2 R )( s + 2 m 2 ) 24 s 2 + ^ g 4 L ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 m 2 2 s : (B.21) Here g L ( R ) = e s w c w ( T 3 Q f s 2w ), ^ g L = g Z for neutrinos and ^ g L = g 2 = p 2 for c harged leptons. g = e s w c w ( T 3 Q s 2w ) with Q = 1 for the upp er en try in the Higgs doublet and Q = 0 for the lo w er en try Since w e ignore EWSB, all gauge b osons ha v e transv erse p olarizations only represen ts either a c harged Higgs b oson ( u isospin 1 2 ) or a neutral Higgs b oson ( d isospin 1 2 ). The previous results allo w us to immediately obtain ( ` 1 ` 1 ) = ( ` +L 1 ` L 1 ) ; ( ` 1 ` 1 Z Z ) = ( ` +L 1 ` L 1 Z Z + Z r + r r ) ; ( ` 1 ` 1 W + W ) = ( ` +L 1 ` L 1 W + W ) ; ( ` 1 ` 1 ` ` ) = ( ` L 1 ` L 1 ` ` ) ; (B.22) ( ` 1 ` 0 1 ` ` 0 ) = ( ` 1 ` 0L 1 ` ` 0 ) = ( ` L 1 ` 0 L 1 ` ` 0 ) ; ( ` 1 ` 0 1 ` ` 0 ) = ( ` L 1 ` 0 L 1 ` ` 0 ) : F or at least one c harged KK lepton in the initial state w e get ( ` +L 1 ` L 1 f f or ` +R ` R ) = N c g 4 ( s + 2 m 2 ) 24 s 2 ; (B.23) ( ` +L 1 ` L 1 ` ` or ` +L ` L ) = ^ g 2 L g 2 (5 s + 2(2 s + 3 m 2 ) L ) 32 2 s 2 + ^ g 4 L ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 m 2 2 s + g 4 ( s + 2 m 2 ) 24 s 2 ; (B.24) ( ` L 1 L 1 f f 0 ) = N c g 4 2 ( s + 2 m 2 ) 96 s 2 ; (B.25) ( ` L 1 L 1 u d ) = g 4 2 ( s + 2 m 2 ) 192 s 2 ; (B.26)

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121 ( ` L 1 L 1 W B 0 ) = g 2 1 g 2 2 ((8 m 4 s 2 4 m 2 s ) L s ( s + 4 m 2 )) 32 2 s 3 ; (B.27) ( ` L 1 L 1 W W 0 3 ) = 5 g 4 2 ( s + 2 m 2 ) 48 s 2 + g 4 2 ( s 2 m 2 L ) 32 s 2 2 g 4 2 ( ( s + 4 m 2 ) + ( s + 2 m 2 ) L ) 64 s 2 + g 4 2 m 2 L 16 s 2 ; (B.28) ( ` L 1 L 1 ` ` ) = g 1 g 3 2 (5 s + 2(2 s + 3 m 2 ) L ) 64 2 s 2 (2 s 2w 1) + g 2 1 g 2 2 ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 m 2 2 s (2 s 2w 1) 2 + g 4 2 ( s + 2 m 2 ) 96 s 2 ; (B.29) ( ` L 1 L 1 ` ` ) = g 1 g 3 2 ( 2 m 2 (4 s 5 m 2 ) L + m 2 s ) 64 2 s 2 m 2 (2 s 2w 1) + s (4 s 3 m 2 ) 64 2 s 2 m 2 g 2 1 g 2 2 (2 s 2w 1) 2 + g 4 2 4 ; (B.30) ( L 1 ` 0L 1 ` 0 ` ) = g 4 2 (4 s 3 m 2 ) 256 m 2 s ; (B.31) ( L 1 ` 0L 1 ` ` 0 ) = g 2 1 g 2 2 (4 s 3 m 2 ) 64 m 2 s (2 s 2w 1) 2 ; (B.32) ( L 1 ` 0L 1 ` ` 0 ) = g 2 1 g 2 2 ( (4 s + 9 m 2 ) + 8 m 2 L ) 64 m 2 s 2 (2 s 2w 1) 2 ; (B.33) ( ` L 1 ` 0L 1 ` ` 0 ) = g 4 2 ( (9 m 2 + 4 s ) + 8 m 2 L ) 256 m 2 s 2 ; (B.34) where g 2 = g 2 1 Y f Y ` R + g 2 2 T 3 f T 3 ` L The ab o v e cross-sections, ( B.13 B.34 ) are consisten t with (B.48) (B.62) and (B.71) (B.74) in [21]. F or one S U (2) W -singlet KK lepton and one S U (2) W -doublet KK lepton w e get ( ` R 1 ` L 1 `` ) = g 4 1 Y 2 e L Y 2 e R 64 m 2 s 2 8 m 2 L + (9 m 2 + 4 s ) ; (B.35) ( ` R 1 ` L 1 ` ` ) = g 4 1 Y 2 e L Y 2 e R 64 m 2 s 4 s 3 m 2 : (B.36) These t w o form ulas ha v e the same structure as ( ` R 1 ` 0 R 1 ` ` 0 ) and ( ` R 1 ` 0 R 1 ` ` 0 ).

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122 B.2 Gauge Bosons The self-annihilation cross-sections of r 1 are ( r 1 r 1 f f ) = N c g 4 1 ( Y 4 f L + Y 4 f R ) 72 s 2 2 5 s (2 m 2 + s ) L 7 s ; (B.37) ( r 1 r 1 ) = g 4 1 Y 4 12 s : (B.38) These t w o cross-sections are iden tical to (A.44) and (A.47) in [21]. F or Z 1 selfannihilation in to fermions and Higgs b osons, ( Z 1 Z 1 f f ) = N c g 4 2 1152 s 2 2 5(2 m 2 + s ) L 7 s ; (B.39) ( Z 1 Z 1 ) = g 4 2 192 s : (B.40) The cross-section for the ab o v e t w o pro cesses are obtained from ( r 1 r 1 f f ) and ( r 1 r 1 ) b y replacing g 1 Y with g 2 = 2, whic h corresp onds to the Z couplings to SM fermions and Higgs b osons. F or the coannihilations of S U (2) W KK b osons in to SM gauge b osons, w e get ( Z 1 Z 1 W + W ) = g 4 2 18 m 2 s 3 2 12 m 2 ( s 2 m 2 ) L + s (12 m 4 + 3 sm 2 + 4 s 2 ) ; (B.41) ( W + 1 W + 1 W + W + ) = g 4 2 36 m 2 s 3 2 12 m 4 ( s 2 m 2 ) L + s (12 m 4 + 3 sm 2 + 4 s 2 ) ; (B.42) ( W + 1 W 1 r r ) = e 4 36 m 2 s 3 2 12 m 4 ( s 2 m 2 ) L + s (12 m 4 + 3 sm 2 + 4 s 2 ) ; (B.43) ( W + 1 W 1 r Z ) = g 2 2 e 2 c 2w 18 m 2 s 3 2 12 m 4 ( s 2 m 2 ) L + s (12 m 4 + 3 sm 2 + 4 s 2 ) ; (B.44) ( W + 1 W 1 Z Z ) = g 4 2 c 4w 36 m 2 s 3 2 12 m 4 ( s 2 m 2 ) L + s (12 m 4 + 3 sm 2 + 4 s 2 ) : (B.45) W e see that the ab o v e v e cross-sections con tain similar expressions up to o v erall factors due to the gauge structure of S U (2) W U (1) Y F or W 1 annihilation in to

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123 other nal states, w e ha v e ( W + 1 W 1 f f ) = N c g 4 2 576 s 2 2 (12 m 2 + 5 s ) L + 2 (4 m 2 + 5 s ) ; (B.46) ( W + 1 W 1 W + W ) = g 4 2 18 m 2 s 2 2 2 m 2 (3 m 2 + 2 s ) L + (11 m 4 + 5 sm 2 + 2 s 2 ) ; (B.47) ( W + 1 W 1 ) = g 4 2 ( s m 2 ) 144 s 2 : (B.48) These three cross-sections are dieren t since they in v olv e s -c hannel Z diagrams. F or r 1 Z 1 and r 1 W 1 in to fermions w e can recycle ( r 1 r 1 f f ) and obtain ( r 1 Z 1 f f ) = N c g 2 1 g 2 2 Y 2 f 288 s 2 2 5(2 m 2 + s ) L + 7 s ; (B.49) ( r 1 W 1 f f 0 ) = N c g 2 1 g 2 2 Y 2 f 144 s 2 2 5(2 m 2 + s ) L + 7 s : (B.50) F or the annihilation of t w o dieren t KK gauge b osons in to Higgs b osons w e ha v e ( r 1 Z 1 ) = g 2 1 g 2 2 192 s ; (B.51) ( r 1 W 1 d u ) = g 2 1 g 2 2 96 s ; (B.52) ( Z 1 W 1 d u ) = g 4 2 288 s ; (B.53) whic h can b e obtained from ( r 1 r 1 ). The cross-section for Z 1 W 1 in to fermions ( Z 1 W 1 f f 0 ) = N c g 4 2 576 s 2 2 (14 m 2 + 5 s ) L + (16 m 2 + 13 s ) ; (B.54) has a dieren t structure compared to other fermion nal states due to an s-c hannel W diagram. The cross-sections for Z 1 W 1 in to gauge b oson nal states ( Z 1 W 1 Z W ) = g 4 2 c 2w 18 m 2 s 2 2 2 m 2 (3 m 2 + 2 s ) L + (11 m 4 + 5 sm 2 + 2 s 2 ) ; (B.55)

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124 ( Z 1 W 1 r W ) = e 2 g 2 2 18 m 2 s 2 2 2 m 2 (3 m 2 + 2 s ) L + (11 m 4 + 5 sm 2 + 2 s 2 ) ; (B.56) can b e obtained from ( W + 1 W 1 W + W ). F or KK gluons w e get ( g 1 g 1 g g ) = g 4 3 64 m 2 s 3 2 8 m 2 ( s 2 + 3 sm 2 3 m 4 ) L + s (34 m 2 + 13 sm 2 + 8 s 2 ) ; (B.57) ( g 1 g 1 q q ) = g 4 3 3456 s 2 2 2(20 s + 49 m 2 ) L + (72 m 2 + 83 s ) ; (B.58) for whic h there are no analogous pro cesses. The cross-sections asso ciated with one gluon and one electro w eak gauge b osons in the initial state ( g 1 r 1 q q ) = g 2 1 g 2 3 ( Y 2 q L + Y 2 q R ) 144 s 2 2 5(2 m 2 + s ) L 7 s ; (B.59) ( g 1 Z 1 q q ) = g 2 2 g 2 3 576 s 2 2 5(2 m 2 + s ) L 7 s ; (B.60) ( g 1 W 1 q q 0 ) = g 2 2 g 2 3 288 s 2 2 5(2 m 2 + s ) L 7 s ; (B.61) are obtained from ( r 1 r 1 f f ), ( r 1 Z 1 f f ) and ( r 1 W 1 f f 0 ) b y simple coupling replacemen ts and accoun ting for the additional color factors. B.3 F ermions and Gauge Bosons Note that in UED, the Dirac KK fermions are constructed out of t w o W eyl fermions with the same S U (2) W U (1) Y quan tum n um b ers while a Dirac fermion in the Standard Mo del is made up of t w o W eyl fermions of dieren t S U (2) W U (1) Y quan tum n um b ers. Therefore the couplings of KK fermions to zero mo de gauge b osons are v ector-lik e. This dierence sho ws up in pro cesses in v olving gauge-b oson couplings with fermions. F or the v ertices whic h in v olv e n = 1 gauge b osons, w e need one KK fermion and one SM fermion in order to conserv e KK n um b er. In this case, there is alw a ys a pro jection op erator asso ciated with the subscript ( L=R ) of the KK fermion. The annihilation cross-sections with S U (2) W -singlet KK fermions

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125 and S U (2) W KK gauge b osons ( Z 1 and W 1 ) are zero: ( f R 1 Z 1 S M ) = 0 ; ( f R 1 W 1 S M ) = 0 : (B.62) The cross-section for S U (2) W -singlet leptons and r 1 is ( r 1 ` R 1 B 0 ` ) = Y 4 ` g 4 1 ( 6 L ) 96 s 2 : (B.63) F or a KK quark and r 1 w e ha v e ( q 1 r 1 q g ) = g 2 1 g 2 3 Y 2 q 1 72 s 2 ( 6 L ) : (B.64) This cross-section is basically the same as ( r 1 ` R 1 B 0 ` ), up to a group factor. In ( q 1 r 1 q g ), the v ertex asso ciated with the SM gluon g con tains a Gell-Mann matrix t aij In the squared matrix elemen t, w e then get [130] 8 X a =1 1 3 3 X i;j =1 t aij t aj i = 1 3 4 3 3 = 4 3 : (B.65) Similarly for the S U (2) W -doublet quarks with Z 1 w e get ( q L 1 Z 1 g q ) = g 2 2 g 2 3 288 s 2 ( 6 L ) ; (B.66) b y replacing the g 1 coupling in ( q 1 r 1 q g ) with g 2 = 2. F or W 1 one should use the coupling g 2 = p 2 instead: ( q L 1 W 1 g q 0 ) = g 2 2 g 2 3 144 s 2 ( 6 L ) : (B.67) F or the S U (2) W -doublet KK leptons and r 1 or Z 1 w e get ( ` L 1 r 1 ` L r = Z ) = g 4 1 1536 s 2 s 2w ( 6 L ) ; (B.68) ( ` L 1 Z 1 ` L r = Z ) = g 4 2 1536 s 2 c 2w ( 6 L ) ; (B.69)

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126 ( ` L 1 r 1 ` W ) = g 2 1 g 2 2 768 s 2 ( 6 L ) : (B.70) The last 7 cross-sections ha v e a similar structure since they all ha v e s and t c hannel diagrams only F or the cross-sections asso ciated with S U (2) W -doublet KK leptons and electro w eak KK gauge b osons in to other nal states, w e ha v e ( ` L 1 Z 1 ` W ) = g 4 2 768 m 2 s 2 26 m 2 L + (23 m 2 + 32 s ) ; (B.71) ( ` L 1 W 1 ` 0 ( r + Z )) = g 4 2 768 m 2 s 2 c 2w m 2 (32 c 2w 6) L + m (24 c 2w 1) + 32 s c 2w ; (B.72) ( ` L 1 W 1 ` L W ) = g 4 2 192 m 2 s 2 3 m 2 L + 4 s ; (B.73) ( ` L 1 W + 1 ` L W + ) = g 4 2 384 m 2 s 2 16 m 2 L + (11 m 2 + 8 s ) ; (B.74) ( ` L 1 W 1 ` L W ) = ( ` 1 W + 1 ` W + ) ; (B.75) ( ` L 1 W + 1 ` L W + ) = ( ` 1 W 1 ` W ) : (B.76) F or KK gluon KK quark annihilation, w e obtain ( g 1 q 1 g q ) = g 4 3 846 m 2 s 2 24 m 2 L + (25 m 2 + 36 s ) : (B.77) B.4 Quarks The annihilation cross-section of t w o KK quarks in to SM quarks of dieren t ra v or ( q 1 q 1 q 0 q 0 ) = g 4 3 ( s + 2 m 2 ) 54 s 2 (B.78) can b e obtained from ( ` +R 1 ` R 1 f f ) b y m ultiplying with the follo wing group factor [130] 1 3 1 3 tr t a t b tr t b t a = 1 9 C ( r ) 2 ab ab = 1 9 1 4 8 = 2 9 : (B.79)

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127 Here C ( r ) = 1 = 2 is the quadratic Casimir op erator for the fundamen tal represen tation of S U (3). KK quark annihilation in to same ra v or SM quarks is giv en b y ( q 1 q 1 q q ) = g 4 3 432 m 2 s 2 2 2 m 2 (4 s 5 m 2 ) L + (6 s 5 m 2 ) s : (B.80) In this pro cess, there are three terms in the squared matrix elemen t since w e ha v e b oth t and u -c hannel diagrams. This pro cess also has an analogy with ( ` R 1 ` R 1 ` ` ). Ho w ev er, eac h term gets a dieren t group factor. The squares of the t and u -c hannel diagrams get the same factor of 2 = 9 but for the cross term w e obtain [130] 1 3 2 tr t a t b t a t b = 1 9 C 2 (2) 1 2 C 2 ( G ) tr ( t a t a ) = 1 9 4 3 3 2 4 3 = 2 9 = 2 27 : (B.81) F or q 1 q 1 annihilation in to same ra v or SM quarks w e get ( q 1 q 1 q q ) = g 4 3 864 m 2 s 2 2 4 m 2 (4 s 3 m 2 ) L + (32 m 4 + 33 sm 2 + 12 s 2 ) ; (B.82) whic h can also b e obtained using the analogy to ( ` +R 1 ` R 1 ` + ` ) and taking in to accoun t group factors. F or the nal state with gluons w e ha v e ( q 1 q 1 g g ) = g 4 3 54 s 3 2 4( m 4 + 4 sm 2 + s 2 ) L + s (31 m 2 + 7 s ) : (B.83) F or dieren t quark ra v ors in the initial state, w e ha v e ( q 1 q 0 1 q q 0 ) = g 4 3 (4 s 3 m 2 ) 288 m 2 s ; (B.84) ( q 1 q 0 1 q q 0 ) = g 4 3 288 m 2 s 2 8 m 2 L + (9 m 2 + 4 s ) : (B.85) The ab o v e t w o cross-sections can b e obtained from ( ` R 1 ` 0 R 1 ` ` 0 ) and ( ` R 1 ` 0 R 1 ` ` 0 ), corresp ondingly b y m ultiplying with the group factor 2 = 9. The

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128 remaining cross-sections are ( q 1 q 0 1 q q 0 ) = ( u R 1 d R 1 u d ) ; = ( u R 1 d L 1 ud ) ; (B.86) = ( u R 1 u L 1 uu ) ; and ( q 1 q 0 1 q q 0 ) = ( u R 1 d R 1 ud ) ; = ( u R 1 d L 1 u d ) ; (B.87) = ( u R 1 u L 1 u u ) : B.5 Quarks and Leptons The cross-sections listed b elo w are mediated b y t or u -c hannel diagrams with KK gauge b osons. F or one KK lepton and one KK quark in the initial state, w e get ( ` L 1 u L 1 `u ) = (4 g 2 1 Y e L Y u L g 2 2 ) 2 (4 s 3 m 2 ) 1024 m 2 s ; (B.88) ( 1 u L 1 u ) = (4 g 2 1 Y e L Y u L + g 2 2 ) 2 (4 s 3 m 2 ) 1024 m 2 s ; (B.89) ( ` L 1 u L 1 d ) = g 4 2 (4 s 3 m 2 ) 256 m 2 s : (B.90) These three cross-sections can b e obtained from ( ` R 1 ` 0 R 1 ` ` 0 ). F or one KK lepton and one KK an ti-quark in the initial state, w e ha v e ( 1 u L 1 ` d ) = g 4 2 256 m 2 s 2 8 m 2 L + (9 m 2 + 4 s ) ; (B.91) ( ` L 1 u L 1 ` u ) = (4 g 2 1 Y e L Y u L g 2 2 ) 2 1024 m 2 s 2 8 m 2 L + (9 m 2 + 4 s ) ; (B.92) ( 1 u L 1 u ) = (4 g 2 1 Y e L Y u L + g 2 2 ) 2 1024 m 2 s 2 8 m 2 L + (9 m 2 + 4 s ) ; (B.93) whic h can b e obtained from ( ` R 1 ` 0 R 1 ` ` 0 ). If one of the particles in the initial state is an S U (2) W -singlet fermion, only r 1 can mediate the pro cess and the

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129 cross-sections can b e obtained from our previous results: ( ` R 1 u L 1 `u ) = ( ` L 1 u R 1 `u ) = ( ` R 1 u R 1 ` u ) ; = ( ` R 1 ` 0 R 1 ` ` 0 ) = ( ` R 1 ` L 1 `` ) ; (B.94) ( ` R 1 u L 1 ` u ) = ( ` L 1 u R 1 ` u ) = ( ` R 1 u R 1 `u ) ; = ( ` R 1 ` 0R 1 `` 0 ) = ( ` R 1 e L 1 ` ` ) : (B.95) B.6 Higgs Bosons The mass terms for the KK S U (2) W -singlets app ear with the wrong sign in the fermion Lagrangian. F or example, the mass term for the KK quarks leads to the follo wing structure for the mass matrix at tree lev el ( u Ln ( x ) ; u Rn ( x )) 0B@ n R m m n R 1CA 0B@ u Ln ( x ) u Rn ( x ) 1CA : (B.96) The corresp onding mass eigenstates u 0Ln and u 0Rn ha v e mass M n = r n R 2 + m 2 : (B.97) These mass eigenstates receiv e dieren t radiativ e corrections whic h lift the degeneracy [28]. The in teraction eigenstates are related to the mass eigenstates b y 0B@ u Ln u Rn 1CA = 0B@ cos r 5 sin sin r 5 cos 1CA 0B@ u 0Ln u 0Rn 1CA ; (B.98) where is the mixing angle b et w een S U (2) W -singlet and S U (2) W -doublet fermions dened b y tan 2 = m n=R : (B.99) This mixing is v ery small except for the top quark. Ho w ev er, ev en with 0 the eect of the rotation ( B.98 ) is presen t in the Y uk a w a couplings through the

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130 redenition u Rn r 5 u Rn It do es not aect the gauge-fermion couplings. W e use the follo wing notation for KK Higgs b osons, 0B@ G +1 H 1 + iG 1 p 2 1CA : (B.100) W e k eep only the top-Y uk a w a coupling and w e also k eep the Higgs self-coupling assuming m h = 120 GeV. Belo w w e list the cross-sections asso ciated with t w o KK Higgs b osons in the initial state. ( H 1 H 1 G + G ) = 1 64 m 2 S 2 s 4w 8 e 2 m 2 2 s 2w L + f (2 s + m 2 ) e 4 +4 m 2 s 2w e 2 + 4 2 m 2 s 4w g ; (B.101) ( H 1 H 1 H H ) = 9 2 32 s ; (B.102) ( H 1 H 1 GG ) = 1 128 m 2 s 2 s 4w c 4w 8 e 2 m 2 s 2w c 2w L + f (2 s + m 2 ) e 4 +4 m 2 e 2 s 2w c 2w + 4 2 m 2 s 4w c 4w g ; (B.103) ( H 1 H 1 Z Z ) = g 4 2 64 s 3 2 c 4w s ( s + 4 m 2 ) + 4 m 2 ( s 2 m 2 ) L ; (B.104) ( H 1 H 1 W + W ) = g 4 2 32 s 3 2 c 4w s ( s + 4 m 2 ) + 4 m 2 ( s 2 m 2 ) L ; (B.105) ( H 1 H 1 t t ) = 3 y 4 t 16 s 2 2 ( s + 2 m 2 ) L 2 s ; (B.106) ( G +1 G +1 G + G + ) = 1 128 m 2 s 2 s 4w c 4w 16 e 2 m 2 s 2w c 2w L + f (2 s + m 2 ) e 4 8 m 2 e 2 s 2w c 2w + 16 2 m 2 s 4w c 4w g ; (B.107) ( G +1 G 1 t t ) = 1 1152 s 2 2 s 4w c 4w 72 s 4w c 2w y 2 t ( 3 sc 2w y 2 t 4 m 2 e 2 ) L 432 s s 4w c 4w y 4 t + s 3 (20 s 4w 12 s 2w + 9) e 4 144 s s 4w c 2w y 2 t e 2 ; (B.108)

