aa r X i v : . [ h e p - ph ] A p r BINARY TETRAHEDRAL FLAVOR SYMMETRY
David A. EbyA dissertation submitted to the faculty of the University of North Carolina at ChapelHill in partial fulfillment of the requirements for the degree of Doctor of Philosophyin the Department of Physics and Astronomy.Chapel Hill2018 Approved by:Y. Jack Ng (Chair)Charles R. EvansJonathan EngelThomas W. KephartAmy L. Oldenberg (cid:13) bstract
DAVID A. EBY: Binary Tetrahedral Flavor Symmetry.(Under the direction of Paul H. Frampton)
A study of the T ′ Model and its variants utilizing Binary Tetrahedral Flavor Sym-metry. We begin with a description of the historical context and motivations for thistheory, together with some conceptual background for added clarity, and an accountof our theory’s inception in previous works. Our model endeavors to bridge two cat-egories of particles, leptons and quarks, a unification made possible by the inclusionof additional Higgs particles, shared between the two fermion sectors and creatinga single coherent system. This is achieved through the use of the Binary Tetrahedralsymmetry group and an investigation of the Tribimaximal symmetry evidenced byneutrinos. Our work details perturbations and extensions of this T ′ Model as weapply our framework to neutrino mixing, quark mixing, unification, and dark mat-ter. Where possible, we evaluate model predictions against experimental results andfind excellent matching with the atmospheric and reactor neutrino mixing angles, anaccurate prediction of the Cabibbo angle, and a dark matter candidate that remainsoutside the limits of current tests. Additionally, we include mention of a numberof unanswered questions and remaining areas of interest for future study. Taken to-gether, we believe these results speak to the promising potential of finite groups andflavor symmetries to act as an approximation of nature.iii cknowledgments
First and foremost, David Eby is grateful to Paul Frampton, his longtime advisor,co-author, and colleague for his guidance and his irreplaceable assistance in com-pleting his research and this dissertation. David is also grateful to the rest of hiscommittee, Y. Jack Ng in particular, for their backing despite trying circumstances.He would like to thank his friends for their support and the department for its coun-sel and instruction. He would also like to express his gratitude to Shinya Matsuzaki,the former group postdoc, for useful discussions and a productive collaboration. Hewishes to thank his Family for their enduring love and, lastly, sends thanks to thoseunmentioned that have aided in his work and his life.iv able of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A and T ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Recent Developments in Neutrino Theory . . . . . . . . . . . . . . . . . 181.4.1 Neutrino Masses and Mixings . . . . . . . . . . . . . . . . . . . 181.4.2 Majorana Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.3 Tribimaximal Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 221.5 The T ′ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.1 A and the Lepton Sector . . . . . . . . . . . . . . . . . . . . . . 241.5.2 The Minimal T ′ Model . . . . . . . . . . . . . . . . . . . . . . . . 31v.5.3 Cabibbo Angle Prediction . . . . . . . . . . . . . . . . . . . . . . 34 T ′ Model Perturbations and Neutrino Mixing . . . . . . . . . . . . . . . . . 37 T ′ Model and Quark Mixing . . . . . . . . . . . . . . . . . . . 51 T ′ M (D) Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Comparison with CKM Data . . . . . . . . . . . . . . . . . . . . . . . . 57 T ′ Quiver Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 T ′ Model Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 T ′ M . . . . . . . . . . . . . . . . . . . . . . 675.2.2 Generalized Seesaw Mechanism . . . . . . . . . . . . . . . . . . 705.3 T ′ Dark Matter Predictions . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.1 Relic Density and WIMP Mass . . . . . . . . . . . . . . . . . . . 725.3.2 Dark Matter Detection . . . . . . . . . . . . . . . . . . . . . . . . 74
A The Higgs Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82B Counting Relativistic Degrees of Freedom . . . . . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 vii ist of Tables Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Character Table of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Kronecker Products for irreps of A . . . . . . . . . . . . . . . . . . . . . . 161.5 Character Table of T ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Kronecker Products for irreps of T ′ . . . . . . . . . . . . . . . . . . . . . . . 185.1 Quark Group Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Lepton Group Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 Higgs Group Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B.1 Relativistic Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . 87viii ist of Figures θ − θ neutrino mixing plane, assuming < θ < ◦ θ − θ neutrino mixing plane, assuming < θ < ◦ δ KM . . . . . . . . . . . . . . 563.2 Data comparison of quark mixing parameter | V td /V ts | . . . . . . . . . . . . 573.3 Data comparison of quark mixing parameter | V ub /V cb | . . . . . . . . . . . . 585.1 WIMP cross-section vs. mass experimental limits . . . . . . . . . . . . . . . 75ix ist of Abbreviations A Tetrahedral Symmetry Group
BSM
Beyond the Standard Model
CKM
Cabbibo-Kobayashi-Maskawa matrix CP Charge-Parity Symmetry IH Inverted Neutrino Mass Hierarchy irrep
Irreducible Representation MR T ′ M Minimal Renormalizable T ′ Model
MSM
Minimal Standard Model NH Normal Neutrino Mass Hierarchy
NMR T ′ M Next-to-Minimal Renormalizable T ′ Model
PMNS
Pontecorvo-Maki-Nakagawa-Sakata matrix
QCD
Quantum Chromodynamics
QED
Quantum Electrodynamics
SUSY
Supersymmetry T ′ Binary Tetrahedral Symmetry Group
TBM
Tribimaximal Mixing
VEV
Vacuum Expectation Value
WIMP
Weakly Interacting Massive Particle Z Cyclic Symmetry Group, of Order 2 x hapter 1Introduction
Particle physics currently stands in transition. After a search lasting decades, theHiggs Boson, the final outstanding prediction of the Standard Model, has been dis-covered. And, despite lingering unexplained questions leading to dozens of inno-vative theories developed in the intervening years since the Standard Model’s cre-ation, the physics community has not settled on a likely successor, or even agreedon a single direction to pursue. Currently, there are numerous theories working ona multitude of problems in the hopes of uncovering the path to a more fundamentalunderstanding of nature.In this dissertation we hope to describe one of the relatively newer areas of study,and, more specifically, to explain the development, evolution, and means for eval-uating our model of Binary Tetrahedral Flavor Symmetry. While this text includesbackground on several topics, we should note that it is written with the expectationthat the reader has a working knowledge of both the particle content and the fun-damental principles of the Standard Model. This is not written with the intention ofbeing a text for instruction or a comprehensive resource, and merely describes theelevant sections of traditional particle theory. For the duration, except where noted,we will be using the natural units of ℏ c = 1 In this first chapter, we give an extended background on several topics. We high-light the successes and ongoing shortcomings of the Standard Model, noting key fea-tures that tie in with our model. Group Theory, in particular, remains a key tool inthe study of nature. We define mathematical groups and describe a series of groups,both familiar and practical examples, for later use. The uncharged leptons, knownas neutrinos, have continued to prove surprising and notoriously difficult to explainin the 80 years since first proposed. Despite these challenges, neutrinos act as anentry point for most modern flavor models, including ours, and it behooves us todescribe recent discoveries and suggested explanations in that area. This includesneutrino masses, mixings, and the proposed Majorana designation. Our backgroundwill conclude with a description of the Minimal Renormalizable T ′ Model (MR T ′ M).This model serves as a starting point for all of our subsequent original work and weinclude both its approach to the dual sectors of quarks and leptons, as well as itspioneering Cabibbo Angle approximation.In the second chapter, and first chapter of original work, we discuss a signifi-cant revision to the previously established model. By rearranging our assumptionsof inputs and variables, we are able to determine new relations between the mixingparameters of quarks and leptons. We describe the manner by which we have de-termined a relation between the neutrino mixing angles as a result of the deviationbetween the experimental data and the prior Cabibbo Angle prediction. We assess es-timates for both individual 3-neutrino mixing values, and correlated values, in lightof the recent groundbreaking experimental results. While no experiments currentlyrunning have the express purpose of validating our model, we compare our predic-tions with the global fits of accumulated neutrino data.The following chapter seeks to extend our model in order to encompass the full2uark mixing matrix, rather than a simplified mixing of the first two quark fami-lies. The Next-to-Minimal Renormalizable T ′ Model (NMR T ′ M) will introduce bothmany new parameters and a greatly expanded potential utility. Though our predic-tions of quark mixing, Cabibbo angle aside, remain limited in comparison with theirneutrino counterparts, we hope this model will better enable testing of our ideas asmixing parameters become better defined in the coming years.Next, we seek to integrate our prior work more closely with the physics of theStandard Model. Where, elsewhere in our research, we are primarily concerned withflavor physics, here we attempt to fuse our work with the famed physics of ( SU ( ) C × SU ( ) L × U ( ) Y ). This is achieved by combining the binary tetrahedral group withmultiple SU ( ) groups, in an arrangement called quartification. While not as datadriven as our other work, this discussion provides a successful test case of unification,and may lay the groundwork for connection with other theories, including GUTs andstring theory.Our final chapter of original work modifies the symmetry groups and particlecontent of our model to create a potential explanation for dark matter. We first discussthe general mechanism that allows for a suitable dark matter particle candidate toarise out of the finite symmetry, as well as the specifics of how these methods areintegrated into our theory. We describe several properties of this new particle andshow how it currently remains outside current testing limits.We end our discussion with some concluding thoughts. These include a descrip-tion of which ongoing and proposed experiments will provide results suitable fortesting our theories in the coming decade, as well as a summary of the limitations ofour model and the remaining outstanding questions, wrapping up with a few reflec-tions on the greater significance of this work.3ot since the years prior to the invention of general relativity and quantum me-chanics, nearly a century ago, has physics been at such a turning point. With the ex-haustion of old theories and a dawning generation of colliders and detectors poisedto discover new physics, we have hope that the next few years will prove just asrevelatory. The initial discovery of flavor physics, while highly remarked upon at the time,was unrecognized for its true significance. In 1936, physicists Carl Anderson and SethNeddermeyer were using cloud chambers to examine the decay products of cosmicradiation that survived long enough (aided by relativistic effects) to reach groundlevel. Cloud Chambers are designed to indicate the curved (due to a magnetic field)trail of electrically charged particles in water droplets, which, via the equations ofcentripetal motion, can determine a particle’s mass. They discovered a particle thathad a mass between that of the electron/positron and the proton (the only othercharged subatomic particles known at the time).[1] One year earlier, Hideki Yukawahad proposed a new particle, dubbed the meson, to mediate the strong nuclear forceand have a mass approximately the same as the newly discovered particle. This ledto the muon’s name, a conjunction of µ (at the time, the symbol used for mesons) andmeson. Later on, physicists would recognize that the muon more closely resembledan unstable, heavier version of the electron, and would repurpose the name mesonto mean bound states of quark-antiquark pairs. While the π meson, not the muon,would turn out to be Yukawa’s predicted particle, both discoveries marked a funda-mental step forward in physics. 4o say the muon was unexpected would be an understatement: Isidor Rabi isreported to have said, ”Who ordered that ? ” upon hearing of its discovery, but themuon acted as a harbinger of the future of particle physics in several ways. It wasthe first indication that the family of leptons, fundamental fermions that do not in-teract via the strong nuclear force, was larger than simply the electron. It was alsothe first indication of flavor, and of forthcoming particle discoveries exhibiting sim-ilar interactions and ever-increasing mass scales. Thus began an 80-year struggle tounderstand and explain this odd corner of physics.[2]Our modern understanding of flavor contains much more variety. We currentlyknow of 6 flavors of quark, organized into 3 families, each of which consists of anup-type quark and a down-type quark. The three up-types, from lightest to heaviest,are named the up, charm, and top quarks. The three down-types, from lightest toheaviest, are named the down, strange, and bottom quarks. Each quark has a fla-vor charge (with corresponding antiquarks given opposing flavor charges), and, asfermions, have a spin of . Leptons, the other fermions, also have spin of and sev-eral flavors, one for each of the three families, each of which contains a single chargedlepton ( e − , µ − , and τ − ) and neutrino ( ν e , ν µ , and ν τ ). Unification
In the 20th century physicists developed several theories to deal with a profu-sion of new phenomena. By the 1930s, the community was aware of 4 fundamen-tal forces: gravity, electromagnetism, the strong nuclear force, and the weak nuclearforce. Gravity had been explained by general relativity, although it continues to resistconsolidation with the other three. Electromagnetism, which had been explained inthe 1800s by Maxwell’s laws, was developed over the 1940s and 50s into the theory of5uantum Electrodynamics (QED), explaining the interaction of electric and magneticfields in terms of charged leptons and photons. The strong nuclear force continuedto advance our understanding of quantum fields during the development of Quan-tum Chromodynamics (QCD), which explained the inner workings of hadrons byproposing a new set of mediating bosons, the gluons. QCD also managed to bringsome order to the ever-growing number of mesons and baryons by reducing them tovarious combinations of the six currently known flavors of quark, in what came to becalled the Eightfold Way.In 1967, it was realized that QED and the weak nuclear force could be unifiedunder the gauge groups ( SU ( ) L × U ( ) Y ).[3] This new electroweak symmetry waslater combined with the SU ( ) C of QCD to form the basis of the Minimal StandardModel (MSM).[4, 5] This model proved to be one of the most successful in the historyof theoretical physics crafting dozens of accurate predictions including the W ± and Z bosons,[6–9] the charm,[10, 11] bottom,[12] and top quarks,[13, 14] as well as thegluon.[15–18] With the discovery of the Higgs boson in 2012, it has now reachedcompletion.[19, 20]
28 Parameters
Key parts of the overall MSM theory are the 28 parameters (sometimes seen num-bering 18, given the original assumption that neutrinos were massless and exhibitedno mixing). These constants have no predicted value, and yet, are part of the the-ory. In a sense, it is a marvel that an accurate description of the known universe(on small scales, excluding gravity) can be described using a single framework withunder thirty measurable quantities. These constants consist of: • • • U ( ) Y , SU ( ) L , and SU ( ) C • CP − violating phase • CP − violating phase(assuming Majorana neutrinos) • Mass of the Z boson • Mass of the Higgs boson • The QCD vacuum angle
These constants and their known values from Ref. 21 have been summarized inthe following table.
