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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. HyeSung 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. HyunChul 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. i1i 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, SungSoo 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 ByoungChul Kim in BNL, Junghan Lee in Mainz and Dr. ByoungIk Hur in cancer treatment center and SeungHwa 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 LeptonJet 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+tfTT .. 58 3.3.2.1 Angular Distributions and Spin Measurements 58 3.3.2.2 Threshold Scans .... . . 60 3.3.2.3 Production CrossSection 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 KaluzaKlein Leptons . . ..... 65 3.3.3.2 KaluzaKlein Quarks . . ..... 67 3.3.3.3 KaluzaKlein 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 KaluzaKlein 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 CROSSSECTIONS .................. .. 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 31 Masses of the KK excitations for R1 500 GeV and AR 20 . 56 32 MSSM parameters for a SUSY study point . . . 58 A1 Fermion content of the Standard Model and the corresponding Kaluza Klein fermions ................... ............ .. 110 A2 Quantum numbers of KK fermions ................ ... 110 A3 Fermions and gauge bosons in the Standard Model . ..... 110 B1 A guide to the formulas in the Appendix B ................ ..118 LIST OF FIGURES Figure page 11 de Broglie's particlewave dii.,iliv ................. . 4 12 An illustration of bulk and brane .................. 5 13 S1/Z2 orbifold . . . . . . .... 6 21 KK states after a compactification on the orbifold . . ..... 12 22 KK number conservation and KK parity ................. .. 14 23 The spectrum of the first KK level at (a) tree level and (b) oneloop 16 24 Qualitative sketch of the level 1 KK spectroscopy . . ..... 17 25 A discovery reach for MUEDs at the Tevatron (blue) and the LHC (red) in the 4 + ST channel ............. . ... .18 31 One loop corrected mass spectrum of the n = 1 and n = 2 KK levels .23 32 Crosssections of n = 2 KK particles at the LHC . . ..... 24 33 Branching fractions of the level 2 "up" quarks versus R1 . ... 26 34 Branching fractions of the level 2 KK electrons versus R1 . ... ..28 35 Masses and widths of level 2 KK gauge bosons . . 30 36 Crosssections for single production of level 2 KK gauge bosons . 33 37 Branching fractions of the n = 2 KK gauge bosons versus R1 ...... .35 38 5a Discovery reach for (a) 72 and (b) Z2 ... . . 38 39 The 72 Z2 diresonance structure in UED with R1 500 GeV . 40 310 Twin diagrams in SUSY and UED .................. .. 43 311 Comparison of dilepton invariant mass distributions . . .... 44 312 A closer look into dilepton invariant mass distributions . ... 46 313 Jetlepton invariant mass distributions ... . .. 50 314 Asymmetries for UED and SUSY .................. ..... 51 315 Asymmetries with relaxed conditions ................ 53 316 The dominant Feynman diagrams for KK muon production . ... 55 317 The dominant Feynman diagrams for smuon production . ... 55 318 Differential crosssection da/dcos O, ..................... 59 319 The total crosssection a in pb as a function of the centerofmass en ergy s near threshold .................. ......... .. 60 320 The muon energy spectrum resulting from KK muon production (314) in UED (blue, top curve) and smuon production (315) in supersymme try (red, bottom curve) .................. ......... 62 321 Photon energy spectrum in e+e p + .. .. . . .64 322 The dominant Feynman diagrams for KK electron production ...... ..66 323 ISRcorrected production crosssections of level 1 KK leptons . ... 67 324 Differential crosssection dar/dcos 0 for UED and supersymmetry . 68 325 ISRcorrected production crosssections of level 1 KK quarks ...... ..69 326 ISRcorrected production crosssections of level 1 KK gauge bosons 71 327 ISRcorrected production crosssections of level 2 KK gauge bosons 72 41 The aterm of the annihilation crosssections for (a) 717  e+e and (b) 7l.i 9 ..... ........ .............. 84 42 The number of effectively massless degrees of freedom and freezeout tem perature . . . . . . . ... .. 86 43 Relic density of the LKP as a function of R1 in the minimal UED model 87 44 Coannihilation effects of (a) 1 generation or (b) 3 generations of singlet KK leptons .................. .. .......... ..89 45 Plots of various quantities entering the LKP relic density computation .91 46 The effects of varying the SU(2)wdoublet KK electron mass ...... ..93 47 The effects of varying KK quarks masses ..... . . 94 48 The effects of varying KK gluon mass ................ . 95 49 The effects of varying EW bosons ................ ...... 96 410 The change in the cosmologically preferred value for R1 as a result of varying the different KK masses away from their nominal MUED values 98 411 The spinindependent direct detection limit from CDMS experiment 99 A1 Dependence of the \\V inii i," angle ~, for the first few KK levels (n 1, 2,.. ,5) on R1 for fixed AR 20 ........ ... ..... 112 A2 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: KaluzaKlein 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 KaluzaKlein 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 KaluzaKlein 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 crosssection measurement are powerful discriminators between the two models. We also calculate the production rates of various KaluzaKlein particles and discuss the associated signatures. A prediction of the models with Universal Extra Dimensions with conserved KKparity is the existence of dark matter. We calculate the relic density of the lightest KaluzaKlein 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 ii. 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), (13) 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 Rparity conserv ing supersymmetric theories. New particles, called superpartners, predicted by the supersymmetry are charged under this Rparity, 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. Rparity 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 energymomentum 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 (15) 000 27R 2A 27R = 5A 27R = 6A Figure 11: de Broglie's particlewave (1dii.il v. As we go around the circle, we must fit an integer multiple of A's in its circumference. Now recall the particlewave 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. 