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Particle phenomenology at the frontiers Winslow, Peter 2013

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Particle Phenomenology at theFrontiersbyPeter WinslowB.Sc. Honors, The University of Winnipeg, 2007M.Sc., The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Peter Winslow 2013AbstractThe Standard Model of particle physics has proven itself as a wildlysuccessful description of the microscopic world. The formulation of thetheory as a renormalizable quantum field theory preserves its predictivepower beyond the first order approximation of perturbation theory, allowingfor precision probes of the underlying quantum structure. Yet, despite allits accomplishments, deep mysteries remain for which the Standard Modeloffers no explanation. This thesis is dedicated to the phenomenologicalstudy, as well as the creation of, a number of well-motivated extensions tothe Standard Model, each originally designed to explain a known flaw.The Randall-Sundrum model employs an extra dimension to exponen-tially suppress all mass scales in the theory, generating the electroweak-Planck hierarchy with no recourse to fine-tuning of parameters. In this way,it can lead to unwanted enhancements of higher dimensional operators in-ducing strongly constrained processes such as proton decay. We investigatethe effects of the Randall-Sundrum model on a similar process, neutron-antineutron oscillation, and find that, in contrast to the example of protondecay, it is naturally suppressed below experimental limits by virtue of theflavor structure of the operators inducing it. We further use these resultsto explore the efficacy of observing neutron-antineutron oscillations at aproposed upcoming high intensity experiment at Fermilab.The D? collaboration has reported strong evidence of CP violation be-yond the Standard Model in measurements of the like-sign dimuon asym-metry at the Tevatron. The nature of the signal suggests the presence ofnew physics in the form of new bosonic particles with masses near the weakscale and new sources of CP violation in the mixing of neutral B mesons.We note that this is exactly what is needed for viable electroweak baryo-iiAbstractgenesis in extensions of the Standard Model. We explore the potential forsimultaneously explaining the dimuon asymmetry and the baryon asymme-try of the universe using a Two-Higgs-Doublet-Model with an unusual flavorstructure.Both the CDF and D? collaborations measure a strong preference for topquarks, produced in pairs at the Tevatron, to propagate in the direction ofthe initial proton beam, in direct conflict with Standard Model predictions.A class of new physics models has been proposed to explain this behaviour.These models employ novel flavor interactions to generate the effect througha Rutherford enhancement in the forward direction. We show that suchflavor interactions can also simultaneously explain a long-standing tensionbetween different measurements of the CKM matrix element Vub throughthe generation of large loop-induced right-handed charge currents.iiiPrefaceThis thesis is based on three published papers.A version of chapter 2 has been published. I performed all the calcula-tions and analyses, made the plots and drafted the manuscript. ProfessorNg provided guidance throughout and comments on the manuscript.A version of chapter 3 has been published. I participated heavily inthe construction of the model and calculation of all flavor constraints. Dr.Tulin provided cross checks on all calculations for flavor constraints. Duringthe time of writing this manuscript, it became known to us that there wereother groups who were simultaneously working on a very similar idea. In aneffort to hasten publication, Dr. Tulin performed most of the calculationsand analysis for the Baryogenesis section of this work during the completionof the paper.A version of chapter 4 has been published. The possible connectionbetween the top quark forward-backward asymmetry and the Vub anomalywas noticed independently by both myself and Professor Ng. I performedall calculations, produced all plots, and drafted the manuscript. ProfessorNg provided guidance throughout and comments on the manuscript.ivPrefaceJournal Papers1. P. T. Winslow and John N. Ng, ?Neutron-Antineutron Oscillations ina Warped Extra Dimension?, Phys.Rev., vol. D81, p. 106010, 2010.2. S. Tulin and P. T. Winslow, ?Anomalous B Meson Mixing and Baryo-genesis?, Phys.Rev., vol. D84, p. 034013, 2011.3. John N. Ng and P. T. Winslow, ?Top Quark Forward-Backward Asym-metry and Anomalous Right-Handed Charge Currents?, Journal ofHigh Energy Physics (JHEP), vol. 1202, p. 140, 2012vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Structure of the Standard Model . . . . . . . . . . . . 31.1.2 Yang-Mills Gauge Theory . . . . . . . . . . . . . . . . 71.1.3 Spontaneous Symmetry Breaking . . . . . . . . . . . 111.1.4 The Standard Model Lagrangian . . . . . . . . . . . . 151.2 Problems in the Standard Model . . . . . . . . . . . . . . . . 201.2.1 The Hierarchy Problem . . . . . . . . . . . . . . . . . 201.2.2 The Baryon Asymmetry of the Universe . . . . . . . 241.2.3 Collider Anomalies . . . . . . . . . . . . . . . . . . . 251.2.3.1 The Like-Sign Dimuon Asymmetry . . . . . 261.2.3.2 The Top Quark Forward-Backward Asym-metry . . . . . . . . . . . . . . . . . . . . . 272 Neutron-Antineutron Oscillations in a Warped Extra Di-mension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29viTable of Contents2.2 The Randall-Sundrum Model . . . . . . . . . . . . . . . . . . 322.2.1 Determining the Metric . . . . . . . . . . . . . . . . . 342.2.2 Solving the Hierarchy Problem . . . . . . . . . . . . . 352.2.3 Chiral Fermions from the Bulk . . . . . . . . . . . . . 372.2.4 Mass Matrices in the 4D Effective Theory . . . . . . 402.3 Proton Decay and n-n Oscillations . . . . . . . . . . . . . . . 412.4 Effective Operator Analysis . . . . . . . . . . . . . . . . . . . 432.5 QCD running of C4Di . . . . . . . . . . . . . . . . . . . . . . 472.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Anomalous B Meson Mixing and Baryogenesis . . . . . . . 553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Flavor Physics Overview . . . . . . . . . . . . . . . . . . . . 583.2.1 B0q -B0q Mixing Overview . . . . . . . . . . . . . . . . 583.2.2 K0-K0 Mixing Overview . . . . . . . . . . . . . . . . 623.2.3 B ? Xs? Decays Overview . . . . . . . . . . . . . . . 633.3 A Top-Centric Two Higgs Doublet Model . . . . . . . . . . . 653.4 Flavor Constraints . . . . . . . . . . . . . . . . . . . . . . . . 673.4.1 B0q -B0q Constraints . . . . . . . . . . . . . . . . . . . . 693.4.2 K0-K0 Constraints . . . . . . . . . . . . . . . . . . . 713.4.3 B ? Xs? Constraints . . . . . . . . . . . . . . . . . . 723.5 Electroweak Baryogenesis . . . . . . . . . . . . . . . . . . . . 753.5.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . 753.5.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . 773.5.3 Charge Transport Dynamics . . . . . . . . . . . . . . 783.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844 The Top Quark Forward Backward Asymmetry, Right-HandedCharge Currents, and Vub . . . . . . . . . . . . . . . . . . . . 864.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2 t-channel Exchange and the AttFB . . . . . . . . . . . . . . . . 894.3 RHCC and the |Vub| Hierarchy . . . . . . . . . . . . . . . . . 934.4 AttFB, RHCC, and the |Vub| Hierarchy . . . . . . . . . . . . . 974.5 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.1 A Leptophobic Z ? . . . . . . . . . . . . . . . . . . . . 994.5.2 A Non-Standard Two Higgs Doublet Model . . . . . . 1044.5.3 Incoherent Top Quark Production . . . . . . . . . . . 1094.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114viiTable of Contents4.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 Conclusions & Outlook . . . . . . . . . . . . . . . . . . . . . . 118Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123AppendixA CP Violation in the SM . . . . . . . . . . . . . . . . . . . . . . 144B Operator Product Expansion and the Principles of EffectiveField Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148C Renormalization Group . . . . . . . . . . . . . . . . . . . . . . 151viiiList of Tables1.1 Gauge representations of the Standard Model particle content. 62.1 Fits of the extra dimensional quark wave function profiles tothe masses and CKM mixing angles. . . . . . . . . . . . . . . 46ixList of Figures1.1 Tree level expansion of the two-point correlation function fora massive gauge field. . . . . . . . . . . . . . . . . . . . . . . 141.2 Virtual 1-loop diagrams for the Higgs mass parameter. . . . . 222.1 The orbifold geometry in the Randall-Sundrum Model. . . . . 322.2 Virtual QCD effects for the anomalous dimensions of effectiveoperators responsible for neutron-antineutron oscillation. . . . 492.3 Frequency of neutron-antineutron oscillations as a function ofthe required warped down mass scale suppression. . . . . . . 512.4 Predicted number of observed neutron-antineutron oscillationevents at Project X at Fermilab. . . . . . . . . . . . . . . . . 543.1 Box diagrams for B0q -B0q mixing arising from W? exchangein the SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Box diagrams for B0q -B0q mixing arising from H? exchange inthe top-charm flavor-violating Two-Higgs-Doublet-Model. . . 683.3 NP magnetic penguin diagrams for b? s?. . . . . . . . . . . 723.4 The parameter space consistent with all flavor observables forthe top-charm flavor-violating Two-Higgs-Doublet-Model. . . 743.5 The left-handed quark and Higgs charge densities in the un-broken phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.6 The computed baryon asymmetry as a function of the phasewall velocity and anomalous flavor-diagonal top quark coupling. 824.1 Left panel: A model-independent fit to the AttFB showing theneed for interference effects. Right panel: The t-channel ex-change of new particles in top-antitop production at the Teva-tron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 A model-independent fit to |Vub| as extracted from inclusive,exclusive, and purely leptonic B meson decays. . . . . . . . . 964.3 1-loop generation of right-handed charge currents in b ? uand t? b transitions. . . . . . . . . . . . . . . . . . . . . . . 97xList of Figures4.4 Electroweak precision constraints and the strength of the right-handed charge current for the leptophobic Z ? model. . . . . . 1024.5 Single top quark production in association with a bb pair inthe leptophobic Z ? model. . . . . . . . . . . . . . . . . . . . . 1034.6 The strength of the right-handed charge current in the non-standard Two-Higgs-Doublet-Model. . . . . . . . . . . . . . . 1064.7 Incoherent production of top-antitop pairs through on-shellproduction and decay of right-handed stop squarks. . . . . . . 1114.8 The strength of the right-handed charge current in the inco-herent production model. . . . . . . . . . . . . . . . . . . . . 113A.1 The Standard Model unitarity triangle. . . . . . . . . . . . . 146xiList of Abbreviations/ET Missing Transverse EnergyAtt?FB Top Quark Forward Backward Asymmetryn-n Neutron-Antineutron2HDM Two-Higgs-Doublet-ModelAdS Anti-deSitterAPV Atomic Parity ViolationBAU Baryon Asymmetry of the UniverseC.L. Confidence LevelCDF Collider Detector at FermilabCKM Cabibbo-Kobayashi-MaskawaCP Charge-ParityEFT Effective Field TheoryEWBG Electroweak BaryogenesisFBA Forward Backward AsymmetryFCNC Flavor Changing Neutral CurrentsHQE Heavy Quark ExpansionIR InfraredKK Kaluza-KleinLEP Large Electron Positron ColliderLHC Large Hadron ColliderxiiList of AbbreviationsLL Leading LogMS Minimal SubtractionMSSM Minimal Supersymmetric Standard ModelnD n DimensionalNLO Next to Leading OrderNP New PhysicsOPE Operator Product ExpansionPMNS Pontecorvo-Maki-Nakagawa-SakataQCD Quantum ChromodynamicsQFT Quantum Field TheoryRG Renormalization GroupRHCC Right Handed Charge CurrentsRS Randall-SundrumSM Standard ModelSSB Spontaneous Symmetry BreakingUV UltravioletxiiiAcknowledgementsThose who know me know that I didn?t start down this path the normalway. I?ve argued with many people over the years as to whether that hashelped or hurt me but one thing, I think, is undeniable. Anyone who takesthis path needs a few lucky breaks in order to make it the whole way. Mylucky breaks have consistently come in the form of the people who havetaken a chance on me despite all the inevitable frustration it entails.First, I would certainly not be anywhere near physics today without theinitial encouragement of Bheesham Dabie and everyone from Argyle. Thankyou also for the scholarship, which allowed me to dedicate most of my timethroughout my undergraduate degree to my studies.I would also like to thank my parents, my sister, Auntie Pat, and all ofmy family for entertaining and encouraging my ?sharp left turn? to pursuethe academic route. Thank you also for understanding and putting up withmy constant absence during this pursuit. I miss you all dearly.My largest thanks, of course, goes to my supervisor and mentor, Profes-sor John Ng. Thank you for teaching me everything I know about quantumfield theory, particle physics, and even a little politics. I think the first thingyou said to me was ?are you going to sink or are you going to swim?? It?staken a while, but I?d like to think that I?ve at least learned to float.I would also like to thank the entire TRIUMF theory group for numer-ous interesting discussions and collaborations over the years. Specifically,I would like to thank David Morrissey for the reference letters and alwayskeeping his door and the possibility of collaboration open to me even thoughI am not his student. I am also deeply indebted to Sean Tulin, who has al-ways been gracious enough to share his ideas, advice, and expertise withme in our continuing collaborations over the years. This has definitely beenone of the more rewarding professional relationships I have had the honorto have. During the last year of my PhD, Travis Martin valiantly tried toencourage me to be more social and live a healthier lifestyle. Although,and unfortunately for me, it never worked, I am still very grateful for theattempt. I would be remiss if I were not to thank all the other TRIUMFpostdocs with whom I have either collaborated or learned from over thexivAcknowledgementsyears: Kristian McDonald, Andy Spray, Abhishek Kumar, and Alejandrode la Puente. Finally, I would like to thank Nikita for all the times we justgot to bounce ideas and questions off of each other. The ?visitors office?sure was lonely before you got there and I hope that more students comesoon so you never have to experience it for yourself.I am ultimately in eternal debt to my beautiful wife, Re?ka, who hassupported me in countless ways throughout this entire process, even when Ididn?t deserve it. I have no idea how to even begin to thank you...I also acknowledge the generous support of the National Science and En-gineering Research Council (NSERC) of Canada through the Post GraduateScholarship program as well as the financial support from the Four-YearFellowship program and various other scholarships from the University ofBritish Columbia.xvJuditnak (For Judit).xviChapter 1IntroductionThe Standard Model (SM) of particle physics, based on the Glashow-Weinberg-Salam theory of electroweak unification [1?3] and quantum chromodynam-ics [4?6], is the distilled knowledge of decades of research. It contains, inprinciple, a highly precise description of all phenomena currently observedin high energy physics experiments. Despite this success, there are a numberof puzzles that continue to plague its status as the ultimate theory of every-thing, indicating that it should be supplanted by some other more funda-mental theory at higher energy scales. The triumph of the SM in explainingcurrently available data imposes strong constraints on any candidate theorythat hopes to replace it. Indeed, a detailed knowledge of the nature of itsfailures has allowed us a glimpse into the possibilities for new physics (NP)that may appear in current and future collider experiments.Within this thesis, we will discuss a number of well-motivated ?exten-sions? to the SM which are designed to ameliorate specific flaws. In orderto investigate these extensions, it is best to proceed on multiple fronts. Thethree main approaches that we employ are often characterized as the threefrontiers of particle physics. They are the cosmic, energy, and intensity fron-tiers. Together, they allow for a multi-faceted approach, fostering a coherentpicture from which a deeper understanding of the underlying physics beyondthe SM can emerge.The cosmic frontier uses a combination of underground-, ground-, andspace-based detectors to capture stable particles carrying information aboutthe cosmos over vast distances. These observations explore the fundamentalphysical laws of the universe through measurements of its large-scale contentand evolution.The energy frontier involves the use of high energy accelerators, such1Chapter 1. Introductionas the Tevatron or the Large Hadron Collider (LHC), to explore the fun-damental constituents of the universe. The subatomic collisions occurringwithin attempt to directly produce particles associated with new phenom-ena such as the origin of mass, with the recently observed Higgs boson, andthe existence of extra dimensions.The intensity frontier seeks to exploit accelerators using high intensitybeams of trillions of particles to perform ultra high precision measurementsof rare phenomena. Exotic particles that do not survive long enough totraverse the cosmos and cannot be produced directly in high energy accel-erators may yet affect these phenomena, producing small, yet significantdeviations in known parameters from their SM predictions. Searching for,and studying these deviations helps to understand and characterize the formof the exotic NP.These frontiers form an interconnected framework that addresses fun-damental questions about the laws of nature simultaneously from multipleangles, ensuring that the whole is more than the sum of the parts and al-lowing for different and unexpected ways to test our best theories as well.Each section of this thesis exploits this framework in various ways.In the first section, we study the Randall-Sundrum model which wasintroduced to address a problem arising in the energy frontier, i.e., the hi-erarchy problem. However, to test this model, we will exploit experimentalsearches for neutron-antineutron oscillations, which lie squarely in the realmof the intensity frontier. We will show that it may be possible to observesuch oscillation events at a proposed upcoming experiment at the Tevatronusing a high intensity beam of cold neutrons.The second section features the dimuon asymmetry, which was measuredin high energy collisions at the Tevatron and found to significantly disagreewith SM predictions. We propose a model to explain this asymmetry and goto great lengths to explore its efficacy when confronted with highly precisemeasurements of observables associated with the intensity frontier. Fur-thermore, we show that our model also has the capacity to simultaneouslyprovide an explanation for a deep mystery of the cosmic frontier, namelythe baryon asymmetry of the universe.21.1. The Standard ModelFinally, in the last section we study a class of models designed to explainanother high energy collider anomaly, i.e., the top quark forward backwardasymmetry. We point out that the interactions that define this class ofmodels are highly advantageous for explaining a long-standing tension in Bmeson decays. Solidifying this possible connection between the energy andintensity frontiers leads to predictions for novel collider signals that can besearched for in the high energy collisions of the LHC.Particle phenomenology is an especially fast-moving area of research.Any idea born into the field, either through an unexpected experimentalobservation or a novel theoretical insight, evolves rapidly over a time scaleof, in some cases, mere days. Because of this, we have added an epilogueto each chapter in order to provide the reader with a contemporary view ofthe ideas presented within.As a final note of caution, although the original models presented withinthis thesis are meant to improve upon the SM, none of them are to be inter-preted as a candidate for a ?theory of everything?. Each of the works thatconstitutes this thesis was done in the spirit of the ?bottom-up? approach,in which a particular flaw of the SM is identified and ameliorated, leavingthe rest for further research.1.1 The Standard ModelThis thesis discusses many extensions to the SM. As such, it behooves usto first revisit it in a more pedagogical manner in order to properly set thestage for our further discussions of how to fix it.1.1.1 Structure of the Standard ModelThe SM is a quantum field theory of fermionic particles (matter) interactingwith each other via the exchange of bosonic particles (forces). All the in-formation we know about the SM and its interactions is summarized in theLagrangian, LSM . This Lagrangian is required to be hermitian (conserva-tion of probability), local (physical phenomena occurring at different points31.1. The Standard Modelin space at a given time must be independent), and invariant under Poincaresymmetry (Lorentz transformations and spacetime translations). One of themain reasons as to why the SM is such a theoretically satisfying theory isthat it is, quite simply, the most general set of renormalizable [7] interac-tions given two basic inputs: the particle content and the gauge symmetry.These must, and have been, gleaned from detailed experimental measure-ments. There currently exists no deep understanding of why the universehas chosen these two inputs1. Thus, in the following, we will simply takethese as observed facts and present them as such.The matter of the universe is understood as being made up of theknown spin-1/2 particles (fermions). The strong, weak, and electromag-netic interactions then arise from the exchange of various spin-1 bosonsassociated with the different symmetries of the SM gauge group, given bySUc(3) ? SUL(2) ? UY (1). The number of gauge bosons associated with agiven symmetry is equal to the number of generators the group has. Thehypercharge group, UY (1), having only one generator, has one gauge bosonassociated with it called the hypercharge boson, B?. The other two groupshave a more complicated structure and require more generators to span theirspaces. The SUL(2) group is associated with weak isospin symmetry andhas three generators corresponding to the ever-familiar Pauli matrices, ?i/2(i = 1, 2, 3), which are then associated with three gauge bosons denoted asW i?. The factor of 1/2 is a normalization factor needed to ensure that thegroup algebra is closed under commutation, i.e.,[?i/2, ?j/2]= iijk?k/2where ijk is the 3-dimensional Levi-Civita symbol. The subscript ?L? ismeant to remind the reader that only the left-handed fermions transformnon-trivially under this particular symmetry. The SUc(3) group is the mostcomplicated with 8 generators, T a (a = 1, 2, . . . , 8), which are related to thewell known Gell-mann matrices, ?a, by a normalization factor of 1/2, i.e.,1Although the requirement of anomaly cancellation does constrain the fermion contentof the theory, this choice is by no means unique, i.e., there are many other choices forwhich this cancellation is satisfied.41.1. The Standard ModelT a = ?a/2. These are given explicitly by?1 =???0 1 01 0 00 0 0??? , ?2 =???0 ?i 0i 0 00 0 0??? , ?3 =???1 0 00 ?1 00 0 0??? ,?4 =???0 0 10 0 01 0 0??? , ?5 =???0 0 ?i0 0 0i 0 0??? , ?6 =???0 0 00 0 10 1 0??? ,?7 =???0 0 00 0 ?i0 i 0??? , ?8 =1?3???1 0 00 1 00 0 ?2??? . (1.1)These 8 generators then correspond to 8 gauge bosons called the gluons, Ga?,while the subscript ?c? is meant to denote color. Any particles that trans-form non-trivially under a given symmetry are said to be ?charged? underthat symmetry. These charged particles will thus experience the force(s)related to that symmetry via Noether?s theorem through the exchange ofthe associated gauge bosons.How a given particle in the SM interacts is understood by specifyingits transformation properties under the symmetries of the SM gauge group.These transformation properties are summarized by the representation of thegauge group under which they transform. The standard accepted method forlabeling the representations of the SU(N) group is with its dimension, e.g., adoublet representation of SUL(2) is labeled as 2 while a triplet (anti-triplet)of SUc(3) is labeled as 3 (3). The singlet representation of SU(N) is denotedas 1. A particle in the singlet representation does not transform under thegroup and thus does not interact with the associated gauge bosons. Therepresentation of a U(1) group is specified by giving the eigenvalue of thegenerator, e.g., the value of the weak hypercharge, Y, for the hyperchargegroup.The gauge bosons of the SM have the following gauge representations:Ga?: (8, 1, 0), W i?: (1, 3, 0), and B?: (1, 1, 0). Note that, since the Ga? (W i?)gauge bosons transform non-trivially under the group SUc(3) (SUL(2)), they51.1. The Standard ModelFermionsGenerations Representation1st 2nd 3rd SUc(3)? SUL(2)? UY (1)(?ee?)L(????)L(????)L( 1, 2, -1/2)e?R ??R ??R ( 1, 1, -1)(ud)L(cs)L(tb)L( 3, 2, 1/6)uR cR tR ( 3, 1, 2/3)dR sR bR ( 3, 1, -1/3)BosonsGa? ( 8, 1, 0)W i? ( 1, 3, 0)B? ( 1, 1, 0)Table 1.1: The gauge representations of the particle content of the SM inthe unbroken phase.also interact through the exchange of gauge bosons, i.e., they have self-interactions. This is in contrast to the UY (1) hypercharge case in whichthe hypercharge boson only interacts directly with charged fermions. Theseself-interactions have drastic consequences for how the strength of the inter-actions change with energy scale. There are 3 ?generations? of fundamen-tal fermions in the SM. Each generation is an exact copy of the other twowith respect to their representations under the SM gauge group, however,they differ significantly in terms of the masses of their constituents. Allgenerations contain both leptons (particles that interact only electroweakly,i.e., only transform non-trivially under SUL(2)?UY (1)) and quarks (parti-cles that also interact strongly, i.e., transform non-trivially under SUc(3)).For each generation there is a charged and neutral lepton as well as a pairof quarks, one of which is denoted as an ?up-type? quark while the otheris denoted as a ?down-type? quark. The SM is a chiral theory, meaning61.1. The Standard Modelthat the left and right handed spinor representations of the fermions havedistinct gauge representations and therefore interact differently from eachother. In each generation, the left-handed leptons and quarks are organizedinto SUL(2) doublets, denoted as L and Q respectively, while their right-handed counterparts transform trivially under the SUL(2) symmetry. Thereason for the odd nomenclature for the quarks is to remind us of their po-sition in the SUL(2) doublets, i.e., the up-type (down-type) quarks occupythe upper (lower) component of the doublet as shown in Table 1.1.There are six leptons and six quarks in total that are currently known.The different generations are organized from lightest (1st) to heaviest (3rd)with their constituents labelled as follows:1st - electron (e?), electron-neutrino (?e), up quark (u), down quark (d)2nd - muon (??), muon-neutrino (??), charm quark (c), strange quark (s)3rd - tau (??), tau-neutrino (?? ), top quark (t), bottom quark (b)In order to understand how their gauge representations dictate their interac-tions with the gauge bosons, we briefly digress to a discussion of Yang-Millsgauge theory in the next section.1.1.2 Yang-Mills Gauge TheoryThe interactions of the SM fermions with the gauge bosons and the dynam-ics arising from them are entirely specified by the principle of local gaugeinvariance. This principle, loosely stated, simply says that the Lagrangianof a gauge theory should be invariant to transformations associated withthe gauge symmetry. The Lagrangian of the SM, LSM , is constructed fromfields associated with the SM particles introduced in the previous sectionand required to be invariant with respect to local gauge transformations.In order to illustrate the principle of local gauge invariance, we consider aYang-Mills gauge theory [8], invariant under the Lie group G. This grouphas N generators, T a (a = 1, 2, . . . , N)2, whose algebra is specified by the2These are not to be confused with the Gell-mann matrices introduced in section 1.1.1.71.1. The Standard Modelcommutation relations[T a, T b]= ifabcT c (1.2)where fabc are called the structure constants of the group G. The structureconstants obey a Jacobi identity,fadef bcd + f bdef cad + f cdefabd = 0, (1.3)and a basis for the generators can always be chosen such that they arecompletely anti-symmetric in their indices.We represent a generic fermion field in the SM as ?(x), which can bedescribed by a 4-component Dirac spinor field which is a function of space-time coordinates x?. Under a local gauge transformation of the group G,?(x) transforms under the action of the local unitary operator, U(x), suchthat?