Description Parameter ValueUp − type quark masses m u , m c , m t . , . , . − type quark masses m d , m s , m b . , , . m e , m µ , m τ , . , . m ν , , < ∼ g , g , g . , . , . CP − phase Θ , Θ , Θ , δ CP . ◦ , . ◦ , . ◦ , .
995 radPMNS angles & CP − phases θ , θ , θ , δ i i =1 , , ∽ . ◦ , ∽ . ◦ , > ∼ . ◦ , δ i =?Electroweak scales M Z , M H . , ∽ θ QCD ∽ Table 1.1: Determined under MS scheme. See Ref. 21 for the individual renormaliza-tion scales used.Despite this compact form, it remains an ongoing effort among physicists to sim-plify and combine these items. These efforts take several forms and fall under severaldifferent searches. 7 .2.3 Lingering Mysteries and BSM Physics Amazingly successful, prescient for its time, a roadmap for 40 years of parti-cle physics, and indisputably incomplete − these all describe the Minimal StandardModel (MSM) of particle physics. Experimentalists have spent much of the pasthalf-century searching for the last pieces of this theory, while also hunting for in-dications of where it falls short. After such a long search, physicists are not entirelyempty handed in their quest for so-called Beyond the Standard Model (BSM) physics,and these problems generally fall into two categories. The first details places whereexperiments have deviated from MSM expectations; the second might be generallyclassified as theoretical inconsistencies of MSM theory, where idiosyncrasies (eitherproblems or coincidences) indicate that we do not yet have a complete understand-ing. Experimental Contradictions
For decades, neutrinos were thought to be massless. This was, in part, a resultof how little information has been gathered in the 80 years since they were initiallydescribed. It should be noted that this is in no way an indication of a lack of intereston the part of the physics community, on the contrary, it is due to the weak natureof neutrino interactions, and the extreme means that experimentalists must go toin order to obtain statistically viable data. Consequently, when the MSM was beingformulated in the 1970s, it was believed that neutrinos were simply massless. Thoughit had been suggested for years that neutrinos might have a small but non-zero mass,or that additional massive neutrinos might be hidden from experiments via somehypothetical mechanism, it was not until 1998 that the Super-Kamiokande NeutrinoDetector was able to measure neutrino flavor oscillation (i.e. that a neutrino couldchange its own flavor, from electron neutrino to muon neutrino, for example).[22]8f that were the end of it, the MSM could be briefly amended to include neutrinomasses with little other consequence. However, there were a number of additionalmysteries borne out of this discovery. There is the question of why neutrino massesare so very light, of why they do not exhibit the nearly diagonal mixing exhibitedby most other fermions, and why, unlike all other fermions, they do not appear tobe Dirac particles. These mysteries provide much of the motivation for the modelsdetailed in later chapters.During the 1970s astronomers began to examine galactic rotation curves, a mea-sure of the relative rotational velocity of galactic plane segments.[23] These curveswere expected to peak at a small radius and have long diminishing tails at higherradii as one traveled further from the galactic core. This would roughly correspondto a quickly spinning area fairly close to the center of the galaxy with the remainderdragging behind. Instead, they observed that the galactic rotation remained fairlyconstant out to large distance.[24] This indicated that the density in most galaxieswas steady to a much greater degree than was visibly indicated. Combined with anobservation of galaxies in the Coma Cluster dating from the 1930s,[25] this missingmass was dubbed Dark Matter, as both indicated evidence for significant mass in ex-cess of what could be visibly observed. Now there are many elements to a galaxythat are not observable from one part of the spectrum or another, but we have con-tinued to catalog a multitude of galaxies that otherwise conform to our establishedmodels, yet still manage to have this unseen and unexplained excess of mass. As-trophysicists have also made use of gravitational lensing to indicate the presence ofdark matter.[26] Gravitational lensing is the practice of measuring the mass of a largestellar body by calculating the curvature of light emitted by an objects on the far sideand bent around by gravity. These studies have resulted, on several occasions, inevidence for dark matter. Though there have been many suggested explanations for9ark matter over the years, the most prominent one in the modern physics commu-nity is the Weakly Interacting Massive Particle (WIMP).[27] These are hypotheticalrare heavy particles that interact only via the weak force (thus shielding them fromdetection by conventional means) and by gravity (corresponding with astronomicalobservations).When Einstein originally formulated general relativity he inserted a cosmologicalconstant, Λ , in order to avoid a dynamic universe size (largely due to philosophicalobjections). Following the discovery of Hubble expansion,[28] it was largely ignoredfor much of the mid-20th century and Einstein, himself, would later term it his ”great-est mistake”. In 1998 and 1999 two teams studying supernovae were able to show thatthe universe’s expansion continues to accelerate.[29, 30] At this point, discussions ofcosmology began reincorporating the cosmological constant, now as an indicationfor and quantification of our universe’s acceleration (should it be constant). DarkEnergy, as this phenomenon has come to be called, is the least understood problemin this section. Suggested explanations have varied widely from undiscovered fun-damental forces to rapidly shifting dark energy densities.[31, 32] It remains of crucialinterest to physicists as estimates of this dark energy suggest it comprises of theenergy in the universe.[33] Theoretical Inconsistencies
One of the most obvious inconsistencies in modern theory is the irreconcilabilityof general relativity and quantum physics. In a way, general relativity is the lastvestige of classical physics, that is to say, physics before the adoption of quantumprinciples. General Relativity creates a smooth geometric interpretation of space andgravity, in seeming contradiction to the bubbling chaos observed on quantum scales.It would make no sense to try and build a classical quantum theory as anything otherthan a toy model. On the other hand, attempts to create theories of quantum gravity10ave proven extremely difficult. The graviton was proposed in the 1930s as a spin-2boson conveying gravity, and one can create a unified theory for empty space withoutmuch effort. Sadly, once one begins to bend space, irreconcilable infinities begin toenter the calculations. There have been several attempts to confront these issues,most notably by string theory,[34] which posits we live in a 10- or 11-dimensionaluniverse, with the unseen dimensions existing in compact spaces.Another inconsistency is the hierarchy problem (sometimes known as the natu-ralness problem). This is the desire to achieve a working theory that explains fun-damental constant values without introducing an arbitrary fine-tuning. Fine-Tuningis the suggestion that sets of constants’ values are, without good justification, sus-piciously coincidental. It can also mean assuming a model will match experimentaldata based on the exact and arbitrary placement of constants. There are whole classesof questions about either conveniently placed or coincidentally canceling fundamen-tal constants. One example would be to ask why the gauge couplings are placedas they are. An explanation that has gathered a significant following in the physicscommunity is Supersymmetry (SUSY), which posits that all fermions have a bosonicsuperpartner and all known bosons have a fermionic superpartner. SUSY allows the 3MSM force gauge couplings to unify at high energies,[35–37] answering one of thesenaturalness problems. There are innumerable variations of SUSY under investiga-tion, given the numerous mechanisms to limit or organize the additional parameters,with the search for the so-called ”Lightest Supersymmetric Particle,” a dark mattercandidate, making up a great deal of the current generation of BSM physics searches.It is noteworthy that string theories typically incorporate SUSY, though for slightlydifferent reasons.Another problem involving fine-tuning arises in Charge-Parity Symmetry ( CP ).It involves two significant physical symmetries: charge conjugation (symmetry under11eversal of electric charge) and parity (symmetry under inversion of one spatial direc-tion). These symmetries are not absolute as the MSM contains mechanisms to allow CP -violation. Notably the weak nuclear force violates parity alone and allows for CP -violation in such phenomena as neutral kaon mixing. The problem enters whenone notices that the QCD contains a term that would allow for CP -violation, but thatexperimentally none has been found as a result of the strong force alone. Meaningthat the QCD vacuum angle, which acts as a measure of QCD CP -violation strength,needs to be extremely small, or just zero. As there is no good theory-based reasonto do this within the MSM, physicists turned to BSM theories, such as axions,[38] toexplain it. Another problem that develops out of CP -violation is matter-antimatterasymmetry. If CP were perfectly preserved all matter and antimatter would havecancelled out in the early universe. As we live in a matter-dominated universe, weare left with the question of how did matter achieve dominance. Some manner of CP -violation seems warranted, but as QCD sources remain ill understood, and weaksources seem poorly equipped for the magnitude of the problem, we are left with anunanswered mystery.Another notable theoretic question is the coincident number of families in bothquarks and leptons. Furthermore, each sector of fermions (excluding neutrinos wheredata remains limited) also exhibits a steeply increasing mass hierarchy. These similar-ities have given rise to the belief that an unexplained symmetry may be giving rise tothe patterned behaviors of the fermions. Coincidentally, as the community discovershow neutrinos break with expectations, they may also serve as the best guide to thisfamily symmetry. Explaining these coincidences also act as primary motivations forour research.There are many other problems under investigation by the physics community.We have simply tried to sketch a few of the best known and most relevant here in12opes of demonstrating the continuing need for research and experimentation to fur-ther our understanding of the universe beyond that offered by the MSM. This Section is largely based on Ref. 39 and Ref. 40
Axioms and Examples
We will begin our discussion of mathematical groups by describing their definingqualities. A group is a collection of operators and a defined operation. Combiningthose, a group containing elements: a , a , . . . , a n , must follow four mathematicalrules: • Closure
The result of operating a group element on any otherwill result in another element of the group : a × a = a • Associativity
The group operation is not dependent on order :( a × a ) × a = a × ( a × a ) • A Unit Element
The group contains an element that can operate on anyelement , including itself , and return that other element : e × a = a × e = a , e × e = e • An Inverse Element
For each group element , a there exists a unique element , ( a ) − , which yields the unit element when the two are operated together : a × ( a ) − = e As a demonstration, we can observe how these principles apply to the Cyclic Sym-metry Group, of Order 2 ( Z ). In addition to being fairly simple and relevant to ourlater discussions, this group should be well-known to readers as the symmetry ofmultiplying positive and negative one ( +1 , − ). We will begin by laying out the Z multiplication table. Closure is easy to demonstrate given the only two products on13 a ba a bb b aTable 1.2: Multiplication Table for Z the table are clearly elements of the group. As this is the symmetry demonstratedby multiplying positive and negative one, we can take the associativity of arithmeticmultiplication to hold for Z as well. Following from that, it is fairly clear that a isthe unit element, while, in this case, a and b are each their own inverse element. Lie Groups
While they are not the primary focus for much of our study, Lie groups are per-haps the best-known symmetry groups to most physicists, and warrant remarkingon. We have already shown a finite symmetry with Z , but Lie groups are differentin that they are continuous symmetries with an infinite number of elements. Someof the easiest examples of continuous symmetries would be rotations about variousaxes. SO ( ) and SO ( ) , the special orthogonal groups, are the symmetry transfor-mations for spherically symmetric 2- and 3-dimensional objects rotating about theircenter.Also of great significance in particle physics are special unitary groups, SU ( N ) .These groups (and their close relatives, the unitary groups, U ( N ) ) form the ( SU ( ) C × SU ( ) L × U ( ) Y ) basis of the MSM. This is, in part, due to their ability to representspinors in quantum mechanics. It is also important to point out that SU ( ) is thedouble cover of SO ( ) . Defining a double cover is difficult without getting overlytechnical, but can be roughly illustrated by saying that the double cover of a groupwill always have two elements representing a single element of the group it covers.14igure 1.1: A Reference Tetrahedron A and T ′ More pertinent to our research are two related groups, the Tetrahedral SymmetryGroup ( A ) and the Binary Tetrahedral Symmetry Group ( T ′ ). Like Z above, theseare both finite non-abelian (the group elements do not necessarily commute) pointgroups. A is rank 12 and, as the name implies, consists of the elements analogous tothe symmetry transformations of a tetrahedron. These transformations fall into threeconjugacy classes: • C the unit element • C a clockwise shift of vertices by 120 ◦ around the center of any of the four faces • C a counter-clockwise equivalent of C • C the three double transpositions of vertices This behavior can be summarized in a group’s character table, where the columnsare the conjugacy classes (with a number listing its size), and the rows are the Ir-reducible Representations (irreps). While there are typically several choices for anygroup’s representation, an irrep is a representation that cannot be reduced any further(for our purposes, one might think of it as the most efficient packaging of a group’s15arious potential behaviors). It is interesting to note that C and C have real charac-ters because they act as their own inverse elements, and C ’s characters list theirrepdimensions. The irreps of dimension 1 are called singlets, while the irrep of dimen-sion 3 is a triplet. The factor of ω = exp (2 πi/ is the complex cube root of unity, andhas a notable function once one observes that three repetitions of any single elementof either C or C become a trivial transformation. Following the A character tableis a second table for the Kronecker products of irreps operating on each other, andachieves a similar practical use as the Z multiplication table above. A C C C C ω ω ω ω