11. Therefore periodicity implies a quantization of momentum along the extra dimen sion, S27wR/ 27n n A = p (1 7) n 27R R Substitute eqn. 17 into eqn. 15, then the energymomentum 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 KaluzaKlein (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. 12, 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 12: 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 spacetime 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 nontrivial 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. 13, 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 KaluzaKlein 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 KKparity. The KKparity adorns the UED scenario with many of the virtues typically associated with supersymmetry: S1 Z2 S /Z2 brane brane j / bulk Z2 Figure 13: 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 KKpartners (those at level 1) must always be pairproduced in collider experiments, which leads to relatively weak bounds from direct searches. 2. The KKparity conservation implies that the contributions to various pre cisely measured lowenergy observables only arise at the loop level and are small. 3. Finally the KKparity 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 Tparity 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 KaluzaKlein (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 [711]. 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. [1318]. 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 Ii 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 crosssections 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 (22) 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) (23) 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( (24) 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 KKtower) 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 6dimensional models have also been built [2325]. 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 Jf7R +iD(x,y)FMDMD(x,y)} (25) 1 anrR LYukawa R dy {A uQ(x, y)U (, y) i72H*(x, y) + AdQ(x, y)D(x, y)H(x, y) 2 JrR +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 uptype quark, the downtype quark and the lepton. Q U A L R L R A, A5 n 3 n 2 n 1 nO Figure 21: 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)wdoublet 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. (A6). 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. 21, where n = 0 corresponds to a \! particle and nonzero 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. 21, 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 offdiagonal entry in a fermion mass matrix (see eqn. B.96). 1 fig. 21 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. A21) 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 R1 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. A22) 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. A24). 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 22: 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, (27) Imnkl = 0. This is the conservation of KaluzaKlein 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. 22, (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. KKparity is always conserved even at higher order. Bulk and brane radiative effects [2628] 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 pairproduced in collider experiments, just like in the Rparity conserving supersymmetry models. KK parity conservation also implies that the contributions to various lowenergy observables [2939] only arise at loop level and are small. As a result, the limits on the scale R1 of the extra dimension from precision electroweak data are rather weak, constraining R1 to be larger than approximately 250 GeV [33]. Fig. 22(b) can be generated by 1 loop corrections with level 1 KK particles, however, KKparity is not conserved in fig. 22(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. A20) and normalization factors (eqn. 27). Of course, KKparity 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 [4042], 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 oneloop radiative corrections to the KK mass spectrum and the brane terms generated by unknown pir, i. at high scales [28]. In fig. 23, the spectrum of the first KK level is shown at tree level (a) and oneloop (b), for R = 500 GeV, AR = 20, and assuming vanishing boundary terms at the cutoff scale A. Fig. 24 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. 23. As indicated in fig. 23, 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 23: The spectrum of the first KK level at (a) tree level and (b) oneloop, for R1 500 GeV, AR 20, mh 120 GeV, and assuming vanishing boundary terms at the cutoff 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 500600 GeV [2022,4749]. KaluzaKlein dark matter offers excellent prospects for direct [5052] or indirect detection [50, 5361]. Once the radiative corrections to the KaluzaKlein 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. 24, an SU(2)wsinglet 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 24: 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 KaluzaKlein 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 101 102 41ZT AR=20 103 . . 0 500 1000 1500 2000 R' (GeV) Figure 25: 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 fb1) as a function of R1, 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 R1 > 500 GeV, sin2 On < 0.01 where 0, is the Weinberg angle for level n. Fig. 25 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 fb1) as a function of R1, 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 (KaluzaKlein 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 Rparity 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 quasistable 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 crosssection times the branching fractions, and so any attempt to measure the couplings from a crosssection 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 BreitWigner 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 partonlevel 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 leptonjet 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 [710] 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 oneloop 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)wdoublet (SU(2)wsinglet) KK fermions are denoted by capital (lowercase) letters. The KK level n is denoted by a subscript. In fig. 31 we show the mass spectrum of the n = 1 and n = 2 KK levels in minimal UED, for R1 500 GeV, AR = 20 and '1\ Higgs boson mass mn 120 GeV. We include the full oneloop 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 [6971]. As a result the spectrum shown in fig. 31 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 31: Oneloop 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 crosssections 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 KKnumber conserving interactions, or through KKnumber violating (but KKparity 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 crosssections are shown in fig. 32 (the crosssections for producing n = 1 KK quarks have been calculated in [13, 72, 73]). In fig. 32a we show the crosssections (in pb) for n = 2 KKquark pair production, while in fig. 32b we show the results for n = 2 KKquark/KKgluon associated production and 103  103 r 2q2q\\ go Q92,gq 10o  00 \\  aga 10 Q Q2 1 \ g S102 102 0a < a P b 94 4 106 I 106 500 1000 1500 2000 500 1000 1500 2000 R1 (GeV) R1 (GeV) Figure 32: Crosssections of n = 2 KK particles at the LHC for (a) KKquark pair production (b) KKquark/KKgluon associated production and KKgluon pair production. The crosssections have been summed over all quark flavors and also include chargeconjugated contributions such as Q2q2, Q2q2, g2Q2, etc. for n = 2 KKgluon pair production. We plot the results versus R1, and one should keep in mind that the masses of the n = 2 particles are roughly 2/R. In calculating the crosssections of fig. 32 we consider 5 partonic quark flavors in the proton along with the gluon. We sum over the final state quark flavors and include chargeconjugated 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. 32 displays a severe kinematic suppression of the crosssections 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 crosssections for the three types of final states with n = 2 particles: quarkquark, quarkgluon, and gluongluon. 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 crosssections for pair production of KK quarks with different SU(2)w quantum numbers. This is partially due to the different masses for SU(2)wdoublet and SU(2)wsinglet quarks (see fig. 31), and the remaining difference is due to the contributions from diagrams with electroweak gauge bosons. Notice that since the crosssections in fig. 32a are summed over charge conjugated final states, the mixed case of Q2q2 contains twice as many quarkantiquark contributions (compare Q22 + Q2q2 to q2q2 or Q2Q2 alone). If we compare the crosssections for n = 2 KK quark production to the crosssections 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, schannel processes. Well above threshold, the UED crosssections 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 righthanded SU(2)wdoublet KK fermions, while in SUSY there are only "lefthanded" SU(2)wdoublet 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 crosssections is /33, while in UED the KKquarks are fermions, and the threshold suppression of the crosssection is only 3. This distinct threshold behavior of the production crosssections further enhances the difference between SUSY and UED. For example, we find that for R = 500 GeV the pair production crosssection for charm KKquarks is about 6 times larger than the crosssection for charm squarks. For processes involving first generation KKquarks, where tchannel diagrams contribute significantly, the effect can be even bigger. For example, up KKquark 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 m101  0.2 U7y lO2^1'   0.  102 L . I . 0.0 1 500 1000 1500 2000 500 1000 1500 2000 R' (GeV) R1 (GeV) Figure 33: Branching fractions of the level 2 "up" quarks versus R1 for (a) the SU(2)wdoublet quark U2 and (b) the SU(2)wsinglet quark u2 In fig. 32 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 (31) where Aa is a \ gauge field and TI is the corresponding group generator. However, (31) is forbidden by gauge invariance, and the lowest order coupling of an n = 2 KK quark to two \ particles has the form [28] Q2VTaPLQOFa (32) 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. 31, 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. 33. Fig. 33a (fig. 33b) is for the case of the SU(2)wdoublet quark U2 (the SU(2)wsinglet quark u2). The results for the "down"type KK quarks are similar. We observe in fig. 33 that the branching fractions are almost independent of R1, 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 (32), only decays which conserve the total KK number n are allowed. The case of the SU(2)wsinglet 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. 33b. The case of an SU(2)wdoublet quark Q2 is much more complicated, since Q2 couples to the (KK modes of the) weak gauge bosons as well, and many more twobody 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. 33a. The branching fractions to the 7 KK modes are only on the order of a few percent. The threshold behavior seen in fig. 33a near R1 400 GeV is due to the finite masses for the \I W and Z bosons, which enter the treelevel 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 R1 (GeV) R1 (GeV) Figure 34: Branching fractions of the level 2 KK electrons versus R1. The same as fig. 33 but for the level 2 KK electrons: (a) the SU(2)wdoublet E2 and (b) the SU(2)wsinglet e2. the mass splitting of the KK modes is due to the radiative corrections, which are proportional to R1, the channels with Wft and Z1 open up only for sufficiently large R1. 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 crosssection. 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 (32), the branching fractions of the n = 2 KK electrons are shown in fig. 34. 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 crosssections 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 KKparity odd, and necessarily decay to the invisible 71. There are four n = 2 KK gauge bosons: the KK *phl ..'i." 72, the KK "Zbc... ii Z2, the KK "Wbc(. ..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 Wboson 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 ..'" i0Z2 W 1000 (a) 10 (b) 500 1000 1500 2000 1000 2000 3000 4000 5000 R1 (GeV) Mass (GeV) Figure 35: Masses and widths of level 2 KK gauge bosons. (a) Masses of the four n 2 KK gauge bosons as a function of R1. (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 Zboson. is that all four of the n = 2 KK gauge bosons are relatively degenerate, as shown in fig. 35a. 