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131 ( G +1 G 1 b b ) = 1 1152 s 2 2 s 4w c 4w 72 s 2w c 2w y 2 t ( 3 ss 2w c 2w y 2 t + e 2 m 2 (4 s 2w 3)) L 432 s s 4w c 4w y 4 t + s 3 (20 s 4w 24 s 2w + 9) e 4 (B.109) 36 s s 2w y 2 t e 2 (4 s 4w 7 s 2w + 3) ; ( G +1 G 1 G + G ) = 1 192 m 2 s 2 2 s 4w c 4w 6 m 2 e 2 s ( e 2 + 4 s 2w c 2w ) L 48 2 m 2 ss 4w c 4w + f e 4 ( m 4 7 sm 2 3 s 2 ) 12 m 2 ss 2w c 2w e 2 g ; (B.110) ( G +1 G 1 GH ) = 1 128 m 2 s 2 s 4w 8 s 2w m 2 e 2 L + f (2 s + m 2 ) e 4 +4 s 2w m 2 e 2 + 4 2 m 2 s 4w g ; (B.111) ( G +1 G 1 GH ) = g 4 Z 48 m 2 s 2 2 24 m 2 c 2w sL f 4(1 2 s 2w ) 2 m 4 + s (92 s 4w 140 s 2w + 47) 24 s 2 c 4w g ; (B.112) ( G +1 G 1 Z Z ) = g 4 Z (1 2 s 2w ) 4 4 s 3 2 4 m 2 ( s 2 m 2 ) L + s ( s + 4 m 2 ) ; (B.113) ( G +1 G 1 r Z ) = e 2 g 2 Z (1 2 s 2w ) 2 2 s 3 2 4 m 2 ( s 2 m 2 ) L + s ( s + 4 m 2 ) ; (B.114) ( G +1 G 1 r r ) = e 4 4 s 3 2 4 m 2 ( s 2 m 2 ) L + s ( s + 4 m 2 ) ; (B.115) ( G +1 G 1 W + W ) = g 4 2 24 s 2 2 6 m 2 L + ( s + 11 m 2 ) ; (B.116) ( G +1 G 1 f f ) = N c g 4 Z 24 s ^ g 2 L + ^ g 2 R ; (B.117) where f represen ts leptons and rst t w o generations of quarks and ^ g 2 L ( R ) = e 2 Q f 2 g 2 Z (1 2 s 2w )( T 3 f Q f s 2w ) :

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132 A n um b er of cross-sections can b e simply related: ( G 1 G 1 G + G ) = ( H 1 H 1 G + G ) ; ( G 1 G 1 GG ) = ( H 1 H 1 H H ) ; ( G 1 G 1 H H ) = ( H 1 H 1 GG ) ; ( G 1 G 1 Z Z ) = ( H 1 H 1 Z Z ) ; ( G 1 G 1 W + W ) = ( H 1 H 1 W + W ) ; (B.118) ( G 1 G 1 t t ) = ( H 1 H 1 t t ) ; ( G +1 G 1 GG ) = ( G +1 G 1 H H ) ; ( G +1 G 1 H H ) = ( G +1 G 1 GG ) ; = 1 2 ( H 1 H 1 G + G ) : The rest of the cross-sections are ( H 1 G 1 H G ) = 1 192 m 2 s 2 2 s 4w c 4w 6 m 2 e 2 s ( e 2 2 s 2w c 2w ) L f ( m 4 7 sm 2 3 s 2 ) e 4 (B.119) +6 m 2 ss 2w c 2w e 4 12 2 m 2 ss 4w c 4w g ; ( H 1 G 1 G + G ) = g 4 Z 48 m 2 s 2 2 24 m 2 c 2w sL + f 4(1 2 s 2w ) 2 m 4 + sm 2 ( 92 s 4w + 140 s 2w 47) + 24 s 2 c 4w g ; (B.120) ( H 1 G 1 W + W ) = g 4 2 96 s 3 2 12 m 2 (2 m 2 + s ) L + s (32 m 2 + s ) ; (B.121) ( H 1 G 1 t t ) = 1 288 s 2 2 c 2w s 2w 54 y 2 t ( e 2 m 2 + c 2w s 2w ( s 2 m 2 ) y 2 t ) L + f 54 sy 4 t c 2w s 2w 27 e 2 y 2 t s + e 2 g 2 Z s 2 (32 s 4w 24 s 2w + 9) g ; (B.122) ( H 1 G 1 f f ) = N c g 2 Z 24 s g 2 L + g 2 R ; (B.123)

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133 ( H 1 G +1 H G + ) = 1 192 m 2 s 2 2 s 4w 6 m 2 e 2 s ( e 2 2 s 2w ) L 12 2 m 2 ss 4w f ( m 4 7 sm 2 3 s 2 ) e 4 + 6 m 2 ss 2w e 4 g ; (B.124) ( H 1 G +1 GG + ) = g 4 Z 48 m 2 s 2 2 12 m 2 (1 2 s 2w )(2 3 2w ) L f (4 m 4 + 47 sm 2 24 s 2 ) c 4w (B.125) 24( m 2 s ) ss 2w c 2w 3 s ( m 2 + 4 s ) s 4w g ; ( H 1 G +1 Z W + ) = g 4 Z 6 s 3 2 6 m 2 L f (1 2 s 2w )(4 m 2 2 ss 2w + s ) + s g + s f 4 s 4w ( s + 11 m 2 ) + (1 2 s 2w )( s + 32 m 2 ) g ; (B.126) ( H 1 G +1 r W + ) = e 2 g 2 2 24 s 2 2 6 m 2 L + (11 m 2 + s ) ; (B.127) ( H 1 G +1 t b ) = 1 64 s 2 2 s 4w 6 s 2w (2 e 2 m 2 + ss 2w y 2 t ) y 2 t L + f s 2 e 4 6 ss 4w y 2 t e 2 12 ss 4w y 4 t g ; (B.128) ( H 1 G +1 f f 0 ) = N c g 4 2 192 s ; (B.129) ( G 1 G +1 GG + ) = ( H 1 G +1 H G + ) ; ( G 1 G +1 H G + ) = ( H 1 G +1 GG + ) ; ( G 1 G +1 Z W + ) = ( H 1 G +1 Z W + ) ; (B.130) ( G 1 G +1 r W + ) = ( H 1 G +1 r W + ) ; ( G 1 G +1 f f 0 ) = ( H 1 G +1 f f 0 ) : B.7 Higgs Bosons and Gauge Bosons The cross-sections in v olving one KK Higgs b oson and one KK gauge b oson are giv en b elo w. They are rather simple compared to the cross-sections from

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134 Section B.6 ( H 1 g 1 t t ) = g 2 3 y 2 t 48 m 2 s 2 2 2 m 2 ( s m 2 ) L + (2 s 5 m 2 ) s ; (B.131) ( H 1 Z 1 Z H ) = g 4 2 96 c 2w s 2 ( L + 4 ) ; (B.132) ( H 1 Z 1 W G + ) = g 4 2 96 m 2 s 2 2 4 m 2 L + s (4 s + m 2 ) ; (B.133) ( H 1 Z 1 t t ) = g 2 2 y 2 t 64 m 2 s 2 2 m 2 L + (4 s 11 m 2 ) ; (B.134) ( G +1 Z 1 Z G + ) = g 4 2 (1 2 s 2w ) 2 96 c 2w s 2 ( L + 4 ) ; (B.135) ( G +1 Z 1 r G + ) = e 2 g 2 2 24 s 2 ( L + 4 ) ; (B.136) ( G +1 Z 1 t b ) = g 2 2 y 2 t 64 m 2 s 2 2 m 2 L + (4 s 11 m 2 ) ; (B.137) ( H 1 r 1 Z H ) = g 2 1 g 2 2 96 c 2w s 2 ( L + 4 ) ; (B.138) ( G 1 r 1 W G + ) = g 2 1 g 2 2 96 s 2 ( L + 4 ) ; (B.139) ( H 1 r 1 t t ) = g 2 1 y 2 t 576 m 2 c 2w s 2 2 2 m 2 (7 s + 8 m 2 ) L + (4 s 43 m 2 ) s ; (B.140) ( G +1 r 1 Z G + ) = g 2 1 g 2 2 (1 2 s 2w ) 2 96 c 2w s 2 ( L + 4 ) ; (B.141) ( G +1 r 1 r G + ) = e 2 g 2 1 24 s 2 ( L + 4 ) ; (B.142) ( G +1 W + 1 G + W + ) = g 4 2 96 m 2 s 2 12 m 2 L + (6 m 2 + 5 s ) ; (B.143) ( H 1 W + 1 G + Z ) = g 4 2 96 m 2 s 2 c 2w m 2 (2 s 2w 2 s 4w ) L f m 2 (4 s 4w s 2w + 1) + (3 s 4w 7 s 2w + 4) g ; (B.144) ( H 1 W + 1 G + r ) = g 2 1 e 2 96 m 2 s 2 2 m 2 L + (4 m 2 + 3 s ) ; (B.145) ( H 1 W + 1 H W + ) = g 4 2 96 s 2 ( L + 4 ) ; (B.146)

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135 ( G 1 W + 1 W + G ) = g 4 2 96 m 2 s 2 2 m 2 L + (4 m 2 + 3 s ) ; (B.147) ( G 1 W + 1 Z G ) = g 4 2 96 m 2 s 2 c 2w m 2 (12 s 4w 15 s 2w + 4) L + f (6 s 4w 3 s 2w + 1) m 2 + s (5 s 4w 9 s 2w + 4) g ; (B.148) ( G 1 W + 1 r G ) = g 2 2 e 2 96 m 2 s 2 12 m 2 L + (6 m 2 + 5 s ) ; (B.149) ( G 1 W + 1 t t ) = g 2 2 64 m 2 s 2 4 m 2 y 2 t L + f e 2 m 2 + 2(4 s 11 m 2 ) y 2 t g ; (B.150) where y t is the top quark Y uk a w a coupling. The cross-sections listed b elo w are obtained from our previous calculations. F or one KK Higgs b oson and one KK gluon w e get ( H 1 g 1 t t ) = ( G 1 g 1 t t ) ; = ( G +1 g 1 t b ) : (B.151) F or one KK Higgs b oson and one Z 1 w e ha v e ( H 1 Z 1 Z H ) = ( G 1 Z 1 Z H ) ; ( H 1 Z 1 W G + ) = ( G 1 Z 1 W G + ) ; = ( G +1 Z 1 W + G ) ; = ( G +1 Z 1 W + H ) ; = ( W + 1 H 1 W + G ) ; (B.152) = ( W + 1 H 1 W + H ) ; ( H 1 Z 1 t t ) = ( G 1 Z 1 t t ) ; ( G +1 Z 1 t b ) = ( H 1 W + 1 t b ) ; = ( G 1 W + 1 t b ) :

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136 F or one KK Higgs b oson and one r 1 w e obtain ( H 1 r 1 Z H ) = ( G 1 r 1 Z G ) ; ( G 1 r 1 W G + ) = ( H 1 r 1 W G + ) ; = ( G +1 r 1 W + G ) ; = ( G +1 r 1 W + H ) ; (B.153) ( H 1 r 1 t t ) = ( G 1 r 1 t t ) ; = ( G +1 r 1 t b ) : F or one KK Higgs b oson and one W 1 w e get ( H 1 W + 1 G + Z ) = ( G 1 W + 1 G + Z ) ; ( H 1 W + 1 G + r ) = ( G 1 W + 1 G + r ) ; ( H 1 W + 1 H W + ) = ( G 1 W + 1 GW + ) ; (B.154) ( G 1 W + 1 Z G ) = ( G 1 W + 1 Z H ) ; ( G 1 W + 1 r G ) = ( G 1 W + 1 r H ) : B.8 Higgs Bosons and F ermions F or the cross-sections b et w een one KK Higgs b oson and one KK S U (2) W singlet fermion, w e ha v e ( H 1 f R 1 f G ) = g 4 1 Y 2 f 32 m 2 s 2 m 2 L + s ; (B.155) ( H 1 t R 1 g t ) = g 2 3 y 2 t 48 s 2 (2 L + 3 ) ; (B.156) ( G +1 t R 1 tG + ) = 1 288 m 2 s 2 2 c 2w 12 c 2w e 2 m 2 y 2 t + m 2 L f +12 c 2w ( m 2 s ) y 2 t e 2 + 9 c 4w ( m 2 s ) y 2 t + 4 se 4 g + s f 4 se 4 9 c 4w m 2 y 4 t g ; (B.157)

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137 ( H 1 f R 1 f G ) = ( G 1 f R 1 f H ) ; = ( G +1 f R 1 f G + ) ; = ( G 1 f R 1 f G ) ; ( H 1 t R 1 g t ) = ( H 1 t L 1 g t ) ; (B.158) = ( G 1 t R 1 g t ) ; = ( G 1 t R 1 g b ) ; = 1 2 ( G +1 b L 1 g t ) : F or the cross-sections b et w een one KK Higgs b oson and one KK S U (2) W -doublet fermion, w e get ( H 1 f L 1 Gf ) = e 2 ( T 3 f c w g 2 2 g 1 s w Y f ) 2 128 m 2 s 2w c 2w s 2 m 2 L + s ; (B.159) ( H 1 f + 1 f G + ) = g 4 2 64 m 2 s 2 m 2 L + s ; (B.160) ( G +1 t L 1 tG + ) = e 2 ( c w g 2 2 g 1 s w Y f ) 2 128 m 2 s 2w c 2w s 2 m 2 L + s + y 4 t 32 s 2 2 m 2 L + s ; (B.161) ( G +1 t L 1 tW + ) = g 2 2 y 2 t L 32 s 2 ; (B.162) ( G 1 b L 1 bG ) = e 2 ( c w g 2 2 g 1 s w Y f ) 2 128 m 2 s 2w c 2w s 2 m 2 L + s + 1 32 c 2w s 2w s 2 2 y 2 t L f se 2 ( c 2w 2 s 2w Y b ) (B.163) + c 2w s 2w ( s m 2 ) y 2 t s 2w c 2w s 2 y 2 t g ;

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138 where T 3 f denotes the fermion isospin. ( H 1 b L 1 Gf ) = 1 2 ( G +1 t L 1 tW + ) ; ( H 1 f L 1 Gf ) = ( G 1 f L 1 H f ) ; = ( G 1 f L 1 G f ) ; ( H 1 f + 1 f G + ) = ( H 1 f 1 f + G ) ; = ( G 1 f + 1 f G + ) ; (B.164) = ( G 1 f 1 f + G ) ; = ( G +1 f 1 f + G ) ; = ( G 1 f + 1 f G ) ; where f stands for an y lepton or quark, except t L 1 and b L 1 and f + ( f ) denotes isospin +1 = 2 (isospin 1 = 2) fermions.

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PAGE 163

BIOGRAPHICAL SKETCH Born in 1974 in Pusan, South Korea, I receiv ed a B.S. degree in 1997 and an M.S. degree in 1999 from Pusan National Univ ersit y I receiv ed a Ph.D. from the Univ ersit y of Florida in 2006. I ha v e receiv ed Korean Graduate Studen t Researc h Aw ard (supp orted b y Univ ersit y of Florida alumni and the New Y ork Times) in 2005, Outstanding In ternational Studen t Aw ard in 2005, Presiden tial Recognition in 2004 in recognition of outstanding ac hiev emen ts and con tribution to the Univ ersit y of Florida, Ph ysics Departmen t T eac hing Assistan t of the Y ear in 2001 and 5 certicates of ac hiev emen t for outstanding academic accomplishmen ts ev ery y ear from 2001 to 2006. I started w orking with Prof. Konstan tin Matc hev in 2002 and am in terested in particle and astroparticle phenomenology and esp ecially in signatures of new ph ysics b ey ond the Standard Mo del. My researc h so far has concen trated on the follo wing areas: collider phenomenology of new ph ysics at the LHC and ILC, phenomenology of dark matter and dev elopmen t of ev en t generators and implemen tation of new mo dels. 149


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PHENOMENOLOGY OF UNIVERSAL EXTRA DIMENSIONS


By
KYOUNGCHUL KONG

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Kyoungchul Kong
















To my family















ACKNOWLEDGMENTS

First and foremost, I am deeply indebted to my advisor, Prof. Konstantin

Matchev, for his time, patience, encouragement, much stimulating advice and

financial support for the research and many academic travels.

My collaborator, Prof. Asesh K. Datta, deserves special thanks for useful

discussions, and his honest advice in lli, -i. and life. I want to thank Dr. Andreas

Birkedal, Dr. Lisa Everett and Dr. Hye-Sung Lee for their useful discussions,

important comments and -ii.-.-i i. ii, on my talks and researches.

I would like to thank Prof. Pierre Ramond for constant support, encourage-

ment and teaching necessary particle 1.li-,-i. courses, and Prof. Richard Woodard

for being a constant source of inspiration and sharing his enthusiasm for li, -i,

with graduate students. I also want to thank Prof. Andrey Korytov, Prof. Rick

Field and Prof. David Groisser for reading my thesis and questions. In my home-

town university, I want to thank Prof. Chang Gil Han, Prof. Deog Ki Hong and

Prof. Hyun-Chul Kim for teaching and their support when I applied for graduate

program in US.

I am grateful for Prof. Michael Peskin, Prof. Jonathan Feng, Prof. Hsin-

Chia Cheng, Prof. Bogdan Dobrescu and Prof. Tim Tait who gave me useful

comments and advice, among many other ]1li,-i. i-1i that I have met at workshops

and seminar visits.

Many thanks should go to Bobby Scurlock and Craig Group for getting me

closer to experiments and computer languages, Karthik Shankar and Sudarshan

Ananth for comments and gossip, and my officemate, Sung-Soo Kim and Taku

Watanabe, for stimulating me with their diligence and comments.









I can not forget my old friends Dr. Suckjoon Jun in AMOLF and Byoung-Chul

Kim in BNL, Junghan Lee in Mainz and Dr. Byoung-Ik Hur in cancer treatment

center and Seung-Hwa Sheen in finance.

I feel a deep sense of gratitude for my parents and brothers who believed in

me although they do not understand what I am doing. But I will never be able to

thank my wife enough, for all her love, support, and friendship and my little son,

Casey, for his genuine smiles that make me the happiest man in the world. I have

been really blessed to have her and Casey beside me.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................ ........ iv

LIST OF TABLES ........ .... .................. viii

LIST OF FIGURES ........ ....................... ix

KEY TO ABBREVIATIONS ................... ....... xii

A BSTR ACT . . . . . . . .. xiii

CHAPTER

1 INTRODUCTION .......... ................. 1

2 UNIVERSAL EXTRA DIMENSIONS ........ ... ........ 9

2.1 Massive Scalar Field in Five Dimensions ............... 9
2.2 Universal Extra Dimensions .................. ... 10
2.3 Comparison between UED and Supersymmetry ........... 18

3 COLLIDER PHENOMENOLOGY ......... ............. 22

3.1 Search for Level 2 KK Particles at the LHC ...... ..... 22
3.1.1 Phenomenology of Level 2 Fermions ....... ..... 23
3.1.2 Phenomenology of Level 2 Gauge Bosons ........... 29
3.1.3 Analysis of the LHC Reach for Z2 and 72 . .... .. 37
3.2 Spin Determination at the LHC .............. .. .. 42
3.2.1 Dilepton Invariant Mass Distributions . . ... 44
3.2.2 Lepton-Jet Invariant Mass Charge Asymmetry ...... ..47
3.3 UED and SUSY at Linear Colliders ..... . . 53
3.3.1 Event Simulation and Data Analysis . . .. 54
3.3.2 Comparison of UED and Supersymmetry in p+t-fTT .. 58
3.3.2.1 Angular Distributions and Spin Measurements 58
3.3.2.2 Threshold Scans .... . . 60
3.3.2.3 Production Cross-Section Determination . 61
3.3.2.4 Muon Energy Spectrum and Mass Measurements 62
3.3.2.5 Photon Energy Spectrum and Radiative Return to
the Z2 ......................... 63
3.3.3 Prospects for Discovery and Discrimination in Other Final
States .............. .... .... ..... 65









3.3.3.1 Kaluza-Klein Leptons . . ..... 65
3.3.3.2 Kaluza-Klein Quarks . . ..... 67
3.3.3.3 Kaluza-Klein Gauge Bosons . . .. 69

4 CO\ IOLOGICAL IMPLICATIONS .............. .. .. 73

4.1 Dark Matter Abundance .................. .... .. 73
4.2 The Basic Calculation of the Relic Density . . 76
4.2.1 The Standard Case ................ .... .. 76
4.2.2 The Case with Coannihilations . . ...... 78
4.2.3 Thermal Average and Nonrelativistic Velocity Expansion 79
4.3 Relic Density in Minimal UED ................... ... 81
4.4 Relative Weight of Different Coannihilation Processes . ... 88
4.4.1 Coannihilations with KK Leptons . . ..... 88
4.4.2 Coannihilations with KK Quarks and KK Gluons . 94
4.4.3 Coannihilations with Electroweak KK Bosons . ... 95
4.5 Other Dark Matter Candidates and Direct Detection . ... 96

5 CONCLUSIONS .................. ............ .. 101

APPENDIX ................... ......... ...... 103

A STANDARD MODEL IN 5D .................. .... .. 103

A.1 Lagrangian of the Standard Model in 5D . . 103
A.2 The Kaluza-Klein Fermions and Gauge bosons . . ... 109
A.3 The Decay Widths of KK Particles ................. .. 111
A.4 Running Coupling Constants in Extra Dimensions . ... 114

B ANNIHILATION CROSS-SECTIONS .................. .. 117

B.I Leptons .................. .............. .. 118
B.2 Gauge Bosons .................. ........... .. 122
B.3 Fermions and Gauge Bosons ................. .. .. 124
B.4 Quarks ................... ... .. ....... 126
B.5 Quarks and Leptons .................. ..... .. 128
B.6 Higgs Bosons .................. ........... .. 129
B.7 Higgs Bosons and Gauge Bosons .............. .. 133
B.8 Higgs Bosons and Fermions ................... .. 136

REFERENCES ...... ....... ................. 139

BIOGRAPHICAL SKETCH ............. . . .. 149















LIST OF TABLES
Table page

3-1 Masses of the KK excitations for R-1 500 GeV and AR 20 . 56

3-2 MSSM parameters for a SUSY study point . . . 58

A-1 Fermion content of the Standard Model and the corresponding Kaluza-
Klein fermions ................... ............ .. 110

A-2 Quantum numbers of KK fermions ................ ... 110

A-3 Fermions and gauge bosons in the Standard Model . ..... 110

B-1 A guide to the formulas in the Appendix B ................ ..118















LIST OF FIGURES
Figure page

1-1 de Broglie's particle-wave dii.,iliv ................. . 4

1-2 An illustration of bulk and brane .................. 5

1-3 S1/Z2 orbifold . . . . . . .... 6

2-1 KK states after a compactification on the orbifold . . ..... 12

2-2 KK number conservation and KK parity ................. .. 14

2-3 The spectrum of the first KK level at (a) tree level and (b) one-loop 16

2-4 Qualitative sketch of the level 1 KK spectroscopy . . ..... 17

2-5 A discovery reach for MUEDs at the Tevatron (blue) and the LHC (red)
in the 4 + ST channel ............. . ... .18