13 3 0 0 -1Table 1.3: Character Table of A with ω = exp(2 iπ/ A
33 3 3 3 1 + 1 + 1 + 3 + 3 Table 1.4: Kronecker Products for irreps of A Most relevant to the models we discuss later is T ′ . This group is rank 24 and isthe double cover of A (though it is interesting to note A is not a subgroup of T ′ ,[41]merely its central quotient). As mentioned above, it is difficult to give a nontechnical16efinition of a double cover, but here it can be taken as the difference between a per-mutation and an oriented permutation. While a simple illustration of a permutationmight be rearranging a set of playing cards, an oriented permutation would also in-clude the possibility that cards shift from face up to face down and back. It contains7 classes, which, in terms of irreps, translate into the three singlets and single tripletof A , as well as an additional three doublets. T ′ C C C C C C C ω ω ω ω ω ω ω ω − − − ω ω − − ω − ω ω ω − − ω − ω
03 3 0 0 3 0 0 − Table 1.5: Character Table of T ′ with ω = exp(2 iπ/ It is worth pointing out that one of the most significant qualities of T ′ is that thesinglet and triplet irreps and their multiplication remain unaltered from A . This,potentially, allows us to expand on ideas originally constructed for A without alter-ation, while allowing us a greater flexibility in model building due to the doublets.17 ′ + 3 1 + 3 1 + 3 2 + 2 + 2 + 3 1 + 3 1 + 3 2 + 2 + 2 + 3 1 + 3 1 + 3 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 1 + 1 + 3 + 3 Table 1.6: Kronecker Products for irreps of T ′ As previously hinted in Table 1.2.2 neutrino masses are not quite the same as otherknown fundamental particles. The alignment between flavor and mass eigenstates istermed, mixing. For quark mixing, flavors and masses are very closely aligned, withonly some incidental mixing between flavors. As of 1998, and the discovery of neu-trino mass, physicists realized that mixing would also occur in neutrinos. Remark-ably, neutrino mixing is not nearly as simple, the matrix which translates betweenneutrino flavor and mass eigenstates is labeled for its developers, the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS).[42, 43] The matrix allows us to see that eachflavor of neutrino exists as a superposition of three mass eigenstates, without any18tate dominating any particle, ν ν ν = U PMNS ν τ ν µ ν e . (1.1)In order to better understand and measure this matrix, we can parametrize the el-ements of the PMNS matrix. This parametrization can be constructed of three (3 × θ is the so-lar angle, θ is the reactor angle, and θ is the atmospheric angle. This parametriza-tion is constructed as follows, U PMNS = − s c c s − s e iδ c c s e − iδ − s c c s , (1.2)which yield the form, U PMNS = + s s − c c s e iδ CP − s c − c s s e iδ CP + c c − c s − s c s e iδ CP + c c − s s s e iδ CP + s c + c c + s c + s e − iδ CP , (1.3)where c and s stand for cos and sin , respectively; so c is equivalent to cos θ . Asseen above, the PMNS matrix is comprised of the three angles ( θ , θ , and θ ), aswell as a phase ( δ CP ). While each of the angles has been measured, to one degree oranother, the CP -violating phase has merely been reported at preferred values, and atpresent even the strictest experimental bounds encompass the entire feasible rangefrom π . For the purposes of this document, and for the sake of simplicity in our19lgebra, we will assume δ CP = 0 . We freely admit that this questionable assumptionmay need to be revisited in the face of future evidence to the contrary; as, indeed,current best-fit approximations place the value closer to δ CP = π .While non-zero neutrino mass has been clearly demonstrated, individual massmeasurements still elude us (though upper bounds to neutrino masses do exist).Instead, experiments have been able to determine difference between the squaresof the neutrino mass eigenstates. Consequently, we can see that for m and m , ∆ m = 0 . , and for m and m , ∆ m = 7 . × − eV . As we do not yetknow the sign for ∆ m , the ordering of the mass states remains unknown. This hasresulted in two Mass Hierarchies: The Normal Neutrino Mass Hierarchy (NH) ex-hibits a natural ordering of m < m < m , while the alternative Inverted NeutrinoMass Hierarchy (IH) places the third mass eigenstate notably lower than the othertwo, m < m < m .[21] As our observations of neutrinos have grown more detailed, there have beenmany unexpected discoveries. Perhaps, the most puzzling to the theory commu-nity is neutrino helicity. In all other observed fermions there exist two variations:a left-handed and right-handed variant. These are determined and named for thehandedness of the projection of the spin vector onto the momentum. Notably, formassless spin- particles, helicity can, then, be interchanged with chirality. However,neutrinos have only ever been observed with left-handedness, while anti-neutrinosare only observed with right-handedness. If we were to assume that neutrinos sharedthis symmetry with other Dirac fermions, then there should be four forms, not two:right- and left-handed variants of both the neutrino and anti-neutrinos. All of thisbegs the question “where are the right-handed neutrinos?”20here have been several suggested solutions to this dilemma. One of the simplest,but least satisfying, is that neutrinos are simply different. Many of our expectationsfor the behavior of neutrinos come from observing other fermions. This bias is largelydue to our proficiency at measuring particles with charges or larger masses than neu-trinos. Nonetheless, neutrinos have been confounding our expectations for decadesand this may simply be another difference from the rest of the MSM. In this case,there simply would be no right-handed neutrino and, disappointingly, no significantnew physics to be discovered.A second possibility is that of sterile neutrinos. This theory holds that right-handed neutrinos exist, but have significantly different properties than their left-handed counterparts. While left-handed neutrinos interact primarily via the weaknuclear force and via gravitation, right-handed neutrinos would interact only viagravitation and, potentially, through some mixing between left- and right-handedtypes. This idea has some backers both among theorists and experimentalists. Anumber of theories require either sterile right-handed neutrinos or a 4th generationof sterile neutrinos. Experimental evidence, by comparison, falls into two camps di-vided over the likely mass of these particles. There have been some indications atterrestrial detectors of sterile neutrinos with mass in the eV range, most notable atLSND,[44] and more recently at MiniBooNE.[45] The other experimental evidencefor sterile neutrinos is largely astrophysical,[46] and suggest a keV mass scale wouldanswer questions about primordial element abundances.The third, and to our mind, most convincing solution to the mystery of the missingright-handed neutrinos are the so-called Majorana neutrinos. Neutrinos aside, allknown fermions fall into the classification of Dirac particles, that is to say they obeythe Dirac Equation,[47] ( i δ − m ) ψ = 0 , (1.4)21nd are not their own anti-particles (by comparison, a photon, a boson, is its own an-tiparticle). Another equation and classification of particle is that of Majorana particlesfrom 1937’s Ref. 48: − 6 δψ + mψ ∗ = 0 , (1.5)where fermions will act as their own anti-particles. While there are no confirmedexamples, many in the community believe neutrinos to be attractive candidates forthis designation. Under many of these proposed models, including ours, a set ofundiscovered, right-handed Majorana neutrinos also exist. Hints to these particles’existence remain a high priority of neutrino detector searches.Given the significant interest on the part of the community, an experimental de-sign was proposed to test whether neutrinos are, indeed, Majorana particles titled,Neutrinoless Double-Beta Decay.[49] In typical beta decay a single neutrino is emit-ted; thus, in the most common variant of double-beta decay, one would observe twoemitted neutrinos. However, if neutrinos are Majorana particles, then they are theirown antiparticle and can coannihilate. This would lead to the rare but statisticallysignificant possibility of neutrinos from two proximate beta decays annihilating withone another, giving the test its name. Neutrinos in this case, as in most cases, re-main difficult to detect, necessitating expensive materials and long running times inorder to build up the statistically necessary evidence. This is further complicated bythe enduring vagaries of neutrino behavior including their mass hierarchy and massscale. In the first five years following the determination that neutrinos had mass andthat their mixing, encapsulated in the PMNS matrix, was nontrivial, there were manyattempts to introduce a flavor symmetry explaining quark and lepton mixing. Many22f these introduced so-called texture zeros. These would be elements of the PMNSmatrix set to zero by the model. Introducing one, or more, of these zeros created astructural stability that benefited many models by making it simpler to explain. Overthe years, there were even several works that attempted to categorize the likelihoodof any of each element being a texture zero and their potential to explain the behav-iors of flavors and families.[50–52]In 2002, a paper postulated a form for the PMNS matrix. This practice, whilenot unusual following on earlier attempts such as the Bi-Maximal and Tri-Maximalmodels,[53,54] proved fairly accurate and was summarily dubbed Tribimaximal Mix-ing (TBM).[55] In it, θ = 0 ◦ , θ = 45 ◦ , and θ = sin − ( √ ) ≃ . ◦ , (1.6)leading to a PMNS matrix of the form: U TBM = − q − q q q q q q − q . (1.7)A noted benefit to this depiction is the rational root form that leads to an ease of in-corporation into theoretical models, and in particular finite symmetry models (giventheir Kronecker products). Indeed, many attempts to utilize various finite or flavorsymmetries since the initial TBM proposal have attempted to show that they are ableto incorporate this symmetry structure.Although theories have successfully demonstrated that there are numerous po-tential paths that all arrive at the TBM form, it should be noted that this form wasmerely a guess at the actual values of PMNS elements. As the years have progressed,we have indeed seen that the initial TBM form may not be correct and that either23 new form should be adopted or additional mechanisms are needed to shift TBMvalues in line with experimental data. µ - τ Symmetry
Another notable feature of TBM is that it exhibits µ - τ symmetry.[56] This can beeasily demonstrated by examining the first and second columns of U and notingthat they are identical. Physically this implies that ν µ and ν τ have identical superpo-sitions of the three neutrino mass eigenstates. Also of note, is that slight breaking ofthe µ - τ symmetry can lead to a similar perturbation as that seen in Chapter 2. T ′ Model
This Section is largely based on the work of Ref. 57 and Ref. 58 A and the Lepton Sector Model Characteristics
Having concluded our historical background, we shall proceed to lie out the mod-els that form the basis of our work. In this chapter we show two derivations of theMajorana mass matrix, M ν , and the conclusions found from relating the two.We shall start by crafting an initial A model comprised of ( A × Z ),[59–69] where24he various particles are assigned to irreps as, ν τ τ − L ν µ µ − L ν e e − L L L (3 , +1) , τ − R (1 , − µ − R (1 , − e − R (1 , − , and N (1)R (1 , +1) N (2)R (1 , +1) N (3)R (1 , +1) . (1.8)As Eq. (1.8) shows, this model, as with all A models, will only include the leptons(including right-handed Majorana neutrinos). Refs. 64–66 have shown, A is notcapable of replicating quark mixing − a mixing typically encapsulated in the Cabbibo-Kobayashi-Maskawa matrix (CKM).[70, 71] In parentheses next to every particle arethe specifications for how that particle rotates, first under A , then under Z . Inthis setup, we have placed the left-handed leptons in a triplet, and the right-handedleptons in various singlets.From here we can proceed to the formation of a Lagrangian. This crucial stepmust be carefully considered, for while we hope to discover new physics in the courseof our investigation, we must tread lightly in order to avoid blatantly contradictinghistoric experimental particle physics data. This, oddly, leads to a middle groundwhere some things are new, but not too many. We will also include the constraint ofusing only renormalizable couplings, and though the A model contains an anomaly,this is subsequently cancelled by the T ′ model discussed later.The Lagrangian for this model is then, L Y = 12 M N (1)R N (1)R + M N (2)R N (3)R ( Y (cid:16) L L N (1)R H (cid:17) + Y (cid:16) L L N (2)R H (cid:17) + Y (cid:16) L L N (3)R H (cid:17) + Y τ ( L L τ R H ′ ) + Y µ ( L L µ R H ′ ) + Y e ( L L e R H ′ ) ) + h . c . . (1.9)In this form it is clear that that many of the classic features of the MSM remain. Wehave also added 2 triplet Higgs scalars (6 doublets under SU ( ) L , 2 triplets under A ) where needed of H (3 , +1) and H ′ (3 , − . These additions are needed in order toensure that each Lagrangian term rotates as a singlet under A . By referencing thisLagrangian, the assignments in Eq. (1.8), and the Kronecker product table in Sec. 1.3,one can see this approach is consistently applied. The factors of have been addedin order to mitigate the identical hermitian conjugates of Majorana mass terms.Our model maintains that the masses of the charged leptons ( e, µ, τ ) emerge fromthe Vacuum Expectation Values (VEVs) of < H ′ > = ( m τ Y τ , m µ Y µ , m e Y e ) = ( M τ , M µ , M e ) . (1.10)If, largely for the sake of simplicity, we then choose a flavor basis where the chargedleptons act as mass eigenstates, we can then separate, at leading order, charged leptonand neutrino masses. We also note that the N ( i )R masses break L τ × L µ × L e symme-try, but will alter the charged lepton masses at the one-loop level only by a factor ∝ Y m i /M R .One of the most notable features of this model are the Majorana neutrinos, whosebenefits were detailed in Sec. 1.4. Given this, we must now further specify the form26f the neutrino mass matrices. First the Majorana mass matrix in typical form, M R = M M M . (1.11)Next is the Dirac mass matrix formed from the Lagrangian Yukawa couplings and ageneric set of VEVs from the other Higgs triplet, < H > = ( V , V , V ) , (1.12) M Dν = Y V Y V Y V Y V Y V Y V Y V Y V Y V . (1.13)The Majorana mass matrix, M ν , is then given by M ν = M Dν M − ( M Dν ) T . (1.14)Defining x ≡ Y /M and x ≡ Y Y /M yields the symmetric form of, M ν = x V + 2 x V V x V V + x ( V + V V ) x V V + x ( V + V V ) x V + 2 x V V x V V + x ( V + V V ) x V + 2 x V V . (1.15) Incorporating TBM and Majorana Neutrinos