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) (33) 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 (34) 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 (33) and (34) 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 (33) is an approximation neglecting the I\! fermion mass mfo. The second line in (34) 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. (33) and (34). The partial widths are proportional to the oneloop 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 (35) 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 (36) 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 (36), 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 (3336) are suppressed, the total width of V2 is much smaller than the width of a typical Z'. This is illustrated in fig. 35b, 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 SMlike 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. 7 104  500 1000 1500 2000 R (GeV) Figure 36: Crosssections for single production of level 2 KK gauge bosons through the KK number violating couplings (35). let us briefly discuss their production. There are three basic mechanisms: 1. Single production through the KK number violating operator. The corresponding crosssections are shown in fig. 36 as a function of R1. 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 DrellYan crosssections in fig. 36 to the mass dependence of the n = 2 pair production crosssections from fig. 32, indeed we see that the former drop less steeply with R1 and become dominant at large R1. On the other hand, the DrellYan processes of fig. 36 are mediated by a KK number violating operator (35) 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 crosssections shown in fig. 36. This is somewhat surprising, since the crosssections scale as the corresponding gauge coupling squared, and one would have expected a wider spread in the values of the four crosssections. This is due to a couple of things. First, for a given R1, the masses of the four n = 2 KK gauge bosons are different, with mn2 > rW2 mZ2 > m72. Therefore, for a given R1, the heavier particles suffer a suppression. This explains to an extent why the crosssection for 72 is not the smallest of the four, and why the crosssection for g2 is not as large as one would expect. There is, however, a second effect, which goes in the same direction. The coupling (35) 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. 32 and 33. 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. 32b. 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 101 101 (a) (b) 102 I ..I I 102 , 500 1000 1500 2000 500 1000 1500 2000 R1 (GeV) R1 (GeV) 100 100 C livi C\ 101 H o 101 102 (c) (d) 102 .... 500 1000 1500 2000 500 1000 1500 2000 R' (GeV) R1 (GeV) Figure 37: Branching fractions of the n = 2 KK gauge bosons versus R1: (a) g2, (b) Z2, (c) W', and (d) '2. also calculated the pair production crosssections 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. 37(ad). Again we observe that the branching fractions are very weakly sensitive to R1, just as the case of figs. 33 and 34. This can be understood as follows. The partial widths (33) and (34) for the KK number conserving decays are proportional to the available phase space, while the partial width (36) for the KK number violating decay is proportional to the mass corrections (see eq. (37)). qq2 q__ lqlt_______ K 11.,jj. I Both the phase space and the mass corrections are proportional to R1, 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 crosssection 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. 37a 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 (37). 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 crosssection is not that much different from the crosssections for 72, Z2 and W' (see fig. 36). 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. 37b and 37c 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 (37). 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 (37). 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. 37d 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 (35). 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. 35b), 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 ..  / PPp'2P / / + / / e 101 / pPPyee e 1 / .. PPZ e (a) (b) S.. 102 I I I 102 200 400 600 800 1000 200 400 600 800 1000 R1 (GeV) R1 (GeV) Figure 38: 5a discovery reach for (a) 72 and (b) Z2. We plot the total integrated liiii i.il' L (in fb1) required for a 5a excess of signal over background in the dielectron (red, dotted) or dimuon (blue, dashed) channel, as a function of R1. In each plot, the upper set of lines labelled "DY" makes use of the single V2 produc tion of fig. 36 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 R1 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 R1 = 1 TeV. The main _\! background to our signal is DrellYan, 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. 38. We plot the total integrated luiii. .iliT L (in fb1) required for a 5a excess of signal over background in the dielectron (red, dotted) or dimuon (blue, dashed) channel, as a function of R1. In each panel in fig. 38, the upper set of lines labelled "DY" only utilizes the single V2 production crosssections from fig. 36. 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. 38a 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 R1. Fig. 38 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 fb1), the LHC will have sufficient statistics in order to probe the region up to R1 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 R1 ~ 500 600 GeV [2022]. 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. 38). If the LKP accounts for only a fraction of the dark matter, the preferred range of R1 is even lower and the discovery at the LHC is easier. From fig. 38 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 fb1), extends up to just beyond R1 = 1 TeV. One should keep in mind that the actual KK masses 50 11 11 40 (a) ppV2tu (b) ppV,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 39: The 72 Z2 diresonance structure in UED with R1 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 nonminimal 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. 