3-1 One loop corrected mass spectrum of the n = 1 and n = 2 KK levels .23

3-2 Cross-sections of n = 2 KK particles at the LHC . . ..... 24

3-3 Branching fractions of the level 2 "up" quarks versus R-1 . ... 26

3-4 Branching fractions of the level 2 KK electrons versus R-1 . ... ..28

3-5 Masses and widths of level 2 KK gauge bosons . . 30

3-6 Cross-sections for single production of level 2 KK gauge bosons . 33

3-7 Branching fractions of the n = 2 KK gauge bosons versus R-1 ...... .35

3-8 5a Discovery reach for (a) 72 and (b) Z2 ... . . 38

3-9 The 72 Z2 diresonance structure in UED with R-1 500 GeV . 40

3-10 Twin diagrams in SUSY and UED .................. .. 43

3-11 Comparison of dilepton invariant mass distributions . . .... 44

3-12 A closer look into dilepton invariant mass distributions . ... 46

3-13 Jet-lepton invariant mass distributions ... . .. 50

3-14 Asymmetries for UED and SUSY .................. ..... 51









3-15 Asymmetries with relaxed conditions ................ 53

3-16 The dominant Feynman diagrams for KK muon production . ... 55

3-17 The dominant Feynman diagrams for smuon production . ... 55

3-18 Differential cross-section da-/dcos O, ..................... 59

3-19 The total cross-section a in pb as a function of the center-of-mass en-
ergy s near threshold .................. ......... .. 60

3-20 The muon energy spectrum resulting from KK muon production (3-14)
in UED (blue, top curve) and smuon production (3-15) in supersymme-
try (red, bottom curve) .................. ......... 62

3-21 Photon energy spectrum in e+e- p-- + .. .. . . .64

3-22 The dominant Feynman diagrams for KK electron production ...... ..66

3-23 ISR-corrected production cross-sections of level 1 KK leptons . ... 67

3-24 Differential cross-section dar/dcos 0 for UED and supersymmetry . 68

3-25 ISR-corrected production cross-sections of level 1 KK quarks ...... ..69

3-26 ISR-corrected production cross-sections of level 1 KK gauge bosons 71

3-27 ISR-corrected production cross-sections of level 2 KK gauge bosons 72

4-1 The a-term of the annihilation cross-sections for (a) 717 -- e+e- and
(b) 7l.i 9 ..... ........ .............. 84

4-2 The number of effectively massless degrees of freedom and freeze-out tem-
perature . . . . . . . ... .. 86

4-3 Relic density of the LKP as a function of R-1 in the minimal UED model 87

4-4 Coannihilation effects of (a) 1 generation or (b) 3 generations of singlet
KK leptons .................. .. .......... ..89

4-5 Plots of various quantities entering the LKP relic density computation .91

4-6 The effects of varying the SU(2)w-doublet KK electron mass ...... ..93

4-7 The effects of varying KK quarks masses ..... . . 94

4-8 The effects of varying KK gluon mass ................ . 95

4-9 The effects of varying EW bosons ................ ...... 96

4-10 The change in the cosmologically preferred value for R-1 as a result of
varying the different KK masses away from their nominal MUED values 98









4-11 The spin-independent direct detection limit from CDMS experiment 99

A-1 Dependence of the \\V inii i," angle ~, for the first few KK levels (n
1, 2,.. ,5) on R-1 for fixed AR 20 ........ ... ..... 112

A-2 Running coupling constants in '1\ (a) and UED (b) . . .... 114















KEY TO ABBREVIATIONS


CDMS: Cryogenic Dark Matter Search

CLIC: Compact Linear Collider

EW: Electroweak

EWSB: Electroweak Symmetry Breaking

ISR: Initial State Radiation

KK: Kaluza-Klein

LHC: Large Hadron Collider

LKP: Lightest KK particle

LSP: Lightest Supersymmetric particle

MSSM: Minimal Supersymmetric Standard Model

mSugra: Minimal Supergravity

MUED: Minimal Universal Extra Dimensions

I\!: Standard Model

SPS: Snowmass Points and Slopes: Benchmarks for SUSY searches

SUSY: Supersymmetry

UED: Universal Extra Dimensions

WIMP: Weakly Interacting Massive Particle

WMAP: Wilkinson Microwave Anisotropy Probe















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

PHENOMENOLOGY OF UNIVERSAL EXTRA DIMENSIONS

By

Kyoungchul Kong

August 2006

Chair: Konstantin T. Matchev
M.,. ir Department: Physics

A major motivation for studying new pir, -i. beyond the Standard Model is

the dark matter puzzle which finds no explanation within the Standard Model.

Models with extra dimensions may naturally provide possible dark matter can-

didates if the theory is compactified at the TeV scale. In this dissertation, the

phenomenology of Universal Extra Dimensions (UED), in which all the Standard

Model fields propagate, is explored. We focus on models with one universal extra

dimension, compactified on an S1/Z2 orbifold. We investigate the collider reaches

for new particles and the cosmological implications of this model.

Models with Universal Extra Dimensions may provide excellent counter exam-

ples for typical supersymmetric theories with dark matter candidates. Therefore we

contrast the experimental signatures of low energy supersymmetry and models with

Universal Extra Dimensions and discuss various methods for their discrimination

at colliders. We first study the discovery reach of the Tevatron and the LHC for

level 2 Kaluza-Klein modes, which would indicate the presence of extra dimensions,

since such particles are guaranteed by extra dimensions but not supersymmetry.

We also investigate the possibility to differentiate the spins of the superpartners









and KK modes by means of a dilepton mass method and the ..i mmetry method

in the squark cascade decay to electroweak (EW) particles. We then study the

processes of Kaluza-Klein muon pair production in universal extra dimensions in

parallel to smuon pair production in supersymmetry at a linear collider. We find

that the angular distributions of the final state muons, the energy spectrum of

the radiative return photon and the total cross-section measurement are powerful

discriminators between the two models. We also calculate the production rates of

various Kaluza-Klein particles and discuss the associated signatures.

A prediction of the models with Universal Extra Dimensions with conserved

KK-parity is the existence of dark matter. We calculate the relic density of the

lightest Kaluza-Klein particle. We include coannihilation processes with all level

one KK particles. In our computation we consider a most general KK particle

spectrum, without any simplifying assumptions. We first calculate the Kaluza-

Klein relic density in the minimal UED model, turning on coannihilations with all

level one KK particles. We then go beyond the minimal model and discuss the size

of the coannihilation effects separately for each class of level 1 KK particles. Our

results provide the basis for consistent relic density computations in arbitrarily

general models with Universal Extra Dimensions.

All these studies not only bring us to deeper understanding of new possibilities

beyond the Standard Model but also provide strong phenomenological backgrounds

and tools to identify the nature of new 1]1l',-i -














CHAPTER 1
INTRODUCTION

The Standard Model of particle 1li, -i. is a theory which describes the

-1 I.. -. weak, and electromagnetic fundamental forces. This theory has been

astonishingly successful in explaining much of the presently available experimental

data. However, the Standard Model still leaves open a number of outstanding

fundamental questions whose answers are expected to emerge in a more general

theoretical framework.

One of the major motivations for pursuing new p1 i-i. beyond the Standard

Model is the dark matter problem which finds no explanation within the Standard

Model. From the accumulated -II ,i1li, --i. .i1 data, we now know that ordinary

matter comprises only about 4% (QB) of the Universe. The remaining 'I,.' are

divided between a inll,--I ii.us form of matter called "dark matter" (22.';, QCDM)

and an even more perplexing entity called "dark E,, ,y;" (7,!', QA). From the

inflationary big bang model,


1 = = A + CDM B (1 1)


is expected where QB is the fractional energy density in '.., ',-. c-.CDM the

fractional energy density in dark matter, and QA the fractional energy density in

dark energy. (The precise measured values are CDM = 0.22o+O., QA = 0.74 0.02,

B = 0.044 +0.02 and =- 1.02 0.02 [1].)

The microscopic nature of the dark matter is at present unknown. Perhaps the

most attractive explanation is provided by the WIMP (weakly interacting massive

particle) hypothesis: dark matter is assumed to consist of hypothetical stable

particles with masses around the scale of electroweak symmetry breaking, whose









interactions with other elementary particles are of the strength and range similar to

the familiar weak interactions of the Standard Model. Such WIMPs naturally have

a relic abundance of the correct order of magnitude to account for the observed

dark matter, making them appealing from a theoretical point of view. The relic

density, QWIMP, of the WIMP dark matter is roughly estimated by

QWIAP )
IWIMP ( t )2 1?7i7 2 (1 2)
102a ) I2 TeV
(1), (1-3)

where mnwiMp is the mass of the WIMP dark matter candidate and the magnitude

of electroweak interaction, a is expected to be of order 0.01. Therefore the relic

density of WIMP dark matter is expected to be of order 1 if the mass scale is

0(1) TeV. The precise relic density including the correct coefficients in the above

equation needs to be calculated using the Boltzmann equation and the result

depends on the particular model. The above estimation tells us that the WIMP

hypothesis can naturally explain all or part of the dark matter. Moreover, many

extensions of the Standard Model contain particles which can be identified as

WIMP dark matter candidates. Examples include supersymmetric models, models

with Universal Extra Dimensions, little Higgs theories, etc.

An excellent candidate for such thermal WIMP arises in the R-parity conserv-

ing supersymmetric theories. New particles, called superpartners, predicted by the

supersymmetry are charged under this R-parity, while the Standard Model particles

are neutral under the symmetry. So the lightest supersymmetric particle (LSP) is

stable and can be a dark matter candidate. The supersymmetric models have other

side benefits:

1. R-parity also implies that superpartners interact only pairwise with <\!

particles, which guarantees that the supersymmetric contributions to low

energy precision data only appear at the loop level and are small.









2. If the superpartners are indeed within the TeV range, the problematic

quadratic divergences in the radiative corrections to the Higgs mass are

absent, being canceled by loops with superpartners. The cancellations are

enforced by the symmetry, and the Higgs mass is therefore naturally related

to the mass scale of the superpartner.

3. The superpartners would modify the running of the gauge couplings at higher

scales, and gauge coupling unification takes places with astonishing precision.

Therefore supersymmetric extensions of the I \ became the primary candidates

for new ]1li,-i. at the TeV scale. Not surprisingly, therefore, the signatures of

supersymmetry at the Tevatron and the LHC have been extensively discussed in

the literature.

However, supersymmetry is not the only model which has WIMP candidates.

Recent developments in string theory have spurred a revival of interest in the

phenomenology of theories with extra spatial dimensions. Some or even all of the

Standard Model particles could also propagate in the extra dimensions and it is

-ii,-'-. -led that a stable particle in the extra dimensional models may be able to

account for the observed dark matter.

The immediate result from the hypothesis of extra dimensions is the existence

of extra particle states. This can be understood easily in the following way. In 4

dimensions, we have the following energy-momentum relation,


E2 p + 2 + m2, (1 4)


where xa, X2, X3 are the coordinates of the usual 3 dimensions, E is an energy of a

particle and m is a mass of a particle. Suppose there was an extra dimension with

a coordinate y; then this relation becomes


E2 p + 2 l 2 2


(1-5)










000
27R 2A 27R = 5A 27R = 6A

Figure 1-1: de Broglie's particle-wave (1dii.il v. As we go around the circle, we must
fit an integer multiple of A's in its circumference.

Now recall the particle-wave d(l.ilil, i

27r
p, = (16)

If the extra dimension is compact, e.g., a circle, then as we go around the circle,
we must fit an integer multiple of A's in its circumference as shown in fig. 1-1.
Therefore periodicity implies a quantization of momentum along the extra dimen-
sion,
S27wR/ 27n n
A = p (1 7)
n 27R R
Substitute eqn. 1-7 into eqn. 1-5, then the energy-momentum relation becomes

E2 -22 2 2 3 2 2 (1 8)
+ +23 +

where f. = + m2 is the effective mass of the particle moving in the extra
dimensions. This translates into a rich and exciting phenomenology at the LHC,
since quantization of the particle momentum along the extra dimension necessarily
implies the existence of whole tower of massive particles, called Kaluza-Klein (KK)
modes or partners. The KK particles within each tower are nothing but heavier
versions of their Standard Model counterpart. A discovery of a compact extra
dimension at a collider can only be made through the discovery of the KK particles
and measurement of their properties. In fig. 1-2, our 4 dimensional spacetime is
one of the two branes, and the space between the two branes is usually referred to
as "the bulk." < \ particles can either freely propagate into the bulk or remain on
the brane. The mass spectrum of the KK partners even encodes information about






5







bulk






brane
(4 dimensional spacetime)

Figure 1-2: An illustration of bulk and brane. 4 dimensional spacetime is shown as
a brane and the space between two brane is called a bulk, where extra dimensions
exist

the space-time geometry: if the extra dimension is flat, the KK masses are roughly

equally spaced [2, 3], and if the extra dimension is warped, the KK mass spectrum

follows a non-trivial pattern [4, 5].

Now consider, for example, the most "democratic" scenario (which has become

known as Universal Extra Dimensions) in which all Standard Model particles

propagate in the bulk. Its simplest incarnation has a single extra dimension

of size R, which is compactified on an S1/Z2 orbifold [6]. In fig. 1-3, we show

S1/Z2 orbifold where the extra dimension is shown as a line in this geometry.

Interestingly, the dark matter puzzle can be resolved in a compelling fashion

in models with Universal Extra Dimensions. A peculiar feature of UED is the

conservation of Kaluza-Klein number at tree level, which is a simple consequence

of momentum conservation along the extra dimension. However, bulk and brane

radiative effects break KK number down to a discrete conserved (qi.ml, il'v, called

KK-parity. The KK-parity adorns the UED scenario with many of the virtues

typically associated with supersymmetry:









S1 Z2 S /Z2

brane brane

j / bulk


Z2

Figure 1-3: S1/Z2 orbifold. A half of the circle (SI) is identified with the other half
with a Z2 symmetry. The geometry becomes a line with two fixed points. The line
between two fixed points represents the bulk.


1. The lightest KK-partners (those at level 1) must always be pair-produced

in collider experiments, which leads to relatively weak bounds from direct

searches.

2. The KK-parity conservation implies that the contributions to various pre-

cisely measured low-energy observables only arise at the loop level and are

small.

3. Finally the KK-parity guarantees that the lightest KK partner is stable, and

thus can be a cold dark matter candidate.

As we will see in the next chapters, the phenomenology of this scenario clearly

resembles that of supersymmetry. In this sense, many of the SUSY studies in the

literature apply, and it is perhaps more important to find methods to distinguish

between the two models. Recently, other models such as little Higgs theory with

T-parity have been proposed as new 1pi, -i, beyond the Standard Model. Our

studies can also apply in the case of little Higgs models since the first level of the

UED model looks like the little Higgs particle spectrum.

Except for its abundance, no other properties of dark matter candidates are

known at present. Therefore it is important to study the properties of new types

of dark matter candidates in the extra dimensional models and compare them with

those in supersymmetry. Then a number of questions can arise: What are the









properties of dark matter candidates in the extra dimensional models? How differ-

ent are they from ones in supersymmetry? Can we see any evidences for the extra

dimensions in dark matter or collider experiments? etc. In this dissertation, we

want to answer at least some of these questions. Hence we investigate the collider

phenomenology and .,--1. .1 li,--i, .il implications of Universal Extra Dimensions.

In chapter 2, we first show a simple example of a Lagrangian in extra dimen-

sions and later introduce the complete model with Universal Extra Dimensions. We

review the basic phenomenology of the UED model, contrasting it with a generic

supersymmetric model as described above. The detailed properties of UED models

are summarized in appendix A.

In chapter 3, we identify two basic discriminators between UED and SUSY,

and proceed to consider each one in turn in the following sections. One of the

characteristic features of extra dimensional models is the presence of a whole

tower of Kaluza-Klein (KK) partners, labelled by their KK level n. In contrast,

N = 1 supersymmetry predicts a single superpartner for each Y \I particle. One

might therefore hope to discover the higher KK modes of UED and thus prove

the existence of extra dimensions. In section 3.1, we study the discovery reach

for level 2 KK gauge boson particles and the resolving power of the LHC to see

them as separate resonances. This study was done by our group for the first time

[7-11]. The other fundamental difference between SUSY and UED is the spin

of the new particles (superpartners or KK partners). Therefore in section 3.2,

we investigate how well the two models can be distinguished at the LHC based

on spin correlations in the cascade decays of the new particles. In particular, we

use the ..--' mmetry variable recently advertised by Barr [12], as well as dilepton

mass distributions. Until recently there were no known methods for measuring

the spins of new particles at the LHC but now the spin determination at the LHC

has become a hot topic in collider ]li, --i. [13-18]. In section 3.3, we contrast the









experimental signatures of low energy supersymmetry and the model of Universal

Extra Dimensions, and this time at a linear collider, discuss various methods for

their discrimination. This was also the first study to contrast SUSY and UED at a

linear collider [10, 19].

In chapter 4, we consider the I-i 1,-i, .i1 implications of the UED. We

calculate the relic density of the KK dark matter and show new results on direct

detection limits [20]. The first calculation of KK dark matter [21] was done in

the past but under the assumption that all KK particles have the same masses.

In addition, only a subset of the relevant coannihilation processes was included.

Therefore in our new calculation [20], we include all possible coannihilation

processes without assuming KK mass degeneracy. A similar calculation about one

particular type of dark matter was done by a group at Princeton [22] independently

and our results are in agreement. We then go beyond the minimal model and

discuss the size of coannihilation effects separately for each class of level 1 KK

particles. This calculation with different types of KK dark matter in nonminimal

UED models was performed by our group only. The annihilation cross-sections for

the dark matter calculation are listed in appendix B. In chapter 5, we conclude.














CHAPTER 2
UNIVERSAL EXTRA DIMENSIONS

2.1 Massive Scalar Field in Five Dimensions

The full Lagrangian of Universal Extra Dimensions is given in appendix A

and here we consider a simple example to illustrate the ]1 li,-i of a theory with

extra compact dimensions. As the simplest example of a Lagrangian in higher

dimensions, we consider the action for a massive scalar filed in 5 dimensions,

S = d4xdy [9M*(x, y)aM(x, y) M2*(, y)P(x, y)] (2 1)

Here M, N = 0, 1, 2, 3, 5 = t, 5, 5 dimensional metric is gMN ( M

(0p, 05), and y is the extra dimensional coordinate (5th component of a Lorentz
index, M). In the case of a circular extra dimension (S1), the 5 dimensional scalar

field D is expressed in terms of an exponential basis as follows:

1 y
4(x, y) = In(X)exp y (2-2)

where R is the radius of the extra dimension. This exponential basis satisfies the

following orthog(n.lilil relation between different modes,

2R 6n,m j2R dyexp i(n m)y) (2-3)
o R

Now we integrate out the extra dimensional coordinate y to get a 4 dimensional

effective theory. Then the action becomes


fd- [a T\9(x)a (x) Rn2 ) ) 0( (2-4)
n -oo









Here cp, is a 4 dimensional scalar field of mass mr = + m2. We started

with one massive scalar field in 5D and compactified this theory on S1. As a

result, we get an infinite number of scalar fields (called a KK-tower) with mass,
In2
mn = + m2, in 4 dimensions. n 0 (the :, ..' mode) corresponds to a
regular massive scalar field in 4 dimensions with mass, m. For the nonzero modes

(or KK modes), the mass comes mostly from the derivative with respect to the
extra dimension (05). Notice that all KK modes have the same spin.

2.2 Universal Extra Dimensions

The models of Universal Extra Dimensions are similar to this example. In

the simplest and most popular version, there is a single extra dimension of size

R, compactified on an orbifold (S1/Z2) instead of circle (S1) [6]. The orbifold can

introduce chiral fermions and project out unwanted 5th components of the gauge

fields (see appendix A). More complicated 6-dimensional models have also been

built [23-25]. The Standard Model is written in 5 dimensions as follows.


Gauge dy BMr MN I N waMN G A NGAMN
2 w-,R 4 4 4 m

LGF =- \ dy\ (O B,- 05B5 1 (OW 05 Wia) 2
1 0-,R (1 2 1
1 (0B A 5) Gr 2
y [ 2 S(OrB
1 jfrR
7FR
Leptons 2 7, dy {L(x, y)rMDML(x, y) + iE(x, y)MDME(, y) }
2 J-_xR
1 _7R
Quarks dy {Q(x, y)PMDMQ(x, y) + U(x, y)FMDMU(, y)
2 J-f7R
+iD(x,y)FMDMD(x,y)} (2-5)
1 anrR
LYukawa R dy {A uQ(x, y)U (, y) i72H*(x, y) + AdQ(x, y)D(x, y)H(x, y)
2 J-rR
+AL(x, y)E(x, y)H(x, y)} ,

Higgs 2- dy 7 (DMH(x, y))t (DMH(x, y)) + Ht(x, y)H(x, y)

(Ht(x, y)H(x, ))2]









where covariant derivatives are defined in the appendix and each Standard Model

field is expressed in terms of cos and sin modes on the orbifold,

H(x, y) = H(x) + v2 H,(x)cos()y


B,(x,y) = B(x) + v B(x) cos( ,




W/(x, y) = P Y r) P
B5(x, y) B (x) sin()





r =1
n~l



WG(x, y) = RWG(x) sin( ( ,
1W5 (,

G(x,y) G (x) + P2ZG ,(x) cos( ) (26)
V'R Rnl nR

G(.x,y) = YoG(x) s+ i) +








n= 1
2 qL() [PLQ2() cos(- )+P0 ()



n 1n

(xy) = R() +v [PR (x) cos(-) + P sin(


R R
EI R}
[P r) Cos(-) + PRdT ( siny









and (G,(z, y), G5(x, y)) are the 5D gauge fields for U(1), SU(2) and SU(3)

respectively. Q(z, y) and L(z, y) are the SU(2) fermion doublets while U(x, y),

D(x, y) and E(x, y) are respectively the generic singlet fields for the up-type quark,

the down-type quark and the lepton.








Q U A
L R L R A, A5

n 3

n 2

n 1
-nO-



Figure 2-1: KK states after a compactification on the orbifold. Q (U) is a fermion
state which is a doublet (singlet) under SU(2) and A is a gauge boson. L/R rep-
resent the chirality of each state. There is one corresponding KK fermion (n / 0)
for each chirality of %I\! fermion (n = 0). Two KK states sharing an arrow make
one Dirac fermion while a I\! Dirac fermion needs one SU(2)w-doublet and one
SU(2)w singlet. A5 is eaten by A. at each KK level after compactification and KK
gauge bosons become massive while a I\ gauge boson remains massless. These
states are equally spaced since all KK states have the same mass m before elec-
troweak symmetry breaking. Crossed states do not exist on an orbifold compactifi-
cation.

The 4 dimensional effective Lagrangian is obtained by integrating out the extra

dimension using orthogon.ilil v relations between these trigonometric functions,

which are given in eqns. (A-6). As a result of the compactification, we find the

following properties of the 4 dimensional effective theory and list them below rather

than showing the actual Lagrangians, which are are quite lengthy.

1. Each SM particle has an infinite number of KK partners. This is

illustrated in fig. 2-1, where n = 0 corresponds to a \! particle and non-zero

modes correspond to KK states. In fact, AR is the number of KK levels

below a cutoff scale, A, since this theory requires a cutoff at high energy.

2. KK particles have the same spin as SM particles. All KK particles at

level n have the same mass, before the Higgs gets a vacuum expectation

value through the EWSB. The EWSB gives masses to Standard Model

particles and changes KK masses to 2 + where m is the mass term









from EWSB. A peculiar feature of UED is that there are two KK Dirac

fermions for each Dirac fermion in the \!. In fig. 2-1, L (R) represents a

left (right) handed chirality of the \ I fermion or KK fermion. In the \!, a

fermion doublet (denoted by Q) is left handed and a fermion singlet (denoted

by U) is right handed. Therefore there is no right handed fermion doublet

and left handed fermion singlet in the \!. However, due to the orbifold

boundary condition, there are two KK states with different chiralities for both

Q and U. These two KK states make one Dirac spinor and therefore there

are two Dirac fermions for each Dirac fermion in \!. In other words, the

zero mode (1\! particle) is either right handed or left handed but the KK

mode (KK particle) comes in chiral pairs. This chiral structure is a natural

consequence of the orbifold boundary conditions. The mass from EWSB

appears as an off-diagonal entry in a fermion mass matrix (see eqn. B.96).