(x)? U(x)?(x) = ei?a(x)Ta?(x) (1.4)where the ?a(x)?s are local transformation parameters that act as a setof coordinates for the abstract N -dimensional group space. The free fieldLagrangian for the spinor field, ?(x), is given by the standard formL = ?(i/? ?m)?. (1.5)For brevity we have dropped the explicit space-time dependence of the spinorfield and have used the Feynman slash notation to represent the contractionof the space-time derivative with the Dirac matrices, i.e., /? = ????. Al-though the mass term is invariant under these transformations, the kineticterm is not due to the fact that the derivative acts on the space-time depen-dent group space coordinates. Invariance is regained by the introduction ofthe quantity known as the covariant derivative,D? = ?? ? igT aAa?, (1.6)81.1. The Standard Modelwhich ensures that D?? transforms identically to ? as long as the realvector fields, Aa?, (the gauge bosons associated with the symmetry group G)transform asT aAa? ? ei?bT b(Aa?T a +ig ??)e?i?cT c (1.7)where the real constant, g, is called the gauge coupling. The gauge fieldsare usually said to transform under the ?adjoint? representation. As withthe fundamental representation for the fermions, the transformation prop-erties of the gauge field are specified by giving the dimension of the adjointrepresentation, e.g., N2 - 1 for SU(N). In this representation, the structureconstants of the group are determined by the generators as (T b)ac = ifabc.The gauge invariant Lagrangian for the spinor field is then given bysimply replacing the partial derivative with the covariant one,L = ?(i /D ?m)? = ?(i/? ?m)? + g???T a?Aa? (1.8)The introduction of the covariant derivative, necessitated by the need forgauge invariance, has forced us to include interactions between the spinorfield and the gauge bosons with strength g. In order for the gauge fields tobe dynamical (propagate in space-time) we must obviously design a uniquekinetic term for them. It turns out that this is given by adding the term? 14F a??F a?? (1.9)to the Lagrangian, where the quantity, F a?? , is called the ?field tensor? andis defined in terms of the covariant derivative asig [D?, D? ] = Fa??T a. (1.10)Using this definition, the field tensor is written in terms of the gauge fieldasF a?? = ??Aa? ? ??Aa? + gfabcAb?Ac? . (1.11)91.1. The Standard ModelThe total classical, interacting, gauge invariant Lagrangian is then given asL = ?14F a??F a?? + ?(i /D ?m)?= ?(i/? ?m)? + 12Aa?(?2??? ? ????)Aa? + g???T a?Aa??gfabc(??Aa?)Ab?Ac? ? g2fabef cdeAa?Ab?Ac?Ad? . (1.12)The Lagrangian now contains interactions between the fermions and thegauge fields as well as both cubic and quartic self-interactions for the gaugefields themselves. Note, however, that these self-interactions depend directlyon the structure constants of the group. A group is called ?abelian? if all thegenerators of the group commute with each other, i.e., if the structure con-stants are exactly zero. The gauge bosons associated with abelian symmetrygroups do not have the self-interactions listed above. This is the reason thatthe hypercharge gauge boson does not have the capacity for self-interactionswhile the gauge bosons of the weak isospin and color groups do, i.e., UY (1)is abelian while SUL(2) and SUc(3) are non-abelian.The most crucial observation to note about the Lagrangian in eq. (1.12)is that there is no mass term for the gauge fields. In fact, any term with theappropriate form for such a mass term, m2Aa?Aa?, is explicitly forbidden bythe principle of gauge invariance as it explicitly breaks the gauge symme-try! As the weak force is observed to be a short range force, an effect thatrequires the gauge bosons that communicate this force to be massive, thiswas an extremely deep problem that had to be understood before a properdescription of the SM could be written down. A method was discoveredin which a mass term for gauge fields can be generated without explicitlybreaking the gauge symmetry. This method is generally known as spon-taneous symmetry breaking (SSB) [9?12] and its application to the SM iscalled the Higgs mechanism after one of the authors who first discovered itsutility in the field of particle physics.In addition to fixing the form of the renormalizable interactions of thegauge fields with the fermions, the gauge symmetry plays a further role inthe construction of particle theories by specifying conserved currents as pre-101.1. The Standard Modeldicted by Noether?s theorem3. The conservation of these currents imposesfurther constraints on the interactions of the theory through a special set ofrelations called the Ward-Takahashi identities [13, 14]. These identities holdeven in the presence of spontaneous breaking of the gauge symmetry and,in particular, require the two-point function for the (massive) gauge fieldsto be transverse, i.e.,???(p) =(??? ? p?p?p2)?(p). (1.13)The fact that the Ward-Takahashi identities are directly due to gauge in-variance and yet remain intact after the gauge symmetry is spontaneouslybroken is the reason why it is often said that spontaneous symmetry break-ing simply ?hides? the gauge symmetry and doesn?t actually ?break? it.In the next section, we will see explicitly how SSB generates masses forthe gauge fields while maintaining the transverse form for the two-pointfunction, thereby upholding the Ward identities.1.1.3 Spontaneous Symmetry BreakingSSB is a special realization of symmetry breaking in a physical system inwhich, although the equations of motion, and therefore the Lagrangian, areinvariant with respect to a set of symmetry transformations, the ground stateof the system is not. To illustrate this, consider a system of complex scalarfields, ?i(x) (i = 1, 2, . . . , N), whose dynamics are described by a Lagrangianwhich is invariant under the transformations of the symmetry group SU(N).These scalar fields transform in the fundamental representation of the group3Noether?s theorem guarantees that a continuous symmetry of a Lagrangian corre-sponds to a conserved current at the classical level. Although there exist a class of sym-metries, called anomalous symmetries, that are broken by quantum effects, we do not dealwith these here as the SM gauge symmetry is not anomalous.111.1. The Standard Modelas?? ei?aTa? where ? =???????1?2...?N??????(1.14)and the Lagrangian is chosen to beL = (D??)?D??? V (|?|2) with V (|?|2) = ??2|?|2 + ?(|?|2)2(1.15)where D? is the covariant derivative from eq. (1.6). The ground stateof the theory, a real constant field configuration denoted as ?0, minimizesthe potential and is called the vacuum expectation value (vev) of the field?. In the case of the potential in eq. (1.15), this is given by |?0|2 =?Ni=1 |?0i|2 =?22? ?v22. Note that this condition only specifies the ?length?of the vector ?0, leaving the direction in field space completely arbitrary.The symmetry is said to be spontaneously broken when the system choosesa particular preferred direction in field space. It is common to choose thisdirection to be aligned with the real part of the N th component of the fieldvector ? by choosing the vevs of the individual fields such that: ?0i = 0(i=1, 2, . . . , N ? 1), Im?0N = 0, Re?N =v?2.The potential can now be simplified by adding a constant term, ?(|?0|2)2,whose only effect is to shift the minimum energy state to V (|?0|2) = 0and highlight the existence of a non-trivial ground state, i.e., V (|?|2) =?(|?|2 ? |?0|2)2. In order to expand about the true vacuum state, it isconvenient to shift the field vector such that?? ? + ?0 =????????1?2...?N +v?2???????. (1.16)121.1. The Standard ModelExpanding the potential in terms of this new shifted field vector then leadstoV (?i) =12(2?v2)Re?2N + 2?vRe?NN?i=1|?i|2 + ?(N?i=1|?i|2)2. (1.17)The mass term for the real part of the field ?N breaks the symmetry in thepattern SU(N)? SU(N?1). Note also that the only field to develop a massis the exact field which absorbed the vev. This is no accident! The appear-ance of massless particles in theories with spontaneously broken continuoussymmetries is required by Goldstone?s theorem [15?17]. Specifically, for ev-ery spontaneously broken continuous symmetry, the theory must contain amassless particle. The number of continuous symmetries of a group is equalto the number of generators it has. Since the original theory had an SU(N)symmetry (N2?1 generators) while the spontaneously broken theory has anSU(N?1) symmetry ((N?1)2?1 generators), the number of spontaneouslybroken symmetries is given by the difference, N2?1?((N?1)2?1) = 2N?1.Since each complex field has two degrees of freedom and the complex partof ?N remains massless, the number of Goldstone particles, indeed, exactlymatches the number of spontaneously broken continuous symmetries.In order to understand the effect of SSB on the kinetic term of our theory,we expand it in terms of the shifted field vector as|D??|2 = |???|2 + igAa?((T a?)????? ????(T a?))+ g2Aa?Ab?((T b?)?(T a?) + (T b?)?(T a?0) + (T b?0)?(T a?))igAa?((T a?0)????? ????(T a?0))+ g2Aa?Ab?(T b?0)?(T a?0).(1.18)The first term is the standard kinetic term for all the complex scalar fieldsin ?, while the next five terms describe how the scalars and gauge fieldsinteract with each other. It is the last three terms, however, that are the131.1. The Standard ModelAa?(p) Ab?(p)= +Figure 1.1: The tree level perturbative expansion of the two-point functionfor a gauge field rendered massive through SSB.most interesting.First, consider the last term. Since ?0 is some constant field configu-ration, the quantity 2g2(T a?0)?(T b?0) acts as a mass matrix for the gaugebosons. Generally speaking, all gauge bosons will now receive masses fromthe vev unless there exist any generators that leave the vacuum invariant,i.e., for which the vacuum has zero charge T a?0 = 0. These generators willnot contribute to the mass matrix and the corresponding gauge bosons will,thus, remain massless.Next, consider the other terms in the last line of eq. (1.18). The onlyscalar fields, ?i, that are involved in these interactions are those that are par-allel to the non-zero components of the vector (T a?0). These are preciselythe fields that are associated with the breaking of a continuous symmetry,i.e., the Goldstone bosons. Treating these interactions perturbatively, thetwo-point function for the massive gauge field is expanded as in Figure 1.1,where both the gauge field mass term and the Goldstone propagation areincluded. The amplitude for this expansion is given by4(ig2???(T b?0)?(T a?0) +(g(T b?0)?p?) ip2(? g(T a?0)p?))= ig2(T b?0)?(T a?0)(??? ? p?p?p2). (1.19)The Goldstone bosons therefore uphold the Ward identity during SSB byproviding the longitudinal components of the polarization tensor. Because4This amplitude is calculated in the Landau gauge, where the Goldstone bosons remainbut are massless. In the R? gauge, the two point function develops a gauge dependent,and therefore non-physical, longitudinal component. Once we choose a gauge, such as theLandau gauge chosen above, the two-point function is rendered transverse.141.1. The Standard Modelof this, they are re-interpreted as the longitudinal spin components of thenow massive gauge bosons. In this sense, observation of the longitudinalspin states of the massive gauge bosons of the SM can be interpreted as theobservation of the remnants of SSB and therefore as indirect evidence forthe existence of a Higgs boson.In the next section, we will put all of this together to write down the SMLagrangian.1.1.4 The Standard Model LagrangianArmed with knowledge of gauge theories, SSB, and the particle content andgauge symmetry of the SM, we can now construct the SM Lagrangian. Thefull Lagrangian is broken up into five main parts asLSM = LKin +LY uk +LHiggs +LGF +LGhost. (1.20)The first part consists of the kinetic terms for each of the fermions and thegauge bosons and is given byLKin = QiLi /DQiL + uiRi /DuiR + diRi /DdiR + Lii /DLi + `iRi /D`iR?14B??B?? ? 14W i??W i?? ? 14Ga??Ga?? (1.21)where i is a generational index. These terms necessarily include the in-teractions between the fermions and the gauge bosons as well as the self-interactions between the gauge bosons described in section 1.1.2. In theweak basis, the general form for the covariant derivative for the quarks andleptons is given by(D?)?? = ????? ? ig1W i?(?i2)??? ? ig2Y B???? ? igsGa?(T a)??(1.22)where g1, g2, and gs are the weak, hypercharge, and strong gauge couplingconstants, i(a) are the adjoint indices of SUL(2) (SUc(3)) and ?, ? are thefundamental indices of SUc(3). The field strength tensors are the same as151.1. The Standard Modelin eq. (1.11) with the structure constants given by 0 (UY (1)), ijk (SUL(2)),and fabc (SUc(3)).In order to write down the Yukawa Lagrangian, we need to know whatthe gauge representation of the Higgs is. We can find this out by the re-quirement that these interactions generate the fermion mass terms. If wetake a general fermion mass term, m??, and expand it in terms of thechiral projections of the fermion field, ?L,R = PL,R? = 12(1 ? ?5)?, thisyields m(?L?R + ?R?L). The Yukawa interactions must then take theform QLHqR for the quarks and LH`R for the leptons. Looking at thegauge representations of the quarks and leptons, we can uniquely identifythe SM Higgs gauge representation as H : (1,2, 1/2), which dictates theform of the Yukawa Lagrangian asLY uk = ??uijQiLH?ujR ? ?dijQiLHdjR ? ?`ijLiH`jR + h.c. (1.23)where ?u,d,`ij are 3?3 matrices in ?flavor? space and H? = i?2H?. This La-grangian is written down in the weak basis and must be rotated to the massbasis before the fermion mass terms can be manifest. Since the ?u,d,` ma-trices are not necessarily hermitian, somewhat more round-a-bout methodsmust be used to diagonalize them. Each matrix is factorized using a singularvalue decomposition, i.e., ?u,d,` = Au,d,` yu,d,` Bu,d,`? where both Au,d,` andBu,d,` are unitary matrices and yu,d,` is a diagonal matrix with real, positiveentries. This diagonalization requires that the fermion fields are rotated inflavor space as follows:uiL ? AuijujL uiR ? BuijujRdiL ? AdijdjL diR ? BdijdjR`iL ? A`ij`jL `iR ? B`ij`jR?iL ? A?ij?jL (1.24)where rotations of any possible right-handed neutrinos have been neglectedas no such particles have yet been observed. After this rotation, the Yukawa161.1. The Standard ModelLagrangian becomesLY uk = ?yuijQiLH?ujR ? ydjkQiLVijHdkR ? y`jkLiUijH`kR + h.c. (1.25)with Q = (uL, V dL)T and L = (`L, U?L)T . The unitary matrices, V =Au?Ad and U = A`?A? , are the Cabibbo-Kobayashi-Maskawa (CKM) [18?20] and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) [21?23] matrices re-spectively. As it will play a larger role in this thesis, the structure of theCKM matrix is discussed in more detail in Appendix A.Anticipating that the mass of a given fermion will be parameterized asmf = yfv?2, we can rewrite the Yukawa Lagrangian in it?s final form interms of the diagonal fermion mass matrices asLY uk = ??2v MuijQiLH?ujR ??2v MdjkQiLVijHdkR??2v M`jkLiUijH`kR + h.c. . (1.26)When the doublet takes on its vev, these terms give both the fermion massesand the fermion-Higgs and fermion-Goldstone interactions.The Higgs sector of the SM is given byLHiggs = |D?H|2 ? V (|H|2) (1.27)with the potential V (|H|2) = ?(|H|2 ? v22)2, as in section 1.1.3. As theHiggs field transforms under SUL(2)?UY (1), we can parametrize it as shownin eq. 1.16,H = 1?2( ?2?+v + h+ i?0)(1.28)where ?+ and ?0 are the Goldstone bosons corresponding to the longitudinalcomponents of theW+ and Z boson. Inserting this doublet into the potentialgenerates a mass for the Higgs field, h, and all interactions between theGoldstone bosons and the Higgs.171.1. The Standard ModelThe kinetic term for the Higgs doublet generates the mass matrix forthe spin 1 weak gauge bosons|D?H|2 ?v28g21(W 1?W 1? +W 2?W 2?)+v28(W 3? B?)(g21 ?g1g2?g1g2 g22)(W 3?B?)(1.29)Defining the charged weak gauge bosons as W?? = 1?2(W1? ?W 2?) and in-troducing the weak mixing angle, ?W , to diagonalize the mass matrix, thisleads to|D?H|2 ?v24g21W?? W?? +v28(A? Z?)(0 00 g21 + g22)(A?Z?)(1.30)where(A?Z?)=(cos ?W sin ?W? sin ?W cos ?W)(B?W 3?)(1.31)with tan ?W = g2/g1. Note that this process leaves the photon masslesswhich, as elaborated on in section 1.1.3, implies that the vacuum expectationvalue of the H doublet has zero electric charge. Rewriting the covariantderivative in eq. (1.22) in terms of the mass basis fields then gives(D?)?? = ????? ? ieW(W+? ?+ +W?? ??)????ieZ(?3/2?Q sin2 ?W)Z???? ? ieQA???? ? igs(T a)??Ga? (1.32)with eW = g1/?2, eZ =?g21 + g22, e = g1 sin ?W , ?? = 12(?1 ? i?2) andQ = ?3/2 + Y . With this electric charge operator, one can explicitly checkthat the Higgs vacuum expectation value is neutral, explaining the presenceof the zero eigenvalue for the photon in the mass matrix in eq. (1.30). Usingthis covariant derivative in the quark and lepton kinetic terms in eq. (1.21)181.1. The Standard Modelleads to the electric, neutral, and charged current interactionsLKin ? eJ?EMA? + eZJ?NCZ? + eWJ?CCW+? + h.c.withJ?EM =?f f i??Qfi J?NC =?f f i??(gfLPL + gfRPR)fiJ?CC = ui??PLVijdjW+? + ?i??PLUij`jW+? (1.33)where f is a generic fermion field with flavor indices i, j and the neutralcurrent couplings are given by gfL = ?3/2 ? Q sin2 ?W , gfR = ?Q sin2 ?W .Note that the electric and neutral current interactions do not depend oneither the CKM or the PMNS matrices. This is directly due to the unitarityof the two matrices and implies that only the charged current can induceflavor-changing interactions connecting different generations of fermions inthe SM. Although they are absent at the tree level, at the loop level thecharged current induces flavor-changing neutral currents (FCNC). Thesecurrents are suppressed by not only the characteristic loop suppression butalso by the GIM mechanism, which is the manifestation of the unitarity ofthe CKM in loop processes [20]. This is seen as a major success for the SMas processes induced by these FCNC are observed to be very rare in natureas well.In the full theory, there is need for further complications in the formof gauge fixing and the introduction of ghost states in order to achievea consistent quantization of the model. In principle, one would start inthe R? gauge, introducing the gauge fixing terms using the Faddeev-Popovtrick [24]LGF = ?12?(??W i? + ig1?(H??i2H0 ?H?0?i2H))2? 12?(??B? + ig22?(H?H0 ?H?0H))2(1.34)where ? parameterizes our choice of gauge. These terms have multiple pur-191.2. Problems in the Standard Modelposes:? To allow for a consistent quantization of the gauge fields by ensuringthat the two-point function of a given gauge field is non-singular.? To cancel all mixing between the gauge fields and the Goldstone bosonsfrom the Higgs kinetic term.? To introduce the gauge-dependent, and therefore non-physical, Gold-stone boson masses.The ghost states arise when attempting to use the Faddeev-Popov trickon a non-abelian gauge theory. Because of the non-abelian nature of thegauge symmetry, it becomes necessary to introduce new fields, called Faddeev-Popov ghosts. However, these fields necessarily have the wrong relationbetween their spin and statistics to be physical particles, i.e., they are anti-commuting spin 0 fields. This means that they can only affect the quantumnature of the theory, existing only as internal lines in loop corrections. Theirpurpose is to cancel the effects of unphysical timelike and longitudinal po-larization states of the gauge bosons. A full review of these aspects of theSM are beyond the scope of this thesis, however, they have been extensivelydetailed in the standard textbooks [25?30].1.2 Problems in the Standard ModelIn the previous sections, we have outlined the principles and particle contentof the SM of particle physics, which has survived almost four decades ofintense experimental scrutiny. In this section, we discuss a selection of someof the current known problems with the SM. Our selection is specificallychosen to motivate the original work contained in the subsequent sectionsof this thesis.1.2.1 The Hierarchy ProblemAt its heart, the hierarchy problem is a question of naturalness. The mass ofthe smallest possible black hole defines what is known as the Planck mass,201.2. Problems in the Standard ModelMPl. This defines the strength of the gravitational force as G ? 1/M2Pl ?10?36, whereas the strength of the weak force is GF ? 1/M2W ? 10?5. Wecurrently have no understanding whatsoever of why the strengths of thesetwo fundamental forces are so different or, put another way, why MW is 16orders of magnitude smaller than MPl. This vast separation of scales, orrather our lack of an explanation for it, is called the hierarchy problem. Thephysical principle of naturalness, as articulated by ?tHooft[31], states thata small parameter is natural only when a symmetry is gained as it is set tozero. In the context of quantum field theory, if a bare parameter is tunedto be small then radiative corrections will lead to a physical value that isrestored to the natural scale of the theory unless there exists a symmetrywhich prevents this. Each time this principle is violated, a hierarchy problemarises.In the SM, there are a number of hierarchy problems. The most fa-mous of which is the gauge hierarchy problem, in which the Higgs potentialdisplays a disturbing sensitivity to NP at the highest mass scales throughquantum corrections to the Higgs mass parameter. The method of consis-tently obtaining physical scalar masses in renormalized perturbation the-ory is described in detail in reference [26] and we will not repeat it here.We will simply state that the physical Higgs mass is given by the relationm2h = m20 +M2(p2)????p2=0, where m0 is the bare mass parameter and M2(p2)is the sum of all 1-particle irreducible corrections to the scalar propagator.We consider the corrections from interactions with another scalar, a gaugeboson, and a fermion with the generic interactionsL Sint = ?g|H|2|?|2 ? ?gvh|?|2 ?g22h2|?|2L Vint = |(?? ? igA?)H|2 ? 2vg2hA?A? + g2h2A?A?L Fint = ?ghff. (1.35)The virtual 1-loop diagrams generated by these interactions are shown inFigure 1.2 and the contributions to M2(p2) are calculated in the Landaugauge (in which electroweak ghost contributions can be ignored), using the211.2. Problems in the Standard Modelh hff+h h?h h??+h hAh hAAFigure 1.2: Virtual 1-loop diagrams that arise when calculating the physicalHiggs mass in the Landau gauge.MS renormalization scheme. The results from the scalar, vector, and fermioninteractions areM2(p2)????Sp2=0m???=g2(4pi)2m2?(log(m2??2)? 1)M2(p2)????Vp2=0mA??=12g2(4pi)2m2A(log(m2A?2)? 1)M2(p2)????Fp2=0mf??=4g2(4pi)2m2f(1? 3 log(m2f?2))(1.36)where ? is an arbitrary momentum scale associated with dimensional regu-larization and we have taken the limit of large masses. The key feature of221.2. Problems in the Standard Modelall the above corrections is that, regardless of the properties of the particlespropagating in the loops, the result is always proportional to the mass. Inthe SM, the top quark contribution dominates due to its large Yukawa cou-pling and mass. However, since these are virtual corrections, any particlethat couples to the Higgs, regardless of how heavy it is, must be included.This makes the Higgs mass extremely sensitive to the heaviest mass scalesin the theory. In principle, we are always free to tune the bare mass term toachieve a weak scale physical Higgs mass. However, if there exist unknownvery heavy particles propagating in these loops, we are necessarily forcedinto a situation in which this requires a high level of fine-tuning to achievea physical weak scale mass.If the SM were the full theory of nature then we expect no further parti-cles and there is no fine-tuning problem as the SM is renormalizable. How-ever, as there are many outstanding problems for which the SM offers noexplanation5, it is widely regarded that it must be an effective theory. Thisnecessarily implies the existence of another scale which corresponds to somenew particle mass and acts as the SM cut-off. The higher this cut-off be-comes, the more fine-tuning is needed. In the worst case scenario, this cut-offwill be O(MPl) with O(1) coupling to the Higgs, requiring a cancellation atthe level of O(1026) in order to achieve a weak scale Higgs mass!The gauge hierarchy problem is unique to fundamental scalar particlesas gauge boson and fermion masses are protected from unreasonably largequantum corrections by gauge invariance and approximate chiral symmetryrespectively. This has motivated the search for new symmetries that mayprovide such protection for fundamental scalars, such as supersymmetry [37]and some formulations of extra dimensions [38].In this thesis, we will concentrate on an especially elegant solution to thehierarchy problem known as the Randall-Sundrum model [39]. In this model,the large hierarchy between the weak and Planck scales arises due to thewarping of a compactified Anti-deSitter (AdS) extra dimensional geometry.5e.g., the strong CP problem [32], the origin of neutrino masses [33], the cosmologicalconstant problem [34], the particle nature of dark matter [35], the baryon asymmetry ofthe universe [36], etc.231.2. Problems in the Standard ModelThis warping manifests itself as an exponential suppression of the massscales within the theory, generating the hierarchy with only O(1) tunings ofparameters. In chapter 2, we will further elaborate on the Randall-Sundrummodel and explore the possibility of testing it through searches for neutron-antineutron oscillations.1.2.2 The Baryon Asymmetry of the UniverseAlthough the CKM paradigm, i.e., the SM description of CP violationthrough the single phase in the CKM matrix, has established itself as thedominant source of CP violation [40], there are many good reasons to be-lieve that this cannot be the full story. The first and most compelling reasoncomes from the cosmic frontier in the form of the baryon asymmetry of theuniverse.Consider the case in which there is no CP violation. Assuming an initialcondition of equal number densities for quark and anti-quark fields6, CP in-variance guarantees that this equality is conserved throughout the history ofthe universe. At high temperatures, T & 2mp, annihilation and productionensures that protons and antiprotons are in equilibrium, i.e., nB = nB ? n? .However, at T . 2mp, production slows, allowing annihilation to take over.If nB = nB while annihilation continues, we would expect there to be essen-tially no matter (or antimatter) nowadays, an expectation that is in seriousconflict with present observations of a non-zero matter density and our ex-istence in general!If CP violation is present, then this can allow for different rates for agiven process and its CP conjugate. If this situation is accompanied by bothbaryon number violation and a departure from thermal equilibrium then anasymmetry in quark and antiquark densities can develop dynamically dur-ing the evolution of the universe [41]. This biased dynamical generation ofmatter over antimatter is called Baryogenesis. If this asymmetry is presentin the early universe then, at T . 2mp, the annihilations continue untilthe antiproton density is completely depleted, leaving a small amount of6This assumption is consistent with the theory of inflation which would dilute anyinitial asymmetry in the rapid expansion process.241.2. Problems in the Standard Modelproton density which should last forever. The present day measurement ofthe baryon-to-entropy ratio Y obsB =(nBs)0= (7.8 ? 0.7) ? 10?11, derivedfrom nucleosynthesis constraints [42], is then actually a CP violating observ-able and representative of the baryon asymmetry,(nq ? nqs), in the earlyuniverse.Interestingly, the SM with the CKM paradigm predicts the asymmetryto be many orders of magnitude below the measured value [43, 44]. Fur-thermore, this basic result is independent of other aspects of the generationof the asymmetry, i.e., the asymmetry is far too small even if the depar-ture from equilibrium is achieved by beyond SM mechanisms [45]. This hasthe extremely strong implication that there must exist other sources of CPviolation beyond the CKM paradigm.There have been many proposed mechanisms to account for this, how-ever, in this thesis we will restrict our interest to the method of electroweakbaryogenesis [46?48] (EWBG), in which the baryon asymmetry is dynam-ically generated during the electroweak phase transition. In chapter 3, wewill discuss a Two-Higgs-Doublet-Model which is designed to simultaneouslyexplain the dimuon asymmtery (a collider anomaly discussed in the next sec-tion) and accommodate the observed baryon asymmetry of the universe.1.2.3 Collider AnomaliesPossibly the most compelling arguments for NP beyond the SM come di-rectly from collider experiments. Within these highly controlled environ-ments, any deviation from SM expectations can be reproduced and studiedfrom different angles by changing the experimental setup. Because of thecontrol that can be exerted over the system under study, any anomaloussignals from existing collider experiments are of great interest. In the fol-lowing, we discuss two such signals which serve to motivate the original workpresented in chapters 3 and 4 of this thesis.251.2. Problems in the Standard Model1.2.3.1 The Like-Sign Dimuon AsymmetryStudies of particle production and decay under the reversal of discrete sym-metries in colliders have yielded considerable insight into the necessary struc-ture of high energy physics. In particular, in 2010, the D? collaboration atthe Tevatron released evidence for an anomalous like-sign dimuon chargeasymmetry in semileptonic b-hadron decays [49]. The asymmetry is definedasA ? N++ ?N??N++ +N?? (1.37)with N++ (N??) representing the number of events in which two muons,passing kinematic selection cuts, have the same positive (negative) charge.The signal is interpreted by assuming the only source of the asymmetry tobe the mixing of neutral B0q mesons that decay semi-leptonically.At the Tevatron, b quarks are produced mainly in bb pairs from an initialproton-antiproton beam with center of mass energy 1.96 TeV. The signalfor the charge asymmetry consists of like-sign muon events with one muonoriginating from direct semi-leptonic b(b?)