We shall now attempt to approach the same matrix from a different direction anddetermine what limits can be placed as a result of assuming the symmetries of TBM.27s seen in Eq. (1.7), this proposed mixing structure takes the form, U TBM = − q − q q q q q q − q , (1.16)and acts to delineate the relation between flavor and mass eigenstates, ν ν ν = U TBM ν τ ν µ ν e . (1.17)Assuming no CP -violation, the Majorana matrix M ν is real and symmetric, andhas the general form of M ν = A B CB D EC E F , (1.18)which, in general can be diagonalized by the PMNS matrix. In order to incorporatesymmetries present in TBM, specifically, one can use U TBM to diagonalize: M diag = m m
00 0 m = U TBM M ν U T TBM . (1.19)Substituting Eq. (1.16) into Eq. (1.19) and solving for M ν leads to a further reduction28f Eq. (1.18) to the three real parameters A, B, C : M ν = A B CB A CC C A + B − C , (1.20)with eigenvalues, m = ( A + B − C ) ,m = ( A + B + C ) ,m = ( A − B ) , (1.21)whose individual assignments can be found by a substitution of Eq. (1.20) back intoEq. (1.19) and multiplying out the right-hand side.Now, by relating the two forms of M ν , Eq. (1.15) and Eq. (1.20), we find threeequations, x V + 2 x V V = x V + 2 x V V , (1.22) x V V + x ( V + V V ) = x V V + x ( V + V V ) , (1.23) x ( V + V V − V V ) + x (2 V V + V + V V − V − V V ) = x V + 2 x V V , (1.24)corresponding to A , C , and A + B − C , respectively.We find no solutions of Eqs. (1.22, 1.23, 1.24) with any of x , x , V , V , V van-ishing. It is straightforward to note that Eq. (1.22) and Eq. (1.23) can only both besatisfied if V = V . Solving Eq. (1.24) further requires (2 V + V )( V − V ) = 0 , sinceit can be shown that x = x is not possible for any hierarchy consistent with experi-ment. In this A model, therefore, only two VEVs of H give TBM.29he first is, < H > = ( V, V, V ) , (1.25)which is studied in Ref. 72. Now by equating Eq. (1.15) and Eq. (1.20) one can findrelations for A , B , and C , from there one can use Eq. (1.21) and the relative values ofEq. (1.25) to find expressions for the neutrino mass eigenstates, A = V ( x + 2 x ) ,B = V ( x + 2 x ) ,C = V ( x + 2 x ) , m = 0 ,m = V (3 x + 6 x ) ,m = 0 . (1.26)Clearly this implies m ≫ m = m = 0 , an inappropriate hierarchy being neitherNH or IH, thus Eq. (1.25) is an unacceptable VEV for < H > in our model.The only other VEV for A is therefore, < H > = ( V, − V, V ) . (1.27)As before, the forms of A , B , and C as well as the masses can be found from thecombination of Eqs. (1.15, 1.20, 1.21) with the relative values of Eq. (1.27), A = V ( x − x ) ,B = V ( x + 5 x ) ,C = V ( − x − x ) , m = x V (6 + 3 y ) ,m = 0 ,m = x V ( − y ) , (1.28)where y = x /x . If we continue in our assumption that m ≃ m (an appealing This VEV, < H > ∝ (1 , , , can be transformed to < H > ∝ (0 , , by an A transforma-tion. The literature distinguishes these designations as the Ma-Rajaskaran and Altarelli-Feruglio basesrespectively. Because < H > ∝ (1 , , could be made consistent with the neutrino masses in most previous A models, due to additional parameters, the alternative of < H > ∝ ( − , , seems not to havebeen previously studied. ∆ m invalidates m < m ), then y is constrained to avalue y = − .We are then left with a strong model preference for the NH, with m ≫ m ≃ m ,making Eq. (1.27) our only feasible VEV for < H > . T ′ Model
History
In 1994, Ref. 73 investigated the simple non-abelian discrete groups up to order31 (they stop before order 32 because of the large number of additional groups atevery power of 2) as potential family symmetries. They began by laying out a setof model-building guidelines, and proposed model assignments for each symmetrygroup. One such group, labeled at the time the Double Tetrahedral Group, was de-tailed as a subset of SU ( ) . There, they detailed a particle assignment set for the sixMSM quarks and the leptons known at the time: tb L cs L ud L t R c R u R b R s R d R ν τ τ − L ν µ µ − L ν e e − L τ − R µ − R e − R (1.29)This particle assignment had the benefits of containing all of the, then known,fundamental fermions. In addition, it labeled the top quark mass as a singlet of the Previously, other A models have shown more flexible in choosing between IH and NH, they alsoinclude more parameters, and, as a result, incur diminished predictivity. m t = 173 . .[21]This model is not without problems in light of later discoveries and data, indeedneutrino masses and non-trivial mixing came as a surprise to much of the community.It would be over a decade before this basic model was fully overhauled to comportwith our modern understanding. T ′ as Flavor Symmetry Following their articulation of a minimal A model, as detailed in earlier, the au-thors of Ref. 57 sought to expand their treatment to include quark mixing by con-verting to the symmetry now titled, the Binary Tetrahedral Group ( T ′ ).[74–77] Asdetailed in 1.3 the connection between A and T ′ is quite special. Since they sharethe singlet and triplet elements and multiplications, the prior A minimal model caneasily be converted to T ′ . Thus, the assignments of the leptons will remain the samein the new model, as will the treatment of the TBM angles and Higgs VEVs. Now,though, T ′ affords the advantage of doublet symmetry elements, which we can useto accommodate the, mostly diagonal, CKM matrix.As before, it is best to keep the top quark in its own singlet as a way of motivatinga higher mass. So, the lighter two families will be placed in doublets: one doublet forboth left handed families labeled Q L , one doublet for the two lighter down-type right-handed quarks labeled S R , and one doublet for the two lighter up-type right-handedquarks labeled C R . The left-handed third family will be in a singlet Q L , while theright-handed third family will be in two self-named singlets. These new assignments32re summarized below, tb L Q L (1 , +1) cs L ud L Q L (2 , +1) , and t R (1 , +1) b R (1 , − c R u R C R (2 , − s R d R S R (2 , +1) . (1.30)As before, the parenthetical numbers state the element assignment under the maingroup algebra first ( T ′ this time) followed by the auxiliary Z .Now, as we prepare to state a Lagrangian for both lepton and quark sectors, weshould note it remains the intent to craft a minimal model using finite symmetry.When the Higgs were chosen for the leptons, they were limited by the particle as-signments and the elements of A . Now that we have progressed to T ′ , doublets areavailable as well. However, in this Minimal Renormalizable T ′ Model (MR T ′ M) onlyHiggs singlets and triplets (under T ′ , they all remain doublets under SU ( ) L ) will beused. With these stipulations in place, we can write down the MR T ′ M Lagrangian: L Y = 12 M N (1)R N (1)R + M N (2)R N (3)R + Y e L L e R H ′ + Y µ L L µ R H ′ + Y τ L L τ R H ′ + Y L L N (1)R H + Y L L N (2)R H + Y L L N (3)R H + Y t ( Q L t R H ) + Y b ( Q L b R H )+ Y C ( Q L C R H ′ ) + Y S ( Q L S R H ) + h.c. . (1.31)33his equation, naturally, now includes several new terms. To complement and com-plete these terms, two new Higgs scalars are required, H (1 , +1) and H (1 , − ,with VEVs of: < H > = m t /Y t , < H > = m b /Y b , (1.32)to provide the masses of the third family. Taken together this allows for the estab-lished quark mass hierarchy of m t ≫ m b > m c,s,d,u .Also notable is the fact that the quark and lepton sectors reuse the same two Higgstriplets. This can serve as the basis for a connection between the families and a uni-fying foundation among all fermions. A note on formalism: We will be parametrizing the CKM matrix in an identical way toour previously described depiction of the PMNS matrix (Sec. 1.4) as is customary given theparametrization we use originated with CKM, and distinguish the angles of the two by using Θ ij for CKM, and θ ij for PMNS. Additionally, we will distinguish between the two CP -violating phases with δ CP for PMNS and δ KM for CKM. Due to our choice to avoid T ′ doublet Higgs terms in this MR T ′ M, an assessmentof the CKM matrix will need to be reduced down to the ( × ) quark mixing matrices,which assume Θ = Θ = 0 . While this is demonstrably inaccurate, it remainsa decent approximation given that both angles are quite small, Θ = 0 . ◦ and Θ = 2 . ◦ .The two remaining nontrivial ( × ) matrices for ( c, u ) and ( s, d ) will hereafter34e denoted U ′ and D ′ , respectively, and calculated using the T ′ complex Clebsch-Gordan coefficients illustrated in Ref. 78. If we divide U ′ by Y C we find, U ≡ (cid:16) Y C (cid:17) U ′ = q ωM τ √ M e − √ ωM e q M µ , where ω = e iπ/ . (1.33)If we take the additional step of setting the electron mass, M e , to zero, U becomesimmediately diagonal and leaves m u , m c , m µ , and m τ free.Next we shall take a look at the ( × ) Cabibbo Matrix. In its general parametrizedform it appears as P ≡ cos Θ − sin Θ sin Θ cos Θ . (1.34)We can use this form to diagonalize the hermetian square of D ′ , after dividing by Y S , D ≡ (cid:18) V Y S (cid:19) D ′ = √ q ω q − √ ω , (1.35) D ≡ DD † = (cid:18) (cid:19) −√ −√ . (1.36)We now have developed the tools needed to solve m d m s = P T D P . (1.37)for the two remaining unknowns, Θ and ( m d /m s ) .The first result, that of the Cabibbo Angle, yields:35 an 2Θ = √ ! , (1.38)which converts to a decimal prediction of Θ = 12 . ◦ (by comparison its experimen-tal value is Θ = 13 . ◦ ± . ◦ ). While this is σ away from the measured value, thisis largely due disparity in precision between quark and neutrino data, and it remainsan adequate first-order prediction. As we shall see, later attempts to adjust the theoryachieve better agreement.As for ( m d /m s ) , the solution to Eq. 1.37 yields a predicted value of ≈ . , com-pared with experimental findings of ( m d /m s ) ⋍ . . While admittedly a poor initialguess, one might suppose that this is due to the assumption of Θ = Θ = 0 , andthat incorporation of mixing between ( d, s ) and b would help matters.This model proved to be an important step, both in demonstrating the viabil-ity of T ′ as a suitable basis for neutrino mixing models, and its superiority over A given it has the ability to address the quark sector. Having achieved a semi-reliableframework to connect the mixing angles of quarks and leptons, we can begin to askquestions of the underlying hypotheses. The A model was constructed to handleneutrinos, and was incapable of addressing quarks. The MR T ′ M model made strongassumptions about the neutrino mixing angles and used them to make moderatelyclose predictions of the well-measured quark angles. As Ch. 2 will show, it bearssome investigation to see if we can use quark mixing to learn about neutrinos.36 hapter 2 T ′ Model Perturbations and NeutrinoMixing
This Chapter is largely based on the work of Ref. 79
So far, the T ′ model has shown itself capable of creating a successful mechanismto link the mixing angles of quarks and neutrinos. Yet, thus far, we have assumedthe exact accuracy of the TBM angles, and their underlying symmetry, in order todevelop a prediction for quark mixing. This was largely due to the historical devel-opment of the theory, from A (which can only function for leptons), to T ′ (which canaccommodate both). But given the fact that T ′ can accommodate both and the mea-sured value of the Cabibbo angle is both highly precise, and not easily reducible to arational form, it may make more sense to assume the measured value of the Cabibboangle and use our framework to develop a prediction for neutrino mixing. Whenthis model was initially developed the neutrino mixing angles had as much as ◦ of uncertainty around the TBM values. Consequently, we will keep the mechanismdiscussed in 1.5, but introduce perturbations.e will begin by redefining the neutrino mixing angles as, θ ij = ( θ ij ) TBM + ǫ k , (2.1)(where ǫ is used for θ , etc.), and proceed to perturb around Eq. (1.38).First, we recall a few salient points about the model in Sec. 1.5 based on A sym-metry. The only important scalar for the present analysis is the triplet H (3 , +1) whose vacuum expectation value in Sec. 1.5 was taken as < H > = ( V , V , V ) = V (1 , − , , (2.2)which linked to the TBM form seen in Eq. (1.7). We shall consider the perturbation < H > = ( V ′ , V ′ , V ′ ) = V ′ (1 , − b, a ) , (2.3)where | a | , | b | ≪ . We shall first consider the calculation of a perturbation around the earlier workin Sec. 1.5 by using Eq. (2.3) in place of Eq. (2.2). The down-type quark ( × ) massmatrix for the first two families ( s, d ) is perturbed to D ≡ (cid:18) V ′ Y S (cid:19) D ′ = √ (2 − b ) q ω q (1 + a ) − √ ω , (2.4)38here, again, ω = exp 2 iπ/ . The hermitian square D ≡ DD † is, to first order in a and b , D ≡ DD † ≃ (cid:18) (cid:19) − b √ − a + b ) √ − a + b ) 3 + 4 a . (2.5)The eigenvalues of Eq. (2.5) satisfy the quadratic equation (9 − b − λ e )(3 + 4 a − λ e ) − − a − b ) = 0 , (2.6)with solutions of λ e ± = (6 ± √
11) + 2 a (cid:18) ∓ √ (cid:19) − b (cid:18) ± √ (cid:19) . (2.7)An eigenvector ( α, β ) has components satisfying (cid:18) βα (cid:19) = − √ √ ! (cid:20) − a √
11 + b √ (cid:21) , (2.8)whose normalization N ( α, β ) satisfies N − = 1 + β /α , (2.9)from which the Cabibbo angle sin Θ = N β/α is sin Θ = s(cid:18) − √ (cid:19) − √ a − b ) ! . (2.10)From this one finds at leading order cos 2Θ ≃ (cid:18) √ (cid:19) (cid:18) a − b ) (cid:19) , (2.11)39nd sin 2Θ ≃ √ √ ! (cid:18) −
311 ( a − b ) (cid:19) , (2.12)where tan 2Θ ≃ ( √ / (cid:18) −