39, 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 fb1 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. 38 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, nonvanishing boundary terms at the scale A can distort the minimal UED spectrum beyond recognition. A priori, in such a relaxed framework the UEDSUSY 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 Tparity [16, 9194] 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 partonlevel 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 310: 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 lookalike 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 leptonjet invariant mass distributions from a particular cascade (see fig. 310), 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 twobody decays like slepton production e+e 2 +tp. 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 lefthanded and righthanded 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 [9194]. 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 311: Comparison of dilepton invariant mass distributions in the case of (a) UED mass spectrum with R1 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 fb1 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+tyi in UED. Both of these processes are illustrated in fig. 310. 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 R1 = 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. 311, we show invariant mass distributions in UED and SUSY for two different types of mass spectrum. In fig. 311(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 pairproduction crosssections 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 fb1 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 (38) 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 (39) 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 onshell decay. From eqn. 38, 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 312: A closer look into dilepton invariant mass distributions. (a) Contour dotted lines represent the size of the coefficient r in eqn. 39. The minimal UED is a blue dot in the upperright 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. 312(a). In fig. 312(a), the contour dotted lines represent the size of the coefficient r in eqn. 39. The minimal UED is blue dot in upperright 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. 311. 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.312(a) and show the corresponding dilepton distributions in fig.312(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.312(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 LeptonJet Invariant Mass Charge Asymmetry Now we look at spin correlations between q and i in fig. 310. 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 leptonquark 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 antisquark. 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 antiquarks in the proton, which in turn leads to a difference in the production crosssections for squarks and antisquarks [12]. Most importantly, we do not know which jet is actually the correct jet in this cascade decay chain. We pairproduce 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/antiquark 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. 313, (d fq dP2 d+ dmj + dml ) d\dm,) d fq + + fq ( ), (310) dm qtd, dm, dmn where P1 (P2) represents distribtuion for a decay from a squark or KK quark (antisquark or antiKK quark) and fq (fq) is the fraction of squarks or KK quarks (antisquarks or antiKK 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 antiparticles. For a UED mass spectrum and SPSla, fq ~ 0.7 [13]. These two distributions in UED (SUSY) are shown in fig. 313(a) (fig. 313(b)) in different colors. The distributions are normalized to L 10fb1 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 (311) (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 nonzero aymmetry, it means that the intermediate particle (ou or Z1) has nonzero 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 antiquarks. 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 nonzero 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 antisquarks. Therefore we expect fq ~ fq ~ 0.5 and the ..'Immetry will be washed out. Our comparison between A+ in the case of UED and SUSY for UED mass spectrum is shown in fig. 314(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: Jetlepton 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 = 10fb1. 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. 310. Overall, fig. 314 shows that although the .i'.,mmetry (eqn. 311) does encode some spin correlations, distinguishing between the specific cases of UED and SUSY appears challenging. Similarly in fig. 314(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. 314(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 314: 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 fb1. conclusion persists in a general '1I .ii and once the combinatorial backgrounds are included [96]. Notice that comparing (a) and (b) in fig. 311, the signs of the two ..i.'mmetries have changed. The difference is the chirality of sleptons or KK leptons. In fig. 311(a) (fig. 311(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. 310). In principle, we can find more kinematic endpoints such as a lower edge, here we are being conservative and take upper edges only [98100]. 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) (312) 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 (313) 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. 315(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 10fb1 and constructed the .11...',,i 1i1i l in SUSY and UED in fig. 315(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 315: 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 fb1 including 10% jet energy resolution. 2minimized 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. 