-hadron decays (b(b?)? ??(?+)X)and the other muon resulting from a B0q -B0q (B0q -B0q ) oscillation followedby a direct semi-leptonic B0q (B0q ) decay. Because B0q -B0q oscillations areenabled by SM W? exchange, CP violation from the CKM matrix biasesthe oscillation, generating a ?wrong sign? semi-leptonic asymmetry given byaqsl =?(B0q ? ?+X)? ?(B0q ? ??X)?(B0q ? ?+X) + ?(B0q ? ??X). (1.38)The like-sign dimuon asymmetry then arises from the wrong sign semi-leptonic decays of both B0d and B0s hadrons as [49]A = (0.506? 0.043)adsl + (0.494? 0.043)assl (1.39)where the numerical coefficients are essentially the production fractions forb ? B0d and b ? B0s respectively [50]. The SM prediction for like signdimuon asymmetry is ASM = (?2.30.5?0.6) ? 10?4 [51] whereas the value261.2. Problems in the Standard Modelmeasured by the D? collaboration is AD? = ?0.00957 ? 0.00251 (stat) ?0.00146 (syst) [49, 52]. This measured asymmetry is in disagreement withthe SM prediction by 3.2? and was considered to be the first evidence foranomalous CP violation in the mixing of neutral B0q mesons. Moreover, thismeasurement was quickly improved upon, with the disagreement betweenSM and observation subsequently rising to 3.9? [53].In chapter 3, we will elaborate on a Two-Higgs-Doublet-Model specif-ically designed to simultaneously accommodate both the like-sign dimuonasymmetry and the observed baryon asymmetry of the universe.1.2.3.2 The Top Quark Forward-Backward AsymmetryThe surprisingly large mass of the top quark has long been thought to implya possible sensitivity to any NP lurking in the electroweak symmetry break-ing sector. Because of this, high precision top quark studies have alwaysbeen viewed as a necessary component of every high energy experimentalprogram. Besides production cross sections, asymmetries have also been ofgreat interest. At LO in the SM, production of top quark pairs (tt?) at theTevatron proceeds through both quark and gluon fusion: qq? ? tt? (? 85%)and gg ? tt? (? 15%) [54]. Both of these production mechanisms are to-tally charge conjugation symmetric relative to the initial pp? beam, leadingto totally symmetric angular distributions for the resulting top and antitopquarks. At higher order in ?s, this is no longer true. At ?3s, some sub-processes can be identified that are not symmetric under charge conjugationwith respect to the incoming pp? beam. Interference between processes thatdiffer under charge conjugation can then lead to a small charge asymmetry,defined asAtt?C =Nt(cos ?)?Nt?(cos ?)Nt(cos ?) +Nt?(cos ?). (1.40)where Nt(cos ?) (Nt?(cos ?)) is the number of top (antitop) quarks in theforward direction, i.e., the direction of the proton beam. Because CP is agood symmetry of perturbative QCD, we are free to do a CP transformationon any of these occupation numbers so that Nt?(cos ?) = Nt(? cos ?), by CP271.2. Problems in the Standard Modelsymmetry. This implies that, as long as the theory under question is onlyperturbative QCD, a charge asymmetry is equivalent to a forward backwardasymmetry, i.e.,Att?FB =Nt(cos ?)?Nt(? cos ?)Nt(cos ?) +Nt(? cos ?). (1.41)This effect has been calculated in the SM, including NLO QCD and leadingelectroweak contributions, to be ASMFB = 0.088? 0.006 [55?58].Both the CDF and D? experiments have measured this asymmetry atthe Tevatron and found it to be significantly larger than the SM predic-tion [59, 60]. Although both collaborations observe a preference for the pro-duction of top quarks in the direction of the proton beam, only CDF observesa strong dependence on the invariant mass of the top quark pairs [59]. Thisdependence has persisted as CDF has recently redone their analysis, usingthe full Tevatron data set of 9.4 fb?1 [61]. Although the parton level asym-metry in the high invariant mass bin (Mtt? > 450 GeV) has decreased in theirmore recent analysis, to AhighFB =(29.5 ? 6.5)% from AhighFB =(47.5 ? 11.4)%,it remains a ?2.5? deviation from the current (NLO QCD including elec-troweak corrections) SM prediction of (AhighFB )SM=(12.9? 0.7)% [58].In section 4, we will explore a possible connection between a class of NPmodels used to explain this asymmetry and a long-standing tension in themeasurements of the CKM matrix element Vub.28Chapter 2Neutron-AntineutronOscillations in a WarpedExtra DimensionIn this chapter, we investigate neutron-antineutron oscillations in the Randall-Sundrum warped extra dimensional scenario. The four dimensional effectivecouplings of the leading order operators that induce the oscillations are cal-culated up to arbitrary O(1) couplings. We also calculate the effects of therenormalization group running from the weak scale to the oscillation scale,mn. We find that the baryon number violating (?B = 2) operators thatare responsible for the oscillations can be geometrically suppressed withinexperimental limits without any appeal to fine tuning of the arbitrary O(1)couplings. This result holds even in the presence of a warped down fourdimensional mass scale suppression as low as a fraction of a TeV.2.1 IntroductionThe SM of particle physics has, for 30 years, enjoyed unequaled success indescribing the results of particle physics experiments. It is, however, not anentirely satisfactory theory due to the fact that it has, to date, left manyunanswered fundamental questions. In particular, it provides no explanationfor the many different hierarchies which have been built into it. The mostfamous of these being the electroweak-Planck hierarchy problem, describedin detail in section 1.2.1. One particularly appealing solution to this problemis the Randall-Sundrum (RS) model [39, 62?64]. Within the context of this292.1. Introductionmodel, the large hierarchy arises due to the warping of a compactified Anti-deSitter (AdS) extra dimensional geometry. This warping manifests itself asa warping factor which exponentially suppresses the mass scales within thetheory, creating an effective hierarchy. Another appealing feature of the RSmodel is its ability to explain the SM fermion mass hierarchies with the samemechanism [65?67]. By promoting all SM fermions to bulk fields, the fermionmass hierarchies are explained in terms of the fermion geography within thewarped extra dimensional space. In such a scenario, the five dimensional(5D) fermion fields are Dirac fields whose wave function localization in theextra dimension is completely characterized by a single O(1) c parameter.By using a Z2 orbifold projection or equivalently by choosing appropriateboundary conditions on the ultraviolet (UV) and infrared (IR) branes, onecan project out the chiral zero modes. The SM fermions are identified withthese chiral zero modes of the bulk fermions and they have exponential wavefunction profiles in the extra dimension. The effective Yukawa couplingsdepend heavily on the wave function overlap of these fermion wave functionswith the Higgs, which is situated on the TeV brane in the extra dimension.Heavy fermions are localized near the IR brane and thereby have a largeoverlap with the Higgs field, while light fermions are localized closer tothe UV brane. In this particular class of RS flavor models, the SM gaugesymmetry is promoted to a bulk symmetry. The 4D Yukawa couplings canthen all be taken to be O(1) while the c parameters are determined by fittingto the fermion masses and the CKM mixing parameters (see, e.g., [68]).One can then ask about the nature of higher mass dimension operatorswithin the context of the RS model, such as those corresponding to protondecay and neutron-antineutron (n-n) oscillation [65, 69?71]. If no extrasymmetry forbids these operators, they will be suppressed by some massscale which is close to or exceeds the UV completion scale of the RS model.If one simply takes this to be the Planck scale, then this would be sufficientto satisfy the experimental constraints. However, the exact same warpingmechanism which reduces the Planck scale to the electroweak scale acts toreduce this mass scale suppression as well.It is well known that proton decay is a problem for the RS model [65, 72].302.1. IntroductionIn order to properly suppress the relevant operators for proton decay, it isnecessary to maintain large separations between the quarks and leptonsin the extra dimension. However, successful mass configurations for thesefields do not allow for such large separations. As opposed to acceptingunnaturally small dimensionless couplings for these operators, it is thoughtthat there may exist an extra symmetry which will forbid these operatorsentirely. However, the exact nature of this symmetry is as yet unknown.The simplest solution is to introduce a UX(1) symmetry where X coulddenote the total baryon number (B), lepton number (L), or their difference(B-L). These symmetries are currently understood to be only accidental,i.e., symmetries that are accidentally present after the SM gauge symmetryare imposed. Discrete symmetries of the ZN type have also been suggested.It is thought that n-n oscillations could present yet another problem forthe RS model due to the fact that the corresponding operators contain onlyquarks of similar mass scales and therefore similar localizations within theextra dimension [70]. In the current work, we analyze the effective strengthof the six quark operators which induce n-n oscillations in the warped RSmodel, assuming that there is no symmetry which a priori forbids them. Forexample, the introduction of a UL(1) or Z3 symmetry would have no effecton the operators which induce n-n oscillations but does forbid the operatorswhich induce proton decay. Discrete symmetries have also been used tostudy Dirac neutrinos in warped models [73]. Previous investigations of theeffective strength of n-n oscillation operators within the context of the 6DArkani-Hamed?Dimopoulos?Dvali (ADD) have yielded a lower bound on themass scale suppression in the observable range MX & (45? 100) TeV [74].The chapter is organized as follows: Section 2.2 briefly reviews the RSmodel including the derivation of the metric and the necessity of the AdS5spacetime, the resolution of the hierarchy problem, and the treatment of theSM fermions on the 5D AdS5 background. In section 2.3, we briefly discussthe problem with proton decay in the RS model and introduce some basicaspects of n-n oscillations. Section 2.4 introduces the relevant operatorswhich induce the n-n oscillations and details the calculation of their cor-responding 4D effective Wilson coefficients. In section 2.5, we present the312.2. The Randall-Sundrum Model? = 0 ? = pi? = ?pi Identify ? with ? ? ? = 0 ? = piFigure 2.1: The orbifold geometry is chosen for the extra dimension in theRS setup. The periodic condition imposes a circular geometry for the extradimension while the identification of ? with ?? changes this to a finite linesegment with end points that are fixed under translation and reflection.enhancement of the Wilson coefficients due to SM QCD 1-loop renormaliza-tion group (RG) running effects. In section 2.6, we present our conclusionsand in section 2.7, we discuss the possibility of observing n-n oscillationevents at the proposed upcoming Project X at Fermilab given our results.2.2 The Randall-Sundrum ModelIn this chapter, we will introduce the basic concepts associated with the RSmodel [39], which features one finite size extra dimension with two branes,one at each end. After a brief description of the metric and the resolutionof the hierarchy problem, we study the low energy effective 4D behaviour ofthe chiral fermions of the SM as they propagate within the 5D spacetime.The RS model was proposed as an alternative, extra dimensional solutionto the hierarchy problem. In this scenario, the weak scale is generatedfrom a large scale, of order the Planck scale, i.e., O(1018GeV ), throughan exponential hierarchy. This exponential connection has its roots in thenon-factorizable, 5D metricds2 = e?2?(?)???dx?dx? ? r2d?2, (2.1)where r and ? ? [?pi, pi] are the radius and angular coordinate of the com-pactified extra dimension. In this chapter, we adhere to the west coast 4D322.2. The Randall-Sundrum Modelmetric signature convention ??? = diag(1,?1,?1,?1). This metric struc-ture is interesting because it departs from factorizability with the introduc-tion of the exponential ?warp factor?, e?2?(?), while retaining 4D Poincareinvariance. The main function of the warp factor is to describe the change in4D length scales as one moves along the extra dimension. Orbifold boundaryconditions are imposed on the extra dimensional angular coordinate. Theseboundary conditions impose periodicity (? ? ? + 2pi) supplemented withthe reflection property (x?, ?)? (x?,??), which defines the S1/Z2 orbifolddisplayed in Figure 2.1 (for a pedagogical introduction to the orbifold see,e.g., [75]). The end points of the orbifold at ? = 0, pi are fixed under theoperations of translation and reflection of the extra dimensional coordinate,allowing them to support (3+1) dimensional branes upon which 4D fieldtheories can be built. However, the true motivation for the orbifold bound-ary conditions is in anticipation of embedding a theory of chiral fermions,i.e., fermions whose left and right handed projections have distinct gaugeinteractions, into the 5D setup [76, 77]. In 4D, fermions fit into the small-est representation of the 4D lorentz group, a 2-component Weyl fermion.This implies that the different chiral projections of a given fermion fieldare independent of each other and we are free to assign different quantumnumbers to them. In contrast to this, the smallest representation of the 5Dlorentz group is a 4-component Dirac fermion, forcing us to assign the samequantum numbers to each chiral projection. Thus, in 5D, fermions are in-trinsically non-chiral. In order to embed the SM, a chiral gauge theory, into5D, one needs a new mechanism to project out the desired chiralities. Thisnew mechanism is the application of non-trivial orbifold boundary conditionswhich, at low energies in 4D, insures that only one of the chiral projectionsof a given fermion survives. This mechanism will become clear in section2.2.3.The ? = 0 (? = pi) brane is denoted as the UV (IR) brane for reasonsthat will become clear and the region between them is called the ?bulk?.The induced 4D metrics on each of the branes are given by expanding the332.2. The Randall-Sundrum Modelbulk metric, GMN (x?, ?), about a Minkowski spacetime asg4D?? = GMN (Y (x))??YM??Y N (2.2)where g4D?? is representative of either 4D brane metric. The YM (x) arethe 5D coordinates that map the 4D branes and are given by YM (x) =?M? x?. Accounting for the propagation of gravity into the compactified fifthdimension, the classical action of the RS model is given byS = SBulk + SUV + SIRSBulk =?d4x? pi?pid??|G|(?? + 2M3R)SUV =?d4x? pi?pid??|gUV |?(?) (LUV ? VUV )SIR =?d4x? pi?pid??|gIR|?(?? pi) (LIR ? VIR) (2.3)where R and M are the 5D Ricci scalar and Planck mass, ? is the cosmo-logical constant, and g ? detg. The Lagrangians on the brane describe thelocalized field content and the brane tensions, VUV , VIR, are necessary toallow for a non-trivial metric.2.2.1 Determining the MetricAssuming a vacuum, i.e., LUV ,LIR = 0, Einstein?s field equations for theaction in 2.3 are?|G|(RMN ?12GMNR)= ? 1M3(?|G|?GMN +?|gIR|gIR?? ??M??NVIR?(?? pi)+?|gUV |gUV?? ??M??NVUV ?(?)). (2.4)342.2. The Randall-Sundrum ModelAssuming that the radius of the extra dimension is stabilized7, i.e., r ? R,the 5D Christoffel symbols are rather simple to work out as they are func-tions of a single variable. The field equations give rise to the two equations???? = r???24M3 (2.5)?2???2 =r12M3 (VIR?(?? pi) + VUV ?(?)) (2.6)The first equation implies that a non-trivial solution can only be achievedif ? < 0, which imposes an Anti-deSitter character for the bulk spacetimeupon us. The second equation relates the brane tensions as ?VIR = VUV =24M3k where the constant, k, is interpreted as the curvature of the extradimension and is related to the cosmological constant as k ????24M3 .Because of the finite curvature within the 5D vacuum, the RS models aresometimes also classified as warped extra dimensional models. Solving eq.(2.5) and imposing the orbifold boundary conditions leads to the solution?(?) = kr|?|, where the integration constants have been omitted as theyjust rescale the 4D coordinates x?. At this point, the theory possesses threefundamental scales, (M,k, 1/r), which, for reasons of naturalness, should allbe taken to be the same order.2.2.2 Solving the Hierarchy ProblemIf r represents distance scales much smaller than the current experimen-tal resolution, spacetime essentially appears 4D at low energies. The 4Deffective field theory description is obtained by integrating out the extra di-mensional dependence of the action. In this limit, we use the metric in eq.(2.1) with the difference that we use the more general 4D metric, g?? , which7The Goldberger-Wise mechanism is often used to stabilize the radius of the extradimension dynamically [78, 79]. This mechanism treats the radius as a scalar field inthe bulk with a non-trivial potential. The radius takes on some constant value when thepotential is minimized, stabilizing the size of the extra dimension. Here we simply assumestability.352.2. The Randall-Sundrum Modelallows for non-trivial fluctuations in the 4D gravitational fieldds2 = e?2kr|?|g??dx?dx? ? r2d?2. (2.7)Using this metric in the 5D Einstein-Hilbert part of the action leads to arelation between the 4D and 5D Planck masses and the curvature scale givenbyM2Pl =M3k(1? e?2krpi), (2.8)implying that gravitational interactions are only weakly effected by r andthat both M,k ? O(MPl). The natural distance scale for the extra dimen-sion is then r ? 1/MPl ? 10?30 mm, far too small for current experimentsto resolve.The solution to the hierarchy problem requires that the electroweak sym-metry breaking sector be localized to the IR brane, where gIR?? = e?2krpig?? .The action on this brane isSIR ??d4x?d??(?? pi)?|gIR|(g??IRD?H?D?H ? ?(|H|2 ? v20)2)=?d4x?|g|e?4krpi(e2krpig??D?H?D?H ? ?(|H|2 ? v20)2)(2.9)where v0 is the Higgs vev. The natural scale for the Higgs vev is the sameas the other three scales of the theory, i.e., O(MPl).In order to obtain a canonically normalized kinetic term, we rescale theHiggs field as H ? e?krpiH. The action then becomesSIR ??d4x?|g|(g??D?H?D?H ? ?(|H|2 ? (e?krpiv0)2)2)(2.10)with the electroweak symmetry breaking scale now set by the product e?krpiv0.This result is actually generic for any mass scale on the IR brane, i.e., forany 4D (5D) mass scale m4D (m5D) the relation between them is m4D =e?krpim5D. In order to numerically connect the Planck scale to the weak362.2. The Randall-Sundrum Modelscale, the warping factor must have a damping effect, r > 1/k, with theproduct of the curvature and the radius given bykr = 1pi ln(MPlv)? 12 (2.11)with v = 246 GeV.No fine-tuning is required for this result, everything is accomplished bythe exponential form of the warp factor which comes from embedding thetheory into a curved AdS5 background. The fact that a large hierarchyof mass scales can be generated from a model with only O(1) hierarchiesbetween the fundamental mass scale parameters is the true power of the RSmodel [39].In the next section, we will discuss the low energy effective theory of achiral fermion allowed to propagate in the bulk.2.2.3 Chiral Fermions from the BulkVarious aspects of the following description of fermions in the RS warpedbackground can be found in [63, 65, 67, 76, 80, 81].The Dirac matrices are only defined on flat spacetimes. They can, how-ever, be employed on curved spacetimes by defining them on the tangentspace. Vielbeins, geometric objects which provide a mapping between thecurved and tangent spaces, are then used to connect spacetime indices, M ,to tangent space indices, a, allowing us to reformulate the Dirac algebra ona curved background. They are defined in terms of the relationGAB = eaAebB?ab (2.12)where ?ab is the flat space metric defined on the tangent space. The inverseVeilbein, EAa , is defined such that EAa eaB = ?AB and EAa ebA = ?ba. In theRS model, these quantities are given by eaA = (e?kr|?|?a?, ?a4) and EAa =(ekr|?|??a , ?4a). The 5D Dirac algebra, defined on the tangent space, can thenbe elevated to a relation in curved space through the action of the inverse372.2. The Randall-Sundrum ModelVielbeins asEAa EBb {?a, ?b} = 2EAa EBb ?ab =? {?A, ?B} = 2GAB. (2.13)In the following, we will use ?a = (??, i?5) as the representation for theDirac matrices.Before moving forward, it will pay to explore the action of the orbifoldreflection symmetry on a 5D fermion. This can be understood from the 5DDirac equation on the tangent space(i?a?a ?m) ?(x?, ?) = 0. (2.14)The reflection symmetry is a discrete Z2 symmetry whose action is describedby the operator AZ as Z?(x?, ?)Z?1 = AZ?(x?,??). The effect on theDirac equation is(i???? ? ?51r??? ?m)AZ?(x?,??) = 0. (2.15)Insisting on invariance of the Lagrangian with respect to this symmetryleads to AZ = ??5 and Zm(?)Z?1 = ?m(??). This type of coordinatedependence for the mass may be achieved by generating it through an in-teraction with a pseudoscalar under AZ. The spinor then transforms underthe orbifold symmetry as Z?Z?1 ? ??5?, which implies that only one ofthe chiral projections can be made even under the symmetry, i.e.,Z(?L?R)Z?1 ? ?(??L?R). (2.16)On the boundaries, the product of the two chiralities must then vanish, i.e.,?L(x?, ?)?R(x?, ?)?????=0,pi= 0.Gravitational interactions induced by the curved AdS5 space can beintroduced by treating gravity as a gauge theory built upon diffeomorphism(general coordinate) invariance, see, e.g., [82]. However, due to the diagonalnature of the metric and a fortuitous cancellation, these interactions have382.2. The Randall-Sundrum Modelno effect on the action [83, 84]. This leaves us with the following action fora Dirac fermion ? on the curved RS backgroundS =?d4x?d??|G|?[ i2EAa ?a???A ?m sgn(?)]?. (2.17)Expanding this action in terms of the left and right handed fermions givesS =?d4xd?re?4kr|?|[ekr|?|(?Li/??L + ?Ri/??R)?m sgn(?)(?L?R + ?R?L)]+12r?d4xd?[?R(e?4kr|?| ??? +???e?4kr|?|)?L??L(e?4kr|?| ??? +???e?4kr|?|)?R]. (2.18)where we have dropped boundary terms as they involve the product of bothchiralities at the fixed points. The equations of motion are given byekr|?|i?????L ?1r????R ? k(c? 2)?R = 0ekr|?|i?????R +1r????L ? k(c+ 2)?L = 0 (2.19)where c = m/k, which is taken to be O(1) by assuming that all mass scalesare O(MPl). Exploiting the periodicity of the coordinate ?, the 5D fieldscan be expanded as?L,R(x, ?) =e3/2kr|?|?r?n?nL,R(x)?nL,R(?) (2.20)where each of the Kaluza Klein (KK) fermion modes, ?nL,R(x), obey a Diracequation, i?????nL,R = mn?nR,L , and the extra dimensional wave functionprofiles are orthonormal? pi?pid? ??nL?mL =? pi?pid? ??nR?mR = ?nm. (2.21)392.2. The Randall-Sundrum ModelThis leads to the equations of motion for the wave function profiles(1r??? + k(c? 1/2))?nR = mnekr|?|?nL(1r??? ? k(c+ 1/2))?nL = ?mnekr|?|?nR (2.22)where the masses, mn, are interpreted as effective KK masses. All SM fieldsare associated with the zero modes of the expansion, which receive no KKmasses, leaving a set of decoupled equations of motion which can be solvedfor the normalized profiles?0L,R = NL,R e(1/2?cL,R)kr|?|with NL,R =?kr(1/2? cL,R)e2krpi(1/2?cL,R) ? 1 . (2.23)Since both profiles are even under the orbifold reflection symmetry one of thenormalizations must vanish, however, which one is totally arbitrary. Thisallows us to generate a chiral theory in the low energy 4D effective theory,i.e., for each SM fermion we insert a 4-component Dirac fermion into the 5Dbulk and then use the freedom to choose the boundary conditions to dropthe undesired chiral zero mode.2.2.4 Mass Matrices in the 4D Effective TheoryThe localization of the fermion wave function profile in the extra dimensionis controlled entirely by the single c parameter, where cR < 1/2 (cR > 1/2)corresponds to a closer proximity to the UV (IR) brane for the right handedzero mode while cL < ?1/2 (cL > ?1/2) corresponds to a closer proximity tothe IR (UV) brane for the left handed zero mode. With the Higgs localizedat the IR brane, we can promote the Yukawa couplings in eq. (1.26) to 5D.For brevity, we only look at one representative Yukawa coupling, however,402.3. Proton Decay and n-n Oscillationsall others follow in a straightforward way. The couplings are?d4xd??|G|?(?? pi)(??2v0M?ijQiL(x, ?)H(x)ujR(x, ?))=?d4x(??2v ?Qi0L(pi)e?krpiM?ij?qj0R(pi)Q0,iL(x)H(x)u0,jR(x))(2.24)from which we can extract the 4D effective mass matricesMij = NQiL NqjR e(1+cQiL ?cqjR )krpie?krpiM?ij . (2.25)The nature of this effective mass matrix is such that the heavy fermions mustbe interpreted as being localized near the IR brane and therefore have a largeoverlap with the Higgs field while the light fermions must be localized nearthe UV brane yielding a small overlap with the Higgs. Another consequenceis that only O(1) differences in the c parameters are now needed to generatethe observed large fermion mass hierarchies, e.g., me/mt = O(10?6). Inparticular, using only O(1) differences between the nine c parameters in thequark sector, it is possible to reproduce the entire set of quark masses andmixing angles. To this extent, there have been a number of parameter setsput forward which simultaneously fit both the observed masses and mixingangles [68, 85, 86].In the next section, we will briefly discuss proton decay in the RS modeland introduce some basic aspects of n-n oscillations.2.3 Proton Decay and n-n OscillationsIn the previous section, we explicitly introduced the zero modes of the KKmode expansion and their corresponding extra dimensional wave functionprofiles. The higher modes in the KK expansion are much more massive asthey receive mass from the effective KK masses, mn, as well. In the lowenergy 4D effective theory, these modes should be integrated out, resultingin higher dimensional operators as discussed in appendix B. Some of these412.3. Proton Decay and n-n Oscillationsoperators can induce proton decay, such as?d4xd??G 1M3 (g1QLQLQLL+ g2ucRucRdcR`cR) (2.26)where the fields QL, L, uR, dR, and `R are the bulk versions of the corre-sponding SM fields. Integrating out the 5D coordinate dependence revealsthe first order approximation to the effective strength with which the zeromodes of the above fields will induce proton decay. Although the effec-tive strength of these operators receives a suppression from the resultingexponential overlaps between the various fields, the 5D Planck scale sup-pression is also warped down and replaced by Me?krpi. The end result ofthese two competing effects is that there is not enough suppression from theresulting wave function overlap to prevent proton decay within the currentlimits and we must therefore concede either the fine tuning of the dimen-sionless couplings or the introduction of a convenient symmetry (such astotal lepton number or the above mentioned Z3) which will forbid the oper-ator entirely [65, 69, 72]. Neither of the above mentioned symmetries forbidthe operators which induce n-n oscillations and the question of whether ornot the RS model can inherently provide the required suppression for thesetransitions from geometry alone is the subject of the present work.The time evolution of an initially slow moving beam of neutrons is de-scribed by the following Schroedinger equation involving the simple 2 ? 2Hamiltoniani~ ??t(nn)=(En ?m?m En)(nn)(2.27)where ?m = ?n|Heff |n? parametrizes the underlying physics and is inter-preted as the frequency of the oscillation [70, 71, 87]. Note that we haveimplicitly assumed that ?m is a pure real number. This assumption elicitsno loss of generality as we are not interested in CP violating effects in theoscillations. We have also neglected any effects from possible decay matri-ces. The off-diagonal elements of these matrices originate from decays intofinal states common to both n and n states. Their absence in the Hamilto-nian corresponds to the assumption that the physics that generates the n-n422.4. Effective Operator Analysisoscillations does not induce any shared decay modes. The effect of the diag-onal elements of the decay matrices is to modify the oscillation probabilityby an overall factor of e??nt [88], where ?n is the neutron lifetime. Sincecurrent experimental time scales can be O((10? 