13 ( a − b ) (cid:19) , (2.13)which, of course, reduces back to TBM values (and the Sec. 1.5 Cabibbo prediction)for a = b = 0 .Next, we will relate the ǫ i neutrino angle perturbations of Eq. (2.1) to the vacuumalignment perturbations a and b of Eq. (2.3).As before we will start with the basic TBM form as seen in Eq. (1.7, 1.16), U TBM = − q − q q q q q q − q . (2.14)using the neutrino mixing angle values given in Eq. (1.6).By utilizing the small-angle approximations and the added-angle trigonometricidentities, one can fashion a practical form of Eq. (2.1), which at first order comes to, • s ≃ q (1 + √ ǫ ) • c ≃ q (1 − ǫ / √ • s ≃ q (1 + ǫ ) • c ≃ q (1 − ǫ ) • s ≃ ǫ • c ≃ Consequently, one may write U ≃ U TBM + δU = U TBM + δU ǫ + δU ǫ + δU ǫ , (2.15)40here δU = − q + q q − q − q − q , (2.16) δU = − q + q − q + q
00 0 1 , (2.17) δU = − q − q − q − q − q + q , (2.18)combine to form δU = 1 √ − ǫ − √ ǫ + ǫ ) ǫ + √ ǫ − ǫ ) −√ ǫ √ ǫ − ( ǫ + ǫ ) −√ ǫ + ( ǫ − ǫ ) 2 ǫ −√ ǫ −√ ǫ √ ǫ . (2.19)By inserting Eq. (2.14) into Eq. (1.19), we arrive at: ( M ν ) TBM = (cid:18) (cid:19) m + 2 m + 3 m m + 2 m − m − m + 2 m m + 2 m + 3 m − m + 2 m m + 2 m . (2.20)Analysis of Eq. (1.19) leads to the full perturbation of, δ ( M ν ) diag = δm δm
00 0 δm = δU ( M ν ) TBM U T TBM U TBM δM ν U T TBM + U TBM ( M ν ) TBM δU T , (2.21)in which U TBM is known from Eq. (2.14) and δU from Eq. (2.19). Since the derivation of ( M ν ) TBM contains further multiplications by the unitary matrix U TBM , we can furthersimplify by eliminating factors of U TBM U T TBM = U T TBM U TBM = . This abbreviatedform is simply: δm δm
00 0 δm = δU U T TBM M diag + U TBM δM ν U T TBM + M diag U TBM δU T . (2.22)To compute δM ν in Eq. (2.22) we start from Eq. (1.15), ( M ν ) TBM = x V + 2 x V V x V V + x ( V + V V ) x V V + x ( V + V V ) x V + 2 x V V x V V + x ( V + V V ) x V + 2 x V V . (2.23)where < H > = ( V , V , V ) , x = Y /M and x = Y Y /M . These variables, in-cluding Yukawa couplings and right-handed neutrino masses, all remain empiricallyunknown. As all terms include either x or x , Eq. (2.23) can be further simplifiedby combining these factors into y = x /x , leaving us to obtain predictions by deter-mining this unknown.We shall now introduce our perturbation of the vacuum alignment, Eq. (2.3), to42q. (2.23), again at first-order only in a and b , to find, δM ν = x ( V ′ ) − a + b ) y a + ( a − b ) y b + (2 a + b ) y a + by ) ( − a + b )(1 + y ) − b + 2 ay . (2.24)By inserting this δM ν into Eq. (2.22) we obtain six equations from the ( × ) sym-metric matrix. In the δm of (I) - (III) a common (but unpredicted) normalization factorhas been omitted. • (I) δm = (2 + y )( a − b ) • (II) δm = 0 • (III) δm = − y ( a − b ) • (IV) ǫ = √ ǫ • (V) a = ǫ m − m y • (VI) ( a + b ) = (cid:16) √ m − m y − (cid:17) ǫ Result (IV) is significant and contains two interpretations. The first is as written,implying that a θ > results in θ > ◦ . The second interpretation is to redefine theangle θ with the transformation θ ⇒ − θ (note that until now, we had assumed θ = 0 , meaning this transformation has no phenomenological affect), which leadsto a θ in the first quadrant. Summarizing these possibilities is analogous to statingthat θ = |√ | (cid:16) π − θ (cid:17) , (2.25)43hich, interestingly, links any non-zero value for θ to the departure of the atmo-spheric neutrino mixing angle θ from maximal mixing at θ = π/ . This is ourmost definite prediction from T ′ , and is independent of phenomenological input.To arrive at further T ′ predictions for the neutrino mixings, θ and θ , we shallrequire additional input.The equation (I) through (III) must be combined with the zeroth-order values m = 3( y + 2) ,m = 0 ,m = − y , ⇛ m = 3( y + 2) + ( a − b )(2 + y ) ,m = 0 ,m = − y − y ( a − b ) . (2.26)It is notable that m = 0 remains even at first order. This arises from the zerostructures in the terms of Eq. (2.21). They are δU ( M ν ) TBM U T TBM , (2.27) U TBM δM ν U T TBM , (2.28) U TBM ( M ν ) TBM δU T
00 0 00 . (2.29)In order to satisfy the criterion that m ≤ m , there are two possibilities, neither ofwhich is particularly satisfying.The first is setting a = − b , though upon closer examination, this fails to meet44he criteria that a, b ≪ , and therefore we discard it.Second is the phenomenological input, originating in Sec. 1.5, that set y = − .This, combined with m ∽ m , gives ( a + b ) = 0 and Eq. (2.13) becomes simply tan 2Θ = (cid:16) √ (cid:17) (cid:0) − a (cid:1) with a = ǫ m , ∆ m ≈ p ∆ m . (2.30)Eq. (2.30) allows us to approximate the size of the perturbation from the experi-mental value reported in Ref. 80, (Θ ) experiment = 13 . ± . ◦ , to identify the limits . < ǫ < . , (2.31)and by using Eq. (2.25), . < ǫ < . . (2.32)The values of Eqs. (2.31, 2.32) lead directly to predictions for the neutrino mixingangles. Substitution of Eqs. (2.31, 2.32) into Eq. (2.1) gives . ◦ ≤ θ ≤ . ◦ , (2.33)and . ◦ ≤ θ ≤ . ◦ , (2.34)It is notable that this creates a theoretically motivated deviation from TBM values,if far larger than experiments indicate. We are inclined to believe that when thisMR T ′ M is fully expanded to account for full 3-family quark mixing, these projectionswill better accommodate experimental data.On the topic of quark and lepton masses, too, we are disappointed with the lackof progress. Although we understand why m t ≫ m b > m c,s,d,u for quarks and why m ≫ m = m for neutrinos, when we look more closely at the details we find that45asses are not quantitatively explained. It is not clear to us whether this will be cor-rected in the ( T ′ × Z ) model by higher order corrections, or by adding T ′ doubletVEVs. In the present work, we take the view that our model has made reliable pre-dictions about mixing angles even when details of the mass spectra are incomplete. This Section is largely based on the work of Ref. 81
Recalling the values of the angles θ and θ listed in the 2010 Review of ParticlePhysics,[82] as they help to illustrate the recent leap in experimental precision forPMNS parameters, . ◦ . θ ≤ . ◦ , . ◦ ≤ θ . . ◦ , (2.35)consistent with vanishing θ and maximal θ .Up to 2011, neutrino mixing angles were all empirically consistent with TBMvalues. However, as the experimental precision has now improved due to recentdata from T2K,[83–88] MINOS,[89–95] Double Chooz,[96–100] Daya Bay,[101, 102]and RENO,[103, 104] this situation has changed dramatically. This is clearly seen inthe global fits of Refs. 105–107; of these we shall primarily use Fogli et al .,[105] butwill also include a limited analysis of Tortola et al .,[106] given its preference for a θ > ◦ . These five remarkable experiments have provided us with a rich new per-spective on the mixing angles. From flavor symmetry, it is then possible to predictquantitatively how departures from the TBM values are related.In this section, we intend to thoroughly investigate the ramifications of the mostpowerful prediction made by the T ′ model, that deviations from the TBM matrix inEq. (2.14) in θ and θ are correlated and independent of the solar neutrino mixing46ngle θ . To do this we shall consider only the projection on the two-dimensional θ - θ plane of the three-dimensional θ - θ - θ space. As a reminder, these pertur-bations stem from the small angle approximation, requiring sin α ∼ α for θ and ( π − θ ) . The data from KamLAND, LBL accelerators (like T2K and MINOS), solar ex-periments, SBL accelerators (such as Double Chooz, Daya Bay, and RENO), andSuper-Kamiokande, as combined in Ref. 105 currently indicate (accounting for CP -violation) sin θ = 0 . +0 . − . with 95% C . L . , (2.36)for a NH, as favored by T ′ .As noted in Sec. 2.3 our perturbed model leads to the linear relationship, , θ = | η | (cid:16) π − θ (cid:17) , (2.37)with a sharp prediction, from Eq. (2.25), of η = √ . Thus resulting in θ = |√ | (cid:16) π − θ (cid:17) . (2.38)Several years ago Super-Kamiokande showed θ > . ◦ ,[111] and current singlemeasurements place it at θ ≃ . ◦ .[112] Once combined in a global fit of ν oscilla-tion, Ref. 105 states the best fit of θ = 38 . ◦ , tantalizingly close to our central valueof θ = 38 . ◦ (or, alternatively, in Ref. 106 a best fit of θ = 51 . ◦ , compared with our This is a < approximation for θ and ( π − θ ) since both angles are less than α = 12 ◦ = 0 . radians with sin α = 0 . . A is also capable of producing Eq. (2.37) with η = √ , though we give preference in this analysisto T ′ for its capacity to explain CKM mixing. It is notable that Eq. (2.37) with η ≃ √ appears en passant in Ref. 108; see also Ref. 109 whichimplies that η ∼ . Another, model-independent correlation was developed in Ref. 110, including thethree PMNS mixing angles and the CP -violating phase. σ . The sameassessment excludes the orange-shaded region at σ . The best fit value for ( θ , θ )is indicated by the star at ( . ◦ , . ◦ ). Extreme values of the linear correlation coef-ficient, η , are indicated by dashed lines at η = 1 . and η = 3 . , while our predictedcorrelation of η = √ is indicated by the solid dark blue line. The combination of ourcorrelation and the experimental value of θ result in a prediction of θ = 38 . , aclose match to its shown best fit value.value of θ = 51 . ◦ ).As shown in Fig. 2.1, the recent experimental data,[105] combined with theory,suggest that ( θ , θ ) are respectively closer to ( . ◦ , . ◦ ) than to ( . ◦ , . ◦ ). Beforethe surge of new data η was unconstrained, ≤ η ≤ ∞ ; with the current global fitdata, we find . ≤ η ≤ . .Fig. 2.2, using a different global analysis created from an alternate weighting ofmuch of the same data,[106] suggests that θ does not lie in the first octant (i.e. that θ > ◦ ). Because our derivation of Eq. (2.37) is not sign dependent, we can alter48igure 2.2: This figure shows a second global analysis by Ref. 106, including many ofthe same sources. The red-shaded region remains excluded at σ , with σ exclusionfor orange. The difference in this figure is the possibility that θ > ◦ . Since manyexperiments are only sensitive to the sin θ , thus leaving the two octants degener-ate, there have been some indications that the assumption θ < ◦ is untrue. As ithappens, our prediction does not distinguish between the octants and gives a best fitat θ = 9 . ◦ and θ = 51 . ◦ , extremely close to the experimental best fit at θ = 51 . ◦ .In this case, it makes more sense to frame η as /η to avoid running through ∞ . Thus,the allowed range for this global fit exist from /η = 1 . to /η = − . .our projection of θ accordingly. Based on this global fit and Eq. (2.38), ( θ , θ )are approximately ( . ◦ , . ◦ ). Since this analysis still allows θ = 45 ◦ , albeit at σ exclusion, which remains analogous to an η of ∞ , it makes more sense, for our secondanalysis, to state limits on /η . As such, /η is here constrained to − . ≤ /η ≤ . .This is in sharp contrast to the previously widespread acceptance of a maximal θ = π/ , which fitted so well with vanishing θ = 0 as in TBM.As the measurement of θ sharpens experimentally, so will our prediction for θ from Eq. (2.38), and an accurate measurement of the atmospheric neutrino mixing’s49eparture from a maximum value will provide an interesting test of Binary Tetrahe-dral Flavor Symmetry.While several paper have suggested links between these angles, ours is singularin tying the cause of this exact correlation to the Cabibbo angle’s deviation from therational form of Eq. (1.38). This suggests to us that the T ′ flavor symmetry, introducedin Ref. 73, should be taken quite seriously. As errors in θ and θ diminish evenfurther, it will be interesting to see how the T ′ prediction of Eq. (2.38) perseveres, asit would inspire further investigation into other mixing angles for quarks and leptons.This, in turn, may show that T ′ , first mentioned in physics as an example of an SU ( ) subgroup,[113] is actually a useful approximate symmetry in the physical applicationof quark and lepton flavors. 50 hapter 3An Expanded T ′ Model and QuarkMixing
This Chapter is largely based on the work of Ref. 114
In the present chapter, we will examine the addition of T ′ Higgs doublet scalars.As anticipated in Sec. 1.5, this allows more possibilities of T ′ symmetry breaking andpermits non-zero values for Θ , Θ and δ KM . We present an explicit ( T ′ × Z ) modeland investigate the CKM angles.Note that we continue to focus on a renormalizable model with few, if any, freeparameters, and prioritize the mixing matrixes rather than the masses, as the formerare more likely to have a geometrical interpretation without adding a surfeit of extraparameters, sadly leaving the masses unpredicted. With that said, the placementof the top quark in a singlet does allow it a much heavier mass in accordance withexperiments.Thus, we shall proceed to develop the Next-to-Minimal Renormalizable T ′ Model(NMR T ′ M). To do this we will introduce one T ′ doublet scalar in an explicit model.his addition, then, allows non-vanishing Θ and Θ to be induced by symmetrybreaking.The possible choices under ( T ′ × Z ) for the new scalar field are: A H (2 , +1) , (3.1) B H ′ (2 , − , (3.2) C H ′ (2 , − , (3.3) D H (2 , +1) , (3.4)allowing the following Yukawa couplings, respectively, A Y Qt Q L t R H + h.c. , (3.5) B Y Qb Q L b R H ′ + h.c. , (3.6) C Y QC Q L C R H ′ + h.c. , (3.7) D Y QS Q L S R H + h.c. . (3.8)This leaves us to choose between multiple candidates for the NMR T ′ M. Largelyto ensure computational simplicity, we opt for the single additional term, D , inspiredby the Chen-Mahanthappa mechanism for CP -violation.[115] We shall choose to52eep Y QS real, allowing CP -violation to arise from the imaginary part of T ′ Clebsch-Gordan coefficients.The VEV for H is taken with the alignment < H > = V (1 , . (3.9) T ′ M (D) Predictions
From the Yukawa term, D , and the vacuum alignment, we can derive the down-quark mass matrix: D = M b √ Y QS V √ Y QS V √ Y S V q ω Y S V q Y S V − √ ω Y S V , (3.10)where ω = e iπ/ , and M b = Y b V .The hermitian squared mass matrix D ≡ DD † for the down-type quarks is then D = M ′ b √ Y S Y QS V V (1 − √ ω ) √ Y S Y QS V V ( ω + √ √ Y S Y QS V V (1 − √ ω ) 3( Y S V ) − √ ( Y S V ) √ Y S Y QS V V ( ω + √ − √ ( Y S V ) ( Y S V ) , (3.11)where M ′ b = M b + ( Y QS V ) .Note that in this model the mass matrix for the up-type quarks is diagonal, sothe CKM mixing matrix arises purely from diagonalization of D in Eq. (3.11). Thepresence of the complex T ′ Clebsch-Gordan in Eq. (3.11) acting as the source of the CP -violating phase, δ KM (Chen-Mahanthappa mechanism). This uses the approximation that the electron mass is m e = 0 ; c.f. Ref. 58.