1 fig. 2-1 shows that the 5th components of 5 dimensional gauge bosons are

eaten by KK gauge bosons and these KK gauge bosons become massive while

the \!I gauge bosons remain massless since there is no 5th component. The

M\! gauge boson can get a mass through EWSB. 2

3. All vertices at tree level satisfy KK number conservation. For each

term in the Lagrangian, we have a A (see eqn. A-21) which is a linear

combination of the Kronecker delta functions. Due to this structure, the



1 Then there could be, in principle, a mixing between two KK Dirac fermions but
the mixing angle is small since R-1 is larger than fermion mass in the \!.
2 Similarly to the fermion case, there is a nonzero contribution to the diagonal
part of the gauge boson mass matrix from EWSB (eqn. A-22) and therefore there
is a mixing between KK partners of U(1) hypercharge gauge boson (B,) and KK
partners of neutral SU(2)w gauge boson (Wi), as in the ,i-\! (see eqn. A-24). This
mixing angle turns out to be small and we ignore it in our analysis.









0 0 1 1


1 2 2 0


1 0 1 2
(a) (b) (c) (d)

Figure 2-2: KK number conservation and KK parity. KK parity is always con-
served in all cases. (a) KK number is conserved and therefore this vertex exists at
tree level. This coupling is the same as a 1\! coupling. (b) KK number is not con-
served and it does not exist at tree level. It is generated at 1 loop. (c) KK number
is conserved and it exists at tree level. This coupling does not involve any 1\! par-
ticle and its magnitude is less than 1\! coupling by d/2. (d) Either KK number or
KK parity are not conserved. It does not exist at any loop.

allowed vertices satisfy one of the following conditions,


m n k = 0, (2-7)

Imnkl = 0.


This is the conservation of Kaluza-Klein number at tree level, which is a

simple consequence of momentum conservation along the extra dimension.

Therefore it is easy to see which vertices are allowed or which vertices are

not. In fig. 2-2, (a) and (c) satisfy KK number conservation and those two

vertices are allowed at tree level. (b) and (d) are not allowed at tree level.

4. KK-parity is always conserved even at higher order. Bulk and brane

radiative effects [26-28] break KK number down to a discrete conserved

quantity, the so called KK parity, (-1)", where n is the KK level. KK

parity ensures that the lightest KK partners (those at level one) are always

pair-produced in collider experiments, just like in the R-parity conserving

supersymmetry models. KK parity conservation also implies that the

contributions to various low-energy observables [29-39] only arise at loop

level and are small. As a result, the limits on the scale R-1 of the extra









dimension from precision electroweak data are rather weak, constraining R-1

to be larger than approximately 250 GeV [33]. Fig. 2-2(b) can be generated

by 1 loop corrections with level 1 KK particles, however, KK-parity is not

conserved in fig. 2-2(d), hence it can never be generated by higher order

corrections.

5. New vertices are basically the same as SM couplings (up to normal-

ization). Vertices which have both \1I and KK particles are the same as

the vertices in the \! if the KK particles are replaced by the corresponding

S\!I particles. Vertices with KK particles only can differ by a factor such as

v2 due to orthogon.llili' relations (eqns. A-20) and normalization factors

(eqn. 2-7). Of course, KK-parity must be always conserved in any case.

This UED framework has been a fruitful p1l:.-round for addressing different

puzzles of the Standard Model, such as electroweak symmetry breaking and

vacuum stability [40-42], neutrino masses [43, 44], proton stability [45] or the

number of generations [46].

To continue the study on the phenomenology of UED model, we need to

know the mass spectrum. It depends on the interplay between the one-loop

radiative corrections to the KK mass spectrum and the brane terms generated by

unknown pir, -i. at high scales [28]. In fig. 2-3, the spectrum of the first KK

level is shown at tree level (a) and one-loop (b), for R- = 500 GeV, AR = 20,

and assuming vanishing boundary terms at the cut-off scale A. Fig. 2-4 shows a

qualitative sketch of the level 1 KK spectroscopy depicting the dominant (solid)

and rare (dotted) transitions and the resulting decay product, based on the mass

spectrum given in fig. 2-3. As indicated in fig. 2-3, in the minimal UED model

(\!UED) defined below, the LKP turns out to be the KK partner 71 (or the KK

partner B1 of hypercharge gauge boson since the Weinberg angle for KK states

is small) of the photon [28] and its relic density is typically in the right ballpark:










650 650 650 650
(a) (b)

t2
600 600 600 Q 600

a b d ati

550 -550 550 -550
tlt2
AoH H'
-----H A --D L ----2,T
AD- Q,u,d L,e bl,ba T7,, e T1
500 Qud b,2 500 500 500

Figure 2-3: The spectrum of the first KK level at (a) tree level and (b) one-loop,
for R-1 500 GeV, AR 20, mh 120 GeV, and assuming vanishing boundary
terms at the cut-off scale A. The figures are taken from Cheng et al. [28].


in order to explain all of the dark matter, the B1 mass should be in the range

500-600 GeV [20-22,47-49]. Kaluza-Klein dark matter offers excellent prospects for

direct [50-52] or indirect detection [50, 5361]. Once the radiative corrections to the

Kaluza-Klein masses are properly taken into account, the collider phenomenology

of the minimal UED model exhibits striking similarities to supersymmetry [62, 63]

and represents an interesting and well motivated counterexample which can "fake"

supersymmetry signals at the LHC.

At hadron colliders, the dominant production mechanisms are KK gluon (gi)

or KK quark (qi or Qi) productions. As shown in fig. 2-4, an SU(2)w-singlet KK

quark (qi) dominantly decays into a jet and a KK photon (7,) while an SU(2)w-

doublet KK quark (Qi) decays into level 1 EW gauge bosons (Zi or WI). Level

1 gauge bosons decay into a KK lepton producing a \!I lepton and later the KK

lepton also produces a \!I lepton. We can notice that this cascade decay looks like

a typical SUSY cascade.

For the purposes of our study we have chosen to work with the minimal

UED model considered in [62]. In UED the bulk interactions of the KK modes

are fixed by the \ Lagrangian and contain no unknown parameters other than
























71

Figure 2-4: Qualitative sketch of the level 1 KK spectroscopy depicting the dom-
inant (solid) and rare (dotted) transitions and the resulting decay product. The
figure is taken from Cheng et al. [62].


the mass, mh, of the \ I Higgs boson. In contrast, the boundary interactions,

which are localized on the orbifold fixed points, are in principle arbitrary, and their

coefficients represent new free parameters in the theory. Since the boundary terms

are renormalized by bulk interactions, they are scale dependent [26] and cannot

be completely ignored since they will be generated by renormalization effects.

Therefore, one needs an ansatz for their values at a particular scale. Like any

higher dimensional Kaluza-Klein theory, the UED model should be treated only

as an effective theory valid up to some high scale A, at which it matches to some

more fundamental theory. The minimal UED model is then defined so that the

coefficients of all boundary interactions vanish at this matching scale A, but are

subsequently generated through RGE evolution to lower scales. The minimal UED

model therefore has only two input parameters: the size of the extra dimension, R,

and the cutoff scale, A. The number of KK levels present in the effective theory is

simply AR and may vary between a few and ~ 40, where the upper limit comes

from the breakdown of perturbativity already below the scale A. Unless specified

otherwise, for our numerical results below, we shall always choose the value of A











102
5a
Tevatron 0
101

100 / LHC

10-1

10-2 41ZT
AR=20
10-3 . .
0 500 1000 1500 2000
R-' (GeV)

Figure 2-5: A discovery reach for MUEDs at the Tevatron (blue) and the LHC
(red) in the 4 + FT channel. A 5a excess or the observation of 5 signal events is
required, and lines show the required total integrated luminosity per experiment (in
fb-1) as a function of R-1, for AR 20. In either case the two experiments are not
combined. The figure is taken from Cheng et al. [62].


so that AR = 20. Changing the value of A will have very little impact on our

results since the A dependence of the KK mass spectrum is only logarithmic. For

R-1 > 500 GeV, sin2 On < 0.01 where 0, is the Weinberg angle for level n. Fig. 2-5

shows the discovery reach for MUEDs at the Tevatron (blue) and the LHC (red) in

the 4 + tST channel. A 5a excess or the observation of 5 signal events is required,

and lines show the required total integrated luminosity per experiment (in fb-1) as

a function of R-1, for AR = 20.

2.3 Comparison between UED and Supersymmetry

We are now in a position to compare in general terms the phenomenology of

UED and supersymmetry at colliders. The discussion of Section 2.2 leads to the

following generic features of UED:

1. For each particle of the Standard Model, UED models predict an infinite3

tower of new particles (Kaluza-Klein partners).


3 Strictly speaking, the number of KK modes is AR, see Section 2.2.









2. The spins of the 1 \ particles and their KK partners are the same.

3. The couplings of the 1 \ particles and their KK partners are equal.

4. The generic collider signature of UED models with WIMP LKPs is missing

energy.

Notice that the defining features 3 and 4 are common to both supersymmetry and

UED and cannot be used to distinguish the two cases. We see that while R-parity

conserving SUSY implies a missing energy signal, the reverse is not true: a missing

energy signal would appear in any model with a dark matter candidate, and even

in models which have nothing to do with the dark matter issue, but simply contain

new neutral quasi-stable particles, e.g. gravitons [2, 64, 65]. Similarly, the equality

of the couplings (feature No. 3) is a celebrated test of SUSY, but from the above

comparison we see that it is only a necessary, but not a sufficient condition in

proving supersymmetry. In addition, the measurement of superpartner couplings

in order to test the SUSY relations is a very challenging task at a hadron collider.

For one, the observed production rate in any given channel is only sensitive

to the product of the cross-section times the branching fractions, and so any

attempt to measure the couplings from a cross-section would have to make certain

assumptions about the branching fractions. An additional complication arises from

the fact that at hadron colliders all kinematically available states can be produced

simultaneously, and the production of a particular species in an exclusive channel

is rather difficult to isolate. The couplings could also in principle be measured

from the branching fractions, but that also requires a measurement of the total

width, which is impossible in our case, since the Breit-Wigner resonance cannot be

reconstructed, due to the unknown momentum of the missing LSP (LKP).

We are therefore forced to concentrate on the first two identifying features as

the only promising discriminating criteria. Let us begin with feature 1: the number

of new particles. The KK particles at n = 1 are analogous to superpartners in









supersymmetry. However, the particles at the higher KK levels have no analogues

in N = 1 supersymmetric models. Discovering the n > 2 levels of the KK tower

would therefore indicate the presence of extra dimensions rather than SUSY. In

this study we shall concentrate on the n = 2 level and in Section 3.1 we investigate

the discovery opportunities at the LHC and the Tevatron (for linear collider studies

of n = 2 KK gauge bosons, see [10, 19, 66, 67]). Notice that the masses of the KK

modes are given roughly by m, ~ n/R, where n is the KK level number, so that

the particles at levels 3 and higher are rather heavy and their production is severely

suppressed.

The second identifying feature the spins of the new particles -also provides

a tool for discrimination between SUSY and UED: the KK partners have identical

spin quantum numbers as their <\ counterparts, while the spins of the superpart-

ners differ by 1/2 unit. However, spin determinations are known to be difficult at

the LHC (or at hadron colliders in general), where the parton-level center of mass

energy ECM in each event is unknown. In addition, the moment of the two dark

matter candidates in the event are also unknown. This prevents the reconstruction

of any rest frame angular decay distributions, or the directions of the two particles

at the top of the decay chains. The variable EcM also rules out the possibility of

a threshold scan, which is one of the main tools for determining particle spins at

lepton colliders. We are therefore forced to look for new methods for spin deter-

minations, or at least for finding spin correlations. Recently it has been -ii,.'-. -Ied

that a charge -....'1 ,, I ry in the lepton-jet invariant mass distributions from a

particular cascade, can be used to discriminate SUSY from the case of pure phase

space decays [12]. The possibility of discriminating SUSY and UED by this method

will be the subject of Section 3.2 (see also [7-10] and [13]).

For the purposes of our study we have implemented the relevant features

of the minimal UED model in the CompHEP event generator [68]. The minimal









Supersymmetric Standard Model (\!SS \ !) is already available in CompHEP since

version 41.10. We incorporated all n = 1 and n = 2 KK modes as new particles,

with the proper interactions, widths, and one-loop corrected masses [28]. Similar

to the '-! case, the neutral gauge bosons at level 1, Z1 and 71, are mixtures of the

KK modes of the hypercharge gauge boson and the neutral SU(2)w gauge boson.

However, as shown in [28], the radiatively corrected Weinberg angle at level 1

and higher is very small. For example, 71, which is the LKP in the minimal UED

model, is mostly the KK mode of the hypercharge gauge boson. For simplicity, in

the code we neglected neutral gauge boson mixing for n > 1.















CHAPTER 3
COLLIDER PHENOMENOLOGY

In this chapter, we consider collider implications of Universal Extra Dimen-

sions at the LHC and a future linear collider. Since the discovery of the first KK

level is discussed in [62], we first focus on the discovery of level 2 KK particles at

the LHC and the Tevatron. We then consider discrimination between supersymme-

try and Universal Extra Dimensions with several different methods at the LHC and

a linear collider.

3.1 Search for Level 2 KK Particles at the LHC

In this section we shall consider the prospects for discovery of level 2 Kaluza-

Klein particles in UED. Our notation and conventions follow those of Ref. [62].

For example, SU(2)w-doublet (SU(2)w-singlet) KK fermions are denoted by

capital (lowercase) letters. The KK level n is denoted by a subscript. In fig. 3-1

we show the mass spectrum of the n = 1 and n = 2 KK levels in minimal UED,

for R-1 500 GeV, AR = 20 and '1\ Higgs boson mass mn 120 GeV. We

include the full one-loop corrections from Cheng et al. [28]. We have used RGE

improved couplings to compute the radiative corrections to the KK masses (see

appendix A.4). It is well known that in UED the KK modes modify the running

of the coupling constants at higher scales. We extrapolate the gauge coupling

constants to the scale of the n = 1 and n = 2 KK modes, using the appropriate

3 functions dictated by the particle spectrum [69-71]. As a result the spectrum

shown in fig. 3-1 differs slightly from the one in [28]. Most notably, the colored

KK particles are somewhat lighter, due to a reduced value of the strong coupling

constant, and overall the KK spectrum at each level is more degenerate.









1200
q. Q2
1000 -H2,G ,G2 e2 L

800

0 600
a 60 ___ L1
a H I,GI,GI ei L,
400

200 R- = 500 GeV, AR = 20

0

Figure 3-1: One-loop corrected mass spectrum of the n = 1 and n = 2 KK levels
in minimal UED including the effect of running coupling constants in the presence
of extra dimensions, for R`-1 500 GeV, AR 20 and mh 120 GeV. We show
the KK modes of gauge bosons, Higgs and Goldstone bosons and first generation
fermions.


3.1.1 Phenomenology of Level 2 Fermions

We begin our discussion with the n = 2 KK fermions. Since the KK mass

spectrum is pretty degenerate, the production cross-sections at the LHC are

mostly determined by the strength of the KK particle interactions with the proton

constituents. As KK quarks carry color, we expect their production rates to be

much higher than those of KK leptons. We shall therefore concentrate on the case

of KK quarks only.

In principle, there are two mechanisms for producing n = 2 KK quarks at

the LHC: through KK-number conserving interactions, or through KK-number

violating (but KK-parity conserving) interactions. The KK number conserving

QCD interactions allow production of KK quarks either in pairs or singly (in

association with the n = 2 KK mode of a gauge boson). The corresponding

production cross-sections are shown in fig. 3-2 (the cross-sections for producing

n = 1 KK quarks have been calculated in [13, 72, 73]). In fig. 3-2a we show the

cross-sections (in pb) for n = 2 KK-quark pair production, while in fig. 3-2b

we show the results for n = 2 KK-quark/KK-gluon associated production and










103 ------------------------ 103 r-----------------------

2q2q\\ go Q-92,gq
10o - -00 \-\ -- aga
10 Q Q2 1 \ g


S10-2 10-2
0a < a
P b
94 4



10-6 I 10-6
500 1000 1500 2000 500 1000 1500 2000
R-1 (GeV) R-1 (GeV)

Figure 3-2: Cross-sections of n = 2 KK particles at the LHC for (a) KK-quark
pair production (b) KK-quark/KK-gluon associated production and KK-gluon pair
production. The cross-sections have been summed over all quark flavors and also
include charge-conjugated contributions such as Q2q2, Q2q2, g2Q2, etc.


for n = 2 KK-gluon pair production. We plot the results versus R-1, and one

should keep in mind that the masses of the n = 2 particles are roughly 2/R. In

calculating the cross-sections of fig. 3-2 we consider 5 partonic quark flavors in the

proton along with the gluon. We sum over the final state quark flavors and include

charge-conjugated contributions. We used CTEQ5L parton distributions [74]

and choose the scale of the strong coupling constant as to be equal to the parton

level center of mass energy. All calculations are done with CompHEP [68] with our

implementation of the minimal UED model.

Several comments are in order. First, fig. 3-2 displays a severe kinematic

suppression of the cross-sections at large KK masses. This is familiar from the case

of SUSY, where the ultimate LHC reach for colored superpartners extends only

to about 3 TeV. Notice the different mass dependence of the cross-sections for the

three types of final states with n = 2 particles: quark-quark, quark-gluon, and

gluon-gluon. This can be easily understood in terms of the structure functions of

the quarks and gluon inside the proton. We also observe minor differences in the

cross-sections for pair production of KK quarks with different SU(2)w quantum

numbers. This is partially due to the different masses for SU(2)w-doublet and









SU(2)w-singlet quarks (see fig. 3-1), and the remaining difference is due to the

contributions from diagrams with electroweak gauge bosons. Notice that since

the cross-sections in fig. 3-2a are summed over charge conjugated final states, the

mixed case of Q2q2 contains twice as many quark-antiquark contributions (compare

Q22 + Q2q2 to q2q2 or Q2Q2 alone).

If we compare the cross-sections for n = 2 KK quark production to the

cross-sections for producing squarks of similar masses in SUSY, we realize that

the production rates are higher in UED. This is due to several reasons. Consider,

for example, s-channel processes. Well above threshold, the UED cross-sections

are larger by a factor of 4 [19]. One factor of 2 is due to the fact that in UED

the particle content at n > 1 is duplicated -for example, there are both left-

handed and right-handed SU(2)w-doublet KK fermions, while in SUSY there are

only "left-handed" SU(2)w-doublet squarks. Another factor of 2 comes from the

different angular distribution for fermions, 1 + cos2 0, versus scalars, 1 cos2 0.

When integrated over all angles, this accounts for the second factor of 2 difference.

Furthermore, at the LHC new heavy particles are produced close to threshold, due

to the steeply falling parton luminosities. In SUSY, the new particles (squarks) are

scalars, and the threshold suppression of the cross-sections is /33, while in UED

the KK-quarks are fermions, and the threshold suppression of the cross-section is

only 3. This distinct threshold behavior of the production cross-sections further

enhances the difference between SUSY and UED. For example, we find that for

R- = 500 GeV the pair production cross-section for charm KK-quarks is about

6 times larger than the cross-section for charm squarks. For processes involving

first generation KK-quarks, where t-channel diagrams contribute significantly, the

effect can be even bigger. For example, up KK-quark production and up squark

production differ by about factor of 8.









100 1.0
dW, (a) (b)
0.8
uZ,
D DIW, 2X 0.6
m10-1
-- 0.2
U7y

lO2^-1'---- -------------- -- 0. -------------------
10-2 L . I . 0.0 1
500 1000 1500 2000 500 1000 1500 2000
R-' (GeV) R-1 (GeV)

Figure 3-3: Branching fractions of the level 2 "up" quarks versus R-1 for (a) the
SU(2)w-doublet quark U2 and (b) the SU(2)w-singlet quark u2-


In fig. 3-2 we have only considered production due to KK number conserving

bulk interactions. The main advantage of those processes is that the corresponding

couplings are unsuppressed. However, the disadvantage is that we need to produce

two heavy particles, each of mass ~ 2/R, which leads to a kinematic suppression.

In order to overcome this problem, one could in principle consider the single

production of n = 2 KK quarks through KK number vi l.l.ii-:. but KK parity

conserving interactions, for example


Q2,2 PLQoA (3-1)


where Aa is a \ gauge field and TI is the corresponding group generator.

However, (3-1) is forbidden by gauge invariance, and the lowest order coupling of

an n = 2 KK quark to two \ particles has the form [28]


Q2VTaPLQOFa (3-2)


Such operators may in principle be present, as they may be generated at the scale

A by the unknown ]li,--i, at higher scales. However, being higher dimensional, we

expect them to be suppressed at least by I/A, hence in our subsequent analysis we

shall neglect them. Having determined the production rates of level 2 KK quarks,









we now turn to the discussion of their experimental signatures. To this end we

need to determine the possible decay modes of Q2 and q2. At each level n, the KK

quarks are among the heaviest states in the KK spectrum and can decay promptly

to lighter KK modes (this is true for the top KK modes [75, 76] as well). As can

be seen from fig. 3-1, the KK gluon is always heavier than the KK quarks, so the

two body decays of KK quarks to KK gluons are closed. Instead, n = 2 KK quarks

will decay to the KK modes of the electroweak gauge bosons which are lighter. The

branching fractions for n = 2 "up"-type KK quarks are shown in fig. 3-3. Fig. 3-3a

(fig. 3-3b) is for the case of the SU(2)w-doublet quark U2 (the SU(2)w-singlet

quark u2). The results for the "down"-type KK quarks are similar. We observe in

fig. 3-3 that the branching fractions are almost independent of R-1, unless one is

close to threshold. This feature will persist for all branching ratios of KK particles

which will be shown later.

Once we ignore the KK number violating coupling (3-2), only decays which

conserve the total KK number n are allowed. The case of the SU(2)w-singlet

quarks such as u2 is simpler, since they only couple to the hypercharge gauge

bosons. Recall that at n > 1 the hypercharge component is almost entirely

contained in the 7 KK mode [28]. We therefore expect a singlet KK quark q2 to

decay to either qiyi or qoy2, as seen in fig. 3-3b. The case of an SU(2)w-doublet

quark Q2 is much more complicated, since Q2 couples to the (KK modes of the)

weak gauge bosons as well, and many more two-body final states are possible.

Since the weak coupling is larger than the hypercharge coupling, the decays to W

and Z KK modes dominate, with BR(Q2 QW 2)/BR(Q2 QoZ2) = 2 and

BR(Q2 Q I2 )/BR(Q2 Q1Zi) = 2, as evidenced in fig. 3-3a. The branching

fractions to the 7 KK modes are only on the order of a few percent. The threshold

behavior seen in fig. 3-3a near R1 400 GeV is due to the finite masses for

the \I W and Z bosons, which enter the tree-level masses of Wf and Z1. Since









1.0 1.0
(a) (b)
0.8 0.8

S 0.6 11X 0.6


17 2
0.2 0.2

0.0 'L 0.0
500 1000 1500 2000 500 1000 1500 2000
R-1 (GeV) R-1 (GeV)

Figure 3-4: Branching fractions of the level 2 KK electrons versus R-1. The same
as fig. 3-3 but for the level 2 KK electrons: (a) the SU(2)w-doublet E2 and (b) the
SU(2)w-singlet e2.


the mass splitting of the KK modes is due to the radiative corrections, which are

proportional to R-1, the channels with Wft and Z1 open up only for sufficiently

large R-1.