1)s) [89] while the neutronlifetime is ?15 minutes, this factor does not result in any significant modi-fication. With these assumptions, the probability of finding an antineutronafter some time t is then given by|?n|n(t)?|2 = 4?m2?E2 + 4?m2 sin2(??E2 + 4?m2t)(2.28)where ?E = En ? En is the energy splitting. Although the evolution timeand energy splitting are controlled by experiment, ?m must be computedfrom first principles. Experimental limits from reactors and matter instabil-ity have produced limits on the off-diagonal component of the Hamiltonianof ?m ? .75?10?32 GeV and ?m ? .6?10?32 GeV respectively [90?92].2.4 Effective Operator AnalysisUsing the operator product expansion, the 5D effective Hamiltonian thatgenerates ?m is given by H5Deff =?iC5DiM7XOi(x, ?), where each effectiveoperator is a SUc(3) ? SUL(2) ? UY (1) gauge invariant, ?B = 2, six-quarkoperator. The Ci?s are the dimensionless Wilson coefficients, and MX isthe 5D mass scale at which a detailed knowledge of the underlying physicsgenerating the operators becomes indispensable. The leading contributionsto n-n oscillations come from six quark operators. Through a series of Fierztransformations [93, 94], the total set of these operators can be reduced to432.4. Effective Operator Analysisfour linearly independent scalar operators given by [74]O1 =(u?TR Cu?R)(d?TR Cd?R)(d?TR Cd?R)T s??????O2 =(u?TR Cd?R)(u?TR Cd?R)(d?TR Cd?R)T s??????O3 =(Qi?TL CQj?L)(u?TR Cd?R)(d?TR Cd?R)ijT a??????O4 =(Qi?TL CQj?L)(Qk?TL CQl?L)(d?TR Cd?R)ijklT a?????? (2.29)where the round brackets imply the contraction of spinor indices, C is thecharge conjugation operator, and Greek and Latin indices represent SUc(3)and SUL(2) degrees of freedom respectively. The color tensors contract theSUc(3) indices into color singlet combinations in two different ways givenby [95]T s?????? = ??? ??? + ??? ??? + ??? ??? + ??? ???T a?????? = ??? ??? + ??? ??? (2.30)where the first tensor is symmetric about the interchanges (?, ?), (?, ?),(?, ?), (??, ??), (??, ??), (??, ??) while the second is anti-symmetric aboutthe interchanges [?, ?] and [?, ?] and symmetric about (?, ?) and (??, ??).These operators can all be easily generalized to 5D by replacing the fermionfields with the corresponding bulk fields.In order to obtain the dimensionless, 4D effective couplings and massscale, we integrate out the extra dimensional dependence to obtain the 4DHamiltonianH4Deff =?iC4DiM54DOi(x) =?iC5DiM7X? pi?pid??|G|Oi(x, ?) (2.31)where C4Di are the 4D effective Wilson coefficients and M4D = MXe?krcpiis the warped down 4D mass scale. The physics that generates these op-442.4. Effective Operator Analysiserators is largely unknown and highly model dependent. However, if theyarise due to integrating out heavy KK modes, as we?ve suggested, then weshould expect MX ? O(k). In our effective field theory approach, it is thenreasonable to parametrize the 5D mass scale as MX = ?k, with ? taken to bea free parameter of the theory. The warped down 4D effective mass scale isthen given by M4D = ?ke?krcpi. This is a convenient choice of parametriza-tion since studies of precision electroweak measurements and FCNC implythat ke?krcpi & 1.65 TeV [96, 97]. This remains the strictest bound on thecurvature scale despite existing direct LHC searches which are currently atthe level ke?krcpi & 0.8 TeV [98].Writing the zero modes of the SM quark bulk fields as specified in eq.(2.23), the effective Wilson coefficients areC4D1 = C5D1(NuR)2(NdR)4 ekrpi(3?2cuR?4cdR)?2(4? cuR ? 2cdR)C4D2 = C4D1C5D2C5D1C4D3 = C5D32(NdR)3(NQL )2NuR ekrpi(3+2cQL?cuR?3cdR)?2(8 + 2cQL ? cuR ? 3cdR)C4D4 = C5D4(NQL )4(NdR)2 ekrpi(3+4cQL?2cdR)?2(4 + 2cQL ? cdR)(2.32)where cuR, cdR, cQL are the c parameters for the right handed up and downquark and the first generation left handed quark doublet. The similaritybetween C4D1 and C4D2 is due to the fact that O1(x, ?) and O2(x, ?) sharethe same overall quark content and differ only in the way in which the spinorand color indices are contracted. The dimensionless Wilson coefficients,C5Di , remain unknown and so, without loss of generality we set them all tounity8.As previously mentioned, there have been a number of numerical fitsmade to the existing observational data. In Table 2.1, we list a set of threedifferent representative configurations of c parameters for the first generationof the left and right handed quarks which fit the data [68].The most modern available calculation of the matrix elements of the 4D8Note that in this limit C4D1 = C4D2 .452.4. Effective Operator AnalysisConfigurationsc parameters I II IIIcQL -0.634 -0.629 -0.627cuR 0.664 0.662 0.518cdR 0.641 0.58 0.576Table 2.1: Numerical Fits of the Quark Masses with CKM Mixing Angleseffective six quark operators, ?n|Oi(x)|n?, are from the MIT bag model [99,100] in reference [94]. Averaging the results of the various fits, we obtainthe values?n|O1(x)|n? = ?5.945? 10?5 GeV6?n|O2(x)|n? = 1.485? 10?5 GeV6?n|O3(x)|n? = ?2.95? 10?5 GeV6?n|O4(x)|n? = 2.22? 10?5 GeV6. (2.33)All of this allows us to write the off-diagonal element of the n-n Hamil-tonian as a function of the c parameters and ? as?m(cQL , cuR, cdR, ?) =C4D1M54D(?n|O1(x)|n?+ ?n|O2(x)|n?)+C4D3M54D?n|O3(x)|n?+C4D4M54D?n|O4(x)|n?. (2.34)The c parameters in Table 2.1, which determine ?m, were determined bymatching to the quark mixing matrices and mixing angles at 1 TeV [101],however, the n-n oscillations take place at the neutron mass scale, ?1 GeV.To obtain a realistic comparison with the data, we need to use the RGequations to run the Wilson coefficients down to the oscillation scale. Thecalculation of these running effects is the subject of the next section.462.5. QCD running of C4Di2.5 QCD running of C4DiThe introduction of higher dimensional effective operators, such as thosein eq. (2.29), introduces new divergences due to the loss of informationinherent in their construction, see e.g., [26]. In order to renormalize thesedivergences, it is necessary to rescale the effective operators as one does witha single field operator. The renormalized operators, Oi, are then expressedin terms of the unrenormalized operators, Oi0, asOi = Z?1Oi Z?3q Oi0 (2.35)where we have included both the new wave function renormalization of theeffective operator, ZOi , and those of the six associated quark fields. Sincethe strong coupling has the most significant running effects between 1 TeVand the neutron mass scale [102], we restrict ourselves to calculating theSUc(3) renormalization effects only. In this limit, all the quark wave functionrenormalizations are the same.The renormalized effective Lagrangian isLeff = ?1M54D?i[C4Di Oi + ?OiC4Di Oi](2.36)where the effective operator counter term is related to the wave functionrenormalization as ?Oi = ZOiZ3q ? 1. The Wilson coefficients must carry acompensating scale dependence to insure that Leff remains scale indepen-dent, i.e., C4Di = ZOiC4Di0 . In direct analogy with eq. (C.2), this leads toa renormalization group equation for the Wilson coefficients which dependson the anomalous dimensions of the corresponding effective operators??C4Di?? = ?OiC4Di (2.37)where ?Oi = Z?1Oi ??ZOi?? . In order to determine the anomalous dimensionsof the effective operators, we note that the operator rescaling in eq. (2.35)implies that the amplitudes generating the operators each follow their own472.5. QCD running of C4DiCallan-Symanzik equation(? ??? + ?Oi + 3?q)Mi = 0 (2.38)where ?q = Z?1q ??Zq?? . The amplitude for the QCD corrections to a giveneffective operator includes the tree level amplitude, all possible vertex cor-rections, all external leg corrections, and all counterterms. The general formof this amplitude isMi = 1 +?j(AjC +Bj + ?Oi) +?j(CiC +Dj ? 6?q) (2.39)where we have chosen a renormalization condition such that the operator isnormalized to unity and Aj , Bj , Cj , Dj represent any possible finite terms.Since all the scale dependence is in the counter terms, the Callan-Symanzikequation gives the relation?Oi = ???? (??Oi + 3?q) . (2.40)Working to first order in ?s =g2s4pi , we find a total of 15 vertex correctiondiagrams in addition to the external leg corrections, depicted in Figure 2.2,giving a total of 21 diagrams for each operator. For completeness, we willinclude a sample calculation of the first diagram in Figure 2.2 for the firstoperator, i.e., O1.In the following, we neglect the external momenta of the constituentquarks in the neutron. The amplitude isM1 =?4pi?sC4D1M54DT s?????? (T a)??(T a)??1D? d4k(2pi)41k4?(u?TR C????u?R)(d?TR C????d?R)(d?TR Cd?R). (2.41)The momentum integral is logarithmically divergent, however, we are only482.5. QCD running of C4Di+ 5 otherpermutationsFigure 2.2: Virtual QCD effects for the anomalous dimensions of effectiveoperators responsible for neutron-antineutron oscillation.interested in the scale dependence and so we make the replacement1D? d4k(2pi)41k4 =i4(4pi)2 ln?2. (2.42)To reduce the tensor structure, we note that???? = 12({??, ??}+ [??, ?? ]) = ??? ? i??? . (2.43)Inserting this, we see that the interference terms between the symmetric ???and antisymmetric ??? vanish and, due to the identity ??? = ? i2????????5,the tensor product [??? ][??? ] vanishes as well. This leaves us withM1 =?i?sC4D14piM54DT s?????? (T a)??(T a)?? ln?2(u?TR Cu?R)(d?TR Cd?R)(d?TR Cd?R)(2.44)The other amplitudes are very similar and differ only in overall sign and colorstructure. Adding all 15 vertex corrections for the first operator togetheralong with the counter term and using the MS renormalization scheme gives492.5. QCD running of C4Dithe result?O1 = ??spi ln?2. (2.45)The quark counterterm can be calculated from the QCD 1-loop correctionsto the quark propagator and is given by ?q = ??s3pi ln?2, which yields avanishing anomalous dimension, i.e., ?O1 = 0. This is the case for the secondoperator as well as the color structure for the two operators is identical9.The calculations for the third and fourth operators are virtually identicaland, although the color structures are very different, the only differenceturns out to be an overall sign in the counter terms, giving the anomalousdimensions ?O3 = ?O4 =4?spi . Solving the renormalization group equationsfor the running of the Wilson coefficients in eq. (2.37), we findC4Di (Q2) = C4Di (?2)[ ?s(?2)?s(Q2)]2pi?s?Oib0 (2.46)where b0 = 11? 23Nf . Since the Wilson coefficients have been calculated atthe TeV scale, all quarks are included in the effective field theory. However,as we run the scale down to the neutron mass scale, this number, Nf , willdecrease giving rise to threshold effects. Accounting for these leads to thefollowing scaling behaviour for the effective Wilson coefficientsC4D1,2 (M24D) = C4D1,2 (GeV2) (2.47)C4D3,4 (M24D) = C4D3,4 (GeV2)[?s(GeV2)?s(m2c)]8/9 [?s(m2c)?s(m2b)]24/25?[?s(m2b)?s(m2t )]24/23 [ ?s(m2t )?s(M24D)]8/7(2.48)where mc, mb, and mt are the masses of the charm, bottom, and topquark respectively and we have used the leading log (LL) approximation9This has been checked explicitly as well.502.5. QCD running of C4DiIIIIII0.0 0.5 1.0 1.5 2.0 2.510-3810-3510-3210-2910-26rdmHGeVLFigure 2.3: The frequency of neutron-antineutron oscillations, ?m, is shownas a function of the required higher dimensional mass scale suppression, ?,for each of the three configurations listed in Table 2.1. Solid (dashed) curvesrepresent the dependence on ? without (with) the QCD 1-loop running effects.The horizontal line represents the experimental limit |?mexp|.for ?s(Q2).The full matrix element of the off-diagonal element of the n-n Hamilto-nian, evaluated at the neutron mass scale, is then given by?m = C4D1 (GeV2)M54D(?n|O1(x)|n?+ ?n|O2(x)|n?)+(C4D3 (GeV2)M54D?n|O3(x)|n?+C4D4 (GeV2)M54D?n|O4(x)|n?)?[?s(GeV2)?s(m2c)]8/9 [?s(m2c)?s(m2b)]24/25 [?s(m2b)?s(m2t )]24/23 [ ?s(m2t )?s(M24D)]8/7(2.49)In Figure 2.3, |?m| is plotted for each of the three configurations consistentwith numerical fits to the quark masses and CKM mixing angles in Ta-ble 2.1. For completeness, we have included both the curves with (dashed)and without (solid) the QCD 1-loop running effects. For configuration III theeffective Wilson coefficients are such that C4D1 dominates by 2 and 5 ordersof magnitude over C4D3 and C4D4 respectively. The 1-loop running effects are512.6. Conclusionstherefore negligible and the two curves are essentially indistinguishable onthe present scale. An upper limit on ? for all three configurations can alsobe obtained by requiring that the value of |?m| be less than the experimen-tal limit |?mexp| = .6? 10?32 [74, 90?92]. Imposing this constraint for eachof the three configurations leads to the following bounds: ?I & 0.240591,?II & 0.568982, and ?III & 1.96185. We can easily turn these bounds on? into bounds on the 4D warped down effective mass scale M4D which arethen given by M I4D & 0.4 TeV, M II4D & 0.94 TeV, and M III4D & 3.24 TeV.This implies that only a relatively small warped down 4D mass suppressionis actually needed to satisfy the currently observed experimental limits.2.6 ConclusionsWe have investigated the effective strength of the linearly independent setof six quark operators which induce neutron-antineutron oscillations withinthe context of the warped Randall-Sundrum model. The overall strengthof the relevant operators arose from the combination of the resultant wavefunction overlap of the six quark fields in the extra dimensional bulk, the 4Deffective warped down mass scale suppression, and, to a lesser extent, QCD1-loop running effects. The 4D effective warped down mass scale suppressionwas parametrized by a dimensionless factor in order to determine the extentof any extra suppression needed beyond the minimum allowed by flavorchanging neutral current constraints. It was determined, for the quark cparameter configurations listed, that the constraints on the dimensionlessfactor are such that the effective warped down mass scale never has tobe greater than O(1) TeV and, in two of the three configurations, is onlyrequired to be a fraction of a TeV even with enhancements due to 1-looprunning effects. The enhancements due to the QCD running were includedbut were determined to not have an overtly large effect on the strength of theoperators as the contributions from the warped geometry far outweighed anyrunning effects. The resultant wave function overlap of the six quark fieldsin the bulk play the most significant role in the suppression of the effectiveoperators. The resultant overlap is sensitively controlled by the c parameters522.7. Epilogueof the quark fields which one determines by fitting the quark masses and theCKM parameters. The reason that these effective operators receive greatersuppression than their proton decay counterparts stems from the simplefact that the n-n transition operators contain more fermion fields whichleads to more negative contributions within the exponential overlap. Thissame simple reasoning should play a significant role in our intuition abouteffective operators of even higher mass dimension which are constructedfrom light fermion fields. The more light fermion fields that are present inthe effective operator the more negative contributions we can expect withinthe resultant exponential wave function overlap. The significance of thisresult is that it shows that the geometric suppression from the warped RSbackground is sufficient to suppress the n-n transition operators to withinthe current experimental limits without any fine tuning while the effectivemass scale can be as low as a fraction of a TeV.2.7 EpilogueNew experimental opportunities to improve measurements of the neutronoscillation frequency may arise in the near future. Specifically, Project X is acurrently-under-development forefront intensity frontier facility at Fermilabwhich has proposed the experimental goal of improving the sensitivity to then-n oscillation frequency by 3-4 orders of magnitude[89]. A high intensitybeam of ultra-cold (vn . 10 m/s) or very-cold (vn . 100 m/s) neutrons willpropagate down an existing 105m vertical shaft at Fermilab. The verticalnature of this shaft avoids gravitational defocusing of the beam and will helpmaintain high intensity during propagation. A high-vacuum, demagnetizedregion will be created within the shaft to avoid unwanted collisions andlarge energy splittings between the n and n states from interactions withthe earth?s magnetic field. Neutrons will freely propagate down the shaft toa detector placed at the end. If an oscillation event occurs within the shaft,then an annihilation event is expected in the detector, creating a multi-pionsignal with total energy ? 2 GeV.In this kind of experimental setup, the number of antineutrons expected532.7. Epilogue>1 events>10 events>50 events2 4 6 8 100.00.20.40.60.81.0FnH??sLx1011THyrsL>1 events>10 events>50 events2 4 6 8 100.00.51.01.52.02.53.0FnH??sLx1012THyrsLFigure 2.4: Predicted number of observed neutron-antineutron oscillationevents at Project X at Fermilab. Left panel: For an ultra-cold neutron sourcewith the specified flux, n-n oscillation events may be observed within a yearat Project X. Right panel: For a very-cold neutron source with the specifiedflux, n-n oscillation events may be observed within 3 years at Project X.in the neutron beam is given in terms of the neutron beam flux, ?n, therunning time of the experiment, T , the time of flight for a neutron to reachthe detector, ttof , and the n-n oscillation frequency, ?m, as [70]Events = ?n T (ttof ?m)2. (2.50)Using the experimental specifications listed above, the average ultra-cold(very-cold) neutron time of flight is ttof = 105m/vn = 10.5s(1.05s). In orderto explore what flux and running time are needed to observe n-n oscillationevents at Project X, we employ the lowest 4D effective mass scale allowedby FCNC and the third c parameter configuration listed in Table 2.1. Forthese choices, the number of n-n oscillation events that should be observedat Project X as a function of the necessary beam flux and running time areillustrated in Figure 2.4. A high flux beam of very cold neutrons is preferableas it allows for a longer time-of-flight, providing more time for oscillationsto occur.54Chapter 3Anomalous B Meson Mixingand BaryogenesisExperimental hints for CP violation beyond the Standard Model CKMparadigm have arisen in the B meson sector. An anomalous dimuon asym-metry was reported by the D? collaboration, while tension also exists be-tween the branching ratio for B+ ? ?+? and the CP asymmetry S?K inB0d ? J/?K. These measurements, disfavoring the Standard Model at the? 3? level, can be explained by NP in both B0d-B0d and B0s -B0s mixing aris-ing from new bosonic degrees of freedom at or near the electroweak scaleand new, large CP-violating phases. These two new physics ingredients areprecisely what is required for electroweak baryogenesis to work in extensionsof the Standard Model. We show that a simple Two-Higgs-Doublet-Modelwith a top-charm flavor violating Yukawa structure can simultaneously ex-plain the B meson anomalies and the baryon asymmetry of the Universe.Moreover, the presence of a large relative phase in the top-charm Yukawacoupling, favored by B0d,s-B0d,s mixing, weakens constraints from rare B de-cays (b ? s?) and the K meson sector (K), allowing for a light chargedHiggs mass of O(100 GeV).3.1 IntroductionPrecision tests of CP violation have shown a remarkable consistency with theSM, where all CP violating observables are governed uniquely by the singlephase in the CKM matrix [103]. However, many well-motivated extensionsof the SM contain new sources of CP violation at the electroweak scale.Furthermore, new CP violation beyond the CKM phase is required to explain553.1. Introductionthe origin of the baryon asymmetry of the Universe.Recent experimental analyses have suggested that the CKM paradigmmay be in trouble. First, the D? collaboration has measured the like-signdimuon asymmetry, interpreted as arising from CP violation in the mix-ing and decays of B0d,s mesons, in excess of the SM prediction at the 3.2?level [49, 52]. This tension grew with added data to 3.9? [53]. Second, thereis tension at the ? 3? level between Br(B+ ? ?+?) and the CP asymmetryS?K [104, 105]. Additionally, CDF and D? have measured the CP asym-metry S?? in B0s ? J/? ?. While their earlier results (each with 2.8 fb?1data) showed a ? 2? deviation from the SM [106?110], this discrepancy hasbeen reduced in their updated analyses with more data (5.2 and 6.1 fb?1,respectively) [111, 112].Although further experimental study is required, taken at face value,these anomalies suggest CP violation from NP in the mixing and/or decayamplitudes of B0d and B0s mesons [113?118]. Recently, the CKMfitter grouphas performed a global fit to all flavor observables, allowing for arbitraryNP in B0d,s-B0d,s mixing amplitudes [119]. They conclude that the SM isdisfavored at 3.4?, while the data seem to favor NP with large CP-violatingphases relative to the SM in both B0d and B0s mixing. As described in ap-pendix B, the effective field theory description of this NP should take thegeneric formLNP ?CdM2NP(b??PLd)(b??PLd) +CsM2NP(b??PLs)(b??PLs) + h.c. (3.1)These operators can arise from new bosonic degrees of freedom at or nearthe weak scale, with new large CP-violating phases [120?123].It is suggestive that the same NP ingredients, new weak-scale bosonsand new CP violation, can also lead to successful EWBG. This scenario, inwhich the baryon asymmetry is generated during the electroweak phase tran-sition [124?126], is particularly attractive since two out of three Sakharovconditions [41] can be tested experimentally. First, a departure from thermalequilibrium is provided by a strong first-order phase transition, proceedingby bubble nucleation. While this does not occur in the SM [127], additional563.1. Introductionweak-scale bosonic degrees of freedom can induce the required phase tran-sition and should be within reach of current colliders. Second, as arguedin section 1.2.2, there must exist new CP violation beyond the SM [43?45, 128]. This CP violation must involve particles with large couplings tothe Higgs boson, since it is the interactions of those particles with the dy-namical Higgs background field that leads to baryon production. Precisiontests, such as electric dipole moment searches [129] and flavor observables,can probe directly CP violation relevant for EWBG. (The third condition,baryon number violation, is provided in the SM by weak sphalerons [130];however, it is difficult to probe experimentally, since the sphaleron rate ishighly suppressed at temperatures below the weak scale.)If we wish to connect eq. (3.1) to EWBG, it is better to generate theseoperators at one-loop, rather than tree-level. Constraints on the mass dif-ferences ?md,s in the B0d,s systems require that M2NP /|Cd| & (500 TeV)2and M2NP /|Cs| & (100 TeV)2 [131]. For tree-level exchange, it seems un-likely that all three Sakharov conditions can be met at once. Sufficientbaryon number generation typically requires couplings & O(10?1), suchthat Cd,s & O(10?2), while a viable phase transition requires MNP . 1TeV. Therefore, EWBG requires M2NP /|Cd,s| . (10 TeV)2, at odds with?md,s constraints. However, if the operators in eq. (3.1) arise at one-looporder, Cd,s will have an additional 1/(4pi)2 loop suppression, allowing forboth large couplings and lighter scale MNP , without conflicting with ?md,sconstraints.In this chapter, we propose a simple Two-Higgs-Doublet-Model (2HDM)to account for both the anomalous CP violation in B0d,s-B0d,s mixing andEWBG. Previous works have studied CP violation in B0d,s-B0d,s mixingwithin a 2HDM [120?123], however, our setup is different in that we assumea top-charm flavor violating Yukawa structure. In this case, the NP B0d,s-B0d,s mixing amplitudes are naturally generated at one-loop order throughcharge current interactions (similar to reference [122]), rather than throughtree-level exchange [120, 121, 123].This chapter is organized as follows: We begin with a brief overview,in section 3.2, of the relevant flavor physics to establish our notation and573.2. Flavor Physics Overviewconventions. Next, in section 3.3, we detail the construction of our model. Insection 3.4, we calculate the NP mixing amplitudes in our model, exploringthe allowed parameter space using the constraints made available to us bythe CKMFitter group. In section 3.5, we discuss EWBG in our model.We focus on the CP violation aspects of EWBG, computing the baryonasymmetry in terms of the underlying parameters of our model by solving asystem of coupled Boltzmann equations. We find that the parameter regionfavored by flavor the constraints can easily account for the observed baryonasymmetry. However, the relevant CP-violating phases for flavor observablesand EWBG is unrelated. In section 3.6, we summarize our conclusions and,in section 3.7, we include a short epilogue outlining our current views onthis line of research.3.2 Flavor Physics OverviewIn this section, we review the relevant aspects of flavor physics that arerequired for the understanding of the original results presented here.3.2.1 B0q -B0q Mixing OverviewThere are eight bottom-flavored pseudoscalarB mesons, i.e., B? = (bu), (ub),B?c = (bc), (cb), B0q = (bq), and B0q = (qb) (q = d, s), which are all eigen-states of the strong and electromagnetic interactions. Although the B0q , B0qmesons are electrically neutral, they differ from each other in weak isospin(T 3(B0q , B0q ) = ?1/2), and bottom-ness and are therefore unambiguouslydifferent particle states. As neither of these quantities is conserved by theweak interaction, we expect them (and their K0 and D0 meson equivalents)to mix under these interactions and any others which break these symme-tries.The standard Schroedinger formalism can be used to describe the timeevolution of a B0q meson, where the initial state, |B0q (0)?, evolves with thetotal Hamiltonian, H, as|B0q (t)? = U(t, 0)|B0q (0)? with U(t, 0) = e?iHt. (3.2)583.2. Flavor Physics OverviewThis Hamiltonian is split up into the sum of the strong and electromagneticinteractions, H0, and the weak interactions, HW , which is responsible forthe mixing and decays of the B mesons. Treating HW as a perturbationand expanding to second order gives the standard resultHeffW = HW +?nHW |n??n|HWmB ? En + i= HW +?nHW |n??n|HW(P 1mB ? En + i? ipi?(mB ? En))(3.3)where the sum runs over all possible virtual transitions to intermediatestates, |n?. The resulting matrix in B0q -B0q space is then parametrized as(?B0q |HeffW |B0q ? ?B0q |HeffW |B0q ??B0q |HeffW |B0q ? ?B0q |HeffW |B0q ?)?(M q11 ? i2?q11 Mq12 ? i2?q12M q?12 ? i2?q?12 Mq11 ? i2?q11)(3.4)where we have assumed CPT invariance (M22,?22 = M11,?11) and allowedfor CP violation (M?12,??12 6= M12,?12). The physical eigenstates of this2?2 Hamiltonian are the light, BqL, and heavy, BqH , mass eigenstates, withmasses (mL, mH) and widths (?L, ?H). These can be written as linearcombinations of B0q and B0q as|BqL? = p|B0q ?+ q|B0q ?|BqH? = p|B0q ? ? q|B0q ? (3.5)with |p|2 + |q|2 = 1. Solving for the eigenvalues and eigenvectors of eq. (3.4)gives(?mq)2 ?14(??q)2 = 4|M q12|2 ? |?q12|2 ?mq??q = 4Re(M q12?q?12)qp = ?2M q?12 ? i?q?12?mq ? i2??q(3.6)where ?mq = mqH ? mqL and ??q = ?qH ? ?qL are the mass and widthdifferences. The mass difference is interpreted as the oscillation frequency.In the Bq systems, it is experimentally well known that |?q12| << |M q12|593.2. Flavor Physics Overviewb?q bq?W? W?u, c, tu?, c?, t?Figure 3.1: Box diagrams for B0q -B0q mixing amplitudes arising from W?exchange in the SM.so that, to leading order, ?mq = 2|M q12| and ??q = 2Re(M q12?q?12)/|M q12| =2|?q12| cos?q with ?q = arg(?M q12/?q12). Because of this, we can also expandthe third relation in eq. (3.6), to obtainqp = ?M q?12|M q12|(1? 12Im(?q12M q12)). (3.7)Note that if there is no relative phase between M q12 and ?q12 then |q/p| = 1.This allows us to define the (wrong-sign) semileptonic CP asymmetry (fordetails see, e.g., [132]), asaqsl = 2(1?????qp????)= Im(?q12M q12)=|?q12||M q12|sin?q =??q?mqtan?q. (3.8)As discussed in section 1.2.3.1, this asymmetry (for the case of muons in thefinal state) has been measured by D? and arises from wrong sign semilep-tonic decays of both B0d and B0s mesons. The total measured asymmetry isgiven by the combination Absl ? 0.5 adsl + 0.5 assl and is in 3.9? disagreementwith the SM predictions [49, 52, 53]. B0q -B0q oscillations are then completelydetermined by three physical quantities: the oscillation frequency ?mq, thedifference of the widths ??q, and the CP phase ?q. In the SM, the dispersiveterm, M q12, of the off-diagonal elements of the mixing matrix is dominantlydetermined by the box diagrams in Figure 3.1, in particular those diagramsinvolving top quark exchange [133, 134]. These diagrams generate ?B = 2operators contained in the HW Hamiltonian which are represented, in the603.2. Flavor Physics Overvieweffective field theory formalism, by the HamiltonianH?B=2q = (V ?tqVtb)2CSMtt (q??PLb)2 + h.c.. (3.9)The CKM factors have been extracted from the Wilson coefficient of theoperator, leavingCSMtt =G2F4pi2M2W ??BS (xt) (3.10)where xq =mq2M2W, mt is the MS top mass, calculated at NLO in QCD andgiven by mt = 165.017 ? 1.348 [135], and ??B = 0.8393 ? 