53n Eq. (3.11) the ( × ) sub-matrix for the first two families coincides with theresult discussed in Sec. 1.5, thereby preserving the successful Cabibbo Angle formula tan 2Θ = ( √ / .For m b the experimental value is . ,[21] although the CKM angles andphase do not depend on this overall normalization.Actually our results depend only on assuming that the ratio ( Y QS V /Y S V ) ≪ ismuch smaller than one.Defining D ′ = 3 D / ( Y S V ) , (3.12)we find D ′ = D ′ Ae − iψ Aξe − iψ Ae iψ −√ Aξe iψ −√ , (3.13)in which we defined the following: D ′ = 3 M ′ b / ( Y S V ) , (3.14) A = r ! (cid:18) Y QS V Y S V (cid:19) | − √ ω | , (3.15) ξ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω + √ − √ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . ... , (3.16) tan ψ = −√
61 + √ − . ... , (3.17) tan ψ = √ √ − . ... , (3.18)54 and ψ have been included to consolidate the imaginary portion of D ′ elementsremoved by the absolute values in A and ξ .To arrive at predictions for the other CKM mixing elements other than the Cabibboangle ( i.e. Θ , Θ , δ KM ) one only needs to diagonalize the matrix D ′ in Eq. (3.13) by D ′ diag = V † CKM D ′ V CKM . (3.19)We now write the mixing matrix as V CKM = V ts V td V cb cos Θ sin Θ V ub − sin Θ cos Θ , (3.20)which, with Eq. (3.13), can be substituted into Eq. (3.19), becoming V cb V ub = 1ˆ D ′ D ′ − −√ −√ D ′ − Ae − iψ Aξe − iψ , (3.21)where ˆ D ′ = ( D ′ − − √ D ′ − √ , while from unitarity it follows that V ts V td = − cos Θ − sin Θ sin Θ cos Θ V ∗ cb V ∗ ub . (3.22)Our strategy is to now calculate the CP -violating Kobayashi-Maskawa phase, δ KM = γ = arg (cid:18) − V ud V ∗ ub V cd V ∗ cb (cid:19) , (3.23)55igure 3.1: The vertical axis is the value of δ KM ≡ γ T ′ in degrees and the horizontalaxis is the value of D ′ defined in the text. The dashed horizontal lines give the σ range for δ KM allowed by the global fit of Ref. 80. The orange area is excluded by σ confidence, while the red region is excluded by σ confidence.and, by using Eqs. (3.20, 3.21), we arrive at the formula in terms of D δ KM = γ T ′ = arg " −√ D ′ − ξe − i ( ψ − ψ ) ( D ′ − − √ ξe − i ( ψ − ψ ) = arg[Γ( D ′ )] . (3.24)From the preceding equations (3.20, 3.21) we find a formula for | V ub /V cb | = | tan Θ csc Θ | , (3.25)using unitarity, Eq. (3.22), and the form for the ratios of CKM matrix elements | V td /V ts | = (cid:12)(cid:12)(cid:12)(cid:12) sin Θ + Γ( D ′ ) cos Θ cos Θ − Γ( D ′ ) sin Θ (cid:12)(cid:12)(cid:12)(cid:12) . (3.26)56 .3 Comparison with CKM Data In Fig. 3.1, we show a plot of γ T ′ versus D ′ using Eq. (3.24) and taking the rangeof experimentally allowed γ ≡ δ KM from the global fit of Ref. 80 prompts us to use avalue D ′ ∽ ± in the subsequent analysis.Figure 3.2: The vertical axis is the value of | V td /V ts | and the horizontal axis is the valueof D ′ defined in the text. The dashed horizontal lines give the value with small errorallowed by the global fit of Ref. 80. The orange area is excluded by σ confidence,while the red region is excluded by σ confidence.Fig. 3.2 shows a plot of | V td /V ts | as a function of D ′ . It requires a value of D ′ ofapproximately 16 which is sufficiently close to that in Fig. 3.1.For the value of | V ub /V cb | there is approximately a factor of between the predic-tion (higher) and the best value from Ref. 80 as seen in Fig. 3.3.Note that once the off-diagonal, third family elements in Eq. (3.11) are taken asmuch smaller than the elements involved in the Cabibbo angle, and that the twoCKM angles and the CP phase are predicted by the present NMR T ′ M.With regard to alternatives to NMR T ′ M( D ), named earlier in Eqs. (3.1, 3.2, 3.3),57igure 3.3: The vertical axis is the value of | V ub /V cb | and the horizontal axis is thevalue of D ′ defined in the text. The dashed horizontal lines give the preferred exper-imental values allowed by the global fit of Ref. 80. The orange area is excluded by σ confidence, while the red region is excluded by σ confidence.the possibilities A and C modify the up-type mass matrix where we take flavor andmass eigenstates coincident. The final possibility B does modify the down-type massmatrix (like D does), but fails to permit the CP -violation we prefer (seen in the Chen-Mahanthappa mechanism), as in the present D model and in Ref. 116.58 hapter 4Quartification This Chapter is largely based on the work of Ref. 117 T ′ Quiver Model
Now that we have managed to construct a functional NMR T ′ M, it may be of inter-est to examine the wider context of fundamental physics. While we have previouslynoted the group symmetries utilized by the MSM, we have not sought to incorpo-rate the ( SU ( ) C × SU ( ) L × U ( ) Y ) groups into our model. In this chapter, we shallattempt to provide a suitable MSM-like framework that is both compatible with theassignments of our T ′ model and allows the particles to rotate under the appropriategroups.For the purposes of this investigation we shall craft a model with SU ( ) N . Thesetypes of group combination are sometimes termed quiver groups, due to the fact thatthe graphs used to diagram the various bifundamental representations resemble aseries of arrows.We will begin by considering a quartification model using SU ( ) ,[118] with bi-fundamental chiral fermions in the usual arrangement of bifundamentals, but findwe are unable to make the necessary charge assignments to recover the requisite T ′ amily symmetry. This will lead us to add a sub-quiver of fermions to accommodate T ′ quartification. We will give each irrep under T ′ a new set of assignments underthe quartification groups comprised of singlets ( ), triplets ( ), and conjugate triplets( ).The quartification gauge group is SU ( ) C × SU ( ) L × SU ( ) ℓ × SU ( ) R , (4.1)which is assumed to break to the standard model at the TeV scale, and includes thecommon groups aligned with color, left-handed particles, leptons, and right-handedparticles, respectively. We choose the family symmetry to be: ( T ′ × Z ) with the mini-mal anomaly-free bifundamental chiral fermions: , ¯3 , ,
1) + (¯3 , , ,
3) + (1 , , ¯3 ,
1) + (1 , , , ¯3)] , (4.2)where we assign the leptons to irreps as follows: (13¯31) ⊃ ν τ τ − L (13¯31) ⊃ ν µ µ − L (13¯31) ⊃ ν e e − L L L (3 , +1) , (113¯3) ⊃ τ − R (1 , − ⊃ µ − R (1 , − ⊃ e − R (1 , − , and N (1)R (1 , +1) N (2)R (1 , +1) N (3)R (1 , +1) . (4.3)60or the left-handed quarks, we make the assignment, (3¯311) ⊃ tb L Q L ( , +1)(3¯311) ⊃ cs L (3¯311) ⊃ ud L Q L ( , +1) . (4.4)Finally we need assignments for the six right-handed quarks. They were previ-ously assigned in Eq. (1.30) as, t R ( , +1) b R ( , − c R u R C R ( , − s R d R S R ( , +1) . (4.5)However, this assignment in not available here since t R and b R are both in the sameirrep, (¯3113) , and likewise for the first and second families. With no alteration of themodel, we can only assign three of the six right-handed quarks. In our attempts tocorrect this problem, we attempted a number of possible alterations, but even addinga fifth SU ( ) (this would have been Quintification) failed to alleviate the problem ofinsufficient irreps to close under known couplings.We therefore need to add an anomaly-free sub-quiver representation, , , , ′ + (1 , , ¯3 , ′ + (3 , , , ¯3) ′ ] , (4.6)61nd proceed to reassign all fermions with Z = − , including the corresponding sub-set in Eq. (4.3), and Eq. (4.5), to this sub-quiver: b R ⊂ (¯3 , , , ′ C R ⊂ (¯3 , , , ′ , τ − R ⊂ (1 , , ¯3 , ′ µ − R ⊂ (1 , , ¯3 , ′ e − R ⊂ (1 , , ¯3 , ′ . (4.7) We shall now introduce a notation for abbreviating the extended group desig-nations of the T ′ Quiver model. For each irrep this notation utilizes a superscriptto denote which SU ( ) was assigned a and a subscript for each SU ( ) assigneda ¯3 . The benefit being that when combined into Yukawa terms, one can check that,for each term, every group in superscript should also be included in subscript onanother (this notation will not apply to the Yukawa couplings, solely the objects ro-tating under our groups). We also list the assignments under T ′ in parenthesis with asuperscript + or − to distinguish between Z = +1 and Z = − , respectively. In ourfirst demonstration, the lepton Yukawas are Σ i =3 i =1 Y ( i ) D L L ℓ (3 + ) N ℓ ( i )R (1 + i ) H RL (3 + ) , (4.8)for the neutrino terms and, Σ i =3 i =1 Y ( i ) ℓ L L ℓ (3 + ) ℓ ℓ ( i )R (1 + i ) H RL (3 − ) , (4.9)62or the charged terms. Where one can clearly see that in each term there is an L , R ,and ℓ in both super- and subscript. Adopting the previous work from Ch. 3, the quarkYukawa couplings are Y t Q CL (1 +1 ) t RC (1 +1 ) H LR (1 +1 ) + Y b Q CL (1 +1 ) b ℓ C (1 − ) H L ℓ (1 − ) + Y QS Q CL (1 +1 ) S RC (2 + ) H LR (2 +3 ) + Y C Q CL (2 +1 ) C ℓ C (2 − ) H L ℓ (3 − ) + Y S Q CL (2 +1 ) S RC (2 +2 ) H LR (3 + ) . (4.10)The Higgs scalar sector is sufficient to break to the MSM and replicate the previ-ously determined mixing matrices (Chs. 2, 3). Note that, for example, the Cabibboangle in Sec. 1.5 follows because, after the breaking of ( SU ( ) ℓ × SU ( ) R ), the H (3 − ) shave a common representation, and can thus act as the appropriate messengers be-tween the charged leptons and the first two families of quarks. The Higgs T ′ doublet, +3 (Eq. 3.8), allows reproduction of the successful CKM matrix derived in Ch. 3.The Higgs VEVs follow a form highly similar to that in Sec. 1.5: < H RL (3 − ) > = ( m τ Y τ , m µ Y µ , m e Y e ) ,< H RL (3 + ) > = V ( − , , ,< H LR (2 +3 ) > = V (1 , , (4.11) < H LR (1 +1 ) > = m t Y t ,< H ℓ L (1 − ) > = m b Y b . We have now shown that it is possible to craft a model that successfully combinesthe predictiveness of the finite group T ′ , with the familiar physics of the MSM. While63his framework contains no additional physics, or new predictions, it demonstratesthat a combination of T ′ and Lie groups is feasible. While at this point it is too earlyto claim that this is a sufficient replacement for the MSM, it is sufficient to note thatthis framework demonstrates a unified symmetry and can act as a proof-of-conceptfor further attempts at unification. 64 hapter 5 T ′ Model Dark Matter
This Chapter is largely based on the work of Ref. 119