We are now in a position to discuss the experimental signatures of n = 2

KK quarks. The decays to level 2 gauge bosons will simply contribute to the

inclusive production of 72, Z2 and W=, which will be discussed at length later

in Section 3.1.2. On the other hand, the decays to two n = 1 KK modes will

contribute to the inclusive production of n = 1 KK particles which was discussed

in [62]. Naturally, the direct pair production of the lighter n = 1 KK modes has a

much larger cross-section. Therefore, the indirect production of n = 1 KK modes

from the decays of n = 2 particles can be easily swamped by the direct n = 1

signals and the \ backgrounds. For example, the experimental signature for an

n = 2 KK quark decaying as Q2 -- Ql'1 (q2 -- qlj1) is indistinguishable from a

single Qi (qi). This is because /71 does not interact within the detector, and there

are at least two additional 7T1 particles in each event, so that we cannot determine

how many 71 particles caused the measured amount of missing energy. The decays

to W1 and Z1 may, however, lead to final states with up to four n = 1 particles,









each with a leptonic decay mode. The resulting multilepton signatures Nf + rT

with N > 5 are therefore very clean and potentially observable. Distinguishing

those events from direct n = 1 pair production would be an important step in

establishing the presence of the n = 2 level of the quark KK tower. Unfortunately,

the n = 2 sample is statistically very limited and this analysis appears very

challenging. We postpone it for future work [77]. Much of the previous discussion

applies directly to the level 2 KK leptons. Assuming the absence of the KK

number violating coupling analogous to (3-2), the branching fractions of the n = 2

KK electrons are shown in fig. 3-4. At each KK level, the KK modes of the weak

gauge bosons are heavier than the KK leptons, therefore the only allowed decays

are to 72 and 71. Just like KK quarks, KK leptons can be produced directly,

through KK number conserving couplings, or indirectly, in W2 and Z2 decays. In

either case, the resulting cross-sections are too small to be of interest at the LHC.

3.1.2 Phenomenology of Level 2 Gauge Bosons

We now discuss the collider phenomenology of the n = 2 gauge bosons V2. As

we shall see, the KK gauge bosons offer the best prospects for detecting the n = 2

structure, since they have direct (but not tree level) couplings to \!I particles,

and can be discovered as resonances, e.g. in the dijet or dilepton channels. This

is in contrast to the case of n = 2 KK fermions, which, under the assumptions

of Sec. 3.1.1, do not have fully visible decay modes. Bump hunting will also help

discriminate between n = 2 and n = 1 KK particles, since the latter are KK-parity

odd, and necessarily decay to the invisible 71.

There are four n = 2 KK gauge bosons: the KK *phl ..'i." 72, the KK

"Z-bc-... ii Z2, the KK "W-bc(-. ..i" W2, and the KK gluon g2. Recall that the

Weinberg angle at n = 2 is very small, so that 72 is mostly the KK mode of the

hypercharge gauge boson and Z2 is mostly the KK mode of the neutral W-boson of

the \ I. An important consequence of the extra dimensional nature of the model











4000 g, 2/ 12 Z
z -..

3000 7- 1


>2000 10
100
2000 ..'" i0-Z2- W

1000 (a) 10 (b)
500 1000 1500 2000 1000 2000 3000 4000 5000
R-1 (GeV) Mass (GeV)

Figure 3-5: Masses and widths of level 2 KK gauge bosons. (a) Masses of the four
n 2 KK gauge bosons as a function of R-1. (b) Total widths of the n 2 KK
gauge bosons as a function of the corresponding mass. We also show the width
of a generic Z' whose couplings to the \ particles are the same as those of the
Z-boson.


is that all four of the n = 2 KK gauge bosons are relatively degenerate, as shown

in fig. 3-5a. The masses are all roughly equal to 2/R. The mass splitting between

the KK gauge bosons is almost entirely due to radiative corrections, which in the

minimal UED model yield the mass hierarchy mi92 > mw2 m z2 > m2. The

KK gluon receives the largest corrections and is the heaviest particle in the KK

spectrum at each level n. The W2 and Z2 particles are degenerate to a very high

degree, due to SU(2)w symmetry. The KK number conserving interactions allow

an n = 2 KK gauge boson V2 to decay to two n = 1 particles, or to one n = 2 KK

particle and one n = 0 (i.e., Standard Model) particle, provided that the decays are

allowed by phase space. For example, the partial widths to fermion final states are

given by
C2 2 2 2 m2 2
( /V2 o) V2 -2 V2


x/(m (mf2 mfO2) (M (m + Mo2) (3-3)

4 C2m t2 1t 1 +
-c g m mv2
c2 2 2 2 m2 h
4 ( M M 2









C22 3
S(V2 i ) 24rT -" 4m (3-4)
V2
3
c2 2mxM2 (in2 m1 2 m2 3
62 m2 mi Knmv2
W2 M2 I V2
3
C2g2r2v2 [Jnv2 Jrf12
627 kM2 mi

where c YNJ/2 for V2 2, C N/2 for V2 m Z2, c = VCKMNJ/ for

V2 W2 and c 1/v2 for V2 92, with Y being the fermion hypercharge in the

normalization Q = T3 + Y/2, VCKM is the CKM mixing matrix, and NJ = 3 for

f q and NJ = 1 for f = Here Jm stands for the total radiative correction to

a KK mass m, including both bulk and boundary contributions [28], m2 2/R,

and g is the corresponding gauge coupling. The first lines in (3-3) and (3-4) give

the exact result, while the last lines are the approximate formulas derived in [62] as

leading order expansions in 8m/m. The second line in (3-3) is an approximation

neglecting the I\! fermion mass mfo. The second line in (3-4) is an alternative

approximation which incorporates subleading but numerically important terms. In

our code we have programmed the exact expressions and quote the approximations

here only for completeness.

Note that the KK number conserving decays of the n = 2 KK gauge bosons

are suppressed by phase space. This is evident from the approximate expressions

in eqs. (3-3) and (3-4). The partial widths are proportional to the one-loop

corrections, which open up the available phase space and allow the corresponding

decay mode to take place. However, not all of the fermionic final states are

available, for example, Z2 and W2~ have no hadronic decay modes to level 1 or 2,

while 72 has no KK number conserving decay modes at all.

The n = 2 KK gauge bosons also have KK number violating couplings which

can be generated either radiatively from bulk interactions, or directly at the scale









A [28]. For example, the operator


foi L fA A (3-5)

couples V2 directly to 1\! fermions fo, and leads to the the following V2 partial

width

c22 mv 2 mf2 1 :2 m
F(V2 f0f0) j= -- Jj 1 ) 1 4-- (3-6)
127 \ 2 M2
c2g22v2 6Jrv22 2
12 m2 M2)

where 6m stands for a mass correction due to boundary terms only [28]. In the

second line we have neglected the 1 \ fermion mass mf,, recovering the result from

Cheng et al. [62].

As we see from (3-6), the KK number violating decay is also suppressed, this

time by a loop factor, and is proportional to the size of the radiative corrections to

the corresponding KK masses. In spite of this suppression, the V2 -- fofo decays

is most promising for experimental discovery. As long as the final state fermions

can be reconstructed, the V2 particle can be looked for as a bump in the invariant

mass distribution of its decay products. In this sense, the search is very similar

to Z' searches, with one major difference. Since all partial widths (3-3-3-6) are

suppressed, the total width of V2 is much smaller than the width of a typical Z'.

This is illustrated in fig. 3-5b, where we plot the widths of the KK particles 72,

W2,, Z2 and g2 in UED, as a function of the corresponding particle mass, and

contrast to the width of a Z' with SM-like couplings. We see that the widths of

the KK gauge bosons are extremely small. This has important ramifications for

the experimental search, since the width of the resonance will then be determined

by the experimental resolution, rather than the intrinsic particle width. In this

sense the width must be included in the set of basic parameters of a Z' search [78].

Before we elaborate on the experimental signatures of the n = 2 KK gauge bosons,














72

0 O 2. 7



10-4 -

500 1000 1500 2000
R (GeV)

Figure 3-6: Cross-sections for single production of level 2 KK gauge bosons
through the KK number violating couplings (3-5).


let us briefly discuss their production. There are three basic mechanisms:

1. Single production through the KK number violating operator.

The corresponding cross-sections are shown in fig. 3-6 as a function of R-1. One

might expect that these processes will be important, especially at large masses,

since we need to make only a single heavy n = 2 particle, alleviating the kinematic

suppression. If we compare the mass dependence of the Drell-Yan cross-sections

in fig. 3-6 to the mass dependence of the n = 2 pair production cross-sections

from fig. 3-2, indeed we see that the former drop less steeply with R-1 and become

dominant at large R-1. On the other hand, the Drell-Yan processes of fig. 3-6 are

mediated by a KK number violating operator (3-5) and the coupling of a V2 to 1 \

particles is radiatively suppressed. This is another crucial difference with the case

of a generic Z', whose couplings typically have the size of a normal gauge coupling

and are unsuppressed [78].

Notice the roughly similar size of the four cross-sections shown in fig. 3-6.

This is somewhat surprising, since the cross-sections scale as the corresponding

gauge coupling squared, and one would have expected a wider spread in the

values of the four cross-sections. This is due to a couple of things. First, for a









given R-1, the masses of the four n = 2 KK gauge bosons are different, with

mn2 > rW2 mZ2 > m72. Therefore, for a given R-1, the heavier particles suffer

a suppression. This explains to an extent why the cross-section for 72 is not the

smallest of the four, and why the cross-section for g2 is not as large as one would

expect. There is, however, a second effect, which goes in the same direction. The

coupling (3-5) is also proportional to the mass corrections of the corresponding

particles:
mv2 Jmf2 (3 7)
mv2 2 f2
Since the QCD corrections are the largest, for V2 72, Z, W2}, the second term

dominates. However, for V2 = 2, the first term is actually larger, and there is a

cancellation, which further reduces the direct KK gluon couplings to quarks.

2. Indirect production. The electroweak KK modes 72, Z2 and W, can

be produced in the decays of heavier n = 2 particles such as the KK quarks

and/or KK gluon. This is well known from the case of SUSY, where the dominant

production of electroweak superpartners is often indirect -from squark and

gluino decay chains. The indirect production rates of 72, Z2 and W2L due to

QCD processes can be readily estimated from figs. 3-2 and 3-3. Notice that

BR(Q2 2 W), BR(Q2 Z2) and BR(q2 -- 72) are among the largest branching

fractions of the n = 2 KK quarks, and we expect indirect production from QCD to

be a significant source of electroweak n = 2 KK modes.

3. Direct pair production. The n = 2 KK modes can also be produced

directly in pairs, through KK number conserving interactions. These processes,

however, are kinematically suppressed, since we have to make two heavy particles

in the final state. One would therefore expect that they will be the least relevant

source of n = 2 KK gauge bosons. The only exception is KK gluon pair production

which is important and is shown in fig. 3-2b. We see that it is comparable in

size to KK quark pair production and q292/Q2g2 associated production. We have














C2 C\2 111,vzv
10-1 -10-1



(a) (b)
10-2 I ..I I 10-2 ,
500 1000 1500 2000 500 1000 1500 2000
R-1 (GeV) R-1 (GeV)
100
100


C livi C\ 10-1
-H o 10-1


10-2
(c) (d)
10-2 ....
500 1000 1500 2000 500 1000 1500 2000
R-' (GeV) R-1 (GeV)

Figure 3-7: Branching fractions of the n = 2 KK gauge bosons versus R-1: (a) g2,
(b) Z2, (c) W', and (d) '2.


also calculated the pair production cross-sections for the electroweak n = 2 KK

gauge bosons and confirmed that they are very small, hence we shall neglect them

in our analysis below. In conclusion of this section, we discuss the experimental

signatures of n = 2 KK gauge bosons. To this end, we need to consider their

possible decay modes. Having previously discussed the different partial widths,

it is straightforward to compute the V2 branching fractions. Those are shown

in fig. 3-7(a-d). Again we observe that the branching fractions are very weakly

sensitive to R-1, just as the case of figs. 3-3 and 3-4. This can be understood as

follows. The partial widths (3-3) and (3-4) for the KK number conserving decays

are proportional to the available phase space, while the partial width (3-6) for the

KK number violating decay is proportional to the mass corrections (see eq. (3-7)).


qq2
q__ lqlt_______


K 11.,jj. I









Both the phase space and the mass corrections are proportional to R-1, which then

cancels out in the branching fraction.

Similarly to the case of n = 2 KK quarks discussed in Sec. 3.1.1, KK number

conserving decays are not very distinctive, since they simply contribute to the

inclusive n = 1 sample which is dominated by direct n = 1 production. The decays

of n = 1 particles will then give relatively soft objects, and most of the energy will

be lost in the LKP mass. In short, n = 2 signatures based on purely KK number

conserving decays are not very promising experimentally -one has to pay a big

price in the cross-section in order to produce the heavy n = 2 particles, but does

not get the benefit of the large mass, since most of the energy is carried away by

the invisible LKP. We therefore concentrate on the KK number violating channels,

in which the V2 decays are fully visible.

Fig. 3-7a shows the branching fractions of the KK gluon g2. Since it is the

heaviest particle at level 2, all of its decay modes are open, and have comparable

branching fractions. The KK number conserving decays dominate, since the KK

number violating coupling is slightly suppressed due to the cancellation in (3-7).

In principle, g2 can be looked for as a resonance in the dijet [79] or tt invariant

mass spectrum, but one would expect large backgrounds from QCD and Drell-

Yan. Notice that there is no indirect production of g2, and its single production

cross-section is not that much different from the cross-sections for 72, Z2 and W'

(see fig. 3-6). Therefore, the inclusive g2 production is comparable to the inclusive

72 and Z2 production, and then we anticipate that the searches for the n = 2

electroweak gauge bosons in leptonic channels will be more promising.

Figs. 3-7b and 3-7c give the branching fractions of Z2 and W,', correspond-

ingly. We see that the decays to KK quarks have been closed due to the large QCD

radiative corrections to the KK quark masses. Among the possible KK number

conserving decays of Z2 and WL, only the leptonic modes survive, and they will









be contributing to the leptonic discovery signals of UED [62]. Recall that the

KK number conserving decays are phase space suppressed, while the KK number

violating decays are loop suppressed, and proportional to the mass corrections

as in (3-7). The precise calculation shows that the dominant decay modes are

Z2 qq and W2 qq'. This can be understood in terms of the large 6mnq

correction appearing in (3-7). The resulting branching ratios are more than 50%

and in principle allow for a Z2/W search in the dijet channel, just like the case of

92. However, we shall concentrate on the leptonic decay modes, which have much

smaller branching fractions, but are much cleaner experimentally.

Finally, fig. 3-7d shows the branching fractions of '2. This time all KK

number conserving decays are closed, and 72 is forced to decay through the KK

number violating interaction (3-5). Again, the jetty modes dominate, and the

leptonic modes (summed over lepton flavors) have rather small branching fractions,

on the order of 2' ., which could be a potential problem for the search. In the

following section we shall concentrate on the Z2 -- +-f and '2 -S +.0- signatures

and analyze their discovery prospects in a Z'-like search [80,81].

3.1.3 Analysis of the LHC Reach for Z2 and 72

We are now in a position to discuss the discovery reach of the n = 2 KK

gauge bosons at the LHC and the Tevatron. We will consider the inclusive pro-

duction of Z2 and 72 and look for a dilepton resonance in both the e+e- and + 1-

channels. An important parameter of the search is the width of the reconstructed

resonance, which in turn determines the size of the invariant mass window selected

by the cuts. Since the intrinsic width of the Z2 and 72 resonances is so small (see

fig. 3-5b), the mass window is entirely determined by the mass resolution in the

dimuon and dielectron channels. For electrons, the resolution in CMS is approxi-

mately constant, on the order of Am,,/m ,, 1% in the region of interest [82]. On











102 ..- 102 DY
/ DY .... / ..
S..DY
101 .. 101 All processes
/ .' All processes
.-100 .100 .. -
/ PP-----p'2-P / /
+-
/ / e
10-1 / p-PPye-e e 1 / .. PP-Z e-
(a) (b)
S..
10-2 I I I 10-2
200 400 600 800 1000 200 400 600 800 1000
R-1 (GeV) R-1 (GeV)

Figure 3-8: 5a discovery reach for (a) 72 and (b) Z2. We plot the total integrated
liiii i.-il' L (in fb-1) required for a 5a excess of signal over background in the
dielectron (red, dotted) or dimuon (blue, dashed) channel, as a function of R-1. In
each plot, the upper set of lines labelled "DY" makes use of the single V2 produc-
tion of fig. 3-6 only, while the lower set of lines (labelled "All p .... --. -") includes
indirect 72 and Z2 production from n = 2 KK quark decays. The red dotted
line marked "FNAL" in the upper left corner of (a) reflects the expectations for a
72 -- e+e discovery at the Tevatron in Run II. The shaded area below R-1 250
GeV indicates the region disfavored by precision electroweak data [33].


the other hand, the dimuon mass resolution is energy dependent, and in prelimi-

nary studies based on a full simulation of the CMS detector has been parametrized

as [83]

Am,- 0.0215 +0.0128 (I .T
mP VI TeV

Therefore in our analysis we impose the following cuts

1. Lower cuts on the lepton transverse moment prT() > 20 GeV.

2. Central rapidity cut on the leptons |r(p)l < 2.4.

3. Dilepton invariant mass cut for electrons myv 2Amee < ree < mV + 2Amee

and muons my 2Am,, < mr, < my2 + 2AmL.

With these cuts the signal efficiency varies from 65% at R- = 250 GeV to 91%

at R-1 = 1 TeV. The main _\! background to our signal is Drell-Yan, which

we have calculated with the PYTHIA event generator [84]. With the cuts listed

above, we compute the discovery reach of the LHC and the Tevatron for the 72


FNAL. .- -


L -- I ------









and Z2 resonances. Our results are shown in fig. 3-8. We plot the total integrated

luiii. .-iliT L (in fb-1) required for a 5a excess of signal over background in the

dielectron (red, dotted) or dimuon (blue, dashed) channel, as a function of R-1.

In each panel in fig. 3-8, the upper set of lines labelled "DY" only utilizes the

single V2 production cross-sections from fig. 3-6. The lower set of lines (labelled

"All pr ....--. -") includes in addition indirect 72 and Z2 production from the

decays of n = 2 KK quarks to 72 and Z2 (we ignore secondary 72 production

from Q2 Z2 -- 2 -- 172). The shaded area below R1 = 250 GeV indicates

the region disfavored by precision electroweak data [33]. Using the same cuts also

for the case of the Tevatron, we find the Tevatron reach in 72 e+e- shown in

fig. 3-8a and labelled "FNAL." For the Tevatron we use electron energy resolution

AE/E = 0.01 e 0.16/v/E [85]. The Tevatron reach in dimuons is worse due to the

poorer resolution, while the reach for Z2 is also worse since mz2 > m2 for a fixed

R-1.

Fig. 3-8 reveals that there are good prospects for discovering level 2 gauge

boson resonances at the LHC. Already within one year of running at low l'uiil. -il

(L = 10 fb-1), the LHC will have sufficient statistics in order to probe the region

up to R-1 750 GeV. Notice that in the minimal UED model, the "good dark

matter" region, where the LKP relic density accounts for all of the dark matter

component of the Universe, is at R-1 ~ 500 600 GeV [20-22]. This region is well

within the discovery reach of the LHC for both n = 1 KK modes [62] and n = 2

KK gauge bosons (fig. 3-8). If the LKP accounts for only a fraction of the dark

matter, the preferred range of R-1 is even lower and the discovery at the LHC is

easier.

From fig. 3-8 we also see that the ultimate reach of the LHC for both 72 and

Z2, after several years of running at high luminosity (L ~ 300 fb-1), extends up

to just beyond R1 = 1 TeV. One should keep in mind that the actual KK masses









50 11 11 40
(a) pp-V2--tu (b) pp-V,-e e
40
7z 30 -
72
30 Z2 Z
:220


10 -
10 -
L=100 fb1 L=100 fb-
0 0
900 950 1000 1050 1100 1150 900 950 1000 1050 1100 1150
M, (GeV) Me, (GeV)

Figure 3-9: The 72 Z2 diresonance structure in UED with R-1 500 GeV, for (a)
the dimuon and (b) the dielectron channel at the LHC with L 100 fb- The \!I
background is shown with the (red) continuous underlying histogram.


are at least twice as large: myv m2 = 2/R, so that the KK resonances can

be discovered for masses up to 2 TeV. While the n = 2 KK gauge bosons are a

salient feature of the UED scenario, any such resonance by itself is not a sufficient

discriminator, since it resembles an ordinary Z' gauge boson. If UED is discovered,

one could then still make the argument that it is in fact some sort of non-minimal

supersymmetric model with an additional gauge structure containing neutral

gauge bosons. An important corroborating evidence in favor of UED would be the

simultaneous discovery of several, rather degenerate, KK gauge boson resonances.

While SUSY also can accommodate multiple Z' gauge bosons, there would be no

good motivation behind their mass degeneracy. A crucial question therefore arises:

can we separately discover the n = 2 KK gauge bosons as individual resonances?

For this purpose, one would need to see a double peak structure in the invariant

mass distributions. Clearly, this is rather challenging in the dijet channel, due

to the relatively poor jet energy resolution. We shall therefore consider only the

dilepton channels, and investigate how well we can separate 72 from Z2.

Our results are shown in fig. 3-9, where we show the invariant mass distri-

bution in UED with R- = 500 GeV, for (a) the dimuon and (b) the dielectron









channel at the LHC with L = 100 fb- We see that the diresonance structure

is easier to detect in the dielectron channel, due to the better mass resolution.

In dimuons, with L = 100 fb-1 the structure is also beginning to emerge. We

should note that initially the two resonances will not be separately distinguishable,

and each will in principle contribute to the discovery of a bump, although with a

larger mass window. In our reach plots in fig. 3-8 we have conservatively chosen

not to combine the two signals from Z2 and 72, but show the reach for each one

separately.

In this section we have discussed the differences and similarities in the hadron

collider phenomenology of models with Universal Extra Dimensions and super-

symmetry. We identified the higher level KK modes of UED and as a reliable

discriminator between the two scenarios. We then proceeded to study the discovery

reach for level 2 KK modes in UED at hadron colliders. We showed that the n = 2

KK gauge bosons offer the best prospects for detection, in particular the 72 and

Z2 resonances can be .. !,,,',I /.1,i discovered at the LHC. Is this a proof of UED?

Not quite -these resonances could still be interpreted as Z' gauge bosons, but

their close degeneracy is a smoking gun for UED. Furthermore, although we did

not show any results to this effect, it is clear that the W2L KK mode can also be

looked for and discovered in its decay to Y\1 leptons. One can then measure mw2

and show that it is very close to mz2 and m.2, which would further strengthen the

case for UED.

Here we only concentrated on the minimal UED model, it should be kept in

mind that there are many interesting possibilities for extending the analysis to a

more general setup. For example, non-vanishing boundary terms at the scale A

can distort the minimal UED spectrum beyond recognition. A priori, in such a

relaxed framework the UED-SUSY confusion can be "complete" in the context of a

hadron collider and a preliminary study is under way to address this issue [14,15].