0.0034 [51, 136]represents the QCD corrections to the effective operators which run the scaledown to the B meson mass scale. The loop function, S (xt), is the standardInami-Lim result [137] which is given by S = 2.35 for the central value ofmt above.This Hamiltonian generates the dispersive term through the relationM q12 =?B0q |H?B=2q |B0q ?2mBq(3.11)where the factor of 1/2mBq is due to a normalization convention of the statevectors. The hadronic matrix elements of the effective operators in eq. (3.9)are in general, very difficult to calculate, however, we will not need themin our calculation of NP effects in B0q -B0q and so we neglect them in ourdiscussion here. For more detail on this subject, we refer the reader to,e.g., [119] and references within.In this thesis, we assume that NP enters only through the mixing, i.e.,only through the box diagrams generatingM q12. Since ?q12 are both generatedthrough tree-level decays, it is a reasonable assumption that NP effects aresub-dominant here. Thus, NP effects are characterized by a change in themagnitude and phase of M q12. These contributions are parametrized in a613.2. Flavor Physics Overviewmodel-independent way through the complex parameter ?q as [51, 119]M q12 = MSM,q12 ?q ?q ? |?q|ei??q , (3.12)where the NP effects shift the CP phase as ?q = ?SMq + ??q .In the next section, we look at K0-K0 mixing and the effect of the NPthere.3.2.2 K0-K0 Mixing OverviewThe mixing of neutral Kaons, with quark content K0 = (sd),K0 = (ds),mirrors that of neutral B meson mixing in many ways with the importantdifference that the exchange of charm as well as top quarks in the box dia-grams is important. This changes the structure of the |?S| = 2 Hamiltoniansuch thatH |?S|=2 =((VcsV ?cd)2CSMcc + 2VtsV ?tdVcsV ?cdCSMct+(VtsV ?td)2CSMtt)(d??PLs)2 + h.c.. (3.13)where the terms correspond to the exchange of two charm quarks, one charmand one top quark, and two top quarks respectively. The peculiar hierarchyof CKM elements in the Hamiltonian is the main reason as to why the charmeffects are now important. The Wilson coefficients are, again, given in termsof the Inami-Lim loop functions as [137]CSMtt =G2F4pi2M2W ??ttS (xt)CSMct =G2F4pi2M2W ??ctS (xc, xt)CSMcc =G2F4pi2M2W ??ccS (xc) (3.14)where the QCD corrections are tabulated in [119]. There are two physicalquantities associated with K0-K0 mixing: the mass difference ?mK and theCP violating quantity K . The derivation of K is somewhat more involved623.2. Flavor Physics Overviewdue to troublesome long-distance contributions [138?140] but, in the end, adefinition can be determined which is largely independent of long-distanceeffects [119]|K | =1?2ImMK12?mK(3.15)withMK12 =?K0|H |?S|=2|K0?2mK= (VcsV ?cd)2M cc12 + 2VtsV ?tdVcsV ?cdM ct12 + (VtsV ?td)2M tt12. (3.16)In analogy with the case of B0q -B0q mixing, NP effects are introduced in amodel independent way by definingM ij12 = MSM,ij12 |?ijK |ei??ijK . (3.17)In the next section, we describe how to constrain the effects of NP in therare B ? Xs? meson decays.3.2.3 B ? Xs? Decays OverviewThe rare decay B ? Xs? is an example of an inclusive decay. In inclusive de-cays, the amplitude involves a sum over all (or at least over a special class) ofaccessible final states, in this case all states with strange quark content, i.e.,Xs. This summation, in the case of heavy mesons, allows the correspondingbranching ratio to be expanded in inverse powers of the heavy quark mass,with the leading term given by the so-called spectator model10. This expan-sion is called the heavy quark expansion (HQE) [141, 142] and, within thisapproximation, the decay of a B meson can be accurately described by the10The main idea behind the spectator model is that, since the massive b quark carriesmost of the momentum of the B hadron, it is a reasonable assumption that its decay ishighly independent of the light quark content, i.e., they act as mere spectators to thedecay event.633.2. Flavor Physics Overviewdecay of the b quark, i.e.,Br(B ? Xs?) = Br(b? s?) +O(1m2b). (3.18)Since there is no tree-level vertex in the SM through which this decay canproceed, it is inherently sensitive to the quantum structure of the SM andtherefore to NP. In the published version of this chapter, we simply use thefull NLO results for the 2HDM in b? s? contained in references [119, 143,144] to constrain our model. However, for the purposes of this thesis, weinclude our original leading log (LL) calculation of the NP effects.The calculation of b? s? is complicated by the effects of operator mixinginduced by the QCD corrections which run the branching ratio down to theb quark mass scale. The full Hamiltonian describing the decay, includingthe operators that enter through mixing, is given by [134]Hb?s?eff = ?GF?2V ?tsVtb(C1(c??PLb)(s??PLc) + C2e4pi2mbs????PRb?F??+ C3gs4pi2 (Ta)??mbs????PRb?Ga??). (3.19)After accounting for the LL QCD running effects down to the b quark massscale, the effective coupling for the b? s? operator is given byCeff2 (?b) = 0.695C2(?W ) + 0.085C3(?W )? 0.158C1(?W ) (3.20)where the numerical coefficients represent the QCD running effects. Tominimize some of the uncertainties involved in the HQE and CKM elements,it is customary to normalize the b? s? branching ratio to the semi-leptonicrate b ? ce?e. The LL calculation of this ratio in terms of the effectiveWilson coefficient Ceff2 is [145]R ? Br(B ? Xs?)Br(B ? Xce?e)? ?(b? s?)?(b? ce?e)=|V ?tsVtb|2|Vcb|26?pif(m2cm2b)???Ceff2???2(3.21)643.3. A Top-Centric Two Higgs Doublet Modelwith f(x) = 1? 8x+ 8x3 ? x4 ? 12x2 lnx. In order to isolate the effects ofNP, we take the ratioRSM+NPRSM =Br(B ? Xs?)SM+NPBr(B ? Xs?)SM=?????1 +Ceff,NP2Ceff,SM2?????2(3.22)where we have assumed that the NP effects in b? ce?e and the total b widthare sub-dominant. By treating Br(B ? Xs?)SM+NP as the experimentalvalue, we can use the results of [146] along with the SM NLO value [143] toderive the boundBr(B ? Xs?)expBr(B ? Xs?)SM=(3.55? 0.24? 0.09)? 10?4(3.6? 0.30)? 10?4 = 0.99? 0.11 (3.23)where, in the experimental value, the first error is experimental while thesecond is a theoretical error associated with a photon shape function usedto extrapolate the branching ratio to different photon energies. Using theLL approximation for Ceff,SM2 = ?0.3 (corresponding to mt = 175 GeV),we can then extract constraints on any NP model11.3.3 A Top-Centric Two Higgs Doublet ModelThe model we propose is based on a general (type III) 2HDM [147?151],where both Higgs fields couple to each SM fermion. In this formulation, onecan perform a basis transformation such that only one Higgs field acquiresa real, positive vev [152]. We denote the two Higgs doublets byH1 =??G+v + h0 + iG0?2?? , H2 =??H+H0 + iA0?2?? , (3.24)11Although it is inconsistent to use NLO SM values for Br(B ? Xs?)SM and LL SMvalues for Ceff,SM2 , this does capture the full behaviour to a reasonably good approxima-tion. It is for this reason, however, that, in the published version of this chapter, we usedthe full NLO results for the SM and the 2HDM.653.3. A Top-Centric Two Higgs Doublet Modelwhere h0, H0 (A0) are the neutral (pseudo)scalars, H? is a charged scalar,G?,0 are the Goldstone modes eaten by the electroweak gauge bosons, andthe vev is v ? 174 GeV. In general, the physical neutral states are admix-tures of h0, H0, A0, depending on the details of Higgs potential, however, weneglect mixing in our analysis so that H1 is exactly the SM Higgs doublet.The most general Yukawa interaction for u-type quarks isLyuk ? uR(yUH1 + y?UH2)QL + h.c. (3.25)where the left-handed quark doublet is QL ? (uL, V dL) and the SU(2)Lcontraction is HiQL ? H+i (V dL) ? H0i uL. The 3 ? 3 Yukawa matrices,yU and y?U , couple the right-handed u-type quarks, uR ? (u, c, t)R, to left-handed u-type, uL ? (u, c, t)L, and d-type, dL ? (d, s, b)L, quarks. Workingin the quark mass eigenstate basis, the matrixyU = diag(yu, yc, yt) = diag(mu,mc,mt)/v (3.26)is a diagonal matrix of SM Yukawa couplings, and V is the CKM matrix.Analogous Yukawa couplings arise for d-type quarks and charged leptons:Lyuk ? ?dR(yDH?1 + y?DH?2)QL ? eR(yLH?1 + y?LH?2)LL + h.c. (3.27)where yD = diag(yd, ys, yb) and yL = diag(ye, y?, y? ) are the SM Yukawacouplings.The NP Yukawa matrices y?U,D,L can be arbitrary. However, the absenceof anomalously large flavor-violating processes and the desire to maintainO(1) couplings for baryogenesis purposes provides strong motivation to in-sure that NP effects in B0-B0 mixing enter predominantly at the 1-looplevel. With these considerations in mind, we assume that flavor violationarises principally in the top sector. Specifically, we takey?U =???0 0 00 0 00 y?tc y?tt??? , y?D,L = 0 . (3.28)663.4. Flavor ConstraintsThat is, we consider a hierarchical structure, in the spirit of reference [151],where the tR-tL and tR-cL couplings are dominant (with |y?tt|  |y?tc|), whileothers are suppressed. The zeros in eq. (3.28) are meant to indicate thesesubleading couplings that, for simplicity, we neglect in our analysis. Inthis setup, the dominant flavor violation effects in meson observables thenarise at the 1-loop level through scalar charge current interactions, i.e., H?exchange. We will discuss these effects in the next section.3.4 Flavor ConstraintsIn the last section, we established that NP effects in B0q -B0q mixing areexpected to enter predominantly through deviations to M q12 which can beparametrized in a model-independent way asM q12 = MSM,q12 ?q (3.29)The consistency of ?md,s with SM predictions constrains |?d,s| ? 1, atthe O(20%) level [119], while the dimuon asymmetry measurement dis-agrees with the SM prediction at 3.9? and requires O(1) NP phases ??q ?arg(?q) [49, 52, 53]. These same phases also enter into other CP asymme-tries due to interference between B0d,s decay amplitudes with and withoutmixing, e.g., the CP asymmetry in B0d ? J/?K0S (B0s ? J/??) decays ismodified by the NP as S?K ? sin(2? + ??d ) (S?? ? sin(??s ? 2?s)), withthe CKM angle ? ? arg(?VcdV ?cbV ?tdVtb) (?s ? ? arg(?VcsV ?cbV ?tsVtb)). Fur-thermore, as emphasized in reference [105], the presence of non-zero ??d canalleviate tension between S?KS and Br(B+ ? ?+?), which is sensitive to ?but not ??d .To quantify these tensions, the CKMfitter group performed a global fitallowing for arbitrary ?d,s (dubbed ?Scenario I?), finding that the SM point(?d = ?s = 1) is disfavored at 3.4? [119]. Moreover, their best fit pointfavors NP CP-violating phases in both B0d and B0s mixing: ??d = (?12+3.3?3.4)?and ??s = (?129+12?12)??(?51.6+14.1?9.4 )?12. In our model, NP effects enters B0d,s12Reference [119] did not include in their fit updated CDF and D? results for S?? [106?673.4. Flavor Constraintsq?1Lq2L q1Lq?2LH+ W+, G+, H+uiu?jFigure 3.2: NP box diagrams arising from charged Higgs exchange for B0q -B0q(and K0-K0) mixing in the top-charm flavor-violating 2HDM.(and K0) observables predominantly through mixing via the box diagramsshown in Figure 3.2. We find that the resulting NP Hamiltonian, in terms ofthe arbitrary external (internal) d-type (u-type) quark fields q1, q2 (ui, uj),is given asHNPeff =GF4?2pi2??B(y?UV )iq1(y?UV )?jq2(V ?iq2Vjq14F1(xH , xi, xj)+(y?UV )jq1(y?UV )?iq216?2GFM2WF2(xH , xi, xj))(q2??PLq1)2 (3.30)where F1, F2 are loop functions of the mass ratios xH , xi = m2H?/M2W ,m2ui/M2W . We also neglect running between the scales mt, mW , and m?H ,instead integrating out these degrees of freedom at a common electroweakscale which allows us to use previous results. Moreover, we have neglected aNP QCD correction factor ??(xH , xt) arising at next-to-leading order [153].The internal top quark contribution dominates in the first term because F1 isproportional to both internal quark masses and in the second term becausethe other internal quark contributions are severely Cabbibo suppressed, al-lowing us to safely take i = j = t.110], which showed improved consistency with the SM over previous results favoring non-zero ??s [111, 112].683.4. Flavor Constraints3.4.1 B0q -B0q ConstraintsFor B0q -B0q mixing we take q1 = b, q2 = q and add the SM Hamiltonian toarrive at the full resultHSM+NPeff = (V ?tqVtb)2G2F4pi2M2W ??BS(xt)(1 +(y?UV )tb(y?UV )?tq?2(V ?tqVtb)2GFM2WS(xt)?(V ?tqVtb4F1(xH , xt) +(y?UV )tb(y?UV )?tq16?2GFM2WF2(xH , xt)))(q??PLb)2(3.31)which allows us to extract ?q in a simple form as?q = 1 + cbqF1(xH , xt)/S(xt) + c2bq F2(xH , xt)/S(xt), (3.32)withcbq ?(y?UV )tb(y?UV )?tq4?2GFm2WVtbV ?tq. (3.33)and the NP loop functionsF1(xH , xt) =xtxH(xH ? 4) log xH(xH ? 1)(xH ? xt)2? xt(xt ? 4)(xt ? 1)(xH ? xt)? xt(xHx2t ? 2xHxt + 4xH ? 3x2t ) log xt(xt ? 1)2(xH ? xt)2F2(xH , xt) =x2H ? x2t ? 2xtxH log(xH/xt)(xH ? xt)3. (3.34)The tRd iLH+ charge current couplings are (y?UV )ti = y?ttVti + y?tcVci, fori = d, s, b. Here, a novel feature arises from the NP CP-violating phaseassociated with y?tc [154]. We can write (y?UV )tq as(y?UV )tq = y?ttVtq(1 +????y?tcVcqy?ttVtq???? ei?q)(3.35)with ?q = arg(y?tcVcqy??ttV ?tq). For q = s this is ?s ? ?tc = arg(y?tcVcsy??ttV ?ts)while, using the Wolfenstein parametrization in appendix A, for q = d we693.4. Flavor Constraintsfind?d = arg(y?tcy??ttVcdV ?td) = arg(y?tcy??ttVcsV ?ts) + arg(VcdV ?tdVcsV ?ts)= ?tc + arg (1? ?+ i?) = ?tc + ? . (3.36)We can summarize these results as(y?UV )tb ' y?ttVtb(y?UV )ts = y?ttVts(1 +????y?tcVcsy?ttVts???? ei?tc)(y?UV )td = y?ttVtd(1 +????y?tcVcdy?ttVtd???? ei(?tc+?)).In the limit that |y?tt|  |y?tc|, we neglect the y?tcVcb term for the first relation,however, y?tc is non-negligible for the other two relations because the y?tt termsare Cabibbo suppressed.Thus, the NP phase that enters (M s12)NP is ?tc, while for (Md12)NP itis (?tc+ ?), due to the different CKM structures of (y?UV )ts and (y?UV )td.The best fit values from the CKMfitter group for ??d,s are quite differentnumerically, but due to this extra ei?, we can explain both ??d,s in terms of thesingle NP phase ?tc. Note that, for y?tc = 0, our model gives ??d,s = 0, since(M q12)NP would have the same complex phase as (Mq12)SM , i.e., (VtbV ?tq)2.Our results for B0d,s-B0d,s mixing are shown in Figure 3.4. Here, we mapthe best fit regions for ?d,s from reference [119] into the parameter spaceof our model. We fix |y?tt| and mH? and evaluate the prefered regions for|y?tc| and ?tc consistent with B0d,s-B0d,s mixing constraints. The blue (red)contours correspond to the best fit regions at 1? (inner) and 2? (outer), for?d (?s). Since ?d,s are quadratic functions of |y?tc|ei?tc , the best fit regionsfor ?d,s each map into two best fit regions in |y?tc|, ?tc parameter space. Asdiscussed in the next section, EWBG favors |y?tt| ? 1 and mH? . 500 GeV.703.4. Flavor Constraints3.4.2 K0-K0 ConstraintsNP contributions to K0-K0 mixing arise in our model through the boxdiagrams in Figure 3.2, where the strongest constraint is due to K . ForK0-K0 mixing, we take q1 = s, q2 = d in eq. (3.30) and add the SMHamiltonian in eq. (3.13) to arrive at the resultHSM+NPeff =G2F4pi2M2W((VcsV ?cd)2??ccS(xc) + 2(VtsV ?tdVcsV ?cd)??ctS(xc, xt)+ (VtsV ?td)2??tt(S(xt) + csdF1(xH , xt) + c2sdF2(xH , xt)))(d??PLs)2 + h.c.(3.37)Inserting this Hamiltonian into eqs. (3.16) and (3.15) gives the result|K |SM+NP = CB?KIm((VcsV ?cd)2??ccS0(xc) + 2(VtsV ?tdVcsV ?cd)??ctS0(xc, xt)+(VtsV ?td)2??tt(S0(xt) + csdF1(xH , xt) + c2sdF2(xH , xt))), (3.38)where NP enters through the coefficients csd defined in eq. (3.33)13. B?K isthe bag parameter associated with the hadronic matrix element andC ?G2FM2W f2KmK12?2pi2?mK, (3.39)which is a function of the Kaon decay constant, fK , has been measured to ahigh degree of accuracy. All SM input parameters in eq. (3.38) are definedand tabulated in reference [119].In the SM, |K |SM = (1.90 ? 0.26) ? 10?3 [155], while experimentally|K |exp = (2.228? 0.011)? 10?3 [156]. Assuming a theoretical error bar asin reference [155], we take the following constraint on our model|K |SM+NP = (2.23? 0.30)? 10?3 . (3.40)13We neglect NP NLO corrections to ?tt.713.4. Flavor ConstraintsbR bL sLH+ H+?uibR bL sLui ui?, gH+Figure 3.3: NP magnetic penguin diagrams contributing to b? s?.At first glance it appears that, since |K |SM < |K |exp, this constraint wouldfavor a small, positive contribution from NP. However, in reality, |K |SMitself is shifted to a central value |K |SM = 2.40 ? 10?3 because the bestfit CKM parameters in the presence of NP in B0d,s-B0d,s mixing (given inTable 11 of reference [119]) are different than in a SM-only fit. As a result,eq. (3.40) favors a small, negative contribution from NP. In Figure 3.4, theparameter region within the dashed dark (light) green contours is consistentwith K constraint in eq. (3.40) at 1? (2?).3.4.3 B ? Xs? ConstraintsWe also implement constraints on our model from b? s?. The NP modifiesthe Wilson coefficients of the magnetic and chromo-magnetic penguin oper-ators, i.e., C2 and C3 in eq. (3.19), through the penguin diagrams in Figure3.3. The new Wilson coefficients are given byCNP2 = ?(y?UV )?is(y?UV )ib2?2GFM2W (V ?tsVtb)F3(xH , xi)CNP3 = ?(y?UV )?is(y?UV )ib2?2GFM2W (V ?tsVtb)F4(xH , xi). (3.41)723.4. Flavor ConstraintsOur choice of Yukawa structure in eq. (3.28) imposes that only the internaltop quark is important, i.e., i = t, so thatCNP2 = ?cbsF3(xH , xt)CNP3 = ?cbsF4(xH , xt) (3.42)with the loop functionsF3(xH , xt) =xH(2xH ? xt)xt ln(xHxt)5(xH ? xt)4+7x2H ? 5xHxt ? 8x2t + 24xHxt ln( xtxH)36(xH ? xt)3F4(xH , xt) =2x2H + 5xHxt ? x2t12(xH ? xt)3+6x2Hxt ln( xtxH)12(xH ? xt)4. (3.43)This leads to the effective Wilson coefficient for the magnetic penguin oper-ator,Ceff,NP2 (?b) = 0.695CNP2 + 0.085CNP3 , (3.44)which, when inserted into eq. (3.22) and subjected to the bounds in eq.(3.23) while taking as inputs the best fit CKM parameters given in Table11 of reference [119], gives rise to parameter regions highly similar to thoseshown in Figure 3.4.In the published version of this chapter, we evaluate SM+NP contribu-tions to BR[B ? Xs?] in our model at NLO following references [143, 144].Adding the experimental and theoretical errors from [146] and [143] inquadrature, we take the following constraint on our model:BR[B ? Xs?]SM+NPE?>1.6 GeV = (3.55? 0.39)? 10?4 . (3.45)In Figure 3.4, the white (light grey) region corresponds to |y?tc|, ?tc parame-733.4. Flavor Constraints?y? tt? = 0.8mH? = 100 GeV-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05-0.10-0.050.000.050.10?y? tc? cos Jtc?y? tc?sinJtc?y? tt? = 1.2mH? = 350 GeV-0.3 -0.2 -0.1 0.0 0.1 0.2-0.15-0.10-0.050.000.050.100.15?y? tc? cos Jtc?y? tc?sinJtcFigure 3.4: Top-charm flavor violation parameter space (|y?tc|, ?tc) consistent withflavor observables, for two choices of |y?tt|, mH? . 68% and 95% CL regions for ?d(?s) from reference [119] shown by blue (red) contours. Region within dark (light)dashed green contours is consistent with K at 68% (95%) CL. Light (dark) greyregion is excluded at 68% (95%) CL from BR(B ? Xs?).ter space consistent with eq. (3.45) at less than 1? (2?), while the dark greyregion is excluded at 2?.Here, we make several important points.? Despite the fact that ??d and ??s are quite different numerically, thereexists regions of parameter space where both NP in B0d-B0d and B0s -B0s can be explained by a single phase ?tc. The 1? best fit regionsfor ?d,s overlap within the parameter space of our model (neglectingcorrelations between ?d and ?s).743.5. Electroweak Baryogenesis? The ?s region that overlaps with the ?d region in Figure 3.4 cor-responds to the ??s = (?51.6+14.1?9.4 )? solution. Therefore, our modelpredicts ??s > 0.? Although b ? s? and K constrain a large parametric region of ourmodel, these two observables are consistent with observation in regionsfavored by B mixing observables.? The presence of a large phase, ?tc, can weaken b ? s? and K con-straints, allowing for the possibility of a light charged Higgs (mH? ?100 GeV).? The values of (|y?tt|,mH?) shown in Figure 3.4 are consistent withRb ? BR[Z ? bb]/BR[Z ? hadrons] at 95% CL [122].Although we have presented only two illustrative values, (|y?tt|,mH?) = (0.8,100 GeV) and (1.2, 350 GeV), in Figure 3.4, there exists a consistency regionbetween all these observables for parameters |y?tt| ? 1, |y?tc| ? 0.05?0.1, and?tc ? 3pi/4, for 100 < mH? < 500 GeV. As we discuss in the next section,EWBG favors |y?tt| ? 1 and mH? . 500 GeV.3.5 Electroweak BaryogenesisEWBG is currently the most compelling mechanism for generating the baryonasymmetry of the universe as it is naturally connected to the electroweakscale. In this section, we provide a short description of the basic paradigmof EWBG followed by a description of our particular setup. We further elab-orate on the charge transport dynamics which reveal how much left-handedcharge is available to source the baryon-to-entropty ratio and conclude bypresenting our results.3.5.1 The BasicsIn EWBG, the production of the baryon asymmetry of the universe generallyoccurs during the electroweak phase transition, when the Higgs field acquires753.5. Electroweak Baryogenesisa vev at T ? 100 GeV. A strongly first-order electroweak phase transitionis needed and occurs when regions of broken phase, ?H? 6= 0, nucleatewithin the surrounding plasma in the symmetric phase, ?H? = 0. Theseregions, called bubbles, expand, collide, and coalesce until only the brokenphase remains. The key to EWBG is that an asymmetric density of baryonscan be generated by the interactions of the surrounding plasma with theexpanding bubble walls in three steps [36]:? As the bubbles sweep through space, the particles in the plasma un-dergo scattering processes within the encroaching bubble walls. If CPviolation is present in the underlying theory that dictates these scat-tering processes then this can lead to a chiral asymmetry in the vicinityof the bubble wall.? These chiral asymmetries can diffuse out in front of the bubble wall,into the symmetric phase, where they are then converted into baryonasymmetries by electroweak sphaleron transitions which are only activein the symmetric phase.? A fraction of the asymmetric baryon density created is then recap-tured back into the broken phase as the bubble wall sweeps by. Sinceelectroweak sphaleron transitions are highly suppressed in the brokenphase, the residual baryon asymmetry is then frozen in.Within the SM alone, the electroweak phase transition is first order only ifthe Higgs mass is far below its observed value, mh . 70 GeV [157]. However,even with a first order phase transition, the SM has an insufficient amount ofCP violation to generate chiral asymmetries large enough to reproduce theobserved baryon-to-entropy ratio [44]. Any extension of the SM that ame-liorates this situation must then incorporate new sources of CP violation,have non-negligible couplings to the SM, and be abundant in the thermalplasma at the time of the electroweak phase transition. Furthermore, onlybosonic particles with couplings to the Higgs enhance the finite temperaturepotential in the required way to achieve a first order phase transition [158].These conditions paint a general picture of new bosonic particles with weak763.5. Electroweak Baryogenesisscale masses and direct CP violating couplings to the SM, matching the de-scription of NP ingredients required to accommodate the B meson anomaliesand possibly rendering them observable at the LHC.3.5.2 FormalismGiven a NP model, viable EWBG requires: (1) the electroweak phase tran-sition must be strongly first order to prevent washout of baryon number,and (2) CP violation must be sufficient to account for the observed baryon-to-entropy ratio Y obsB ? 8 ? 10?11. EWBG in a 2HDM has been studiedmany times previously [159?165]. Most recently, reference [166] showedthat a strong first order phase transition can occur in a type-II 2HDM formh0 . 200 GeV and 300 . mH0 . 500 GeV. Although our 2HDM is notexactly the same as in reference [166], we assume that a strong first ordertransition does occur14.We now study baryon number generation during the phase transition.The dynamical Higgs fields during the transition give rise to a spacetimedependent mass matrix M(x) for, e.g., u-type quarks:Lmass = ?uRMuL + h.c. , M = yU v1(T ) + y?U v2(T ) (3.46)where v1,2(T ) ? ?H01,2?T 6=0 are the vevs at finite temperature T ? 100 GeV.At zero temperature, when v1(T ), v2(T )? v, 0, we recover the usual T = 0masses. However, if v2(T ) 6= 0, then CP-violating quark charge density canarise from y?U , as we show below. Left-handed quark charge, in turn, leadsto baryon number production through weak sphalerons. In previous studies,CP asymmetries were generated by a spacetime-dependent Higgs vev phase,arising from CP violation in the Higgs sector [159?166]. Here, we assumethat the Higgs potential is CP-conserving, such that v1,2(T ) do not havespacetime-dependent phases and can be taken to be real.Is it plausible that v2(T ) 6= 0 during the phase transition? Follow-14The phase transition can also be further strengthed or modified by the presence ofscalar gauge singlets [167, 168] or non-renormalizable operators [169].773.5. Electroweak Baryogenesising [121], the most general potential for H1,2 can be written asV = ?(H?1H1 ? v2)2 +m2H2H?2H2 + ?1H?1H1H?2H2+?2H?1H2H?2H1 + [?3(H?1H2)2 + ?4H?1H2H?2H2+?5H?2H1(H?1H1 ? v2) + h.c.] + ?6(H?2H2)2. (3.47)Our basis choice, i.e., ?H02 ?T=0 = 0, requires that no terms linear in H2 sur-vive when H01 ? v. The same statement does not hold at T 6= 0 due tothermal corrections to V . First, since we expect v1(T ) 6= v, terms linear inH2 appear proportional to ?5. Second, top quark loops generate a contribu-tion to the potential (yty?ttT 2H?1H2/4 + h.c.), given here in the high T limit,also linear in H2. A proper treatment of this issue requires a numericalevaluation of the bubble wall solutions of the finite T Higgs potential, whichis beyond the scope of this project. Here, we treat tan?(T ) ? v2(T )/v1(T )as a free parameter15, and we work in the ?(T )  1 limit. Intuitively, weexpect ?(T ) to be suppressed in the limit m2H2  T 2, since the vev will beconfined along the ?H02 ? = 0 valley.3.5.3 Charge Transport DynamicsThe charge transport dynamics of EWBG are governed by a system of Boltz-mann equations of the form n?a = SCPa +Da?2na +?b ?abnb [170]. Here nais the charge density for species a. The CP-violating source SCPa generatesnon-zero na within the expanding bubble wall, at the boundary betweenbroken and unbroken phases, due to the spacetime-varying vevs v1,2(T ).The diffusion constant Da describes how na is transported ahead of the wallinto the unbroken phase, where weak sphalerons are active. The remainingterms describe inelastic interactions that convert na into charge density ofother species b, with rate ?ab. Our setup of the Boltzmann equations followsstandard methods, described in detail in reference [171].Following reference [170], we assume a planar bubble wall geometry, withvelocity vw  1 and coordinate z normal to the wall. The z>0 (z<0) region15Although, in the type III 2HDM in which we work, the usual tan? is not physical atT = 0, the angle ?(T ) between the T = 0 and T 6= 0 vev directions is physical.783.5. Electroweak Baryogenesiscorresponds to the (un)broken phase. We look for steady state solutions inthe rest frame of the wall that only depend on z. Therefore, we replacen?a ? vwn?a and ?2na ? n??a, where prime denotes ?/?z. We adopt kinkbubble wall profilesv(T )/T = ? [1 + tanh(z/Lw)]/(2?2) , (3.48)?(T ) = ?? [1 + tanh(z/Lw)]/2 , (3.49)where v(T )2 ? v1(T )2 +v2(T )2. We take ? = 1.5, wall width Lw = 5/T , andT = 100 GeV as reference [166] found viable first-order phase transitionswith 1 < ? < 2.5 and 2 < LwT < 15, depending on the Higgs parameters.For definiteness, we take mH2 = 400 GeV; however, our analysis does notaccount for the crucially important mH2-dependence of the bubble profiles.Specializing to our 2HDM, the complete set of Boltzmann equations isvwn?qa = Dqn??qa + ?3a(SCPt + ?yQy + ?mQm)? 2?ssQssvwn?ua = Dqn??ua ? ?3a(SCPt + ?yQy + ?mQm) + ?ssQssvwn?da = Dqn??da + ?ssQss (3.50)vwn?H = DHn??H + ?yQy ? ?hQhwith linear combinations of charge densitiesQy ?nu3ku3? nq3kq3? nHkH, Qm ?nu3ku3? nq3kq3(3.51)Qss ?3?a=1(2nqakqa? nuakua? nuakua), Qh =nHkH. (3.52)The relevant densities are the ath generation left(right)-handed quark chargesnqa (nua , nda), and the Higgs charge density nH ? nH1 +nH2 (we treat H1,2as mass eigenstates in the unbroken phase). We assume that (Cabibbo un-suppressed) gauge interactions are in equilibrium, as are Higgs interactionsthat chemically equilibrate H1,2 (provided by ?3,4,5 quartic couplings in V ).Lepton densities do not get sourced and can be neglected. The k-factors aredefined by na = T 2ka?a/6, with chemical potential ?a.793.5. Electroweak BaryogenesisIn the eqs. (3.50), we take these transport coefficients as input:SCPt ? 0.1?Nc |yty?tt| sin ?tt v(T )2vw?(T )? T (3.53)?m ? 0.1?Nc|ytv1(T ) + y?ttv2(T )|2 T?1 (3.54)?y ? 27?232pi2 ?sy2t T + 9|y?tt|2T(mH22piT)5/2 e?mH2/T (3.55)?ss ? 14?4sT , Dq ? 6/T , DH ? 100/T . (3.56)We compute the CP-violating source SCPt and relaxation rate ?m, arising fortL,R only, following the vev-insertion formalism [172, 173] (explicit formu-lae can be found in [174]).16 The sole source of CP violation here is thephase ?tt ? arg(y?tt), which is not the same phase that enters into B0d,s-B0d,smixing.17 The dimensionless numerical factors (0.1), obtained following ref-erence [173], arise from integrals over tL,R quasi-particle momenta, takingas input are the thermal masses (tabulated in [176]) and thermal widths(?tL,R ? 