314(b) disappeared in fig. 315(b) after including jet energy resolution. From fig. 314, 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 KaluzaKlein muon pair production in universal extra dimensions in parallel to smuon pair production in supersymmetry, accounting for the effects of detector resolution, beambeam 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 crosssection 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 KaluzaKlein 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 oneloop 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. (314) below (see also fig. 316). In complete e l e+ 2, Z2 + + (a) (b) Figure 316: 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 317: 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. (315). The dominant diagrams in that case are shown in fig. 317. In principle, there are also diagrams mediated by 7,, Z, for n = 4, 6,... but they are doubly suppressed by the KKnumber violating interaction at both vertices and the KK mass in the propagator and here can be safely neglected. However, 72 and Z2 exchange (fig. 316b) may lead to resonant production and significant enhancement of the crosssection, 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 KKnumber preserving and the KKnumber 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 711, (314) Table 31: Masses of the KK excitations for R1 = 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, (315) 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 (314) in UED from smuon pair production (315) in supersymmetry. We also determine the masses of the produced particles and test the model predictions for the production crosssections in each case. We first fix the UED parameters to R = 500 GeV, AR = 20, leading to the spectrum given in Table 31. The ISRcorrected signal crosssection 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 105 GeV1. 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 centerofmass 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 YokoyaChen approximation [103,104]: This analysis has backgrounds coming from [\! utppvt final states, which are mostly due to gauge boson pair production W+W Lt+tPv,,, ZZ0 p+t ,p and from e+e  W+W, v6, e+e ZoZoe,, followed by muonic decays. The background total crosssection is ~20 fb at s = 3 TeV. In addition to its competitive crosssection, 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 dilepton 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 crosssection, 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. I1 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 32: 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 KaluzaKlein 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 32. We then simulate both reactions (314) and (315) with CompHEP and pass the resulting events through the same simulation and reconstruction. The ISRcorrected signal crosssection 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. (317) 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 318: Differential crosssection 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 ISRcorrected 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 yellowshaded 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 (317) and (318) are sufficiently distinct to discriminate the two cases. However, the polar angles 0 of the original KKmuons 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 318a). In fig. 318b 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 319: The total crosssection a in pb as a function of the centerofmass energy s near threshold for e+e  Pft [ P+lii. 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 fb1 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 crosssections 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 crosssection 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 31. We account for the anticipated CLIC centreofmass energy spread induced both by the energy spread in the CLIC linac and by beambeam 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 319). 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 crosssection 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 ab1 shared in three points, when the particle widths can be disregarded. 3.3.2.3 Production CrossSection Determination The same analysis can be used to determine the crosssection for the process e+e  *+p fT. The \!I contribution can be determined independently, using antitag cuts, and subtracted. Since the crosssection 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 crosssection 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 + + o0o 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 320: The muon energy spectrum resulting from KK muon production (314) in UED (blue, top curve) and smuon production (315) in supersymmetry (red, bottom curve). The UED and SUSY parameters are chosen as in fig. 318. The plot on the left shows the ISRcorrected 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 yellowshaded histogram is the signal only. 3.3.2.4 Muon Energy Spectrum and Mass Measurements The characteristic endpoints of the muon energy spectrum are completely determined by the kinematics of the twobody 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 ISRcorrected expected distributions for the muon energy spectra at the generator level in fig. 320a, using the same parameters as in fig. 318. 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 ) (319) 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. 320a 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 cutoff by the solenoidal field bending the lepton before it reaches the electromagnetic 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. 320b 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 centerofmass 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 schannel narrow resonances, a sharp peak in the photon energy spectrum appears whenever one of the mediating schannel particles is onshell. In case of supersymmetry, only Z and 7 particles can mediate smuon pair production and neither of them can be close to being onshell. 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. 316b. 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 (320) 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,2like 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. 321. On the left we show the ISRcorrected 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+ftfT 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 crosssections for some of the other main processes of interest, and discuss how they could be analyzed. 