The UED collider phenomenology is also very different in the case of a "fat"

brane [86, 87], charged LKPs [88] or KK graviton superwimps [89, 90]. Notice that

Little Higgs models with T-parity [16, 91-94] are very similar to UED, and can also

be confused with supersymmetry.

3.2 Spin Determination at the LHC

The fundamental difference between SUSY and UED is first the number of

new particles and second, the spins of new particles. The KK particles at n = 1

are analogous to superpartners in supersymmetry. However, the particles at the

higher KK levels have no analogues in N = 1 supersymmetric models. Discovering

the n > 2 levels of the KK tower would therefore indicate the presence of extra

dimensions rather than SUSY. However these KK particles can be too heavy to

be observed. Even if they can be observed at the LHC, they can be confused with

other new particles [10, 11] such as Z' or different types of resonances from extra

dimensions [25].

The second feature -the spins of the new particles -also provides a tool

for discrimination between SUSY and UED: the KK partners have identical spin

quantum numbers as their \ counterparts, while the spins of the superpartners

differ by 1/2 unit. However, spin determinations are known to be difficult at the

LHC (or at hadron colliders in general), where the parton-level center of mass

energy ECM in each event is unknown. In addition, the moment of the two dark

matter candidates in the event are also unknown. This prevents the reconstruction

of any rest frame angular decay distributions, or the directions of the two particles

at the top of the decay chains. The variable ECM also rules out the possibility

of a threshold scan, which is one of the main tools for determining particle spins

at lepton colliders. We are therefore forced to look for new methods for spin










SUSY: q

UED: Q1
Z0
7i

Figure 3-10: Twin diagrams in SUSY and UED. The upper (red) line corresponds
to the cascade decay q -- qO q->_ -- q LN -2o in SUSY. The lower (blue) line
corresponds to the cascade decay Qi -- qZ1 --- q t -- qg+t-' l in UED. In either
case the observable final state is the same: qf+ -OTr.


determinations, or at least for finding spin correlations1 The purpose of this

section is to investigate the prospects for establishing supersymmetry at the LHC

by discriminating it from its look-alike scenario of Universal Extra Dimensions

by measuring spins of new particles in two models2 As discussed before, the

second fundamental distinction between UED and supersymmetry is reflected in

the properties of the individual particles. Recently it has been -i.-'.- -1. .1 that a

charge .. 'mmetry in the lepton-jet invariant mass distributions from a particular

cascade (see fig. 3-10), can be used to discriminate SUSY from the case of pure

phase space decays [12] and is an indirect indication of the superparticle spins

(A study of measuring sleptons spins at the LHC can be found in [17]). It is



1 Notice that in simple processes with two-body decays like slepton production
e+e 2- +t-p.- the flat energy distribution of the observable fi-
nal state particles muonss in this case) is often regarded as a smoking gun for the
scalar nature of the intermediate particles (the smuons). Indeed, the smuons are
spin zero particles and decay isotropically in their rest frame, which results in a flat
distribution in the lab frame. However, the flat distribution is a necessary but not
sufficient condition for a scalar particle, and UED provides a counterexample with
the analogous process of KK muon production [19], where a flat distribution also
appears, but as a result of equal contributions from left-handed and right-handed
KK fermions.
2 The same idea can apply in the case of little Higgs models since the first level
of the UED model looks like the new particles in little Higgs models [91-94].









1500o- I I I I 1000
UED500, L 10 fb SPSla, L = 10 fb1
1250 UED UED
SSUSY or PS 8 SUSY or PS
1000 600

S 750 -
oo : } 400 -
500 -
Z0 200 -

S(a) I (b)
010 20 30 0 20 40 60 80
M1 M11

Figure 3-11: Comparison of dilepton invariant mass distributions in the case of
(a) UED mass spectrum with R-1 500 GeV (b) mass spectrum from SPSla. In
both cases, UED (SUSY) distributions are shown in blue (red). All distributions
are normalized to L 10 fb-1 and the error bars represent statistical uncertainty.


therefore natural to ask whether this method can be extended to the case of

SUSY versus UED discrimination. Following [12], we concentrate on the cascade

decay q -> qO q- -- q -f o in SUSY and the analogous decay chain

Qi -- qZ1 q- f --- qg+t-yi in UED. Both of these processes are illustrated

in fig. 3-10. Blue lines represent the decay chain in UED and red lines the decay

chain in SUSY. Green lines are \ I particles.

3.2.1 Dilepton Invariant Mass Distributions

First we will look for spin correlations between the two _\! leptons in the

final state. In supersymmetry, the slepton is a scalar particle and therefore there is

no spin correlation between the two \_! leptons. However in UED, the slepton is

replaced by a KK lepton and is a fermion. We might therefore expect a different

shape in the dilepton invariant mass distribution. To investigate this, we first

choose a study point in UED (SPSla in mSugra) with R-1 = 500 GeV taken from

Cheng et al. [28, 62] and then we adjust the relevant MSSM parameters (UED

parameters) until we get a matching spectrum. So the masses are exactly the same

and they can not be used for discrimination and the only difference is the spin. In









fig. 3-11, we show invariant mass distributions in UED and SUSY for two different

types of mass spectrum. In fig. 3-11(a), all UED masses are adjusted to be the

same as the SUSY masses in SPSla (mo = 100 GeV, ml/2 = 250 GeV, Ao = -100,

tan = 10 and pf > 0) while in fig. 3 11(b) the SUSY masses are replaced by KK

masses for R- = 500. In both cases, UED (SUSY) distributions are shown in blue

(red). Squark/KK quark pair-production cross-sections are taken from Smillie et
al. [13] and the relevant branching fractions are obtained from Cheng et al. [62]

for UED and [95] for SUSY. All distributions are normalized to L = 10 fb-1 and

the error bars represent statistical uncert.iiiil v. In supersymmetry, the distribution

is the same as the one in the case of pure phase space decay since the slepton has

no spin. As we can notice, the two distributions are identical for both UED and

SUSY mass spectrum even if the intermediate particles in UED and SUSY have

different spins. The minor differences in the plot will completely disappear once the

background, radiative corrections and detector simulation are included.

The invariant mass distributions for UED and SUSY/Phase space can be

written as [13,96]

dN
Phase Space :d 2rn
dlrn
dN
SUSY : -N 2 (3-8)
dmt
dN 4(y + 4z)
UED : 4- +rm
Sdmi (1+ 2z)(2 + y)

where the coefficient r in the second term of the UED distribution is defined as

(2 y)(1 2z)
r (3-9)
y + 4z

h = n-' is the rescaled invariant mass, y = and z = 2 are the

ratios of the masses involved in the decay. y and z are less than 1 in the case of

on-shell decay. From eqn. 3-8, there are two terms in UED. The first term is a










1.0 ... 3 .0
-0.4 U ED ..
0.8. .-" 2.5 Good point: r=-0.3
0.8 0..3ood
-0.2 2.0
SPhase space or SUSY
0.6 0.1 .... ...
N 0 1.5 0 1 .
S......... 0 S SUSY
0 .4 ... ... 0
. .. ------ .- 0 :2 :
0.2 o.a

. . . .. . ( a )( b )
0.0 I -'" 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Y Mi

Figure 3-12: A closer look into dilepton invariant mass distributions. (a) Contour
dotted lines represent the size of the coefficient r in eqn. 3-9. The minimal UED
is a blue dot in the upper-right corner since y and z are almost 1 due to the mass
degeneracy. The red dots represent several snowmass points: SPSla, SPSlb, SPS5
and SPS3 from left to right. The green line represents gaugino unification so all
SUSY benchmark points are close to this green line. (b) The dashed line represents
the dilepton distribution in SUSY or pure phase space. The solid cyan (magenta)
line represents the dilepton distribution in UED for r = -0.3 (r = 0.7).


linear in in like phase space and the second term is proportional to na3. So we see

that whether or not the UED distribution is the same as the SUSY distribution

depends on the size of the coefficient r in the second term of the UED distribution.

The UED distribution becomes exactly the same as the SUSY distribution if

r = 0.5. Therefore we scan the (y, z) parameter space, calculate the coefficient

r and show our result in fig. 3-12(a). In fig. 3-12(a), the contour dotted lines

represent the size of the coefficient r in eqn. 3-9. The minimal UED is blue dot

in upper-right corner since y and z are almost 1 due to the degeneracy in the

masses while red dots represent several snowmass points [97]: SPSla, SPSlb, SPS5

and SPS3 from left to right. The green line represents gaugino unification so all

SUSY benchmark points are close to this green line. As we see r is small for both

MUED and snowmass points and this is why we did not see any difference in the

distributions from fig. 3-11. If the mass spectrum is either narrow (\IUED mass

spectrum) or generic mSugra type, the dilepton distributions are very similar









and we can not tell any spin information from this distribution. However away

from the the mSugra model or MUED, we can easily find the regions where this

coefficient r is large and the spin correlation is big enough so that we can see a

difference in shape. We show two points (denoted by 'Good' and 'Better') from

fig.3-12(a) and show the corresponding dilepton distributions in fig.3-12(b). For

the 'Good' point, the mass ratio is m, : mi : mo = 9 : 10 : 20 and for the

'Better' point, mo : mi : mo = 1 : 2 : 4. In fig.3-12(b), the dashed line

represents dilepton distribution in SUSY or pure phase space and the solid cyan

(magenta) line represents the dilepton distribution in UED for r = -0.3 (r = 0.7).

Indeed for larger r, the distributions look different but background and detector

simulation need to be included. Notice that in the mSugra model, the maximum of

the coefficient r is 0.4.

3.2.2 Lepton-Jet Invariant Mass Charge Asymmetry

Now we look at spin correlations between q and i in fig. 3-10. In this case,

there are several complications. First of all, we don't know which lepton we need

to choose. There are two leptons in the final state. One lepton, called 'near', comes

from the decay of xo in SUSY or Z1 in UED, while the other lepton, called 'far',

comes from the decay of i in SUSY or fl in UED. One can form the lepton-quark

invariant mass distributions mrq. The spin of the intermediate particle (Z1 in

UED or o in SUSY) governs the shape of the distributions for the near lepton.

However, in practice we cannot distinguish the near and far lepton, and one has

to include the invariant mass combinations with both leptons (it is impossible to

tell near and far leptons event by event but there can be an improvement on their

selection [96].). Second, we do not measure charge of jets (or quarks). Therefore we

do not know whether a particular jet (or quark) came from the decay of squark or

anti-squark. This doubles the number of diagrams that we need to consider. These

complications tend to wash out the spin correlations, but a residual effect remains,








which is due to the different number of quarks and anti-quarks in the proton,
which in turn leads to a difference in the production cross-sections for squarks
and anti-squarks [12]. Most importantly, we do not know which jet is actually
the correct jet in this cascade decay chain. We pair-produce two squarks (or KK
quarks) particles and each of them produces one jet. Once ISR is included, there
are many jets in the final state. For now, as in [13], we will assume that we know
which jet is the correct one we need to choose. One never knows for sure which is
the correct jet although there can be clever cuts to increase the probability that we
picked the right one [96]. There are two possible invariant distributions in this case:

(-)) with a positively charge lepton and ( ) with a negatively charged
lepton. In principle, there are 8 diagrams that need to be included (a factor of
2 from quark/anti-quark combination, another factor of 2 from the two different
leptons with different chiralities, a factor of 2 from the ambiguity between near and
far leptons).
For this study, as in the dilepton case, we first start from a UED mass
spectrum and adjust the MSSM parameters until we get a perfect match in the
spectrum. In this case, Z1 does not decay into right handed leptons. There are 4
contributions and they all contribute to both ( T)d and (--)d distributions
which are in fig. 3-13,

(d fq dP2 d+
dmj + dml ) d\dm,)
d fq + + fq ( ), (3-10)
dm qtd, dm, dmn
where P1 (P2) represents distribtuion for a decay from a squark or KK quark

(anti-squark or anti-KK quark) and fq (fq) is the fraction of squarks or KK quarks
(anti-squarks or anti-KK quarks) and by definition, fq + fq 1. This q(I.,nl il'
fq tells us how much squarks or KK quarks are produced compared to their
anti-particles. For a UED mass spectrum and SPSla, fq ~ 0.7 [13]. These two









distributions in UED (SUSY) are shown in fig. 3-13(a) (fig. 3-13(b)) in different

colors. The distributions are normalized to L 10fb-1 and the very sharp edge

near mqe 60 GeV (mqe 75 GeV) is due to the near (far) letpon. However, once

background and detector resolutions are included, the clear edges are smoothed

out.

Now with these two distributions, a convient quantity, '...i ,,ll, I ry' [12] is

defined below

(/d a t (\ d a \)
+- dmq+ \dmqt-
S (-) ae (3-11)
(dm)ql+ + \dm)ql-

Notice that if fq fq = 0.5, () ( ) and +- becomes zero.

This is the case for pure phase space decay. So zero ..-. .-, I ry means we don't

obtain any spin information from this decay chain, i.e., if we measure non-zero

aymmetry, it means that the intermediate particle (ou or Z1) has non-zero spin.

So for this method to work, fq must be different from fq. So this method does not

apply at pp collider such as the Tevatron since a pp collider produces the same

amount of quarks and anti-quarks. The spin correlations are encoded in the charge

.. -,11111 vI I ry [12]. However, even in a pp collider such as the LHC, whether or not we

measure non-zero i. -.-,,,n,. I ry depends on parameter space, e.g.., in the focus point

region, gluino production dominates and gluino produces equal amounts of squarks

and anti-squarks. Therefore we expect fq ~ fq ~ 0.5 and the .-.'I-mmetry will be

washed out.

Our comparison between A+- in the case of UED and SUSY for UED mass

spectrum is shown in fig. 3-14(a). We see that although there is some minor

difference in the shape of the .'-.-i ., 1i I ry curves, overall the two cases appear

to be very difficult to discriminate unambiguously, especially since the regions

near the two ends of the plot, where the deviation is the largest, also happen to

suffer from poorest statistics. Notice that we have not included detector effects









1500 1 1500
UED, L 10 fb SUSY, L = 10 fb 1
1250 1250

1000 1000 ql

^ 750 I 750

500 500

250 250 I
(a) (b)
0 0
0 20 40 60 0 20 40 60
Mq1 Mql

Figure 3 13: Jet-lepton invariant mass distributions. (-)q+ (blue) and (-)
(red) in the case of (a) UED and (b) SUSY for UED mass spectrum with
R`-1 500 GeV. q stands for both a quark and an antiquark, and N(qc+) (N(q-))
is the number of entries with positively (negatively) charged lepton. The distri-
butions are normalized to = 10fb-1. A very sharp edge near mq ~ 60 GeV
(mqe ~ 75 GeV) is due to near (far) letpon. Once background and detector resolu-
tions are included, the clear edges are smoothed out.


or backgrounds. Finally, and perhaps most importantly, this analysis ignores

the combinatorial background from the other jets in the event, which could be

misinterpreted as the starting point of the cascade depicted in fig. 3-10. Overall,

fig. 3-14 shows that although the .i-'.,mmetry (eqn. 3-11) does encode some spin

correlations, distinguishing between the specific cases of UED and SUSY appears

challenging.

Similarly in fig. 3-14(b), we show the ..i-'.mmetry in UED and SUSY for

a mass spectrum from the SPSla point in the mSugra model. In this case, the

mass spectrum is broad compared to the UED spectrum and o in SUSY (Z1 in

UED) does not decay into left handed sleptons (SU(2)w KK letpons). Unlike

the narrow mass spectrum, in this study point with larger mass splitting, as

expected in typical SUSY models, the .i 'mmetry distributions appear to be more

distinct than the case shown in fig. 3-14(a), which is a source of optimism. These

results have been recently confirmed in [13]. It remains to be seen whether this











UED500, L 10 fb SPSla, L = 10 fb
UED UED
o SUSY o SUSY



02 02
0 0 1 T P h a s e s p a c e
Phase space I

-02 -02

(a) (b)
-0.4 -0.4
0 20 40 60 0 100 200 300
Mql M"

Figure 3-14: Asymmetries for UED and SUSY are shown in blue and red, respec-
tively, in the case of (a) UED mass spectrum with R- = 500 GeV and (b) SPSla
mass spectrum. The horizontal dotted line represents pure phase space. The error
bars represent statistical uncert.iiil i, with = 10 fb-1.


conclusion persists in a general -'1I .ii- and once the combinatorial backgrounds

are included [96]. Notice that comparing (a) and (b) in fig. 3-11, the signs of the

two ..i-.'mmetries have changed. The difference is the chirality of sleptons or KK

leptons. In fig. 3-11(a) (fig. 3-11(a)), left handed sleptons or SU(2)w doublet KK

leptons (right handed sleptons or SU(2)w singlet KK leptons) are onshell and the

..i-. mmetry starts out positive (negative) and ends negative (positive). By looking

at the sign of the -....' ,i,'1 I ry, we can see which chirality was onshell.

What we did so far was, first we choose a study point in one model and fake

parameters in other models until we see perfect match in the mass spectrum.

However not all masses are observable and sometimes we get less constraints than

the number of masses involved in the decay. So what we need to do is to match

endpoints in the distributions instead of matching mass spectrum and ask whether

there is any point in parameter space which is consistent with the experimental

data. In other words, we have to ask which model fits the data better. We consider

three kinematic endpoints: nmq, rnq and mer (see fig. 3-10). In principle, we can

find more kinematic endpoints such as a lower edge, here we are being conservative









and take upper edges only [98-100]. In case of an onshell decay of X and these

three kinematic endpoints are written in terms of masses

mqu = mq (1 -x)(1 y- yz)

mq = mq (1 -x)(1 z) (3-12)

mff = mq x( 1- y)(1 z)

where mq is squark mass or KK quark mass and x = y = )

and z = x, 2 are the ratios between masses in the cascade decay chain. By

definition, x, y and z are less than 1.

We are now left with 2 free parameters: fq and x and solve for y, z and mq in

terms of two free parameters. We minimize x2,

X2 i 2 (3-13)
i= 1 i
between the two ..-v.. 1 i ii, in the (x, fq) parameter space to see whether we can

fake a SUSY ..i-. mmetry in the UED model. x, is the theory prediction and pi is

the experimental value with uncertainty ai. Xof = )2/n is the 'reduced' X2 or X2

for n degrees of freedom.

Our result is shown in fig. 3-15(a). We found the minimum X2 is around 3 in

the region where all KK masses are the same as the SUSY masses in the decay and

fq is large. This means that X2 is minimized when we have perfect match in mass
spectrum. The red circle is the SPSla point.

Now since we don't have experimental data yet, we generated data samples

from SPSla assuming 10fb-1 and constructed the .1-1...',,i 1i1i l in SUSY and UED

in fig. 3-15(b). We included 10% jet energy resolution. Red dots represent data

points and the red line is the SUSY fit to the data points and the blue lines are

the UED fits to the data points for two different f,'s. For SUSY, X2 is around

1 as we expect. We can get better X2 for UED from 9.1 to 4.5 by increasing fq.





















0.5 0 100 200 300 400
0.2 0.4 0.6 0.8 1.0 Mq V)
X M1q (GeV)

Figure 3-15: Asymmetries with relaxed conditions. (a) The contour lines show X2
in the (x, fq) parameter space and the red dot represents the SPSla point. X2 is
minimized when fq ~ 1 and x is the same as for SPSla. (b) Red dots represent
the data points with statistical error bars generated from SPSla with L = 10
fb-1 including 10% jet energy resolution. 2-minimized UED (SUSY) fits to data
are shown in blue (red). Since data was generated from SUSY, small X2 in the
SUSY fit is expected. X2 in the UED fits is 9.1 (blue sold) and 4.5 (blue dotted) for
fq = 0.7 and fq1=, respectively.


It is still too big to fit the experimental data. So our conclusion for this study is

that a particular point like SPSla can not be faked through the entire parameter

space of UED. However we need to check whether this conclusion will remain the

same when we include the wrong jets which have nothing to do with this decay

chain [96]. Notice that the clear edge at mq, ~ 300 GeV in fig. 3-14(b) disappeared

in fig. 3-15(b) after including jet energy resolution. From fig. 3-14, we see that

SUSY has a larger .'.-, ....I. I ry.

3.3 UED and SUSY at Linear Colliders

Universal Extra Dimensions and supersymmetry have rather similar exper-

imental signatures at hadron colliders. The proper interpretation of an LHC

discovery in either case may therefore require further data from a lepton collider.

In this section we identify methods for discriminating between the two scenarios

at the linear collider. We will consider 3 TeV Compact Linear Collider (CLIC).

We study the processes of Kaluza-Klein muon pair production in universal extra









dimensions in parallel to smuon pair production in supersymmetry, accounting for

the effects of detector resolution, beam-beam interactions and accelerator induced

backgrounds. We find that the angular distributions of the final state muons, the

energy spectrum of the radiative return photon and the total cross-section measure-

ment are powerful discriminators between the two models. Accurate determination

of the particle masses can be obtained both by a study of the momentum spectrum

of the final state leptons and by a scan of the particle pair production thresholds.

We also calculate the production rates of various Kaluza-Klein particles and discuss

the associated signatures.

3.3.1 Event Simulation and Data Analysis

In order to study the discrimination of UED signals from supersymmetry,

we have implemented the relevant features of the minimal UED model in the

CompHEP event generator [68]. The MSSM is already available in CompHEP since

version 41.10. All n = 1 KK modes are incorporated as new particles, with the

proper interactions and one-loop corrected masses [28]. The widths can then be

readily calculated with CompHEP on a case by case basis and added to the particle

table. Similar to the '!l case, the neutral gauge bosons at level 1, Z1 and 71, are

mixtures of the KK modes of the hypercharge gauge boson and the neutral SU(2)w

gauge boson. However, it was shown in [62] that the radiatively corrected Weinberg

angle at level 1 and higher is very small. For example, 71, which is the LKP in the

minimal UED model, is mostly the KK mode of the hypercharge gauge boson. For

simplicity, in the code we neglect neutral gauge boson mixing for n > 1.

In the next section we concentrate on the pair production of level 1 KK

muons e+e- Ptt and compare it to the analogous process of smuon pair

production in supersymmetry: e+e- /+1-. In UED there are two n = 1

KK muon Dirac fermions: an SU(2)w doublet ft1 and an SU(2)w singlet /f,

both of which contribute in eqn. (3-14) below (see also fig. 3-16). In complete










e l e+ 2, Z2



+ +

(a) (b)
Figure 3-16: The dominant Feynman diagrams for KK muon production
e+e- Pi, in Universal Extra Dimensions. The black dot represents a KK-
number violating boundary interaction [28].



Z /





Figure 3-17: The dominant Feynman diagrams for smuon production
e+e > m+/ in supersymmetry.


analogy, in supersymmetry, there are two smuon eigenstates, [L and ,R, both of

which contribute in eqn. (3-15). The dominant diagrams in that case are shown in

fig. 3-17. In principle, there are also diagrams mediated by 7,, Z, for n = 4, 6,...

but they are doubly suppressed by the KK-number violating interaction at both

vertices and the KK mass in the propagator and here can be safely neglected.

However, 72 and Z2 exchange (fig. 3-16b) may lead to resonant production and

significant enhancement of the cross-section, as well as interesting phenomenology

as discussed below in Section 3.3.2.5. We have implemented the level 2 neutral

gauge bosons 72, Z2 with their widths, including both KK-number preserving

and the KK-number violating decays as in Ref. [62]. We consider the final state

consisting of two opposite sign muons and missing energy. It may arise either from

KK muon production in UED


e+e- C +11 ->Lt 71-1,


(3-14)









Table 3-1: Masses of the KK excitations for R-1 = 500 GeV and AR = 20 used in
the analysis.