As mentioned in Sec. 1.2, dark matter remains one of the leading mysteries inmodern physics. And, while the community has yet to reach a consensus on an ex-planation, there have been no shortages of suggested ideas. Fortunately, there havebeen a number of clues that have allowed us to better understand dark matter and,consequently, rule some possibilities out. Of course the true answer, need not be anysingle theory mentioned here, or elsewhere, and could be a combination of several,but most theories, and our calculations for this chapter, will assume (if only for sim-plicity) that our suggested candidate is the sole contributor.The Weakly Interacting Massive Particle (WIMP) remains the best-known sug-gestion to this problem for several reasons. First is the so-called WIMP-miracle,which notes that a particle with the appropriate relic abundance to explain dark mat-ter would need a cross-section no larger than one typically seen on the weak scale.Additionally, this theory would indicate there are heavy, undiscovered particles (acommon element in many BSM models, including ours) who have had a significantmpact on cosmological development. Many of these WIMP candidates arise out ofR-parity conserving SUSY models, and typically come about as the lightest remainingSUSY particle. A WIMP is, as the name suggests, a rarely forming but massive stableparticle, capable of interacting with know matter only by weak interactions and grav-ity. This would be an example of cold dark matter (non-relativistic), and would likelyhave gained its current stability via the mechanism called thermal freeze-out. As theuniverse cooled, higher energy particles or interactions would become less preferreduntil all that remained was a supply of dark matter. As the universe expanded the re-maining annihilations would grow fewer as the particles diminished in number andwere spread out.Other ideas for dark matter include axions, a suggestion of Ref. 38 intended tosolve the ”Strong CP Problem”, and Massive Compact Halo Objects (sometimes ab-breviated MACHOs). Searches for these objects continue, and there are numerousgroups continuing to investigate these ideas and even more exotic theories.Two additional topics of interest are some theories that have fallen out of favor.The first of these, Modified Newtonian Dynamics (MOND),[120] attempted to alterNewton’s law of gravity to better accommodate the astronomical observations (anappealing idea since, to this point, there has been no short-range proof of dark mat-ter) rather than resorting to the ”missing mass” hypothesis that underlies this chapter.However, following the observation of the bullet cluster in Ref. 26 and some failuresto explain galactic rotation curves, MOND has largely fallen out of favor. Anotheridea to explain observations has been hot dark matter (relativistic), primarily fromneutrinos. While, at one time, there was considerable interest that neutrinos, alwaysdifficult to detect, were a significant cause of dark matter, this assumption led to sig-nificant changes to large-scale astronomical structure formation. As a consequence,they are now believed to play a relatively minor part of the universe’s mass.66 .2 The Valencia Mechanism and an Augmented Model
An ingenious new mechanism involving A model building has been discoveredby a research group based in Valencia, Spain.[121, 122] Their research uses A , whosedouble cover is central to our present work, to add a small number of extra scalarfields, one of which, by virtue of a discrete Z analogous to R-symmetry in SUSY,gives rise to stable dark matter.Their original model assigned all standard model leptons to different singlets of A , with the right-handed neutrinos and one of the newly added Higgs as the onlytriplets (their model’s other Higgs was a singlet). These assignments were unconven-tional, as most A models, like the T ′ model discussed in Sec. 1.5, utilize triplets inthe lepton assignments.In Refs. 121,122 a particular generator of A was used to give rise to a Z subgroupof A and stabilized the WIMP. This Z established a particle sector that is discretefrom the MSM particles and inaccessible to it, except via the weak nuclear force andpotentially gravity.Since A alone has proved incapable of accommodating quarks in a like mannerto leptons,[64–66] the Valencia group relegated the quark sector to ”future work”.An alternative approach, that we pursue, is to replace their A group with T ′ , allow-ing the incorporation of quarks, a prediction of the Cabibbo angle, and controllabledeviations from the TBM angles. T ′ M To accommodate the quark sector, we adopt the ( T ′ × Z ) model formulated inSec. 1.5 and further analyzed in Ch. 2. This section will establish an extended modelincluding elements of the Valencia Mechanism by incorporating a second Z , whichwe will label Z ′ for clarity, while also adding scalar fields and heavy right-handed67eutrinos that are odd under this new group; the lightest resultant odd scalar willbe our dark matter WIMP. This model contains a global symmetry of ( T ′ × Z × Z ′ )restricting the Yukawa couplings. One key difference from Ref. 121, 122 is that our Z ′ will not be subgroup of our added T ′ and is instead an exterior addition.The quark assignments below are unchanged from Eq. (1.30), and denote Q L = (cid:18) tb (cid:19) L , Q L = (cid:18) cs (cid:19) L & (cid:18) ud (cid:19) L , C R = c R & u R , and S R = s R & d R . By setting all quarks tobe even under Z ′ , past T ′ predictions are preserved.Quarks Q L Q L t R b R C R S R T ′ Z + + + − − + Z ′ + + + + + + Table 5.1: Quark Group AssignmentsThe leptons of Eq. (1.8) are retained unchanged, even under Z ′ , again keeping allthe previous successes in Ch. 2. Inspired by Ref. 121, 122, we have incorporated anadditional triplet of right-handed neutrinos, N T . This triplet is odd under Z ′ and isbelow summarized with the other lepton assignments.Leptons L L τ R µ R e R N (1)R N (2)R N (3)R N T T ′ Z + − − − + + + + Z ′ + + + + + + + − Table 5.2: Lepton Group AssignmentsThe Higgs sector is also mostly the same as in Sec. 1.5, being Z ′ -even, with anadded Z ′ -odd, T ′ -triplet, H ′′ . The five Higgs irreps of T ′ are shown in the following68able. Note that this makes for a total of 11 doublets under the gauge group SU ( ) L ,one of which may serve as the MSM Higgs.[123]Higgs H H H H ′ H ′′ T ′ Z + − + − + Z ′ + + + + − Table 5.3: Higgs Group AssignmentsThe resultant Lagrangian and Yukawa couplings are: L Y = 12 M N T N T + 12 M N (1)R N (1)R + M N (2)R N (3)R + Y e L L e R H ′ + Y µ L L µ R H ′ + Y τ L L τ R H ′ + Y L L N (1)R H + Y L L N (2)R H + Y L L N (3)R H + Y L L ( N T H ′′ ) + Y L L ( N T H ′′ ) ′ + Y t ( Q L t R H ) + Y b ( Q L b R H )+ Y C ( Q L C R H ′ ) + Y S ( Q L S R H ) + h.c. . (5.1)It is interesting to note that the terms containing the new right-handed neutrinotriplet N T , and new Higgs H ′′ , result in a multiplication of ( × × ) under T ′ . Sur-prisingly, this results in only two ( ) singlets,[124] producing two additional Yukawacouplings, Y and Y . This will prove important to our implementation of the Type-Iseesaw mechanism. Intriguingly, should these new Yukawa couplings prove com-plex, they can naturally lead to leptogenesis. It is notable that one decay mode of the triplet N T is into a light neutrino and dark matter. .2.2 Generalized Seesaw Mechanism At this point, we can summarize the VEVs of our model’s Higgs as follows, < H > = ( V , V , V ) ,< H ′ > = ( m τ Y τ , m µ Y µ , m e Y e ) ,< H ′′ > = (0 , , ,< H > = ( m t Y t ) , < H > = ( m b Y b ) . (5.2) < H ′ > is tied to the charged lepton masses and remains disconnected from theneutrinos assuming the charged leptons are mass eigenstates. < H ′′ > must have atleast one component without a VEV in order to create stable dark matter, but mustalso have 3 identical values in order for Z ′ to commute with ( T ′ × Z ), hence threezeroes. < H > remains in general form to be specified using the seesaw mechanism.As seen in Sec. 1.5, we can use the TBM form to generate a symmetry, M diag = U TBM M ν U T TBM ,M ν = U T TBM m m
00 0 m U TBM ,M ν = A B CB A CC C A + B − C . (5.3)Next we will implement a generalized Type-I Seesaw Mechanism (the (3 , form,defined by 3 families and 6 SU ( ) singlet fields),[125] first noting the key equation in70ef. 126, given earlier in Eq. (1.14), which shows another way to determine M ν , M ν = M D ν M − ( M D ν ) T . (5.4)The Dirac and Majorana mass matrices below are based on a generalized form ofthose used in Ref. 57. Due to the 6 right-handed neutrino states, the Majorana matrixenlarges to × , while the Dirac matrix becomes × . The zero elements of the Diracmass matrix are determined by the VEVs of H ′′ . M D ν = Y V Y V Y V Y V Y V Y V Y V Y V Y V , M R = M M M M M M . (5.5)These alterations to the seesaw mechanism will result in the following version ofEq. (1.21), m = A + B − C = − x ,m = A + B + C = 0 ,m = A − B = 6 x + 3 x . (5.6)As these mass equation remain essentially unchanged, they show that the addition ofa neutrino triplet to the MR T ′ M does not change the results of the seesaw mechanismand preserves the predictions of Chs. 1, 2, 3. Consequently, the VEVs of H will revertto the form, < H > = V (1 , − , . 71 .3 T ′ Dark Matter Predictions
The T ′ WIMP candidate is the lightest state with an assignment of Z ′ = − . The Z ′ odd states being N T and H ′′ . The neutrino triplet N T , in particular, is expected tobe very heavy from the seesaw mechanism discussed in the Sec. 5.2. It decays into an H ′′ and a lepton, making it a good candidate for the leptogenesis mechanism.[127]The WIMP candidate is therefore a superposition of the CP -even neutral scalarscontained in H ′′ , which has three SU ( ) L doublets: H ′′ (1) = h +1 h + iA , H ′′ (2) = h +2 h + iA , H ′′ (3) = h +3 h + iA . (5.7)This set includes 6 charged scalars, 3 neutral CP -even scalars, and 3 neutral CP -oddscalars. Our dark matter candidate will be a superposition of the three real Z ′ -odd, CP -even, neutral scalar states: Φ W IMP = αh + βh + γh . (5.8)An evaluation of the dark matter candidate coefficients, α , β , and γ , requires knowl-edge of the coefficients in the Higgs scalar potential, shown in Appendix A, and isbeyond the scope of this discussion. One of the most significant properties of a proposed particle is its mass, and byfollowing the treatment laid out in Ref. 128 we can use the measured relic abundanceto determine this property, M Φ .Beginning with the standard definition of dark matter abundance Ω c = ρ/ρ cr ,and the standard assumptions that our particle is a cold relic (that freeze-out will72ccur when the particle is no longer relativistic), and that we will be focusing on thedominant s-wave coannihilation into MSM QED vector bosons, we can state Ω c = M Φ s Y ∞ ρ cr , where ρ cr , = 3 H πG , s = 2 π g ⋆ T . (5.9) g ⋆ is a count of the number of relativistic degrees of freedom and is a common partof cosmological statistics. We have included a detailed discussion of the calculationto retrieve this factor in Appendix B, but for our purposes g ⋆ = 119 . and g ⋆ =65 / ≈ . . After plugging in and dividing both sides by a scale factor of χ =100 km / s / Mpc (typical in reporting of astronomical results) we find, Ω c h = 2 Gg ⋆ χ (cid:18) πT (cid:19) M Φ Y ∞ , (5.10)which includes the cosmic microwave background temperature T = 2 . ◦ K , andthe gravitational constant G = 6 . × − GeV − .[21]Next we need to determine Y ∞ which can be approximated as Y ∞ ∽ H ( M Φ ) x f s h σ A | v |i x (cid:12)(cid:12)(cid:12)(cid:12) x =1 where s = 2 π g ⋆ T , H ( M Φ ) = H ( x ) M T . (5.11)In this notation x ≡ M Φ /T , with x f being defined at the freeze-out temperature( x f > ∼ for a cold relic). Next, H ( x ) can be obtained from the 2nd Friedman Equa-tion by assuming a flat universe: H ( x ) = r πG ρ where ρ = π g ⋆ T . (5.12)Combining these equations leads to Y ∞ = r Gπg ⋆ x f M Φ h σ A | v |i and Ω c h = (cid:18) π √ (cid:19)(cid:18) g ⋆ √ g ⋆ (cid:19)(cid:18) T G x f χ h σ A | v |i (cid:19) . (5.13)73n Ref. 128 an approximation for x f is established (accurate within 5% for any coldrelic value) x f = ln (cid:20)r g ⋆ G M Φ h σ A | v |i π (cid:21) −
32 ln (cid:20) ln (cid:26)r g ⋆ G M Φ h σ A | v |i π (cid:27)(cid:21) , (5.14)For the annihilation cross-section, we will turn to Ref. 129, which lists a generalform that we can customize. Recognizing that our dark matter candidate is a realscalar, inhabits an SU (2) doublet, and, like the MSM Higgs, has a hypercharge of Y = 1 / we see that, h σ A | v |i ≃ g + ( g ′ ) + 6 g ( g ′ ) + 4 λ πM , (5.15)where g and g ′ are the gauge coupling constants, defined as g = √ πα/ sin θ W and g ′ = g tan θ W , respectively. Rather than solve the Higgs scalar potential (detailed inAppendix A), we make the assumption that the quartic coupling constant, λ , yields avery small contribution.Now that all the pieces are in place, we can note that sin θ W = 0 . for the on-shell scheme,[21] and recent data from Ref. 33 sets Ω c h = 0 . . This leads to acalculation of M Φ ≈ .