0.15g2sT [177]). The top Yukawa rate ?y comes from processesH1tL ? tRg and H2 ? tRtL [176, 178].The strong sphaleron rate ?ss [179] plays a crucial role in EWBG inthe 2HDM [180], discussed below, and Dq,H are the quark and Higgs dif-fusion constants [181]. The relaxation rate ?h is due to Higgs charge non-conservation when the vev is non-zero. For simplicity, we set ?h = ?m [170];we find deviations from this estimate lead to . O(1) variations in our com-puted YB.We have omitted from eq. (3.50) additional Yukawa interactions inducedby ytc (e.g., H2 ? tRcL) because we find they have negligible impact onYB. Moreover, CP-violating sources from ytc do not arise at leading orderin vev-insertions. Therefore, ytc plays no role in our EWBG setup (thisconclusion may not hold beyond the vev-insertion formalism).Thus far, we have neglected baryon number violation; this is reasonablesince the weak sphaleron rate ?ws ? 120?5wT [182] is slow and out of equi-16Although there exist more sophisticated treatments, the reliability of quantitativeEWBG computations remains an open question (see discussion in [175]).17The reparametrization invariant phase is ?tt ? arg(y?tty?t v?1v2), but we have adopteda convention where v1,2(T ) and yt are real and positive.803.5. Electroweak Baryogenesisnq1+nq2+nq3nq3nq1+nq2nH-80 -60 -40 -20 0-101234z?LwchargedensityHGeV3LFigure 3.5: Left-handed quark and Higgs charge densities in the unbrokenphase (z < 0) for |y?tt| = 1, ?tt = 0.18, vw = 0.05, ?? = 10?2, givingYB ? 9? 10?11.librium. Therefore, we solve for the total left-handed charge nL ??a nqafrom eqs. (3.50), neglecting ?ws, and then treat nL as a source for baryondensity nB, according tovwn?B ?Dqn??B = ?(3?wsnL +RnB)h , (3.57)with the relaxation rate R = (15/4)?ws [183]. The sphaleron profile h(z)governs how ?ws turns off in the broken phase [184]. Since the energy of theT = 0 sphaleron is Esph ? 4MW /?w , we take [185]h(z) = exp(?Esph(T )/T ) , Esph(T ) = Esphv(T )/v . (3.58)Effectively, this cuts off the weak sphaleron rate for relatively small valuesof the vev: v(T, z)/T & g2/(8pi). In Figure 3.5, we show the spatial chargedensities resulting from a numerical solution to eqs. (3.50) for an examplechoice of parameters giving YB ? 9 ? 10?11. In general, the individualcharge densities have long diffusion tails into the unbroken phase (z < 0).However, nL is strongly localized near the bubble wall (z=0), due to strongsphalerons, thereby suppressing nB [180]. This effect can be understood asfollows: at the level of eqs. (3.50), B is conserved, implying?a(nqa +nua +nda) = 0; additionally, strong sphalerons relax the linear combination of813.5. Electroweak Baryogenesis?y? tt?=1.0?y? tt?=0.5?y? tt?=0.20.001 0.002 0.005 0.010 0.020 0.050 0.1000100200300400500600700vw Hwall velocityLHYB?YBobs L?Hsin? ttD?LFigure 3.6: Computed baryon asymmetry YB, normalized with respect toY obsB ? 9?10?11 and (sin ?tt??), as a function of vw and |y?tt|. Vertical axisshows (sin ?tt??)?1 required for viable EWBG.densitiesQss ? (1/Nc)?a(nqa ? nua ? nda) (3.59)to zero. These considerations imply that nL ? 0 if strong sphalerons are inequilibrium. In Figure 3.5, we see that strong sphalerons are equilibratedand nL vanishes for z . ?10Lw. Since nL is non-zero only near the wall, it isimportant to treat the weak sphaleron profile accurately in this region, ratherthan with a simple step function. Nevertheless, despite this suppression,EWBG can account for Y obsB . (We also note the significant Higgs charge nHin the broken phase. Although we neglect lepton Yukawas here, it is possiblethat nH could be efficiently transfered into left-handed lepton charge via y?L,thereby driving EWBG without suffering from strong sphaleron suppression,analogous to reference [176].) In Figure 3.6, we show how large YB can bein our model. The most important parameters are ??, y?tt, and vw (wefind YB is not strongly sensitive to Lw or ?). The vertical axis shows the(inverse) value of ?? ? sin ?tt required for successful EWBG (YB = Y obsB ),for different values of |y?tt| and vw. Our main conclusion is that our modelcan easily account for the baryon asymmetry of the universe ? even if ?? isas small as 10?3 ? 10?2, provided the NP Yukawa coupling has magnitude823.6. Conclusions|y?tt| & 0.2, with O(1) phase. Moreover, |y?tt| ? 1 is preferred by consistencywith flavor observables.3.6 ConclusionsThe dimuon asymmetry reported by D? [49] and the branching ratio BR(B ???) [104, 105] seem to disfavor the CKM paradigm of CP violation in theSM at the ?(3-4)? level. Although more experimental scrutiny is required,taken at face value, these anomalies can be accounted for by NP in bothB0d-B0d and B0s -B0s mixing [119]. Such NP would involve new weak-scalebosonic degrees of freedom and new large CP-violating phases. These twoingredients are precisely what is required for viable EWBG in extensions ofthe SM.We proposed a simple 2HDM that can account for these B meson anoma-lies and the baryon asymmetry of the universe. An interesting feature of oursetup is a top-charm flavor-violating Yukawa coupling of the NP Higgs dou-blet. The large relative phase of this coupling can explain both the dimuonasymmetry and tension in BR(B ? ??). Although top-charm flavor vi-olation can give potentially large contributions to b ? s? and K , i.e.,less CKM-suppressed than SM contributions, these bounds are weakenedin precisely the same region of parameter space consistent with B0d,s-B0d,sobservables by a single NP top-charm phase.We also discussed EWBG. We showed that, provided a strong first-ordereletroweak phase transition occurs, our model can easily explain the observedbaryon asymmetry of the Universe. CP violation during the phase transi-tion is provided by the relative phase in the flavor-diagonal tL-tR Yukawacoupling y?tt to the new Higgs, and the relevant phase is not related to thetop-charm CP phase entering flavor observables. However, flavor observablesand baryogenesis both require |y?tt| ? 1. Additionally, baryon generation isdependent on a parameter ?? related to the shift in the ratio of Higgsvevs across the bubble wall. We expect ?? to be suppressed in the limitmH?  mW . However, we showed that the charged Higgs state H? can belight (mH? ? 100 GeV) without conflicting with flavor observables due to833.7. Epiloguethe large top-charm phase in our model (as opposed to the limit mH? > 315GeV from b? s? in a type-II 2HDM [143, 156]).It would be interesting to explore the consequences of our model forHiggs- and top-related CP-violating and flavor-violating observables mea-surable in colliders, and also for rare decays such as K ? pi??. Additionally,a more robust analysis of EWBG requires an analysis of the finite tempera-ture effective potential in a Type-III 2HDM, addressing the phase transitionstrength and bubble wall profiles.3.7 EpilogueThe work presented within this section was originally motivated by twointeresting discrepancies with the SM. First, there was significant disagree-ment between the measured value of the branching ratio Br(B ? ??) andthe value obtained from global fits to the unitarity triangle. This discrep-ancy was mainly driven by the CP asymmetry, S?K = sin 2?, as measuredin B0d ? J/?K0S decays [105]. Second, the D? collaboration reported alarge deviation of the semileptonic CP asymmetry in Bd,s decays, i.e., thedimuon asymmetry, from its SM prediction [49, 52, 53]. Accommodation ofthe dimuon asymmetry requires large negative values of the model indepen-dent NP phase ??s as well as somewhat smaller but still negative values ofthe corresponding ??d [119]. Moreover, the required range of the ??d phasealso improves agreement between the CP asymmetry, S?K , and Br(B ? ??)by shifting the SM CP phase as S?K = sin(2? + ??d) [119]. These resultsimply that we should also expect a large negative CP phase in Bs ? J/??decays as it is also shifted as S?? = sin(??s ? 2?s), with 2?s ? 2.2?.Shortly after publication of the work presented within this section, theLHCb collaboration presented a measurement of this CP phase, finding??s?2?s = (?0.1?5.8?1.5)?, in excellent agreement with the SM, ??s = 0,and strongly disfavoring a large ??s phase [186, 187]. Without such a large??s phase, the dimuon asymmetry cannot be understood from NP in B0q -B0q mixing alone. Although there have been some suggestions that allowingNP in the decay matrices, ?d,s12 , may yet help to explain the D? dimuon843.7. Epilogueasymmetry [188], with such strong evidence of agreement with the SM inBs ? J/?? decays, the feeling that any NP is still needed is now muchreduced. The remaining strong motivation for NP comes from the tensionbetween the measured and fitted values of Br(B ? ??). However, even morerecently, the Belle collaboration has updated and improved their analysis oftheir measurement of this branching ratio [189]. Their new results now indi-cate excellent agreement with the fitted values of Br(B ? ??) and thereforestrongly diminish the case for NP in B0q -B0q mixing altogether [190].On a separate note, in our EWBG analysis we assumed, based on resultsin reference [166], that a first order phase transition does occur within theparameter range we are interested in. However, soon after publication ofthe work presented in this section, reference [191] studied the possibilityof simultaneously accommodating the dimuon asymmetry and the baryonasymmetry of the universe via EWBG in type I and II 2HDM. Their resultsare in tension with those of [166] in that they find more difficulty in obtaininga large enough baryon asymmetry. This is partly due to the numerous flavorand collider constraints that they have imposed, however, there is also anextra finite temperature suppression effect which arises from numericallycalculating the bubble wall profiles. Although these results were obtainedfor type I and II 2HDM, while we have instead proposed a type III 2HDM,they emphasize the need for a more careful analysis of the electroweak phasetransition and full treatment of the bubble wall profiles.85Chapter 4The Top Quark ForwardBackward Asymmetry,Right-Handed ChargeCurrents, and VubRecent measurements of the top quark forward backward asymmetry at theTevatron could hint at new physics with an unexpected flavor structure. Thesignificance of such an abnormal flavor structure in alleviating the tensionbetween the various measurements of |Vub| in B meson decays via right-handed charge currents is studied. Specifically, we elaborate on how thenew flavor changing interactions in the top sector can generate loop-induced,right-handed charge currents which are just the right size to simultaneouslyexplain the tension in the measurements of |Vub| and escape the tight indirectbounds from B ? Xs?.4.1 IntroductionThe top quark forward backward asymmetry (AttFB) has been measured byboth the D? and CDF collaborations at the Fermilab Tevatron collider [59?61, 192, 193], with both collaborations finding significant tension betweentheir measurements and current SM predictions [55?58]. Furthermore, theCDF collaboration has reported a dramatic dependence of the size of theasymmetry on the tt invariant mass Mtt [59], which has persisted in lightof a new analysis using the full Tevatron data set [61]. Specifically, in the864.1. Introductionhigh invariant mass region, Mtt > 450 GeV, the parton level asymmetryin the tt rest frame was found to deviate from the next to leading order(NLO) SM prediction by 3.4? [59], an effect which has, more recently, beenreduced to 2.5? with the full data set [61] and improved SM calculations [58].Although the D? collaboration does confirm the possibility of an interestingdeviation from the SM in the tt FBA [60], they find no statistically significantenhancement of the asymmetry in the high invariant mass region.Many NP models have been proposed to explain the asymmetry (fora recent review see, e.g., [194]), each of which can be organized accordingto whether the NP amplitudes are characterized by s-channel or t-channelexchange of new particles. In this thesis, we will concentrate on the latterpossibility. In this class of models, a large AttFB is achieved by the t-channelexchange of NP mediators in the pair production of top quarks, resultingin a Rutherford peak in the forward direction at high invariant mass. Themain characteristic of these models is the presence of large flavor changingcouplings which directly connect the first generation quarks to the top quarkvia the NP mediator (scalar or vector). In the case of a vector mediator,these flavor changing couplings are generally chosen to be right-handed inorder to avoid strong constraints from electroweak precision data and flavorphysics. Furthermore, considering the strong constraints on FCNC in thedown quark sector, this mediator should either only couple to up-type quarksor have flavor diagonal couplings in the down quark sector. However, evenin the latter case, care must be taken as flavor diagonal couplings to bbcan lead to large deviations in precision electroweak observables, as we willdiscuss later.Although many flavor observables provide extremely stringent constraintsdue to their good agreement with the SM, the flavor sector is not with-out its own anomalies. For instance, it has been known for some timenow that there exists tension between the inclusive and exclusive determi-nations of |Vub| [156, 195?197]. Experimental determinations of |Vub| arebased on the semi-leptonic decays of the B meson. The inclusive valueof |Vub|, |Vub|incl, is extracted from measurements in which only the fi-nal state lepton is detected, i.e., B ? Xu`?, while the exclusive value,874.1. Introduction|Vub|excl, requires the identification of a particular hadronic final state, e.g.,B ? pi`?. The inclusive and exclusive results do not agree and insteaddiffer by ? 2.5?, with |Vub|incl = (4.25 ? 0.15exp ? 0.20th) ? 10?3 [198] and|Vub|excl = (3.25 ? 0.12exp ? 0.28th) ? 10?3 [199]. The branching ratio forB ? ?? is also a useful observable, as it is over-all proportional to |Vub|2and therefore sensitive to this deviation. Br(B ? ??) is within 1? of theexperimental value if |Vub|incl is used whereas it deviates by ?3? if we, in-stead, use |Vub|excl [105]. Using experimental results for Br(B ? ??) [196]and lattice results for the B decay constant fB [200], we can also extracta purely leptonic value of |Vub|, |Vub|lep = (5.14 ? 0.57) ? 10?3, which dif-fers from |Vub|incl and |Vub|excl by 1.4? and 2.9? respectively. A hierarchybetween different measurements of |Vub| then starts to present itself in theform |Vub|excl < |Vub|incl < |Vub|lep.Recently, it has been realized that anomalous right-handed charge cur-rents (RHCC) in the quark sector can alleviate the tension between thedifferent determinations of |Vub| [201?203]. This is made possible by thefact that the interference between the mixing matrices associated with theleft- and right-handed charge currents are highly suppressed in the inclusivedecays whereas they enter with much less suppression and opposite sign intothe exclusive and leptonic B decays. In the most recent effort, the authors ofreference [203] considered RHCC with a corresponding mixing matrix, V R,generated by an enlargement of the gauge symmetry. In order to explain the|Vub| hierarchy, Re(V Rub/V Lub) was required to be ? O(20%). In the SM, thetree level W? couplings to fermions are strictly left-handed by construction,however, small effective RHCC are generated though quantum loop effects.The relative strength of such currents in the SM is O(?g4piVf ?fmfmf ?M2W)where mf and mf ? are the masses of the external fermion states, ?g is therelevant coupling, and we have also included the relevant CKM element. Inthe b? u transitions of interest, the dominant SM RHCC are QCD gener-ated and are O(10?12), far too small to explain the observed hierarchy!In the current chapter, we explore the role of the abnormal flavor struc-ture hinted at by the AttFB in enhancing the strength of anomalous loop-884.2. t-channel Exchange and the AttFBinduced RHCC in decays involving b ? u transitions which ameliorate thetension in the different measurements of |Vub|. The same flavor couplingswhich enhance RHCC in b ? u transitions also have the potential to gen-erate large RHCC in t? b transitions, which are subject to strong indirectconstraints from rare B ? Xs? decays [204, 205]. However, we will showthat RHCC in t ? b transitions are CKM (as well as chiral) suppressedwhile, in b ? u transitions, they are CKM (and chiral) enhanced. Thisallows us to choose large, i.e., O(1), couplings to alleviate the |Vub| tensionand remain unconstrained by the strong indirect B ? Xs? constraints. Asan added benefit, the region of parameter space that explains the |Vub| hier-archy is also favored by the AttFB, allowing for a simultaneous explanation ofboth. This provides a possible case for an unexpected connection betweenthe tensions in the energy (AttFB) and intensity (|Vub| hierarchy) frontiers.The layout of the chapter is as follows: in section 4.2, we briefly reviewthe motivation for and general structure of the t-channel class of models forexplaining the AttFB. In 4.3, we elaborate further on RHCC as an explanationof the |Vub| hierarchy and discuss the necessary bounds on the right-handedmixing matrix to alleviate the |Vub| tension. In section 4.4, we make the con-nection between the NP generating the AttFB and the |Vub| hierarchy and,in section 4.5, we demonstrate the effectiveness of the connection using aselection of some of the more highly cited NP models which were proposedto explain the AttFB. This selection includes a leptophobic Z ?, a scalar me-diator with a non-standard flavor structure in its Yukawa couplings, and asomewhat less SM based on t-channel driven, incoherent production of topquarks. Finally, in section 4.6, we conclude and in section 4.7 we discuss amore contemporary view of the results presented here.4.2 t-channel Exchange and the AttFBPossibly the most important piece of phenomenological information aboutthe AttFB is that any NP model introduced to explain it is naturally ex-pected to also affect other observables associated with the top quark. Inparticular, the excellent agreement between experiment and SM predictions894.2. t-channel Exchange and the AttFBfor the tt production cross section, ?(tt), as well its differential counterpart,d?(tt)/dMtt, presents a significant obstacle for any NP model. The con-straint this imposes on NP can be formulated in a model-independent wayoriginally put forward by reference [206].Let ?SM,NPF,B be the forward (F ) and backward (B) SM and NP tt pro-duction cross sections, defined as?SM,NPF =? cos ?=1cos ?=0d?SM,NP (tt)d cos ? ?SM,NPB =? cos ?=0cos ?=?1d?SM,NP (tt)d cos ? .(4.1)The total amplitude for top quark pair production is the sum of the SMand NP amplitudes, i.e., M = MSM +MNP . The SM cross sections areformed from the square of the SM amplitude whereas the NP cross sectionsare formed from both the interference terms as well as the square of the NPamplitude. This is significant as, if the interference terms dominate, then?NPF,B can take on either sign whereas, if pure NP terms dominate, ?NPF,B mustbe positive. Thus, any preference for ?NPF,B < 0 signals an interference effect.The total SM or NP cross sections are given by the sum ?SM,NP =?SM,NPF + ?SM,NPB while the measured cross section should correspond tothe sum of the SM and NP cross sections and can be parametrized as?Tot = ?SM + ?NP . (4.2)The AttFB can also be written in a model-independent way asAttFB =?SMF + ?NPF ? ?SMB ? ?NPB?SMF + ?NPF + ?SMB + ?NPB= Att,SMFB?SM?Tot +Att,NPFB?NP?Tot (4.3)where we have definedAtt,SMFB =?SMF ? ?SMB?SMF + ?SMBAtt,NPFB =?NPF ? ?NPB?NPF + ?NPB. (4.4)To achieve the model-independent fit, we take the further step of normalizing904.2. t-channel Exchange and the AttFB-1.0 -0.5 0.0 0.5 1.0-0.6-0.4-0.20.00.2sFNPsBNPqq?tt?NPFigure 4.1: Left panel: A model-independent fit to the high invariant massbin of the AttFB showing the need for interference effects. The normalizedforward (?F ) and backward (?B) components of the NP cross section areshown. The contours correspond to the boundaries of the 1? (solid), 2?(dashed), and 3? (dotted) regions. Right panel: The diagram describing thet-channel exchange of NP particles in top-antitop production at the Tevatron.the total and NP cross sections by the SM cross section such that?NPF,B ??NPF,B?SM , ?Tot ? ?Tot?SM = 1 + ?NPF + ?NPB ,AttFB =Att,SMFB + ?NPF ? ?NPB1 + ?NPF + ?NPB(4.5)with Att,NPFB now also defined in terms of the normalized NP cross sections.Using these quantities, a simple ?2 fit to the top quark pair productioncross section [207] and AttFB [59] at high invariant mass provides meaningfulconstraints on the variables ?NPF,B. The result of this fit is shown in theleft panel of Figure 4.1, in which a clear preference for negative ?NPB ispresent, implying that the AttFB must be generated through an interferenceeffect. In order to generate large interference terms, tree level exchange ofNP particles is required and can proceed through either the s-channel orthe t-channel depending on the nature of how the NP interacts with the topquark. Since the dominant production of top quark pairs at the Tevatron is914.2. t-channel Exchange and the AttFBthrough QCD, this limits the type of NP possible in the following way:If the interference proceeds through s-channel exchange then the colorrepresentation of the NP particle must transform the same way as a gluon,i.e., as an octet. If this is true then, in order to insure a non-zero AttFB, onerequires chiral couplings to the top quark. This description matches that ofan axi-gluon which is why there has been much interest in this possibilityin the literature [208?212].If the interference proceeds through t-channel exchange, as in the rightpanel of Figure 4.1, then the color representation of the NP particle is lessconstrained, yielding many different options. Furthermore, the general be-haviour of the AttFB seems to naturally coincide with the observed depen-dence on the invariant mass of the top quark pairs. The dominant produc-tion mechanism of the top quark pairs at the Tevatron is through qq fusion.The nature of the t-channel exchange implies that the parton level crosssection must incorporate a propagator such that?(tt) ? 1((pq ? pt)2 ?m2NP )2=1(m2t ?m2NP ? 2pq ? pt)2(4.6)where pq, pt are the 4-momenta of the incoming light quark and outgoing topquark and mNP is the NP particle mass. In the center of mass frame, the4-momenta are pq = (?s?/2,?s?/2) and pt = (?s?/2, ~pt), leading to pq ? pt =s?/4(1??1? 4m2t /s? cos ?)with ? acting as the scattering angle of the topquark in the final state. Noting that M2tt= s?, in the high invariant massregion the cross section takes the form?(tt) ? 1(m2t ?m2NP ?M2tt sin2 ?/2)2 . (4.7)This implies that, as long as the NP particle mass is comparable to thetop quark mass, there exists a Rutherford peak in the high invariant massregion which acts to bias the top quarks in the forward direction. This biassuggests that the AttFB should be positive and increasing for increasing Mtt,as observed by CDF [60, 61], and has thus generated much interest in thesemodels [206, 212?225]. The general characteristics of any t-channel model924.3. RHCC and the |Vub| Hierarchyare then:? A color representation that insures non-trivial interference with theSM QCD production of top quark pairs in qq fusion.? Light masses, i.e., O(mt), so as to allow for the proper behaviour ofAttFB in the high invariant mass regime.? Tree level flavor changing couplings that directly connect the first gen-eration quarks to the top quark, as pictured in the right panel of Figure4.1.In the following section, we will explore RHCC in more detail and theirpossible role in alleviating the tensions generating the |Vub| hierarchy.4.3 RHCC and the |Vub| HierarchyIn the language of effective field theory, the effect of any anomalous, loop-induced, chiral charge currents are felt strictly through the presence of gaugeinvariant, dimension six operators of the formcNP,Rij?2ORij =cNP,R1,ij?2uiR??djRH??? (iD?H)? + h.c.cNP,Lij?2OLij =cNP,L1,ij?2Q?iLH?? (i /DH)?QjL?+cNP,L2,ij?2Q?iL?? (?a/2) ?? QjL?H?? (?a/2)?? (iD?H)? + h.c. (4.8)Flavor indices are denoted as i, j while the fundamental and adjoint SU(2)Lindices are denoted as ?, ?, ?, ? and a respectively. The SM Higgs doubletis H while H? = i?2H? and the scale of the NP that gives rise to thesenew operators is denoted by ?. After spontaneous symmetry breaking androtating to the quark mass basis, the effective operators in eq. (4.8) alterthe charge current sector of the SM lagrangian such thatLW? = eWui??(V Lij PL + V Rij PR)djW+? + h.c. (4.9)934.3. RHCC and the |Vub| HierarchyThe matrix V Lij has contributions from the tree level SM CKM matrix aswell as from the NP coefficients cNP,L1 , cNP,L2 , whereas V Rij is sourced solelyby the cNP,R1 coefficient, i.e.,V Lij = Vij +v22?2(Au?cNP,L1 Ad)ij+v2?2(Au?cNP,L2 Ad)ijV Rij =v22?2(Bu?cNP,R1 Bd)ij. (4.10)The SM charged current is reproduced by setting cNP,L1,ij = cNP,L2,ij = cNP,R1,ij =0. We emphasize here that, in the cNP,L1,ij , cNP,L2,ij 6= 0 limit, V L is not the SMCKM matrix and its elements can therefore take on different values.The effects of RHCC in t ? b transitions have been investigated interms of their effects on observables which are sensitive to the tbW? vertexstructure. In particular, the branching ratio of the rare B ? Xs? decaysstrongly, although indirectly, constrain any alterations to this structure.The constraints on V Rtb are particularly tight due to an enhancement by afactor mt/mb [226]. A relatively recent study has determined that theseconstraints imply the 95% C.L. upper and lower bounds ?0.0007 ? V Rtb ?0.0025 [204, 205] as long as one assumes no other anomalous couplings. Ifone drops these assumptions the possibility of cancellations which couldloosen the constraints arise. We will assume that these bounds are robustand that their implications should be taken seriously.In b? u transitions, this charged current modifies the relevant 4-Fermiinteraction for semileptonic B meson decays asL = ?2?2GF `??PL?`u??(V LubPL + V RubPR)b. (4.11)For inclusive decays, B ? Xu`?, we again make use of the spectator modelto allow us to calculate in terms of the quark transition, b? u`?. The decayrate is then easily calculated as?(b? u`?) = G2Fm5b576pi3 f1(xu)|VLub|2(1 +????V RubV Lub????2? 2xuf2(xu)f1(xu)Re(V RubV Lub))(4.12)944.3. RHCC and the |Vub| Hierarchywhere xu = mu/mb, f1(x) = 1 ? x2(2 + x(16 ? 16x ? 2x3 + x5)), andf2(x) = 1 + x2(?12 + x(16 ? 3x + 2x3)). When evaluated numerically,2xuf2(xu)/f1(xu) = 4 ? 10?4 and so can be neglected, implying that theinclusive decays are only quadratically sensitive to the RHCC. Because ofthis, we can identify |Vub|expincl =?|V Lub|2 + |V Rub|2.The exclusive decay rate for the process, B ? pi`?, is calculated as?(B ? pi`?) = 12mB?d?|?pi`?|L |B?|2 (4.13)where d? is the phase space measure. As both the B and pi mesons arepseudoscalars, the hadronic matrix element is only sensitive to the vectorcurrent, i.e., ?pi|u??(V LubPL + V RubPR)b|B? = 1/2(V Lub + V Rub)?pi|u??b|B?. Theexperimental value for |Vub|excl can then be compared with |V Lub + V Rub|.Finally, for the purely leptonic decay mode, B ? ??, the final hadronicstate is simply the vacuum. Since the B is a pseudoscalar but is decaying toa scalar state (the vacuum is parity symmetric), the matrix element is onlysensitive to the axial current, i.e., ?0|u??(V LubPL + V RubPR)b|B? = 1/2(V Rub ?V Lub)?0|u??b|B?, implying the relation |Vub|explep = |V Rub ? V Lub|.It is clear that we should expect the strength of the RHCC to be smallin comparison to its left-handed counterpart. Because of this, we expandthe above relations to first order in V Rub to obtain|Vub|excl = |V Lub|?1 + 2Re(V Rub/V Lub)|Vub|lep = |V Lub|?1? 2Re(V Rub/V Lub)|Vub|expincl = |V Lub|. (4.14)Performing a simple ?2 fit to the data using the above functional forms leadsto the fit in Figure 4.2 from which the following 1? bounds can be derived|V Lub| = (4.266? 0.183)? 10?3 Re(V RubV Lub)= ?0.211? 0.049. (4.15)In [203], bounds on both the left and right-handed mixing matrices weredetermined in a similar fashion in s ? u, b ? c, and b ? u transitions.Our results agree well within error with theirs from b? u transitions. The954.3. RHCC and the |Vub| Hierarchy3.0 3.5 4.0 4.5 5.0-0.5-0.4-0.3-0.2-0.10.00.10.2?VubL?x10-3ReHVubR?VubLLFigure 4.2: A model-independent fit to |Vub| as extracted from inclusive,exclusive, and purely leptonic B decays.results of their fits to the left-handed mixing matrix in s ? u and b ? ctransitions are|V Lus| = 0.2248? 0.0009 |V Lcb | = (40.7? 0.6)? 10?3 (4.16)We assume a Wolfenstein parametrization of the left-handed mixing ma-trix such that the first two relations in eq. (4.16) uniquely determine theparameters ? and A. In order to determine the rest of the Wolfenstein pa-rameters we note that a large CP-violating phase in Bs mixing, as hinted atby recent Tevatron experiments, can be accommodated partly by assumingthat |V Rtd | ? 0 [203]. In this limit, we can attribute the measured value,|V exptd | = (8.4? 