3.3.3.1 KaluzaKlein Leptons We first turn to the discussion of the other KK lepton flavors. The KK 7 leptons, T, are also produced in schannel diagrams only, as in fig. 316, hence the T  production crosssections are very similar to the p t+~ case. The final state will be T+rfT, 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 tchannel diagram shown in fig. 322c. As a result, the production crosssections for KK electrons can be much higher than for KK muons. We illustrate this in fig. 323, where we e+ e e+ e + ef \ 7 Z 72, Z2 e e eI e e1 (a) (b) (c) Figure 322: The dominant Feynman diagrams for KK electron production, e+e e1eI, the same as fig. 316. show separately the crosssections for SU(2)w doublets (solid lines) and SU(2)w singlets (dotted lines), as a function of R1. (For the numerical results throughout section 3.3.3, we always fix AR = 20.) At low masses (i.e. low R1) the ee+c crosssections 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 crosssection for R1 ~ 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 onshell 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. 316 and 322). 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 crosssections3 Because of the higher production rates, the e+eTf event sample will be much larger and have better statistics than [+ t4T. The e+e C T final state has been recently advertised as a discriminator between UED and supersymmetry in [108]. However, the additional tchannel diagram (fig. 322c) has the effect of not only enhancing the overall crosssection, 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 101 b 102 I I , 250 500 750 1000 1250 1500 R1 (GeV) Figure 323: ISRcorrected production crosssections of level 1 KK leptons (ei in red, ft1 in blue) at CLIC, as a function of R1. 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. 324. 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 31. We then choose a supersymmetric spectrum with selection mass parameters as in Table 32. 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. 318, where the underlying shapes of the angular distributions were very distinctive (see eqs. (317) and (318)), the main effect in fig. 324 is the uniform enhancement of the forward scattering crosssection, which tends to wash out the spin correlations exhibited in fig. 318. 3.3.3.2 KaluzaKlein Quarks Level 1 KK quarks will be produced in schannel via diagrams similar to those exhibited in fig. 316. The corresponding production crosssections are shown in fig. 325, as a function of R1. We show separately the cases of the SU(2)w R = 500 GeV 100 Ec, 3 TeV 101 ee ee ey,7 102 e eme Ce Xii 103 1.0 0.5 0.0 0.5 1.0 COSOe Figure 324: Differential crosssection dor/dcosOe for UED and supersymmetry. The same as fig. 318 (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 crosssections for uf and df cut off at a smaller value of R1. Since singlet production is only mediated by U(1) hypercharge interactions, the singlet production crosssections 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 (righthanded) 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 crosssection will provide further evidence of the particle spins (see section 3.3.2.2). The only major difference with respect to the [p+tJT final state discussed in section 3.3.2, is the absence of the 10  b 2 103 250 500 750 1000 1250 1500 R' (GeV) Figure 325: ISRcorrected production crosssections of level 1 KK quarks at CLIC, as a function of R1 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 lefthanded 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 KaluzaKlein Gauge Bosons The ISRcorrected production crosssections for level 1 electroweak4 KK gauge bosons (Wt, Z1 and 71) at a 3 TeV e+e collider are shown in fig. 326, as 4 The level 1 KK gluon, of course, has no treelevel couplings to ee. a function of R1. The three relevant processes are W,+W ZIZ1 and Z171 (7171 is unobservable). In each case, the production can be mediated by a tchannel exchange of a level 1 KK lepton, while for W,+W1 there are additional schannel diagrams with 7, Z, 72 and Z2. Z1 and W1f are almost degenerate [28], thus their crosssections cut off at around the same point. The analogous processes in supersymmetry would be the pair production of gauginolike 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 R1, 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 (treelevel) couplings. In fig. 327 we show the corresponding crosssections for the case of the neutral level 2 gauge bosons, as a function of R1, For low values of R1, 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 101 e+eWW 5 e+eZZl b 102 5 103 250 500 750 1000 1250 1500 R1 (GeV) Figure 326: ISRcorrected production crosssections of level 1 KK gauge bosons at CLIC, as a function of R1. or dilepton resonance is known, the collider energy can be tuned to enhance the crosssection 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 multiTeV e+e linear collider such as CLIC5 We studied in detail the process of pair production of muon partners in the two theories, KKmuons 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 R1 > 250 GeV for the case of 1 extra dimension, one would need an ILC centerof mass energy above 500 GeV in order to pairproduce the lowest lying KK states of the minimal UED model. 101 e+e>Z Z, e+e y2Z2 b ee > Z2 103 104 250 500 750 1000 1250 1500 R1 (GeV) Figure 327: ISRcorrected production crosssections of level 2 KK gauge bosons at CLIC, as a function of R1. 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 ab1 of integrated lliiliir. il' Once the masses of the partners are known, the measurement of the total crosssection serves as an additional crosscheck 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 centerofmass energy at the CLIC multiTeV collider allows one to unambiguously identify the nature of the new ll, vi 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 KaluzaKlein 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)wsinglet 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 crosssections 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 cutoff scale values of the so called boundary terms, which are not fixed by known \ I ',l1i, . 