Particle Mass
11 515.0 GeV
11f 505.4 GeV
71 500.9 GeV

with 71 being the LKP, or from smuon pair production in supersymmetry:

C+C- -- k + --p+ p+-x, (3-15)


where 4o is the lightest supersymmetric particle. We reconstruct the muon energy

spectrum and the muon production polar angle, aiming at small background from

S\! processes with minimal biases due to detector effects and selection criteria.

The goal is to disentangle KK particle production (3-14) in UED from smuon

pair production (3-15) in supersymmetry. We also determine the masses of the

produced particles and test the model predictions for the production cross-sections

in each case.

We first fix the UED parameters to R- = 500 GeV, AR = 20, leading to

the spectrum given in Table 3-1. The ISR-corrected signal cross-section in UED

for the selected final state [ +-Y1j 1 is 14.4 fb at = 3 TeV. Events have been

generated with CompHEP and then reconstructed using a fast simulation based

on parametrized response for a realistic detector at CLIC. In particular, the

lepton identification efficiency, momentum resolution and polar angle coverage

are of special relevance to this analysis. We assume that particle tracks will be

reconstructed through a discrete central tracking system, consisting of concentric

layers of Si detectors placed in a 4 T solenoidal field. This ensures a momentum

resolution 6p/p2 = 4.5x 10-5 GeV-1. A forward tracking system should provide

track reconstruction down to ~- 100. We also account for initial state radiation

(ISR) and for beamstrahlung effects on the center-of-mass energy. We assume that









muons are identified by their penetration in the instrumented iron return yoke of

the central coil. A 4 T magnetic field sets an energy cutoff of 5 GeV for muon

I. '- '- i I .

The events from the CompHEP generation have been treated with the Pythia

6.210 parton shower [101] and reconstructed with a modified version of the SimDet

4.0 program [102]. Beamstrahlung has been added to the CompHEP generation.

The liulii,. -il v spectrum, obtained by the GuineaPig beam simulation for the

standard CLIC beam parameters at 3 TeV, has been parametrised using a modified

Yokoya-Chen approximation [103,104]:

This analysis has backgrounds coming from [\! utpp-vt final states, which are

mostly due to gauge boson pair production W+W- Lt+tP-v,,, ZZ0 -p+-t ,p

and from e+e -- W+W-, v6, e+e- ZoZoe,, followed by muonic decays.

The background total cross-section is ~20 fb at s = 3 TeV. In addition to its

competitive cross-section, this background has leptons produced preferentially at

small polar angles, therefore biasing the angular distribution. In order to reduce

this background, a suitable event selection has been applied. Events have been

required to have two muons, missing energy in excess to 2.5 TeV, transverse energy

below 150 GeV and event sphericity larger than 0.05. In order to reject the ZoZ

background, events with di-lepton invariant mass compatible with Mzo have also

been discarded. The underlying 77 collisions also produces a potential background

to this analysis in the form of yy -+ ptp-. This background has been simulated

using the CLIC beam simulation and Pythia. Despite its large cross-section, it

can be completely suppressed by a cut on the missing transverse energy EmS2 >

50 GeV. Finally, in order to remove events with large be. I-1 il1.11]Ii.-. the event

sphericity had to be smaller than 0.35 and the acolinearity smaller than 0.8. These

criteria provide a factor 30 background suppression, in the kinematical region of

interest, while not significantly biasing the lepton momentum distribution.









Table 3-2: MSSM parameters for the SUSY study point used in the analysis. This
choice of soft SUSY parameters in CompHEP leads to an exact match between the
corresponding UED and SUSY mass spectra.

MSSM Parameter Value
1t 1000 GeV
M1 502.65 GeV
_f 1005.0 GeV
if,. 512.83 GeV
i ,,, 503.63 GeV
tan/ 10


3.3.2 Comparison of UED and Supersymmetry in p[+ [-TT

In order to perform the comparison of UED and MSSM, we adjusted the

MSSM parameters to get the two smuon masses .,;. and .;, and the lightest

neutralino mass M0o matching exactly those of the two Kaluza-Klein muons M/

and Ms and of the KK photon MI, for the chosen UED parameters. It must be

stressed that such small mass splitting between the two muon partners is typically

rather accidental in supersymmetric scenarios. The supersymmetric parameters

used are given in Table 3-2. We then simulate both reactions (3-14) and (3-15)

with CompHEP and pass the resulting events through the same simulation and

reconstruction. The ISR-corrected signal cross-section in SUSY for the selected

final state jf 44f is 2.76 fb at = 3 TeV, which is about 5 times smaller than

in the UED case.

3.3.2.1 Angular Distributions and Spin Measurements

In the case of UED, the KK muons are fermions and their angular distribution

is given by

d( cos1 + os2 (3 16)
d os 0 ED E 21
Assuming that at CLIC the KK production takes place well above threshold, the

formula simplifies to:

( dc ) -I cos2 0. (3-17)
d cos0 UED












0.012 ------------------------- 1 1---------- ---------------
800 200
R 500 GeV00 8 200
0100 Ecu 3 TeV 700175

po co0 600 150
,3l 0 008 -
ee ep+ : 500 125
S006- 400 100
o

b 004 300 75
+ e +e -j ~o 200 50
0.002
100 25
.000o 0 0
-1.0 -0.5 o.o 0.5 1.0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
cosO/ cos 90

Figure 3-18: Differential cross-section dc/dcosO,, for UED (blue, top) and super-
symmetry (red, bottom) as a function of the muon scattering angle 0,,. The figure
on the left shows the ISR-corrected theoretical prediction. The two figures on the
right in addition include the effects of event selection, beamstrahlung and detector
resolution and acceptance. The left (right) panel is for the case of UED (supersym-
metry). The data points are the combined signal and background events, while the
yellow-shaded histogram is the signal only.


As the supersymmetric muon partners are scalars, the corresponding angular

distribution is

(d cs d 1 cos2 0. (318)
d cos 0 sUY

Distributions (3-17) and (3-18) are sufficiently distinct to discriminate the two

cases. However, the polar angles 0 of the original KK-muons and smuons are not

directly observable and the production polar angles 0Q of the final state muons are

measured instead. But as long as the mass differences MlI MI, and I3; M1,

respectively remain small, the muon directions are well correlated with those of

their parents (see figure 3-18a). In fig. 3-18b we show the same comparison after

detector simulation and including the Y\ background. The angular distributions

are well distinguishable also when accounting for these effects. By performing a X2

fit to the normalised polar angle distribution, the UED scenario considered here










10 10 U





10 2
lo t












1010 1015 102 1 1025 1030 1035 1040 1045 1050 1010 1015 1020 1025 1030 1035 1040 1045
Ws (GeV) E (GeV)

Figure 3-19: The total cross-section a in pb as a function of the center-of-mass
energy s near threshold for e+e- -- Pft -[ P+li-i. Left: the threshold on-
set with (line, blue) and without (dots) beamstrahlung effects. Right: a threshold
scan at selected points. The green curve refers to the reference UED parameters
while for the red (blue) curve the mass of /f (/f) has been lowered by 2.5 GeV.
The points indicate the expected statistical accuracy for the cross section determi-
nation at the points of maximum mass sensitivity. Effects of the CLIC luminosity
spectrum are included.


could be distinguished from the MSSM, on the sole basis of the distribution shape,

with 350 fb-1 of data at s 3 TeV.

3.3.2.2 Threshold Scans

At the e+e- linear collider, the muon excitation masses can be accurately

determined through an energy scan of the onset of the pair production threshold.

This study not only determines the masses, but also confirms the particle nature.

In fact the cross-sections for the UED processes rise at threshold oc 4 while in

supersymmetry their threshold onset is oc i3, where 3 is the particle velocity. Since

the collision energy can be tuned at properly chosen values, the power rise of the

cross-section can be tested and the masses of the particles involved measured. We

have studied such threshold scan for the e+e- -- i p[+1 --+ +-/l1i process at s

S1 TeV, for the same parameters as in Table 3-1. We account for the anticipated

CLIC centre-of-mass energy spread induced both by the energy spread in the CLIC









linac and by beam-beam effects during collisions. This been obtained from the

detailed GuineaPig beam simulation and parametrised using the modified Yokoya-

Chen model [103, 105]. An optimal scan of a particle pair production threshold

consists of just two energy points, sharing the total integrated Ilili.-ili v in equal

fractions and chosen at energies maximising the sensitivity to the particle widths

and masses [106]. For the UED model scan we have taken three points, one for

normalisation and two at the maxima of the mass sensitivity (see figure 3-19).

Inclusion of beamstrahlung effects induces a shift of the positions of these maxima

towards higher nominal s values [107]. From the estimated sensitivity dlo/dM

and the cross-section accuracy, the masses of the two UED muon excitations can

be determined to 0.11 GeV and 0.23 GeV for the singlet and the doublet states

respectively, with a total liiiiiril'. i of 1 ab-1 shared in three points, when the

particle widths can be disregarded.

3.3.2.3 Production Cross-Section Determination

The same analysis can be used to determine the cross-section for the process

e+e- -- *+p -fT. The \!I contribution can be determined independently, using

anti-tag cuts, and subtracted. Since the cross-section for the UED process at

3 TeV is about five times larger compared to smuon production in supersymmetry,

this measurement would reinforce the model identification obtained by the spin

determination. This can be quantified by performing the same \2 fit to the muon

polar production angle discussed above, but now including also the total number of

selected events. Since the cross-section depends on the mass of the pair produced

particles, we include a systematic uncertainty on the prediction corresponding to a

0.05 % mass uncert.iiilr v, which is consistent with the results discussed below. At

CLIC the absolute luminosity should be measurable to 0(0.1 %) and the average

effective collision energy to 0(0.01 %).












0.5 140
o 600 -
R= 500 GeV 120
0.4 E = 3 TeV
500 -
B 100
S8400
- 0.3 80

W:L 300 60
0.2
b 200 40
e+e -gC /, 1
0.1
100 20
+- + -o0-o
0.0 0 0
0 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100
E, (GeV) p (GeV)

Figure 3-20: The muon energy spectrum resulting from KK muon production
(3-14) in UED (blue, top curve) and smuon production (3-15) in supersymmetry
(red, bottom curve). The UED and SUSY parameters are chosen as in fig. 3-18.
The plot on the left shows the ISR-corrected distribution, while that on the right
includes in addition the effects of event selection, beamstrahlung and detector res-
olution and acceptance. The data points are the combined signal and background
events, while the yellow-shaded histogram is the signal only.


3.3.2.4 Muon Energy Spectrum and Mass Measurements

The characteristic end-points of the muon energy spectrum are completely

determined by the kinematics of the two-body decay and hence they don't depend

on the underlying framework (SUSY or UED) as long as the masses involved are

tuned to be identical. We show the ISR-corrected expected distributions for the

muon energy spectra at the generator level in fig. 3-20a, using the same parameters

as in fig. 3-18. As expected, the shape of the E, distribution in the case of UED

coincides with that for MSSM. The lower, Emi,, and upper, Emx,, endpoints of the

muon energy spectrum are related to the masses of the particles involved in the

decay according to the relation:


1 M ( 1
Emax/min = 1 (1 ) (3-19)
2 \









where 31,; and MAo are the smuon and LSP masses and = 1/(1 2) 12 with

3 /1 A/Eeam, is the / boost. In the case of the UED the formula is

completely analogous with .,, replacing 1; and M, replacing M0o.

Due to the splitting between the ,L and iR masses in MSSM and that be-

tween the pf and /ft masses in UED, in fig. 3-20a we see the superposition of two

box distributions. The left, narrower distribution is due to p/ pair production in

UED (fR pair production in supersymmetry). The underlying, much wider box

distribution is due to p pair production in UED (hL pair production in supersym-

metry). The upper edges are well defined, with smearing due to beamstrahlung

and, but less importantly, to momentum resolution. The lower end of the spectrum

has the overlap of the two contributions and with the underlying background.

Furthermore, since the splitting between the masses of the pt pj- and that of 71

is small, the lower end of the momentum distribution can be as low as 0(1 GeV)

where the lepton identification efficiency is cut-off by the solenoidal field bending

the lepton before it reaches the electro-magnetic or the hadron calorimeter [109].

Nevertheless, there is sufficient information in this distribution to extract the mass

of the 71 particle, using the prior information on the /ft and /(l masses, obtained

by the threshold scan.

In fig. 3-20b we show the muon energy distribution after detector simulation.

A one parameter fit gives an uncert.iil v, on the 71 mass of 0.19 Statt.) 0.21 (syst) GeV,

where the statistical uncertainty is given for 1 ab-' of data and the systematics

reflects the effect of the uncert.iiilr on the p/ masses. The beamstrahlung intro-

duces an additional systematics, which depends on the control of the details of the

lialiiii. -ilIv spectrum.

3.3.2.5 Photon Energy Spectrum and Radiative Return to the Z2

With the e+e- colliding at a fixed center-of-mass energy above the pair

production threshold a significant fraction of the KK muon production will proceed










4 30

R_ t-------- j-------
SRR1 = 1350 GeV
3 25
SECM = 3 TeV



10
2 15 +-




100 125 150 175 200 225 250 120 140 160 18 200 22 24
E, (GeV) Ephoton (GeV)

Figure 3 21: Photon energy spectrum in ~+e -- pt- for R`-1 1350 GeV,
AR = 20 and EcM = 3 TeV before (left) and after (right) detector simulation. The
acceptance cuts are E, > 10 GeV and 1 < 08 < 1790. The mass of the Z2 resonance
is 2825 GeV.

through radiative return. Since this is mediated by s-channel narrow resonances, a

sharp peak in the photon energy spectrum appears whenever one of the mediating

s-channel particles is on-shell. In case of supersymmetry, only Z and 7 particles

can mediate smuon pair production and neither of them can be close to being

on-shell. On the contrary, an interesting feature of the UED scenario is that t1,

production can be mediated by Z,n and 7,> KK excitations (for n even) as shown in

fig. 3-16b. Among these additional contributions, the Z2 and 72 exchange diagrams

are the most important. Since the decay Z2 -- /i1/iL is allowed by phase space,

there will be a sharp peak in the photon spectrum, due to a radiative return to the

Z2. The photon peak is at

E = ECM 1 Z (3-20)
2 E CM

On the other hand, M.2 < 2 1,, so that the decay '2 1ti/ li is closed, and

therefore there is no radiative return to 72. Notice that the level 2 Weinberg angle

is very small [28] and therefore Z2 is mostly W,2-like and couples predominantly to









fI and not ,f. The photon energy spectrum in e+e- p1 7 for R- = 1350

GeV, AR = 20 and EcM = 3 TeV is shown in fig. 3-21. On the left we show the

ISR-corrected theoretical prediction from CompHEP while the result on the right in

addition includes detector and beam effects. It is clear that the peak cannot be

missed.

3.3.3 Prospects for Discovery and Discrimination in Other Final States

Previously in section 3.3.2 we considered the p+ft-fT final state resulting

from the pair production of level 1 KK muons. However, this is not the only signal

which could be expected in the case of UED. Due to the relative degeneracy of

the KK particles at each level, the remaining n = 1 KK modes will be produced

as well, and will yield observable signatures. In those cases, the discrimination

techniques which we discussed earlier can still be applied, providing further

evidence in favor of one model over the other. In this section we compute the

cross-sections for some of the other main processes of interest, and discuss how they

could be analyzed.

3.3.3.1 Kaluza-Klein Leptons

We first turn to the discussion of the other KK lepton flavors. The KK 7-

leptons, T-, are also produced in s-channel diagrams only, as in fig. 3-16, hence

the T- -- production cross-sections are very similar to the p t+~ case. The final

state will be -T+r-fT, and it can be observed in several modes, corresponding to

the different options for the T decays. However, due to the lower statistics and the

inferior jet energy resolution, none of the resulting channels can compete with the

discriminating power of the pJ JIIfT final state discussed in the previous section.

The case of KK electrons is more interesting, as it contains a new twist.

The production of KK electrons can also proceed through the t-channel diagram

shown in fig. 3-22c. As a result, the production cross-sections for KK electrons

can be much higher than for KK muons. We illustrate this in fig. 3-23, where we










e+ e e+ e -+ ef
\ 7 Z 72, Z2

e- e eI e- e1

(a) (b) (c)
Figure 3-22: The dominant Feynman diagrams for KK electron production, e+e-
e1eI, the same as fig. 3-16.

show separately the cross-sections for SU(2)w doublets (solid lines) and SU(2)w

singlets (dotted lines), as a function of R-1. (For the numerical results throughout

section 3.3.3, we always fix AR = 20.) At low masses (i.e. low R-1) the ee+c

cross-sections can be up to two orders of magnitude larger, compared to the case of

l+[l. Another interesting feature is the resonant enhancement of the cross-section
for R-1 ~ 1450 GeV, which is present in either case (e or p) for the SU(2)w

doublets (solid lines), but not the SU(2)w singlets (dotted lines). The feature

is due to the on-shell production of the level 2 Z2 KK gauge boson, which can

then decay into a pair of level 1 KK leptons (see diagram (b) in figs. 3-16 and

3-22). Since the Weinberg angle at the higher (n > 0) KK levels is tiny [28],

Z2 is predominantly an SU(2)w gauge boson and hence does not couple to the

SU(2)w singlet fermions, which explains the absence of a similar peak in the

ef and pf cross-sections3 Because of the higher production rates, the e+e-Tf

event sample will be much larger and have better statistics than [+ t-4T. The

e+e- C T final state has been recently advertised as a discriminator between UED

and supersymmetry in [108]. However, the additional t-channel diagram (fig. 3-22c)

has the effect of not only enhancing the overall cross-section, but also distorting



3 One might have expected a second peak closeby due to 72 resonant production,
but in the minimal UED model the spectrum is such that the decays of 72 to level
1 fermions are all closed.










10
100~ + + Ecu = 3 TeV




10-1
b 10-2 I I ,





250 500 750 1000 1250 1500
R-1 (GeV)

Figure 3-23: ISR-corrected production cross-sections of level 1 KK leptons (ei in
red, ft1 in blue) at CLIC, as a function of R-1. Solid (dotted) lines correspond to
SU(2)w doublets (singlets).


the differential angular distributions discussed previously in Section 3.3.2.1, and

creating a forward peak, which causes the cases of UED and supersymmetry to

look very much alike. We show the resulting angular distributions of the final

state electrons in fig. 3-24. For proper comparison, we follow the same procedure

as before: we choose the UED spectrum for R- = 500 GeV, which yields KK

electron masses as in Table 3-1. We then choose a supersymmetric spectrum with

selection mass parameters as in Table 3-2. This guarantees matching mass spectra

in the two cases (UED and supersymmetry) so that any differences in the angular

distributions should be attributed to the different spins. Unlike fig. 3-18, where the

underlying shapes of the angular distributions were very distinctive (see eqs. (3-17)

and (3-18)), the main effect in fig. 3-24 is the uniform enhancement of the forward

scattering cross-section, which tends to wash out the spin correlations exhibited in

fig. 3-18.

3.3.3.2 Kaluza-Klein Quarks

Level 1 KK quarks will be produced in s-channel via diagrams similar to those

exhibited in fig. 3-16. The corresponding production cross-sections are shown

in fig. 3-25, as a function of R-1. We show separately the cases of the SU(2)w










R = 500 GeV
100 Ec, 3 TeV


10-1 ee ee -e-y,7


10-2

e eme Ce Xii
103
-1.0 -0.5 0.0 0.5 1.0
COSOe

Figure 3-24: Differential cross-section dor/dcosOe for UED and supersymmetry.
The same as fig. 3-18 (left panel), but for KK electron production e+e-> e+ e,
with 09 being the electron scattering angle.


doublets uf and df and the SU(2)w singlets uf and df. In the minimal UED

model, the KK fermion doublets are somewhat heavier than the KK fermion

singlets [28], so naturally, the production cross-sections for uf and df cut off

at a smaller value of R-1. Since singlet production is only mediated by U(1)

hypercharge interactions, the singlet production cross-sections tend to be smaller.

We notice that usuf is larger by a factor of 22 compared to dfd>, in accordance

with the usual quark hypercharge assignments.

The observable signals will be different in the case of SU(2)w doublets and

SU(2)w singlets. The singlets, us and df, decay directly to the LKP 71, and

the corresponding signature will be 2 jets and missing energy. The jet angular

distribution will again be indicative of the KK quark spin, and can be used to

discriminate against (right-handed) squark production in supersymmetry, following

the procedure outlined in section 3.3.2.1. The jet energy distribution will again

exhibit endpoints, which will in principle allow for the mass measurements dis-

cussed in section 3.3.2.4. A threshold scan of the cross-section will provide further

evidence of the particle spins (see section 3.3.2.2). The only major difference with

respect to the [p+t-JT final state discussed in section 3.3.2, is the absence of the














10- -


b
2

10-3

250 500 750 1000 1250 1500
R-' (GeV)

Figure 3-25: ISR-corrected production cross-sections of level 1 KK quarks at CLIC,
as a function of R-1


monochromatic photon signal from section 3.3.2.5, since Z2 is too light to decay to

KK quarks. In spite of the many similarities to the dimuon final state considered

in section 3.3.2, notice that jet angular and energy measurements are not as clean

and therefore the lepton (muon or electron) final states would still provide the

most convincing evidence for discrimination. The signatures of the SU(2)w doublet

quarks are richer -both uD and df predominantly decay to Z1 and W/V which in

turn decay to leptons and the LKP [62]. The analogous process in supersymmetry

would be left-handed squark production with subsequent decays to xo or xt, which

in turn decay to tL and 4o. In principle, the spin information will still be encoded

in the angular distributions of the final state particles. However, the analysis is

much more involved, due to the complexity of the signature, and possibly the

additional missing energy from any neutrinos.

3.3.3.3 Kaluza-Klein Gauge Bosons

The ISR-corrected production cross-sections for level 1 electroweak4 KK

gauge bosons (Wt, Z1 and 71) at a 3 TeV e+e- collider are shown in fig. 3-26, as


4 The level 1 KK gluon, of course, has no tree-level couplings to ee-.









a function of R-1. The three relevant processes are W,+W ZIZ1 and Z171 (7171

is unobservable). In each case, the production can be mediated by a t-channel

exchange of a level 1 KK lepton, while for W,+W1 there are additional s-channel

diagrams with 7, Z, 72 and Z2. Z1 and W1f are almost degenerate [28], thus

their cross-sections cut off at around the same point. The analogous processes

in supersymmetry would be the pair production of gaugino-like charginos and

neutralinos. The final states will always involve leptons and missing energy,

since W1 and Z1 do not decay to KK quarks. In conclusion of this section, for

completeness we also discuss the possibility of observing the higher level KK

particles and in particular those at level 2. For small enough R-1, level 2 KK

modes are kinematically accessible at CLIC. Once produced, they will in general

decay to level 1 particles and thus contribute to the inclusive production of level

1 KK modes. Uncovering the presence of the level 2 signal in that case seems

challenging, but not impossible.