0.6)? 10?3 [156], entirely to the left-handed mixing matrixwhich determines the full set of Wolfenstein parameters. This allows us tocalculate Re(V Lub) = A?3? = (1.7 ? 0.6) ? 10?3 which, when inserted intoeq. (4.15) leads to the range of allowed values for the RHCCRe(V Rub) = (?3.6? 1.5)? 10?4. (4.17)In what follows, we will interpret these bounds as the size of the loop-inducedRHCC required to explain the |Vub| hierarchy. In the next section, we will964.4. AttFB, RHCC, and the |Vub| HierarchybR uRW?bR bL tL tRNPtR bRW+uR uL bL bRNPFigure 4.3: Left panel: RHCC are generated at the 1-loop level in b ? utransitions by NP originally designed for the AttFB. Right panel: RHCC aregenerated at the 1-loop level in t ? b transitions by NP originally designedfor the AttFB.further elaborate on the possible connection between the AttFB, RHCC, andthe |Vub| hierarchy.4.4 AttFB, RHCC, and the |Vub| HierarchyThe t-channel models generically include flavor changing couplings in theup-type quark sector, however, if we are to introduce this NP into b ? utransitions then we need to couple it to the down-type quark sector as well.Stringent constraints from FCNC processes force these couplings to be flavordiagonal. In an effort to be conservative, we only introduce a flavor diagonalbb coupling. This allows us to generate RHCC in b? u transitions throughthe loop diagrams shown in the left panel of Figure 4.3. A simple order ofmagnitude estimate of the strength of this current givesgbbgtu(4pi)2 VLtbmbmtm2NP?gbbgtu10?4 for mNP ? O(102 GeV), which represents a 109 enhancementover the SM RHCC! Assuming an O(1) loop function, this indicates a veryfavorable situation for explaining the |Vub| hierarchy while maintaining O(1)size NP couplings, exactly what is needed to simultaneously explain AttFB.The introduction of the flavor diagonal bb coupling also has unintendedconsequences:First, RHCC in t? b transitions, shown in the right panel of Figure 4.3,are necessarily also generated. These are subject to tight indirect constraintsfrom the rare B ? Xs? decays [204, 205]. However, in contrast to the974.4. AttFB, RHCC, and the |Vub| Hierarchysituation in b ? u transitions, the size of these RHCC is highly CKMand chiral suppressed and is estimated asgbbgtu(4pi)2 VLubmbmum2NP? gbbgtu10?12 formNP ? O(102 GeV), well within the indirect bounds for O(1) NP couplings.Second, the flavor diagonal couplings to bb modify the Zbb vertex at the1-loop level, shifting the SM tree level couplings as gbL,R ? gb,SML,R + ?gbL,R.These shifts can be detected through their effects on the observable Rb,defined asRb =?(Z ? bb)?(Z ? Hadrons) =(gb,SML )2 + (gb,SMR )2?i=u,d,c,s,b((gi,SML )2 + (gi,SMR )2) . (4.18)It is standard to normalize the Z ? bb width with the total hadronic widthas this leads to the cancellation of many QCD and electroweak corrections,thereby magnifying the sensitivity to NP. The corresponding shift in Rb dueto the shift in the tree level SM couplings can be parametrized to first orderin ?gbL,R as?Rb = Rb ?RSMb = 2RSMb (1?RSMb )(gb,SML ?gbL + gb,SMR ?gbR)((gb,SML )2 + (gb,SMR )2) . (4.19)Assuming the ratio m2b/m2NP is negligible, only the right-handed Zbb cou-pling is shifted due to the polarization of the NP vertex which acts to weakenthe constraint from Rb.Although the discussion in this section has followed the example of avector t-channel mediator, we will show in subsequent sections that scalarand fermion mediators tend to lead to the same basic conclusions regardless.In the next section, we will explore three benchmark models chosen toillustrate the possible connection between the AttFB and the |Vub| hierarchy.For each one of these models, the observable Rb provides an importantphenomenological constraint.984.5. The Models4.5 The ModelsIn this section, we explore the effectiveness of three benchmark models, allpreviously appearing in the literature, in simultaneously explaining the AttFBand the |Vub| hierarchy. The first is a spin 1 example in the form of a lep-tophobic Z ? [213] while the second is a spin 0 example with a novel Yukawastructure [227]. The third example consists of the t-channel exchange ofa spin 1/2 particle in the on-shell pair production of scalar top partners.These scalars then promptly decay into top quark pairs and missing trans-verse energy, mimicking a tt production signal [228]. Each of these modelshas been put forward as a possible explanation of the AttFB but we now showthat they may also simultaneously explain the |Vub| hierarchy.4.5.1 A Leptophobic Z ?Our spin 1 benchmark model is that of a new neutral Z ? boson. This isa well studied extension of the SM and arises naturally in NP scenariosinvolving the breaking of new gauge symmetries. A leptophobic version ofthe Z ? model has been studied recently in the context of the AttFB [213,220?222, 229?231]. In this context, the Z ? is given flavor changing, chiralcouplings to uR and tR. Any coupling to uL and tL must necessarily arisefrom a gauge invariant interaction involving the first and third generationquark doublets which, in the mass basis, leads also to a interaction of theform fLV ?udVtbdL??bLZ ??. Constraints on the B0d-B0d oscillation frequency,?md, restrict the size of the left handed coupling such that fL < 3.5 ?10?4 (MZ?/100 GeV) [222] and it is therefore usually neglected completely.If the Z ? only interacts with uR and tR then its only available decay modeis Z ? ? tRuR, uRtR which will lead to too many same sign top events atthe Tevatron through the production mechanisms uu ? Z ?Z ? ? ttuu andgu? tZ ? ? ttu. In order to avoid this, a small flavor diagonal coupling toup-type quarks, Uui,R??ui,RZ ??, is usually introduced [213, 220] to open upother more favorable decay modes. As long as the Z ? is lighter than the topquark the new flavor diagonal decay mode to uu will be the preferred one.However, U cannot be too large either as it is subject to dijet constraints994.5. The Modelsat the Tevatron and can, in conjunction with the large tu coupling, enhancethe rate for the normally GIM and loop suppressed rare top decay t? gu.Additionally, if the Z ? is much lighter than the top quark (MZ? . 120 GeV)then this could lead to a large branching ratio for the decay t? Z ?u whichwould manifest itself as a large difference in the measurement of the totaltt production cross section as measured in the lepton + jets and dileptonchannels. A greater proximity of the Z ? mass to the top quark mass willsuppress the t? Z ?u branching ratio and prevent these large discrepancies.Aside from the Tevatron, the new flavor changing couplings of the Z ?also necessarily lead to ample production of same-sign top quarks at theLHC. In references [221, 229, 230], the prediction for the same-sign topquark production cross section were obtained by requiring consistency witha large AttFB and the measured tt production cross section at the Tevatronfor a heavy Z ? (MZ? ? 200 GeV). Following this, the CMS collaborationsearched for same-sign top production and found none, disfavoring the heavyZ ? explanation of the AttFB at greater than 2? [232]. We therefore focus ourattention on the still phenomenologically viable light Z ? interpretation, i.e.,MZ? . 200 GeV.Although a flavor diagonal coupling to lighter up type quarks was pre-viously employed to avoid too many same sign top events at the Tevatron,we see no reason why this flavor diagonal coupling need be in the up-typequark sector. In particular, since we are interested in the effect of such aflavor changing Z ? on b ? u transitions in B meson decays we instead optfor a flavor diagonal coupling to bb. In addition to providing an alternativedecay mode for the Z ? in order to avoid too many same sign top quark pairsat the Tevatron, this coupling also avoids dijet constraints due to the partonluminosity suppression of the Z ? production mode and does not lead to anyenhancements of the rare top decay t ? gu. It is, however, constrained bythe observable Rb, as will be discussed below. Aside from the constraints,the introduction of the new flavor diagonal coupling also leads to an interest-ing signature of single top quark production with an associated bb resonancewhich may be searched for at both the Tevatron and the LHC.From the above arguments, we arrive at the following phenomenologi-1004.5. The Modelscally motivated lagrangianL = gutu??PRtZ ?? + gbbb??PRbZ ?? + h.c. (4.20)Note that these types of interactions need not be generated by charging SMfields under a new U ?(1). Instead, the Z ? can couple to the quark fields viahigher dimensional effective operators [233].In principle, the new Z ? is free to undergo kinetic mixing with the hy-percharge gauge boson which, after spontaneous symmetry breaking, willinduce Z-Z ? mixing. Adherence to constraints from the oblique parametersand the need to avoid the generation of any large effective leptonic couplingsto the Z ? demand that such mixing be highly suppressed (. 10?3 [234]). Wetherefore tune the tree level mixing parameter such that the overall strengthof the mixing is negligible, i.e., we choose the tree level parameter such thatthe sum of the tree level plus 1-loop contributions fall below the necessarybound.The 1-loop corrections to the Zbb couplings are comprised of contribu-tions from Z-Z ? mixing through a b quark loop, mass renormalization of theexternal b quark lines, and the vertex correction. The Z-Z ? mixing only af-fects the oblique parameters and does not correct the vertex. Furthermore,the effect of the mass renormalization of the external b quark lines is to shiftthe mass of the b quark to its physical value. The vertex correction, in theMS scheme, is given by?gbR = ?gb,SMR?bb2pi[52+1xZ?(32+1xZ)lnxZ+(1 +2xZ+1x2Z)(lnxZ ln(1 + xZ) + Li2(?xZ))](4.21)where xZ = M2Z/M2Z? and ?bb = g2bb/4pi. Using the SM calculated valuesgSMbL = ?0.4208, gSMbR= 0.0774, RSMb = 0.21578?0.00010, and the measuredvalue Rexpb = 0.21629? 0.00066 [156], we find the 1? bounds ?1.6? 10?4 <?Rb < 1.18 ? 10?3. Since ?Rb depends on the square of the coupling, onlythe absolute value |gbb| is constrained. The corresponding bounds on the1014.5. The ModelsExcluded100 120 140 160 180 200 220 2400.00.51.01.52.02.5MZ'?gbb?100 120 140 160 180 2000.00.20.40.60.81.01.21.4MZ'gutFigure 4.4: Left panel: The constraint from Rb = ?(Z ? bb)/?(Z ?Hadrons). The red region is excluded. Right panel: The strength of theRHCC for two values of gbb. The Vub hierarchy is accommodated in both theblue (gbb = ?1) and red (gbb = ?1.5) regions. The dot indicates the best fitpoint of reference [213] for accommodating the AttFB.(MZ? , |gbb|) space is shown in the left panel of Figure 4.4.The RHCC in b ? u transitions is generated from the diagram in theleft panel of Figure 4.3 with a virtual Z ? propagating in the loop. The resultis finite due to the two mass insertions and leads to the RHCCV Rub =2gutgbbV Ltb(4pi)2?xtxbxW(Li2(1? xt + xW )? Li2(1? xt)). (4.22)In order to accommodate the |Vub| hierarchy, the two couplings must haveopposite sign. In the right panel of Figure 4.4, both the blue (gbb=?1) andred (gbb=?1.5) regions in (MZ? , gut) space alleviate the tension in the |Vub|hierarchy. The best fit point for explaining the AttFB as well as avoiding largebranching ratios for the decays t? Z ?u in the Z ? model, from reference [213],is indicated by the presence of a dot. We caution that reference [213] doesnot specify the sign of gut and so the plot does have a reflection in thenegative half of the (MZ? , gut) plane. However, the relevant information canbe understood from either half-plane. The conclusion that is gained fromthis plot is that, with gbb . ?1, both AttFB and the |Vub| hierarchy can besimultaneously explained.In concluding this section, we note that the presence of the bbZ ? inter-1024.5. The ModelsugtZ ? bb?tugutZ ? b?b?LHCH7 TeVL?TevH1.96 TeVL100 120 140 160 180 200100101102103MZ'HGeVL?HpbLFigure 4.5: Left panel: The dominant (resonant) diagrams for single topproduction in association with a bb pair from on-shell production and decayof the Z ?. Right panel: The inclusive cross sections for this signal at theLHC and the Tevatron for gut = gbb = 1.action also opens up the possibility of a new and interesting hadron collidersignature in the form of a single top quark in association with a bb pair.The uniqueness of the signal is bolstered by the fact that the invariant massspectrum of the bb pair should contain a resonance peak representing theon-shell production and decay of the Z ?. This signal has no more missingtransverse energy (/ET ) than already associated with the top quark and thedominant production modes are shown in the left panel of Figure 4.5. Asfar as we can tell, there have been no searches for this particular signal.The dominant SM backgrounds are off-shell W? decays in association witha b. These are both sub-dominant to the NP signal due to the added phasespace and CKM suppression. In the right panel of Figure 4.5, we display theinclusive cross sections for the resonant production as a function of the Z ?mass at both the Tevatron and the LHC. The signal events were generatedusing MADGRAPH 5 version 1.3.29 [235] with the CTEQ6L1 parton distri-bution functions [236]. Both cross sections are calculated for gut = gbb = 1,however, the rates for other values of the couplings can be obtained by?(gut, gbb) = ?(gut = 1, gbb = 1)g2utg2bb.1034.5. The ModelsAlthough the cross sections in Figure 4.5 are quite large, we expectthese rates to reduce drastically once application of kinematic cuts, detectorsimulation, and b-tagging efficiencies have been applied. We also brieflynote here that, although the signal does not match any of the SM singletop production topologies, it should still be possible to use inclusive singletop production cross section measurements as a constraint on the signal.However, as our goal for this section was simply to demonstrate a possibleviable connection between the AttFB and the tension in the |Vub| hierarchy,we leave a more detailed analysis of the collider signatures for future work.4.5.2 A Non-Standard Two Higgs Doublet ModelFor our spin 0 benchmark model, we study a variation of a 2HDM with non-standard Yukawa couplings to the quark sector, a scenario which has beenstudied by many authors recently [206, 212, 215, 216, 222, 224, 227, 230, 237?241]. There are eight possible color representations that the scalar may havewhich are capable of producing the desired interference effect with the SM,however, only one is a color singlet. As this representation seems to havethe least amount of tension with the measured tt production cross sectionand invariant mass distribution [230], we choose to focus our efforts on it.Reference [227] put forward a model based on this representation whichincludes a highly non-trivial Yukawa structure to reproduce the required cou-plings. The color singlet scalar was given non-trivial weak isospin and hyper-charge quantum numbers as well, resulting in the representation (1,2,?1/2).With this choice, the scalar can be parametrized as ? =(?0??)with theYukawa interactionsL ? ?XijQiL?ujR + X?ijQiL??djR + h.c. (4.23)where i, j are flavor indices. The conclusions of reference [227] are as follows:? Flavor constraints imply that couplings involving the first generationquark doublet should be avoided.1044.5. The Models? An O(1) coupling connecting the top quark to either the up quark orthe down quark is required for a large AttFB. However, if both couplingsare simultaneously allowed then cancellations arise, preventing a largeAttFB.? Although a dRtL?+ coupling can generate a large AttFB, this cannot bedone without running into cross section constraints.? A large AttFB is generated for |?| & 0.6 and m?0 . 130 GeV.Heeding these lessons, we choose the form of the Yukawa matrix, Xij , asXij =???Vub 0 0Vcb 0 0Vtb 0 0??? (4.24)For our purposes, we also wish to introduce the flavor diagonal bb couplingin the down quark sector, which we do by choosing the following form forthe X?ij Yukawa matrixX?ij = gbb????0 0 Vub0 0 Vcb0 0 Vtb??? . (4.25)In addition to providing the necessary couplings to generate NP b ? utransitions, the form of this matrix also insures that no tree level FCNCappear in the down-type quark sector. The phenomenological model wework with is then described by the LagrangianL = ??0?q=u,c,t VqbqLuR + ???bLuR + gbb?+?q=u,c,t VqbqLbR?gbb???0bLbR + h.c. (4.26)Considering LEPII searches we also restrict ourselves to a lower limit ofm?0 ? 100 GeV. This model is free from indirect B ? Xs? constraints as fla-vor constraints disfavor a large coupling to the right-handed top quark [227]and our choice for the X?ij matrix does not couple the scalar to the s quark.1054.5. The ModelsbR uRW?bL???0?bR uRW?tL?0?+100 105 110 115 120 125 130-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0mf0gbbFigure 4.6: Left panel: The RHCC in the non-standard 2HDM is generatedby 1-loop diagrams involving both neutral and charged scalars. Right panel:The |Vub| hierarchy is explained in both the red (? = 0.6) and blue (? = 0.75)shaded regions. The yellow (? = 0.6), darker yellow (? = 0.7), and orange(? = 0.75) shaded regions are consistent with the Rb bounds. The regions ofoverlap are consistent with a large AttFB [227].The diagrams that generate b? u transitions are shown in the left panelof Figure 4.6. The resulting RHCC is finite, due to an exact cancellation oflogarithmic divergences between the two diagrams, and given byV Rub =?gbb(4pi)2? 10dx? 1?x0dy ln((1? x? y)(1? yxW ) + xxt + yx?0(1? x? y)(x?0 ? yxW ) + y).(4.27)with xi = m2i /m2?? . This RHCC decreases with increasing m?? , how-ever, oblique parameter constraints, specifically the T parameter, restrict thesplitting between the neutral and charged scalar masses to be |m???m?0 | .110 GeV [227]. In order to reduce the number of free parameters, main-tain O(1) gbb couplings, and obtain a more numerically favorable situa-1064.5. The Modelstion for the RHCC, we take m?? = m?0 + 110 GeV, giving the range210 GeV . m?? . 240 GeV.The authors of reference [227] found Rb bounds to be negligible, how-ever, in light of the new flavor diagonal bb coupling we have added, thesebounds must be revisited anew. Since all eight diagrams generating Rb aretopologically similar to the diagrams for the RHCC, we do not list themhere. Both shifts in the Zbb couplings are logarithmically divergent and werenormalize these divergences using the MS scheme at the scale ? = MZ .These shifts are given by?gbL =? 10dx? 1?x0dy[? g2bb2(4pi)2 ln( x0Z(1? x? y)(1? yx0Z) + y)? ?2(1? 4s2W )2(4pi)2 ln( x?Z(1? x? y)(1? yx?Z ) + y)? 2?2s2W3(4pi)2(ln( x?Z1? x? y + xyx?Z)? xyx?Z1? x? y + xyx?Z)+g2bbs2W3(4pi)2(ln( x0Z1? x? y + xyx0Z)? xyx0Z1? x? y + xyx0Z)],?gbR =? 10dx? 1?x0dy[? g2bb2(4pi)2 ln( x0Z(1? x? y)(1? yx0Z) + y)? g2bb(1? 2s2W )2(4pi)2 ln( x?Z(1? x? y)(1? yx?Z ) + y + xx?t)+g2bb2(4pi)2((1? 4/3s2W )(ln( x?Z1? x? y ? xyx?Z + (x+ y)x?t)? 1+xyx?Z1? x? y ? xyx?Z + (x+ y)x?t)? 2s2W3x?tx?Zxyx?Z1? x? y ? xyx?Z + (x+ y)x?t)? g2bb(1? 2/3s2W )2(4pi)2(ln( x0Z1? x? y + xyx0Z)? 1 + xyx0Z1? x? y + xyx0Z)(4.28)where x0,?Z = M2Z/m?0,? , x0,?t = m2t /m?0,? . We find that there exist regions1074.5. The Modelsof overlap between the parameter space favored by the |Vub| hierarchy andallowed by the Rb bounds, which we present in the right panel of Figure 4.6.The red (? = 0.6) and blue (? = 0.75) shaded regions show the parame-ter space which is consistent with the required RHCC to explain the |Vub|hierarchy while the yellow (? = 0.6), darker yellow (? = 0.7), and orange(? = 0.75) shaded regions are consistent with the Rb bounds. We also notehere that the regions of overlap in Figure 4.6 are consistent with a largeAttFB according to reference [227].The new interactions of our model open up new decay modes for thetop quark which can be constrained by the total top width. There are twopossible decay modes, t? ?+b and t? ?0u. The first is an off-shell processdue to the ?? mass range we have chosen. Since the ?? dominantly decaysinto the bu final state, the first decay is given by?t?bbu =5g2bb?2mt3072pi3(6/5xt? + 4/5(xt? ? 1)(3xt? ? 1) coth?1(1? 2xt?)? 1)(4.29)where xt? = m2??/m2t . The rate for the t? ?0u mode is an on-shell process.Since the ?0 decays to bb with a branching ratio close to unity, the decay is,to a good approximation, given by ?(t? b?0 ? bbu) ? ?(t? b?0) where?t??0u =|?|2mt96pi(1?m2?0m2t)2. (4.30)The total width of the top quark at LO is then given by ?t = ?SMt +?t??0u + ?t?bbu, where we take ?SMt = 1.3 GeV. The CDF collaborationhas performed a direct measurement of the top quark width, setting an upperlimit of ?t < 7.6 GeV [242]. The D? collaboration has also recently releasedan indirect determination of the total width of the top quark by combiningthe measurements of the single top t-channel production cross section andthe branching ratio Br(t ? Wb) as measured in tt events, leading to themore stringent result ?t = 2.00+0.47?0.43 [243]. However, even with this morestringent limit, the effect of the new decay modes cannot yet be observed.1084.5. The ModelsThe model also allows for non-standard single top production. In theSM, the dominant single top production mode is through t-channel exchangeof a W?, which has been measured by the D? collaboration [244] and foundto be in good agreement with the SM [245]. The signal for this productionmode at the Tevatron is pp ? tbq where tbq ? tbq + btq. The t-channelproduction mode depends on a b quark from the vacuum. Since the vacuumis flavor symmetric, this necessarily leads to a b quark in the final state,which explains the signal tbq. In our model, all single top production modesinvolve too many b quarks in the final state to apply to this signal.As mentioned in the previous section, there exist stringent constraintsassociated with the non-observation of a significant amount of same-sign topquark events at both the Tevatron and the LHC. In our model there is nosuch signal. The same-sign top final state would have to be produced byt-channel exchange of the neutral ?0. However, since the scalar is complex,this exchange can produce tt pairs only.We also note briefly that dijet constraints are not relevant as all pro-cesses leading to this signal are either highly CKM suppressed or have verylittle parton luminosity support while the SM backgrounds are enormous incomparison.As before, the introduction of the new bb coupling also leads to the newsignal discussed in the previous section, i.e; single top with a bb pair froman on-shell decay of a ?0 and no more /ET than already associated withthe top quark. As Br(?0 ? bb) ? 1, the total cross section is given by?(ug ? t?0). For illustrative purposes, we choose m?0 = 120 GeV, whichleads to ? = ?2g2bb43.8 pb at the LHC for?s = 7 TeV and ? = ?2g2bb0.60 pbat the Tevatron with?s = 1.96 TeV. The signal events were generated withMADGRAPH 5 [235] with the CTEQ6L1 parton distribution functions [236].Again, we postpone a more involved analysis of the signal for future work.4.5.3 Incoherent Top Quark ProductionIn the minimal supersymmetric SM (MSSM), loop-induced RHCC in b? utransitions can be generated from gluino-squark loops [202]. In this case, the1094.5. The Modelsflavor violation resides in the squark sector as opposed to the quark sectorand is due to flavor violating squark mass insertions. These insertions arisebecause the rotations in flavor space that diagonalize the SM quark massesdo not necessarily diagonalize the squark masses or the Higgs-squark-squarktri-linear couplings. As well as flavor violation, these rotations also inducechiral mixing between squark states after the Higgs doublets acquire theirrespect vevs, generally leading to non-diagonal 6?6 mass matrices for boththe up and down type squarks. In reference [246], bounds on the flavor off-diagonal elements of these mass matrices were derived by requiring that nolarge cancellations between tree level SM CKM elements and MSSM loopcorrections occur. Since the SM CKM elements reside completely in the left-handed quark sector, this leaves squark mixing effects that only affect theright-handed quark sector relatively unconstrained. This freedom was ex-ploited in reference [202] to show that RHCC, generated from gluino-squarkloops, can explain the |Vub| hierarchy. If one instead adopts the hypothesisof flavor-blindness, then both squark mass matrices are diagonal and thesize of the flavor violating RHCC are drastically reduced, making a super-symmetric explanation of the |Vub| hierarchy unattainable. In this section,we treat the flavor and chiral off-diagonal elements of the squark mass ma-trices, written in the notation of reference [246] as ?q?,XYij with i, j and X,Yacting as flavor and chiral indices respectively, as perturbative interactions.We also define the dimensionless quantities ?q?,XYij ? ?q?,XYij /m2q? .It is well known that the MSSM alone cannot address the AttFB [247, 248].To address the AttFB in a SUSY-type scenario, one needs to add explicit flavorchanging couplings between the first generation quarks and the top. Onepossibility is to allow for R-parity violation [249]. However, in this section wewill focus on a SUSY-inspired model put forward in reference [228]. In thisscenario, one adds a light, gauge singlet, Majorana fermion to the MSSMthat interacts with the right-handed stop and all the SM up-type quarks.With b ? u transitions in mind, we modify this model to allow the singletstate to interact with the right-handed sbottom squark and SM down-type1104.5. The Modelsuu??t??Rt?R tt???Figure 4.7: The dominant diagram for incoherent production of the tt finalstate in association with extra /ET in the form of extra Majorana singlets.The t-channel exchange of these singlets in the stop pair production processgenerates a large AttFB in the tt final state.quarks. Specifically, we take the LagrangianL = LMSSM + ?i/???m??c?+?q=u,c,t y?qqRt?R?c+?q?=d,s,b y?q?q?Rb?R?c + h.c. (4.31)With these interactions, the tt final state is produced from the on-shell pairproduction of right-handed stops which promptly decay into top quarks ac-companied by the light invisible singlet states, as shown in Figure 4.7. Asthe light singlet states are totally neutral, they register experimentally asmissing tranverse energy (/ET ). A large asymmetry in the tt final state isachieved by the t-channel exchange of these light singlet states in the stoppair production process. This type of production mechanism does not inter-fere with the SM tt production and is therefore called incoherent production.Due to this lack of interference, the model space must lie within the firstquadrant of the left panel of Figure 4.1 and can therefore only attain at best2? agreement with the AttFB. However, we caution the reader that the leftpanel of Figure 4.1 is modified when more recent measurements of the AttFBat high invariant mass are used, allowing now for a ? 1? fit to the data.As these singlet states are Majorana fermions, this model constitutes a spin1/2 benchmark model for the connection between the |Vub| hierarchy andthe AttFB.1114.5. The ModelsRequiring small mass splittings between the top quark and the right-handed stop limits the /ET to experimentally acceptable values. Further-more, the scalar character of the stop minimizes NP contributions to the ttproduction cross section and therefore weakens constraints18. Flavor con-straints from D0-D0 and B0d,s-B0d,s mixing imply that the y?c, y?d, y?s couplingsmust be small and can therefore be neglected. Since the stop masses areheavier than the top quark mass there are no new top decay modes openedup. Although same sign top events at the LHC can occur through t-channelexchange of the Majorana singlet state and subsequent t?R decays, this isseverely suppressed by the ratio (m?/mt?R)2, leading to cross sections wellbelow current bounds.According to reference [228], a large AttFB requires 200 GeV . mt?R .205 GeV, y?u & 1.5, y?t ? 4, and m? ? 2 GeV. We further choose all squarkmasses except for the right-handed stop to be degenerate. The dominantdiagram generating the RHCC in b ? u transitions is shown in the leftpanel of Figure 4.8. A similar diagram also generates t? b transitions. Theresulting RHCC are given byV Rub = ?y?uy?b4(4pi)2 ?t?,RL33 ?b?,LR33(1? x2t?+ 2xt? lnxt?(1? xt?)3)V Rtb = ?y?ty?b4(4pi)2 ?t?,LR33 ?b?,RL33(1? x2t?+ 2xt? lnxt?(1? xt?)3). (4.32)Taking the mass insertions to be ? 1 and choosing the y?u, y?t,mt?R to achieveagreement with the AttFB, the regions consistent with both the constraintsfrom t? b transitions and the explanation of the |Vub| hierarchy are shownin the blue (y?u = 1.5) and red (y?u = 2) shaded regions in the right panel ofFigure 4.8.With singlet masses of ? 2 GeV, the flavor diagonal bRb?R? couplingcan open up new invisible decay modes of the ?(1S) meson, i.e., ? ? ??.These proceed through a 4-Fermi interaction generated from integrating out18Top partners of higher spin have higher production cross sections due to simple spin-state counting [250].1124.5. The ModelsbR b?R b?L t?L t?R uR?W? 300 400 500 600 700 800 900 10000.00.20.40.60.81.0mq?y?bFigure 4.8: Left panel: The dominant diagrams in the incoherent productionmodel for generating the RHCC in b ? u transitions. Right panel: TheRHCC in b ? u transitions is consistent with the explanation of the |Vub|hierarchy in the blue (y?u = 1.5) and red (y?u = 2) regions while the yellowregion is consistent with the experimental limit on ?? invisible decays.the b?R. The decay rate can be estimated as ? y?4bm5?