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 cutoff 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 crosssections 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 hyperchargemediated processes which have already been considered. * We will keep the exact value of each KK mass in our formulas for all an nihilation crosssections. This will render our analysis selfconsistent. All calculations of the LKP relic density available so far [21, 48, 49], have com puted the annihilation crosssections 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 crosssections, 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 crosssections behave properly. Notice that the assumption of exact mass degeneracy overestimates the corresponding crosssections 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 temperaturedependent g, (the total number of effectively massless degrees of freedom, given by eq. (46) below) and include subleading corrections (419) in the velocity expansion of the annihilation crosssections. The availability of the calculation of the remaining coannihilation processes is important also for the following reason. Coannihilations with SU(2)wsinglet KK leptons were found to reduce the effective annihilation crosssection, 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)wsinglet 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)wsinglet 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 crosssection 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(jmTj2emT (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. (42) that the density neq is Boltzmannsuppressed. 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 (43) where X (44) 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 (45) 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 (46) i=bosons i fermions The freezeout temperature, xF, is found iteratively from 1 45 g mnMp,(a+6b/xF) \ XF In c(c+2) ,8 ( (47) 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 schannel 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 (46) 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 selfannihilation crosssection, 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)) (49) ij .geff N geff () = g( Ai)3/2 exp(.xA) (410) i 1 Ai mi (411) 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)x2d, (413) I = 2x beff(x)3dx (414) JXF The corresponding formula for XF becomes ( n 2)45eff(xF) mMpl(aff(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. (49) and (416), one gets N aeff(x) aj (l+ A)3/2(1 Aj)3/2 exp(x(Ai + Aj)), (417) ij Y9eff N beff(x) = bjJ (1+ A)3/2(+ A)3/2 exp(x(A + A)) (418) 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 bb a (419) 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 crosssection 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 energymomentum relation, p = mv + O(v2) and E = m + mv2 O(4). In the CM frame, above equation becomes f lj 3d) ,_, 'E1/TeE2/Ta lr (avrel/ = f d3d3 e 3 V2El/TE2 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 crosssection 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 (423) 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) (424) 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 crosssection 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 oneloop 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 crosssections between any pair of n = 1 KK particles. Our code has been subjected to numerous tests and crosschecks. 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 nexttolightest KK particles are the singlet KK leptons and their fractional mass difference from the LKP is2 AR1, = m1  m 0.01 (430) mr1 Notice that the Boltzmann suppression eAgRF 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 (431) 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)wsinglet Dirac fermion, which has in principle both a lefthanded and a righthanded 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)wdoublet KK leptons tL1 couple weakly, and the KK quarks ql and KK gluon gl have strong couplings. Therefore, their corresponding annihilation crosssections 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 crosssections 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)wdoublet KK leptons might be numerically significant, since their mass splitting in MUED is ~ 3% and the corresponding Boltzmann suppression factor is only e0.0325 ~ eo.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 aterm and bterm of o(Qiti f ff)v as follows 322q 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 41: The aterm of the annihilation crosssection for (a) 7171 e+e and (b) ~7171 + *, as a function of the mass of the tchannel 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 (432) and (436), while the red dotted lines correspond to the approximations (433) and (437). 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) (438) (439) 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 ] (436) (437) 0 0o v arises as a result of the mass degeneracy assumption. In fig. 41 we show the a terms of the annihilation crosssection for two processes: (a) y7171 e+e and (b) 7171  00*, as a function of the mass of the tchannel 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 (432) and (436), while the red dotted lines correspond to the approximations (433) and (437) in which the mass difference between the tchannel particles and the LKP has been neglected. We see that the approximations (433) and (437) 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 temperaturedependent g, function as defined in (46). The relevant value of g, which enters the answer for the LKP relic density (412) is g,(TF), where TF = mr,/xp is the freezeout temperature. In fig. 42a we show a plot of g,(TF) as a function of R1 in MUED, while in fig. 42b we show the corresponding values of xF. In fig. 42a 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 R1 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. 42b 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. 45c below) and therefore xF, in accordance with (415). We are now in a position to discuss our main result in MUED. In fig. 43 we show the LKP relic density as a function of R1 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 R1 (GeV) R1 (GeV) Figure 42: The number of effectively massless degrees of freedom and freezeout temperature. (a) g,(TF) and (b) xz as a function of R1 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. 43 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 Tdependent g, according to (46) and includes the relativistic correction to the bterm (419). 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. 42a, 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. 41), and further increases the calculated Qh2. This can be easily understood from the tchannel mass dependence exhibited in (432) and (436). The tchannel 