We choose to concentrate on the case of the level 2 KK gauge bosons (V2),

which are somewhat special in the sense that they can decay directly to \!I

fermions through KK number violating interactions. Thus they can be easily

observed as dijet or dilepton resonances. In principle, there are two types of

production mechanisms for level 2 gauge bosons. The first is single production

e+e- -- V2, which can only proceed through KK number violating (loop sup-

pressed) couplings. The second mechanism is e+e -- V2V2 pair production which

is predominantly due to KK number conserving (tree-level) couplings. In fig. 3-27

we show the corresponding cross-sections for the case of the neutral level 2 gauge

bosons, as a function of R-1, For low values of R-1, pair production dominates, but

as the level 2 gauge boson masses increase and approach ECM, single production

becomes resonantly enhanced. Thus the first indication of the presence of the level

2 particles may come from pair production events, but once the mass of the dijet













10-1 e+e--WW-
5
e+e--ZZl

b 10-2
5


10-3

250 500 750 1000 1250 1500
R-1 (GeV)

Figure 3-26: ISR-corrected production cross-sections of level 1 KK gauge bosons at
CLIC, as a function of R-1.


or dilepton resonance is known, the collider energy can be tuned to enhance the

cross-section and study the V2 resonance properties in great detail. Supersymmetry

and Universal Extra Dimensions are two appealing examples of new pi-, -i. -, at the

TeV scale, as they address some of the theoretical puzzles of the \ !. They also

provide a dark matter candidate which, for properly chosen theory parameters,

is consistent with present cosmology data. Both theories predict a host of new

particles, partners of the known \ particles. If either one is realized in nature,

the LHC is expected to observe signals of these new particles. However, in order to

clearly identify the nature of the new pir, -i. -. one may need to contrast the UED

and supersymmetric hypotheses at a multi-TeV e+e- linear collider such as CLIC5

We studied in detail the process of pair production of muon partners in the two

theories, KK-muons and smuons respectively. We used the polar production angle

to distinguish the nature of the particle partners, based on their spin. The same




5 Similar studies can also be done at the ILC provided the level 1 KK particles
are within its kinematic reach. Since precision data tends to indicate the bound
R-1 > 250 GeV for the case of 1 extra dimension, one would need an ILC center-of-
mass energy above 500 GeV in order to pair-produce the lowest lying KK states of
the minimal UED model.












10-1

e+e->Z Z,

e+e- -y2Z2
b ee -> Z2
10-3


10-4
250 500 750 1000 1250 1500
R-1 (GeV)

Figure 3-27: ISR-corrected production cross-sections of level 2 KK gauge bosons at
CLIC, as a function of R-1.


analysis could be applied for the case of other KK fermions, as discussed in sec-

tion 3.3.3. We have also studied the accuracy of CLIC in determining the masses of

the new particles involved both through the study of the energy distribution of final

state muons and threshold scans. An accuracy of better than 0.1% can be obtained

with 1 ab-1 of integrated lliiliir. -il' Once the masses of the partners are known,

the measurement of the total cross-section serves as an additional cross-check on

the hypothesized spin and couplings of the new particles. A peculiar feature of

UED, which is not present in supersymmetry, is the sharp peak in the ISR photon

energy spectrum due to a radiative return to the KK partner of the Z.

The clean final states and the control over the center-of-mass energy at the

CLIC multi-TeV collider allows one to unambiguously identify the nature of the

new ll, v-i signals which might be emerging at the LHC already by the end of this

decade.















CHAPTER 4
COY \ BIOLOGICAL IMPLICATIONS

4.1 Dark Matter Abundance

In this chapter, we now focus on the cosmological implications of Universal

Extra Dimensions. We revisit the calculation of the relic density of the lightest

Kaluza-Klein particle (LKP) in the model of Universal Extra Dimensions. The

first and only comprehensive calculation of the UED relic density to date was

performed in [21]. The authors considered two cases of LKP: the KK hypercharge

gauge boson B1 and the KK neutrino vl. The case of B1 LKP is naturally obtained

in MUED, where the radiative corrections to B1 are the smallest in size, since

they are only due to hypercharge interactions. The authors of [21] also realized

the importance of coannihilation processes and included in their analysis coanni-

hilations with the SU(2)w-singlet KK leptons, which in MUED are the lightest

among the remaining n = 1 KK particles. It was therefore expected that their

coannihilations will be most important. Subsequently, Refs. [48, 49] analyzed the

resonant enhancement of the n = 1 (co)annihilation cross-sections due to n = 2 KK

particles.

Our goal in this chapter will be to complete the LKP relic density calculation

of Ref. [21]. We will attempt to improve in three different aspects:

We will include coannihilation effects with all n = 1 KK particles. The moti-

vation for such a tour de force is twofold. First, recall that the importance of

coannihilations is mostly determined by the degeneracy of the corresponding

particle with the dark matter candidate. In the minimal UED model, the KK

mass splitting are due almost entirely to radiative corrections. In MUED,

therefore, one might expect that, since the corrections to KK particles other









than the KK leptons are relatively large, their coannihilations can be safely

neglected. However, the minimal UED model makes an ansatz [28] about

the cut-off scale values of the so called boundary terms, which are not fixed

by known \ I ',l1-i, -. and are in principle arbitrary. In this sense, the UED

scenario should be considered as a low energy effective theory with a mul-

titude of parameters, just like the MSSM, and the MUED model should be

treated as nothing more than a simple toy model with a limited number of

parameters, just like the "minimal sup. I;, .1 il1" version of supersymmetry,

for example. If one makes a different assumption about the inputs at the

cut-off scale, both the KK spectrum and its phenomenology can be modified

significantly. In particular, one could then easily find regions of this more

general parameter space where other coannihilation processes become active.

On the other hand, even if we choose to restrict ourselves to MUED, there

is still a good reason to consider the coannihilation processes which were

omitted in the analysis of [21]. While it is true that those coannihilations

are more Boltzmann suppressed, their cross-sections will be larger, since

they are mediated by weak and/or strong interactions. Without an explicit

calculation, it is impossible to estimate the size of the net effect, and whether

it is indeed negligible compared to the purely hypercharge-mediated processes

which have already been considered.

* We will keep the exact value of each KK mass in our formulas for all an-

nihilation cross-sections. This will render our analysis self-consistent. All

calculations of the LKP relic density available so far [21, 48, 49], have com-

puted the annihilation cross-sections in the limit when all level 1 KK masses

are the same. This approximation is somewhat contradictory in the sense

that all KK masses at level one are taken to be degenerate with LKP, yet

only a limited number of coannihilation processes were considered. In reality,









a completely degenerate spectrum would require the inclusion of all possi-

ble coannihilations. Conversely, if some coannihilation processes are being

neglected, this is presumably because the masses of the corresponding KK

particles are not degenerate with the LKP, and are Boltzmann suppressed.

However, the masses of these particles may still enter the formulas for the

relevant coannihilation cross-sections, and using approximate values for

those masses would lead to a certain error in the final answer. Since we are

keeping the exact mass dependence in the formulas, within our approach

heavy particles naturally decouple, coannihilations are properly weighted,

and all relevant coannihilation cross-sections behave properly. Notice that

the assumption of exact mass degeneracy overestimates the corresponding

cross-sections and therefore underestimates the relic density. This expectation

will be confirmed in our numerical analysis in Section 4.3.

We will try to improve the numerical accuracy of the analysis by taking into

account some minor corrections which were neglected or approximated in [21].

For example, we will use a temperature-dependent g, (the total number of

effectively massless degrees of freedom, given by eq. (4-6) below) and include

subleading corrections (4-19) in the velocity expansion of the annihilation

cross-sections.

The availability of the calculation of the remaining coannihilation processes

is important also for the following reason. Coannihilations with SU(2)w-singlet

KK leptons were found to reduce the effective annihilation cross-section, and

therefore increase the LKP relic density. This has the effect of lowering the range

of cosmologically preferred values of the LKP mass, or equivalently, the scale of

the extra dimension. However, one could expect that coannihilations with the

other n = 1 KK particles would have the opposite effect, since they have stronger

interactions compared to the SU(2)w-singlet KK leptons and the B1 LKP. As a









result, the preferred LKP mass range could be pushed back up. For both collider

and astroparticle searches for dark matter, a crucial question is whether there

is an upper limit on the WIMP mass which could guarantee discovery, and if

so, what is its precise numerical value. To this end, one needs to consider the

effect of all coannihilation processes which have the potential to enhance the LKP

annihilations. We will see that the lowering of the preferred LKP mass range in the

case of coannihilations with SU(2)w-singlet KK leptons is more of an exception

rather than the rule, and the inclusion of all remaining processes is needed in order

to derive an absolute upper bound on the LKP mass.

4.2 The Basic Calculation of the Relic Density

4.2.1 The Standard Case

We first summarize the standard calculation for the relic abundance of a

particle species X which was in thermal equilibrium in the early universe and

decoupled when it became nonrelativistic [21, 110, 111]. The relic abundance is

found by solving the Boltzmann equation for the evolution of the X number density

n
dn
= -3Hn- (av)(n2 n (41)
dt
where H is the Hubble parameter, v is the relative velocity between two X's, (tv) is

the thermally averaged total annihilation cross-section times relative velocity, and

neq is the equilibrium number density. At high temperature (T > m), neq T3
(there are roughly as many X particles as photons). At low temperature (T < m),

in the nonrelativistic approximation, ,eq can be written as
3
(eq g(jmTj2e-mT (42)
(2 )3

where m is the mass of the relic X, T is the temperature and g is the number

of internal degrees of freedom of X such as spin, color and so on. We see from

eq. (4-2) that the density neq is Boltzmann-suppressed. At high temperature, X









particles are abundant and rapidly convert to lighter particles and vice versa. But

shortly after the temperature T drops below m, the number density decreases

exponentially and the annihilation rate F (uv)n drops below the expansion

rate H. At this point, X's stop annihilating and escape out of the equilibrium

and become thermal relics. (av) is often approximated by the nonrelativistic

expansionI

(v} a+b(v2) + ((v)) ~a +6b/x+O O2 (4-3)

where

X- (4-4)
T
By solving the Boltzmann equation analytically with appropriate approximations

[21, 110, 111], the abundance of X is given by

X h2 1.04 x 109 XF 1 (4 5)
R-- (4-5)
Mp, V () a + 3b/xF

where the Planck mass Mp = 1.22 x 1019 GeV and g. is the total number of

effectively massless degrees of freedom,


(.(T)= + gY (4-6)
i=bosons i fermions

The freeze-out temperature, xF, is found iteratively from

1 45 g mnMp,(a+6b/xF) \
XF In c(c+2) ,8 ( (4-7)
V8 2 Vg(xx

where the constant c is determined empirically by comparing to numerical solutions

of the Boltzmann equation and here we take c = as usual. The coefficient in



1 Note, however, that the method fails near s-channel resonances and thresh-
olds for new final states [112]. In the interesting parameter region of UED, we
are always sufficiently far from thresholds, while for the treatment of resonances,
see [48, 49].









the right hand side of (4-6) accounts for the difference in Fermi and Bose statistics.

Notice that g, is a function of the temperature T, as the thermal bath quickly gets

depleted of the heavy species with masses larger than T.

4.2.2 The Case with Coannihilations

When the relic particle X is nearly degenerate with other particles in the

spectrum, its relic abundance is determined not only by its own self-annihilation

cross-section, but also by annihilation processes involving the heavier particles.

The previous calculation can be generalized to this "coannihilation" case in

a straightforward way [21, 111, 112]. Assume that the particles Xi are labeled

according to their masses, so that mi < mj when i < j. The number densities

ni of the various species Xi obey a set of Boltzmann equations. It can be shown

that under reasonable assumptions [112], the ultimate relic density n of the lightest

species X1 (after all heavier particles Xi have decayed into it) obeys the following

simple Boltzmann equation

du
-3Hn- (a ff )(n2 n), (4 8)
dt -H -e

where
N
aeff (x) yj J(1+ Ai)3/2(1 + Aj)3/2 exp(-x(A + A)) (4-9)
ij .geff
N
geff () = g( Ai)3/2 exp(-.xA) (4-10)
i 1
Ai mi (4-11)
m1

Here aij = (XiXj SM), gi is the number of internal degrees of freedom

of particle Xi and n = i1, ni is the density of X1 we want to calculate. This

Boltzmann equation can be solved in a similar way [21,112], resulting in

S 1.04 x 109 XF 1 (
Mph2 ( 3IbXF









with


la = F j aeff(x)x-2d, (413)

I = 2x beff(x)3dx (4-14)
JXF

The corresponding formula for XF becomes

( n 2)45eff(xF) mMpl(a|ff(xF)+rCl' f(x)/x1F) (415)
V 8 273" ,(x)x

Here aeff and beff are the first two terms in the velocity expansion of oeff


aeff(x) v = aeff(x) + beff(x) v2 +0(4) (4 16)

Comparing eqs. (4-9) and (4-16), one gets

N
aeff(x) aj (l+ A)3/2(1 Aj)3/2 exp(-x(Ai + Aj)), (4-17)
ij Y9eff
N
beff(x) = bjJ (1+ A)3/2(+ A)3/2 exp(-x(A + A)) (4-18)
ij .9eff

where aij and bi6 are obtained from oaijv = a + bijv2 + (v4).

Considering relativistic corrections [110, 113, 114] to the above treatment

results in an additional subleading term which can be accounted for by the simple

replacement
1
b-b- a (4-19)
4
in the above formulas, which will be explained in detail in next section.

4.2.3 Thermal Average and Nonrelativistic Velocity Expansion

The thermally averaged cross-section times relative velocity is defined as [110],

( N vd3pld~,l- :-E1T -E2/TV
(f redl = p1d,_,' T T T (4 20)
(13 d )pdl:-ElIT -E21T








where vre, is the relative velocity between two incoming particles. Since the
relic particle decouples from the equilibrium when the particle is nonrelativistic,
we can use nonrelativistic energy-momentum relation, p = mv + O(v2) and
E = m + mv2 O(4). In the CM frame, above equation becomes

f lj 3d) ,_, '-E1/Te-E2/Ta -lr
(avrel/ =
f d3d3 e 3 V2-El/T-E2
d3 Vd3 .- (V2+v) )
f d3Vd3 (V2 +2(421)
fi 1 T,
3 3 r'- l4 __,r el
fddVre e- -1
where V = +i2, 1rel = 71- 2 and El+E2 2m- mM2(v uv2) 2m+ (V2+Ve )
are used and x is x = ". Now all we need to do is to expand (Tvrez in terms of vrel
and integrate over it. The cross-section is given by

a = 2E 2F- (2 +)41'-- (. (4 22)

Now we define function w(s) using above equation,

w(s) = 2E12 J (21")44 (Pl + '-_ (27 1)3 2 (4-23)

Now in CM frame we expand -w(s) in terms of s around 4m2 with s =4m2
mV2 + O(v) and E m + m M ) (2VCM rel),

avrel = w(s) (4-24)
1
I W (S)
w(4m2) ( 4M2) 0(el)
(m+ imvl M O(
Wo mot + (v)
0 4 ) (4 25)
m2 (1+ v, + (

(W ('O W ) o 2 1 +, + ,4
e + 4e









where


wo = w(4m2)

wo 4m2 d (426)
S(dws ) 4r2

Let us define two coefficients in the velocity expansion as a and b,

wo
m2 '
w0 wo
b 4n (427)
4m2

Therefore the thermally averaged cross-section is


(avrel) = a+b(v1) + (0(v 1))

a +6bx + O(2) (4 28)


However the full relativistic calculation gives us [110]

wo 3
(uvrel) = 2n2 (w -2w )+0(x 2)

a + 6(b- -a)x + O(x2) (4 29)
4

So we expand uvrel in terms of relative velocity in the nonrelativistic limit to

get two coefficient a and b and we substitute b by b !a to recover relativistic

correction [113,114].

4.3 Relic Density in Minimal UED

For the purposes of our study we have implemented the relevant features of

the minimal UED model in the CompHEP event generator [68]. We incorporated

all n = 1 and n = 2 KK modes as new particles, with the proper interactions

and one-loop corrected masses [28]. Similar to the \ case, the neutral gauge

bosons at level 1, Z1 and 71, are mixtures of the KK modes of the hypercharge

gauge boson and the neutral SU(2)w gauge boson. However, as shown in [28],









the radiatively corrected Weinberg angle at level 1 and higher is very small. For

example, 71, which is the LKP in the minimal UED model, is mostly the KK mode

of the hypercharge gauge boson. Therefore, for simplicity, in the code we neglected

neutral gauge boson mixing for n = 1. We then use our UED implementation

in CompHEP to derive analytic expressions for the (co)annihilation cross-sections

between any pair of n = 1 KK particles. Our code has been subjected to numerous

tests and cross-checks. For example, we reproduced all results from Servant et

al. [21]. We have also used the same code for independent studies of the collider

and astroparticle signatures of UED [10, 19, 61,115] and thus have tested it from a

different angle as well.

The mass spectrum of the n = 1 KK partners in minimal UED can be found,

for example, in fig. 1 of [62]. In MUED the next-to-lightest KK particles are the

singlet KK leptons and their fractional mass difference from the LKP is2


AR1, = m1 -- m 0.01 (4-30)
mr1

Notice that the Boltzmann suppression

e-AgRF --0.01.25 C -0.25


is not very effective and coannihilation processes with iR1 are definitely important,

hence they were considered in [21]. What about the other, heavier particles in the

n = 1 KK spectrum in MUED? Since their mass splitting from the LKP


A, M m "1 (4-31)




2 In this chapter we follow the notation of [21] where the two types of n =
Dirac fermions are distinguished by an index corresponding to the chirality of their
zero mode partner. For example, tR1 stands for an SU(2)w-singlet Dirac fermion,
which has in principle both a left-handed and a right-handed component.









are larger, their annihilations suffer from a larger Boltzmann suppression. However,

the couplings of all n = 1 KK partners other than R1l are larger compared to

those of 71 and fR1. For example, SU(2)w-doublet KK leptons tL1 couple weakly,

and the KK quarks ql and KK gluon gl have strong couplings. Therefore, their

corresponding annihilation cross-sections are expected to be larger than the cross-

section of the main 7171 channel.

We see that for the other KK particles, there is a competition between

the increased cross-sections and the larger Boltzmann suppression. An explicit

calculation is therefore needed in order to evaluate the net effect of these two

factors, and judge the importance of the coannihilation processes which have been

neglected so far. One might expect that coannihilations with SU(2)w-doublet KK

leptons might be numerically significant, since their mass splitting in MUED is

~ 3% and the corresponding Boltzmann suppression factor is only e-0.0325 ~ e-o.75

In our code we keep all KK masses different while we neglect all the masses of

the Standard Model particles. As an illustration, let us show the a and b terms for

7171 annihilation only. For fermion final states we find the a-term and b-term of

o-(Qiti -f ff)v as follows

322q a n fR ) (4-32)

87aQN( ( y 4 87ia12 (95 (
S 92 (433)



4 a N7 m ll2 + 14M 2 13m
b 4 1 71 fLl Ll
27 fL ( m2 )4
f fL fR1
11m4 + 14m2m 13m4
Y4 i71 lL fLl34)
+YfR -(m t+ l)4- (4 34)

ffl 7N( + 4 1 19(435)
f 71 1










0.6

-


S0.4

+

o 0.2




0.0
500


900 1000


0.030


0.025


- 0.020


c 0.015


0.010
5C


600 700 800
m. (GeV)


Figure 4-1: The a-term of the annihilation cross-section for (a) 7171 e+e and
(b) ~7171 -+ *, as a function of the mass of the t-channel particless. We fix the
LKP mass at m, 500 GeV and vary (a) the KK lepton mass mrn me, or
(b) the KK Higgs boson mass mr,. The blue solid lines are the exact results (4-32)
and (4-36), while the red dotted lines correspond to the approximations (4-33) and
(4-37).


where gi is the gauge coupling of the hypercharge U(1)y gauge group, al = and

N, = 3 for f = q and N, = 1 for f = Yf is the hypercharge of the fermion f.

For the Higgs boson final states we get


2ira Y4i
a 2 0
i 9m

27a 2Y4
3m2
i 'Y2^


(1 41


(m21 +m )2


3m2
71


10
i



( 121m


S12m2
- 71


+ 140m6m 162m4 60,,,_
2 24



*1


15m80) (4-38)


(4-39)


In the limit where all KK masses are the same (the second line in each formula

above), we recover the result of [21]. Notice the tremendous simplification which


600 700 800
me.. = m,e, (GeV)


m l=500 GeV


m =m7






(b)


900 1000


2m2 + 37m42
-Y i pi


]


(4-36)


(4-37)


0
0o


v









arises as a result of the mass degeneracy assumption. In fig. 4-1 we show the a

terms of the annihilation cross-section for two processes: (a) y7171 e+e- and (b)

7171 -- 00*, as a function of the mass of the t-channel particless. We fix the LKP

mass at m, = 500 GeV and vary (a) the KK lepton mass m,re = me,, or (b) the

KK Higgs boson mass mo,. The blue solid lines are the exact results (4-32) and

(4-36), while the red dotted lines correspond to the approximations (4-33) and

(4-37) in which the mass difference between the t-channel particles and the LKP

has been neglected. We see that the approximations (4-33) and (4-37) can result

in a relatively large error, whose size depends on the actual mass splitting of the

KK particles. This is why in our code we keep all individual mass dependencies.

Another difference between our analysis and that of Ref. [21] is that here we

shall use a temperature-dependent g, function as defined in (4-6). The relevant

value of g, which enters the answer for the LKP relic density (4-12) is g,(TF),

where TF = mr,/xp is the freeze-out temperature. In fig. 4-2a we show a plot of

g,(TF) as a function of R-1 in MUED, while in fig. 4-2b we show the corresponding

values of xF. In fig. 4-2a one can clearly see the jumps in g, when crossing the

bb, W+W-, ZZ and hh thresholds (from left to right). The tt threshold is further

to the right, outside the plotted range. As we shall see below, cosmologically

interesting values of Qh2 are obtained for R-1 below 1 TeV, where g,(TF) = 86.25,

since we are below the W+W- threshold. The analysis of Ref. [21] assumed a

constant value of g, = 92, which is only valid between the W+W- and ZZ

thresholds. The expert reader has probably noticed from fig. 4-2b that the values

of Xz which we obtain in MUED are somewhat larger than the xz values one would

have in typical SUSY models. This is due to the effect of coannihilations, which

increase geff (see fig. 4-5c below) and therefore xF, in accordance with (4-15).

We are now in a position to discuss our main result in MUED. In fig. 4-3 we show

the LKP relic density as a function of R-1 in the minimal UED model. We show









100 30

90 28

S80 26

70 24

60 22
(a) (b)
50 20* *** 0 .
500 1000 1500 2000 2500 3000 3500 500 1000 1500 2000 2500 3000 3500
R-1 (GeV) R-1 (GeV)

Figure 4-2: The number of effectively massless degrees of freedom and freeze-out
temperature. (a) g,(TF) and (b) xz as a function of R-1 in MUED.


the results from several analyses, each under different assumptions, in order to

illustrate the effect of each assumption. We first show several calculations for the

academic case of no coannihilations. The three solid lines in fig. 4-3 account only

for the y7171 process. The (red) line marked "a" recreates the analysis of Ref. [21],

assuming a degenerate KK mass spectrum. The (blue) line marked "b" repeats

the same analysis, but uses T-dependent g, according to (4-6) and includes the

relativistic correction to the b-term (4-19). The (black) line marked "c" further

relaxes the assumption of KK mass degeneracy, and uses the actual MUED mass

spectrum.

Comparing lines "a" and "b," we see that, as already anticipated from

fig. 4-2a, accounting for the T dependence in g, has the effect of lowering g,(xz),

a eff(xF), and correspondingly, increasing the prediction for Qh2. This, in turns,

lowers the preferred mass range for q7. Next, comparing lines "b" and "c," we see

that dropping the mass degeneracy assumption has a similar effect on creff(xF) (see

fig. 4-1), and further increases the calculated Qh2. This can be easily understood

from the t-channel mass dependence exhibited in (4-32) and (4-36). The t-channel