/(2pim4b?R), whereas thetotal Upsilon width is ?? = 54?10?6 GeV [156]. The current experimentallimits on the invisible branching ratio are Br(? ? invisible) < 3 ? 10?4 atthe 90% C.L. [156], leading to the allowed (yellow) region shown in the rightpanel of Figure 4.8.Chiral mass insertions such as those featured in b ? u transitions alsoallow for a NP contribution to Rb. Since the interactions only involve theright-handed quark sector, only the gb,SMR coupling is shifted as?gbR =y?2b (1? 2/3s2W )2(4pi)2 ?b?,LR33 ?b?,RL33? 10dx? 1?x0dy xy(x+ y ? xyxb?Z)5(4.33)where xb?Z = M2Z/m2b? . We find that the parameters of this model remainunconstrained from Rb.Taking into account all previous remarks, we conclude that this modelcan simultaneously account for the large AttFB and the |Vub| hierarchy whilepassing all constraints provided that y?b . 0.4 and mq? & 400 GeV.1134.6. Conclusions4.6 ConclusionsRecent measurements of the AttFB at the Tevatron are in disagreement withprecise SM predictions. This has inspired many NP models to explain theanomaly. One class of models put forward are the t-channel models, wherenew flavor changing interactions between the first generation quarks andthe top quark are mediated by NP particles exchanged in the t-channel oftt production at the Tevatron. This type of exchange generates a Ruther-ford peak in the forward direction at high invariant mass of the top quarkpairs, leading to an increase of AttFB with invariant mass which matches thatobserved by the CDF collaboration.Meanwhile, in the B sector, a long-standing tension between measure-ments of |Vub| in different decay channels of the B meson may be resolvedby the existence of small, yet significant, RHCC. With the addition of aflavor diagonal bb coupling, the same NP that explains the AttFB may alsoexplain the tension between |Vub| measurements by generating the necessaryRHCC at the 1-loop level. We have studied the effect of such new flavorphysics on the size of loop-induced RHCC for three separate models. Eachmodel represents a benchmark example of a t-channel model involving theexchange of a NP particle of spin 1, spin 0, and spin 1/2. In each case,we see that the new flavor physics introduced to accommodate a large AttFBnaturally generates RHCC in b? u transitions of the correct size to explainthe |Vub| tension. Furthermore, although RHCC are also generated in t? btransitions, the same mechanism which leads to enhancements of RHCC inb ? u transitions tends to suppress them in t ? b transitions, avoidingstringent indirect constraints from Br(B ? Xs?).This situation comes about for various reasons. For a spin 1 exchange,the flavor changing coupling provides CKM (as well as chiral) enhancementsin b ? u transitions while concurrently providing CKM (as well as chiral)suppression in t ? b transitions. For spin 0 exchange, the non-standardYukawa structure that provides a large AttFB forces both t ? b transitionsand meson-antimeson oscillations to share a similar parameter dependence.The tight constraints on the frequency of these oscillations then also suppress1144.7. Epiloguethe t? b transitions while the b? u transitions remain relatively free. Thespin 1/2 case is less clever than fortunate. Both the RHCC in b ? u andt ? b transitions are identical up to different couplings. Numerically, aRHCC that explains the |Vub| tension falls within the indirect bounds fromBr(B ? Xs?). In this case, this agreement is what allows for a simultaneousexplanation.Aside from a simultaneous explanation of the AttFB and |Vub| tension, thespin 1 and spin 0 benchmark models also lead to a new signal in single topproduction which can be searched for at hadron colliders. However, we haveleft a more detailed exploration of such a signal for future work.4.7 EpilogueRHCC are an attractive theoretical explanation for the |Vub| tension as theyonly mildly affect the inclusive value, while entering with opposite signinto the exclusive and leptonic values. This naturally accommodates theobserved hierarchy of different measurements, with the leptonic value, ex-tracted from the Br(B ? ??) branching ratio, as the largest. Indeed, thedisagreement between experiment and the SM prediction for Br(B ? ??)from global fits to flavor observables has fuelled much interest in NP [105].However, this excitement has proven to be short-lived as, soon after publi-cation of the work presented in this section, the Belle collaboration releasedan updated analysis of their measurement of Br(B ? ??) [189]. After allimprovements in the analysis were accounted for, the new measured value isonly 40% of its original value and in excellent agreement with the SM predic-tion, leaving little to no room for NP [190]. With such a reduced value, theorder of the |Vub| hierarchy is now |Vub|lep < |Vub|excl < |Vub|incl, completelyinconsistent with the pattern prescribed by RHCC. Although the tensionbetween the inclusive and exclusive measurements of |Vub| remains a mys-tery, the new Belle analysis suggests that the possible connection betweenthe AttFB and the |Vub| tension through RHCC is conclusively severed.The AttFB continues to evade explanation, however, some progress hasbeen made. Shortly after publication of the work presented here, low energy1154.7. Epilogueprecision tests of atomic parity violation (APV) were shown to stronglydisfavor certain t-channel models [251]. APV arises at the atomic level frommixing between different energy levels with opposite parities. At the morefundamental level, it is due to parity violating electron-quark interactions,proceeding from low energy SM Z exchange. NP at the weak scale modifiesthese interactions by generating anomalous couplings between the SM Z andthe first generation quarks. Both the leptophobic Z ? and the non-standard2HDM generate particularly large modifications as they directly connect thefirst generation quarks to the top quark, insuring its presence in the 1-loopdiagrams generating the anomalous couplings. These couplings must thentake the form?2NP(4pi)2m2tm2NP, (4.34)where ?NP is the flavor changing coupling and mM is the mass of the lightNP mediator. The proportionality to the top mass can be understood as fol-lows: There are two diagrams at the 1-loop level that generate the anomalouscoupling, a vertex correction and a wavefunction correction for the externalquark line. If these diagrams have no sensitivity to electroweak symmetrybreaking then the ward identity enforces a cancellation between them. Anynon-zero anomalous coupling must therefore be proportional to a parame-ter which is sensitive to electroweak symmetry breaking. In the case of theZ ? and non-standard 2HDM, this is the top quark mass. Since a success-ful description of the AttFB requires both m2t /m2NP and ?NP to be ? O(1),the anomalous couplings are ? O(10?2), i.e., well within range of currentand future APV experimental sensitivities [252]. This directly leads to thestrong constraints described in reference [251].In contrast to this, the incoherent production model is not constrainedby APV. The reason for this is that it does not directly couple the firstgeneration quarks to the top quark. Instead, the 1-loop diagrams involveonly the Majorana singlet and the stop, neither of which receive their massfrom electroweak symmetry breaking. This lack of sensitivity leads to anexact cancellation at the 1-loop level, enforced by the Ward identity. At the1164.7. Epilogue2-loop level, the presence of the top mass is restored, however, its effects arenow more efficiently loop-suppressed. This extra suppression is sufficientto guarantee that the incoherent production benchmark model remains freefrom APV constraints, allowing it to remain a possible explanation of theAttFB.117Chapter 5Conclusions & OutlookIn this final chapter, we summarize the key results obtained in this thesis,briefly discuss the modern view of them, and also list future directions.In chapter 2, we have explored the effects of the Randall-Sundrum warpedextra dimensional scenario on neutron-antineutron oscillations. We have cal-culated the 4D effective Wilson coefficients of the leading order operatorsthat induce neutron-antineutron oscillations up to arbitrary O(1) couplingsand calculated the full renormalization group running of these operatorsfrom the weak scale down to the oscillation scale. Our findings indicate thatthese oscillations can be easily geometrically suppressed to within experi-mental limits without any appeal to fine tuning of Wilson coefficients evenfor warped down mass scales of, in some cases, O(500 GeV). Our resultsalso improve intuition of the general behaviour of higher dimensional op-erators in the RS model based on the flavor structure of the operators inquestion. Furthermore, based on proposed design specifications of a possi-ble upcoming intensity frontier experiment at Fermilab, we have shown thatobservation of up to 50 oscillation events may be possible within the spanof 1 year given a beam of ultra-cold neutrons with flux ? 1012 neutrons/s.An on-going issue in the phenomenology of RS models is to understandtheir behaviour at the quantum level. In this thesis, we have only dealt withthe zero-modes of the Kaluza-Klein (KK) expansions of fields, however, inprinciple there exists an entire tower of 4D KK modes for each 5D field.All of these modes enter into loop calculations, leading to a new type ofdivergence based on the infinite sum over the KK modes. Because of this,a complete analytic calculation of loop-mediated phenomena, such as theFCNC decay B ? Xs?, remains an open problem.118Chapter 5. Conclusions & OutlookIn chapter 3, motivated by the anomalous dimuon asymmetry [49] andtension in the branching ratio Br(B ? ??) [104, 105], we explored thepossibility of new physics involving new weak-scale bosonic degrees of free-dom and new large CP-violating phases in both B0d-B?0d and B0s -B?0s mixing.Noting that these two ingredients are precisely what is required for viableelectroweak baryogenesis in extensions of the SM, we proposed a simpleTwo-Higgs-Doublet-Model with a novel top-charm flavor violating Yukawastructure to account for both the B meson anomalies and the baryon asym-metry of the universe. We showed that the relative phase of this couplingweakens normally strong bounds from neutral K meson mixing and rareB ? Xs? decays in exactly the region of parameter space that accom-modates both the dimuon asymmetry and Br(B ? ??). The phase thatprovides the necessary CP violation during the electroweak phase transi-tion is not the same phase that enters into flavor observables. Despite this,we compute the charge transport equations which determine the amount ofleft-handed charge available to source the baryon density, concluding thatour model can easily generate the required baryon-to-entropy ratio. Un-fortunately, soon after publication of this result, two separate experimentalresults were released which cast doubt on the validity of the dimuon asymme-try [186, 187] and found a much-reduced tension in Br(B ? ??) [189, 190],drastically diminishing the motivation for new physics in the B meson sectoraltogether.Irregardless of the motivation from B meson anomalies, the problem ofexplaining the baryon asymmetry of the universe persists and electroweakbaryogenesis remains the most motivated mechanism of its creation. Withthe recent observation of a new Higgs-like boson by the CMS [253] andATLAS [254] collaborations at the LHC, the focus has shifted from the Bmeson sector directly to the electroweak symmetry breaking sector itself.In particular, extended scalar sectors are a natural extension to the SMelectroweak symmetry breaking sector. They also incorporate the necessaryingredients for electroweak baryogenesis in that they can contain 1) novelsources of CP violation and 2) new undiscovered particles whose dynamics119Chapter 5. Conclusions & Outlookcan aid in inducing a strong first-order electroweak phase transition. Pre-liminary investigation into this line of research has shown that new sourcesof CP violation in the scalar potential mix electrically neutral scalar andpseudoscalar states, allowing us to interpret the new boson as a scalar-pseudoscalar admixture. This modifies the interaction of the boson withthe top quark in such a way that it enhances the rate that it decays into thedi-photon state by an amount directly related to the size of the CP violationin the potential. This is especially pertinent as such an enhancement in thedi-photon signal rate has been observed by both the experimental collabo-rations at the LHC [253, 254]. Understanding whether the size and natureof the CP violation consistent with current LHC data is sufficient for elec-troweak baryogenesis purposes is the subject of eager on-going investigation.In chapter 4, we identified a possible connection between the anomaloustop quark forward backward asymmetry at the Tevatron and a long-standingtension between different measurements of the CKM matrix element |Vub| inB meson decays. Model independent fits to the data suggest that t-channelexchange of new particles, with novel interactions connecting the top quarkto the first generation quarks, may be responsible for the forward backwardasymmetry. We point out that these novel interactions have the unintendedbut welcome side effect of generating right-handed charge currents whichare exactly the correct size to explain the observed hierarchy of measure-ments of |Vub| in B meson decays. In order to illustrate the generality of thisconnection, we explore it in detail for three benchmark models, each distin-guished by the intrinsic spin of the new particle exchanged in the t-channel.These are a leptophobic Z ? (spin 1), a Two-Higgs-Doublet-Model with a non-standard Yukawa structure (spin 0), and an extension to the MSSM in whichtop quarks are produced incoherently through on-shell production and decayof right-handed stops (spin 1/2). In each case, we showed that both the for-ward backward asymmetry and the hierarchy of |Vub| measurements can beaccommodated within the same region of parameter space, while stringentelectroweak precision constraints are simultaneously avoided. Moreover, theinteractions which establish this connection lead to a new collider signal in120Chapter 5. Conclusions & Outlooksingle top quark production which can be searched for at the LHC. Regret-tably, soon after publication of this work, the Belle collaboration releasedtheir result of a much-reduced Br(B ? ??) [189]. The effect of this resultis to reshuffle the order of the observed |Vub| hierarchy in such a way thatright-handed charge currents may no longer play a role in its explanation,effectively severing the possible connection between the forward backwardasymmetry and the |Vub| tension.Although the connection to the |Vub| hierarchy is gone, the forward back-ward asymmetry remains. The standard paradigm of the t-channel modelsis in tension with stringent atomic parity violation constraints, however, theincoherent production models are not part of the standard paradigm. Theyevade these tight constraints by maintaining a separation between the firstgeneration quarks and the top quark and thus remain an interesting pos-sible explanation of the forward backward asymmetry. These models alsogive rise to interesting NP signatures in the single top sector at the LHC.The manifestation of the light singlet states as /ET generates a mono-topsignal while the nature of the stop production mechanism generates a largelepton charge asymmetry in single top decays. Calculating predictions forthese signatures based on a global fit to the top quark data from both theTevatron and the LHC is the subject of on-going research.The research presented in this thesis contributes to the larger on-goingsearch for the end of the SM and the new physics that lies beyond it. Thereare many strong reasons to support the notion that there must be an endto the SM. We have discussed two of these in some detail, i.e., the hierarchyproblem and the baryon asymmetry of the universe, and mentioned othersonly in passing. As our theoretical understanding of the SM and its manywell-motivated extensions improve, the day in which the SM breaks inchesever closer. Indeed, as the LHC begins its long ? 2 year shut-down theimmense amounts of data that have been collected will be poured over,searching for any hint of new physics. The next decade, with the proposedupgrades to upwards of 13 TeV beam energies, will certainly be crucial inthis endeavour.121Chapter 5. Conclusions & OutlookIn conclusion, the unambiguous identification of new physics beyondthe SM will likely require many complimentary observations from each ofthe cosmic, energy, and intensity frontiers. 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Although this is a very interesting and de-veloping field, it is beyond the scope of this thesis and so, in what follows,we will focus our attention solely on the CKM matrix.The CKM matrix connects the weak eigenstates of the SM down-typequarks to their corresponding mass eigenstates,???d?s?b???? =Vij? ?? ????Vud Vus VubVcd Vcs VcbVtd Vts Vtb??????dsb??? (A.1)For 3 generations, the CKM matrix contains 4 physically significant param-eters, i.e., 1 complex phase with 3 real parameters. It is worth noting thatthere would be no complex phases in the quark sector if there were ? 2generations.The significance of the complex phase in the CKM matrix is that itprovides a mechanism for CP violation in the SM, which can be seen byconsidering the transformation of the charged current under the action of19Nonperturbative QCD effects can also induce an interaction at the level of the La-grangian which violates CP, however, the strength of the interaction is highly constrainedby the neutron electric dipole moment. This is related to the strong CP problem [32] andwe will neglect this source in our discussion.144Appendix A. CP Violation in the SMcharge-conjugation (C) and parity (P),J?CC = eWui??PLVijdjW+? + eWdi??PLV ?ijujW??CP?? eWdi??PLVijujW?? + eWui??PLV ?ijdjW+? , (A.2)i.e., CP is a good symmetry only if the elements of the CKM matrix arereal. In general, CP is violated in the SM if and only if [255]Im(det[?d?d?,?u?u?])6= 0 (A.3)where ?u,d are the Yukawa matrices introduced in eq. 1.23.The standard parametrization [256] for these matrices, recommended bythe Particle Data Group [54], is given byV =???c12c13 s12c13 s13e?i??s12c23 ? c12s23s13ei? c12c23 ? s12s23s13ei? s23c13s12s23 ? c12c23s13ei? ?s23c12 ? s12c23s13ei? c23c13???(A.4)with cij = cos ?ij , sij = sin ?ij . Another, more transparent parametrization,is the Wolfenstein parametrization. This is obtained by defining ? ? s12,A?2 ? s23, and A?3(?? i?) ? s13e?i?, which leads toV =???1? 12?2 ? A?3(?? i?)?? 1? 12?2 A?2A?3(1? ?? i?) ?A?2 1???+O(?4). (A.5)The unitarity of the CKM implies the 9 relations?j V ?jiVjk = ?ik. For eachof the relations corresponding to i 6= k, the sum of three complex quantitiesvanish, giving rise to a geometric representation in the form of a trianglein the complex plane. These are called the unitarity triangles and are veryuseful tools for visually representing the consistency of the multitude ofexperimental constraints on the various elements of the CKM. The trianglewith i = b and k = d is of particular interest as all terms in the sum are ofthe same order of magnitude. Dividing the relation by the element of the145Appendix A. CP Violation in the SM(0, 0) (1, 0)(?, ?)?? ?|V ?tbVtdV ?cbVcd||V ?ubVudV ?cbVcd |Figure A.1: The SM unitarity triangle.sum that is pure real we end up with the relation,V ?ubVudV ?cbVcd+V ?tbVtdV ?cbVcd+ 1 = 0 =? (?+ i?) + (1? ?? i?)? 1 = 0(A.6)shown graphically in Figure A.1. There are two non-trivial lengths????V ?ubVudV ?cbVcd???? =??2 + ?2????V ?tbVtdV ?cbVcd???? =?(1? ?)2 + ?2 (A.7)and three angles? ? arg(? V?tbVtdV ?ubVud)? ? arg(?V?cbVcdV ?tbVtd)? = arg(?V?ubVudV ?cbVcd). (A.8)Measurements of flavor-changing processes and unitarity constrain the mag-nitudes of CKM elements and therefore the sides of the triangles while theangles ?, ?, and ? are related to CP violation and can be probed throughvarious CP asymmetries in meson decays.Although there are many different ways in which the CKM matrix canbe parametrized, there is a parametrization independent quantity that can146Appendix A. CP Violation in the SMbe defined called the Jarlskog invariant [255]. This quantity, JCKM , corre-sponds to 1/2?(area of the unitarity triangle), is defined asIm(VijVklV ?ilV ?kj)= JCKM3?n,m=1ikmjln, (A.9)and has the following form in the standard parametrization,JCKM = c12c23c213s12s23s13 sin ?. Translating the condition in eq. A.3 intothe mass basis gives a more intuitive notion of the necessary conditions forphysically meaningful CP violation in the SM,?m2tc?m2tu?m2cu?m2bs?m2bd?m2sdJCKM 6= 0 (A.10)where ?m2ij = m2i ?m2j . The conditions are? No two quarks can be degenerate in mass.? None of the three mixing angles in the standard parametrization canbe zero or pi/2.? The complex phase cannot be zero or pi.Experimentally, CP violation was first discovered in neutral Kaon decays [257]but has now been observed in many rare processes [54]. Furthermore, theaccuracy of SM predictions is O(20%) [40], implying that these conditionsare, indeed, satisfied in the SM to a good approximation.147Appendix BOperator Product Expansionand the Principles ofEffective Field TheoryIn this section, we present the basic aspects of effective field theories (EFT),which are used extensively in chapters 2 and 3. By invoking the renormal-ization group, effective field theories are well-suited to deal with multi-scaleproblems, which frequently occur in nature. Furthermore, they allow for amodel-independent approach. For more detailed treatments, there are manyexcellent reviews available, e.g., see [258, 259].Consider a QFT with some fundamental scale, M , associated with it.This scale could be the mass of a heavy particle or some characteristicmomentum transfer and suppose that we are only interested in derivingpredictions for energies E << M . The construction of an EFT, valid onlyfor this low energy range, then follows from three steps:? Fix a cut-off ? < M , after which the EFT is no longer valid and thefull (UV complete) theory must be used. Divide the constituent fieldsof the theory into their low and high momentum fourier modes basedon this cut-off.? As these modes cannot be produced directly in experiments at energiesE < ?, integrate out the high momentum modes only from the pathintegral of the theory. This leaves a modified action which is non-localon distance scales < ??1.? This non-local action can then be expanded as an infinite linear com-148Appendix B. Operator Product Expansion and the Principles of Effective Field Theorybination of local operators of increasing dimension D constructed fromthe low momentum modes of the fields. This expansion is called theoperator product expansion (OPE) and is given by SE<? =?ddxLeffwith the effective LagrangianLeff =?i,DC(D)iMD?dO(D)i . (B.1)Each term in the linear combination consists of a finite sum over allpossible D-dimensional operators consistent with the symmetries of the lowenergy theory. The coefficients, C(D)i , are called Wilson coefficients andmust appear with appropriate powers of the scale M in order for the effectiveLagrangian to main the correct units. As operators with increasing massdimensions are increasingly suppressed by higher powers of M , the termswith D < 4 are the most important at low energies. This changes theinterpretation of non-renormalizable operators in a given theory. From thisview, they originate from integrating out unknown heavy degrees of freedomand so are of great interest since, by studying their coefficients, we maygain information about the more complete high energy theory. Turningthis around, we may also use the OPE to understand the phenomenologyof some proposed high energy theory by studying its effects on low energyphenomena. We start by writing down all possible operators, with so farunknown coefficients, up to some mass dimension that are consistent withthe given symmetries and depend on only the field content available in thelow energy theory. For E < ?, the amplitudes of the EFT, after integratingout certain high energy modes, have to agree with those in the full theory,i.e.,?out|Lfull|in? =?i,DC(D)i (?2)MD?d ?out|O(D)i (?2)|in? (B.2)where we have included the characteristic scale dependence from dimen-149Appendix B. Operator Product Expansion and the Principles of Effective Field Theorysional regularization. By calculating the matrix elements in the full theorywe can determine the corresponding Wilson coefficients in the EFT, whichare then useful for phenomenological applications. The most familiar exam-ple of this procedure is from nuclear beta decay in the SM. This process is in-duced by a single dimension 6 four-fermion operator, Leff =GF?2VudO(6) =GF?2Vude??(1 ? ?5)?eu??(1 ? ?5)d, in the low energy EFT. The coefficientis the famous g Fermi coupling which has mass dimension -2. In the fulltheory, this operator is generated through W? exchange asiM = ? ie2W4Vud1q2 ?M2We??(1? ?5)?eu??(1? ?5)d=ie2W4M2WVude??(1? ?5)?eu??(1? ?5)d+O( q2M2W). (B.3)Each element in the expansion of the momentum will correspond tohigher dimensional operators involving derivatives in the EFT. In order toreproduce the full result, all of these operators should be accounted for in theEFT amplitude. However, identifying the two amplitudes at first order inthe expansion, yields the famous resultGF?2=e2W4M2W. This process is calledmatching. In chapter 2, we use this method to determine the effects ofthe Randall-Sundrum model on non-renormalizable, dimension 9 operatorsgenerating neutron-antineutron oscillations.150Appendix CRenormalization GroupIn eq. (B.2) of the previous section, the mass scale, ?, appeared in match-ing the amplitudes of the full and effective theories. This scale dependencearises during dimensional regularization where it parametrizes the variationin mass dimension of the Lagrangian parameters in arbitrary spacetime di-mensions. In renormalized perturbation theory, the bare Lagrangian is splitinto a renormalized Lagrangian plus a set of counter-terms which are fixed,order by order in perturbation theory, by imposing some renormalizationconditions on certain correlation functions. These conditions are imposed atsome arbitrary mass scale, corresponding to ?, and define the physical cou-plings, masses, and field strength renormalizations of the theory, (?i,mi, Zi),by specifying exactly how the UV divergences are subtracted from the corre-lation functions. However, since ? is completely arbitrary, the renormaliza-tion conditions could have also been imposed at some other scale ??, leadingto different physical parameters, (??i,m?i, Z ?i). In order for this to be thesame theory, the relation between (?i,mi, Zi) and (??i,m?i, Z ?i) must be fixedby the relation between ? and ??. This suggests that a given theory doesnot have one unique set of physical parameters but, instead, is associatedwith a collection of physical parameters which vary continuously (or ?flow?)with scale, implying a constrained dependence of the renormalized Green?sfunctions on (?, ?i(?),mi(?), Zi(?)). The Callan-Symanzik equations arethe mathematical expressions of these constraints. They insure that a shiftin the arbitrary mass scale, ?, is compensated for by a corresponding shiftin the parameters of the Lagrangian.To derive the renormalization group equations for the couplings andmasses, recall that the renormalized parameters, (?i,mi), are obtained byrescaling the bare parameters, (?0i ,m0i ), as151Appendix C. Renormalization Group?i(?) = Z?1?i (?)?0i mi(?) = Z?1/2mi (?)m0i . (C.1)Since all scale dependence is introduced through dimensional regulariza-tion, the bare parameters do not depend on scale, i.e.,d?0id? =dm0id? = 0.This leads directly to the renormalization group equations?d?id? = ??i ?dmid? = ?mi?mi (C.2)where ?i = ?iZ?1?i ?dZ?id? and ?mi =12Z?1mi?dZmid? . Note that ?i and ?miare both finite as the divergences in Z?i and Zmi cancel in the ratios.The Callan-Symanzik equation is derived by considering the truncated,connected n-point Green?s function of a given theory with fields ?1, ?2, ?3, . . . .The Green?s function is essentially a matrix element of the time orderedproduct of n field operators and is proportional to the corresponding Feyn-man amplitude. When these fields are renormalized they are rescaled, im-plying that the amplitude is itself rescaled asM(?) = ?iZ?pi/2i (?)M0 (C.3)where there are p1 ?1 fields, p2 ?2 fields, etc. in the Green?s function.The Callan-Symanzik equation is then derived from the fact that the bareamplitude is scale-independent, i.e.,dM0d? = 0 so that(? ??? ??i?i???i??imi?mi??mi+?ipi?i)M = 0 (C.4)where ?i =12Z?1i ??Zi?? . In chapter 2, we use the Callan-Symanzik equa-tion, as presented here, for the OPE to derive the running of Wilson co-152Appendix C. Renormalization Groupefficients corresponding to higher dimensional operators inducing neutron-antineutron oscillations.153

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