84 TeV
A thorough discussion of the techniques and evidence for dark matter detectioncould fill volumes, so we will opt here for only a brief and superficial analysis. But itremains the case that currently there is little to no evidence for dark matter beyondthe astrophysical evidence resulting in its discovery and confirmation.Dark Matter detection usually falls into two categories: direct and indirect detec-tion. Direct Detection would be any method of interacting with dark matter itself and74igure 5.1: Generated by Ref. 130, this figure details the current and projected limitson WIMP dark matter masses and cross-section. We have indicated our rough pre-diction for the T ′ dark matter candidate with a star. While this analysis is not gearedspecifically towards our candidate, being designed for a WIMP arising out of SUSY,these limits should still be roughly applicable.includes searches from the LHC at CERN, as well as nuclear recoil experiments deepunderground. Indirect Detectors search for signs of dark matter decay or annihila-tion. Most recently the PAMELA experiment and results from the Alpha MagneticSpectrometer generated a great amount of excitement after announcing an excessof high-energy positrons,[131, 132] leading some to suggest that they had seen darkmatter decay products. While the cause of the anomaly remains uncertain, and couldsimply be a local astrophysical source, it remains a promising sign.In addressing direct detection, we can combine Eqs. (5.13, 5.14, 5.15) to get a firstorder estimate for the coupling λ . Then using the derivation from Ref. 121, σ el (nucleon) ≈ λ × (cid:18) M H (cid:19) × (cid:18) M Φ (cid:19) × (cid:18) × − cm (cid:19) , (5.16)75ith the most recent measured value of the Higgs mass M H ≈
125 GeV , we candevelop obtain an order-of-magnitude estimate for the dark matter-nucleon cross-section for our model of ∽ × − cm .As Fig. 5.1 shows, our prediction remains below the most stringent limits placedby current (or proposed) detectors. That said, each successive generation of detectorhas pushed dark matter cross-section limits further down, with the current best limitsfound at . We hope that in the coming years detectors improve to the pointwhere they will find our proposed WIMP.76 hapter 6Conclusions This Section is largely based on data found in the work of Ref. 133
Part of the reason for the rapid advances in understanding of neutrinos over thepast decade has been the continually growing number of neutrino experiments. Inthis section we will try and mention key current and future experiments and whataspects of our model they affect.Most of the recent excitement in neutrino physics has been over the rapid ex-perimental precision in measurements of θ . These measurements primarily comefrom the Daya Bay, Double Chooz, and RENO collaborations. While any of theseexperiments might receive updates in the future, they currently stand poised to havemeasured σ (sin (2 θ )) < by 2015. In addition, because θ is large, they may beable to provide ∆ m measurements.Next are θ and ∆ m . These parameters are currently being studied by theSuper-Kamiokande and IceCube experiments, with Minos+ set to join in 2013. Look-ing to the future, the proposed INO and PINGU experiments may join this searchnear the end of the decade. As these are key to our predictions, we will be closelybserving any new results.The mass hierarchy question remains an unanswered for the time being. NO ν Awill begin taking data in late 2013 and may soon have some relevant data on thissearch. If, by the end of the decade, this factor remains unsettled, Daya Bay II, INO,or the LBNE may be able to make a final determinationThe CP -violating phase, δ CP , has long been a mystery of the PMNS matrix, aswell as having one of the poorest constraints of the 28 MSM parameters. Though sim-ply assuming a value of simplifies calculations, it may yet have a non-zero value.Experiments have shown indications of what that value might be, but as their σ spreads always contain the entire region π , it remains a secondary concern. Cur-rently T2K and NO ν A are attempting to measure δ CP , with Hyper-K (an update ofSuper-Kamiokande) and the LBNE to take the baton at the end of the decade. As we conclude, in the interest of candor, we note what work remains to be doneand what limitations our model continues to face.Perhaps we should first mention the scope of our model. We have never claimedthat the T ′ model in its current form can act as a grand unified theory. As such we seekto use it to better understand the mixing and masses of the leptons. We have madesome movement toward generating a true unified theory in Ch. 4, by demonstratingthat such a model is possible in a Quiver Theory, but these are simply first steps.Ideally, the completed flavor symmetry should be reconcilable with the Lie Groupsthat make up the MSM and more holistic theories.Another outstanding issue remains a satisfying inclusion of the full CKM matrix.Thus far we have shown that the ( × ) Cabibbo matrix offers increased certainty butdecreased utility, whereas our attempts to examine the full CKM matrix are stymied78y the added complexity. As fermion mixing is so integral to our model, this lack ofcoherence may be creating a number of problems. As mentioned earlier, our modelproduces an inaccurate value for ( m d /m s ) and only the vaguest checks against theCKM element values. To a degree this is due to by the lack of higher order correctionsto the model, and by having multiple choices in forming the NMR T ′ M from Ch. 3.Another issue remains the solar mass splitting. While experiments have clearlyshown that the neutrino mass eigenstates, m and m , have a separation of roughly ∆ m = 7 . × − , our models, thus far, have maintained these mass states areequal. As with the other listed issues, there are a number of potential perturbationsthat might be introduced to compensate for this initial assumption, but we are leftwith the dual problems of altering the neutrino properties while maintaining thesymmetry and properly motivating this change without simply fitting to the data.One last area of uncertainty is that of the assumptions we have made about neu-trino properties, namely the NH, Majorana behavior and others. Many models havea point of rigidity, a place where strong assumptions have been made that cannotbe modified or altered without undoing the theory. In the past few years, a com-mon tripping point for other theories has been θ = 0 . Even the original Valenciawork was inflexible on this point (unlike our variation in Ch. 5).[121] For our model,that point of rigidity may indeed be our assumption of the normal hierarchy. Whilean inverted hierarchy would in no way undermine the potential of our T ′ theory,our choice of VEVs and our use of shared Higgs between leptons and quarks do notleave much room for alteration. At this point there remains no preferred hierarchy,only time, and experimental results, will tell if this assumption is borne out.On the same note, we should also mention our model’s inability to predict theindividual neutrino masses. Although only upper bounds for these masses exist, itwould be preferable to suggest a value for experiments to reach for. In addition, wehave always assumed that δ CP = 0 . While this was done largely in the interest of79implifying the algebra to a solvable degree, it may not be the case. If this proves truewe will need to accommodate this, and deal headlong with that more complicatedreality. Having now explained how our model works, both its capabilities and limita-tions, we should make some comments about our work in the context of the greaterphysics community. The work we have presented here is admittedly flawed, at timesincomplete, and overly vague. And yet the number of completely verified and unani-mously accepted theories in particle physics can likely be counted at less than a dozenfor a generation. The real question to ask is: does this investigation advance our col-lective understanding of either the mathematical principles underlying our models,or the physical systems we are trying to describe. To both, we would answer: yes.We have managed to use finite group symmetries to successfully explain a num-ber of features of the quark and lepton mixing, we would hope that this model,and those like it, would prove the benefits to incorporating these ideas into futureattempts at unification. In addition, the model as stated has proven remarkably re-silient. We managed to predict that θ would prove to be nonzero, and that θ wouldbe non-maximal and correlated together. While our model has limitations and mayprove incorrect in the long run, the accuracy we have seen so far is immensely excit-ing. We would hope that it might prove the spark for new discoveries as we pass intoan era of BSM physics. 80 ppendices ppendix AThe Higgs Scalar Potential This Appendix is adapted from Appendix A in Ref. 119
As stated in Ch. 5, below is the Higgs scalar potential up to quartic order, con-sisting of 218 terms and 77 hermitian conjugates. We will use , , , to represent thethree singlet representations of T ′ ; additionally and will be used to distinguishthe two triplet products of two contracted T ′ triplets.We have studied assiduously the set of equations ∂V /∂v i , where the v i are theVEVs, and the related requirements for a local minimum of positive Hessian eigen-values. We find, after careful calculation, that the VEVs in Eq. (5.2) are allowed with-out fine-tuning.Without further assumptions, one cannot determine the superposition coefficients α , β , and γ from Eq. (5.8). It may be fruitful to seek an additional assumption toincrease our model’s predictivity. For the dedicated reader who wishes to pursuethis interesting question, we provide the complete Higgs potential below. V = µ H H † H + µ H H † H + µ H H † H + µ H ′ H ′ † H ′ + µ H ′′ H ′′ † H ′′ + λ [ H † H ] + λ [ H † H ] + λ [ H † H ] [ H † H ] + λ [ H † H † ] [ H H ] + λ [ H † H † ] [ H H ] + λ [ H † H ] [ H † H ] + λ [ H † H ] [ H ′ † H ′ ] + λ [ H † H ] [ H ′′ † H ′′ ] λ [ H † H ] [ H † H ] + λ [ H † H ] [ H ′ † H ′ ] + λ [ H † H ] [ H ′′ † H ′′ ] + λ ([ H † H † ] [ H H ] + h.c. ) + λ ([ H † H † ] [ H ′ H ′ ] + h.c. )+ λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. )+ λ ([ H † H † ] [ H H ] + h.c. ) + λ ([ H † H † ] [ H ′ H ′ ] + h.c. )+ λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. )+ λ [ H † H ] [ H † H ] + λ [ H † H ′ ] [ H ′ † H ] + λ [ H † H ′′ ] [ H ′′ † H ] + λ [ H † H ] [ H † H ] + λ [ H † H ′ ] [ H ′ † H ] + λ [ H † H ′′ ] [ H ′′ † H ] + λ ([ H † H ′ ] [ H † H ] + h.c. )+ λ ([ H † H ] [ H † H ] + h.c. ) + λ ([ H † H † ] [ H H ] + h.c. )+ λ ([ H † H ] [ H † H ] + h.c. ) + λ ([ H † H † ] [ H H ] + h.c. )+ λ ([ H † H ] [ H ′ † H ′ ] + h.c. ) + λ ([ H † H ′ ] [ H † H ′ ] + h.c. )+ λ ([ H † H † ] [ H ′ H ′ ] + h.c. ) + λ ([ H † H ′ † ] [ H H ′ ] + h.c. )+ λ ([ H † H ] [ H ′ † H ′ ] + h.c. ) + λ ([ H † H ′ ] [ H † H ′ ] + h.c. )+ λ ([ H † H † ] [ H ′ H ′ ] + h.c. ) + λ ([ H † H ′ † ] [ H H ′ ] + h.c. )+ λ ([ H † H ] [ H ′′ † H ′′ ] + h.c. ) + λ ([ H † H ′′ ] [ H † H ′′ ] + h.c. )+ λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. ) + λ ([ H † H ′′ † ] [ H H ′′ ] + h.c. )+ λ ([ H † H ] [ H ′′ † H ′′ ] + h.c. ) + λ ([ H † H ′′ ] [ H † H ′′ ] + h.c. )+ λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. ) + λ ([ H † H ′′ † ] [ H H ′′ ] + h.c. )+ λ ([ H † H ′ ] [ H ′ † H ′ ] + h.c. ) + λ ([ H † H ′ † ] [ H ′ H ′ ] + h.c. )+ λ ([ H † H ′ ] [ H ′ † H ′ ] + h.c. ) + λ ([ H † H ′ † ] [ H ′ H ′ ] + h.c. )+ λ ([ H † H ′ ] [ H † H ] + h.c. ) + λ ([ H † H ] [ H ′ † H ] + h.c. )+ λ ([ H † H ′ † ] [ H H ] + h.c. ) + λ ([ H † H † ] [ H ′ H ] + h.c. )+ λ ([ H † H ′ ] [ H † H ] + h.c. ) + λ ([ H † H ] [ H ′ † H ] + h.c. )+ λ ([ H † H ′ † ] [ H H ] + h.c. ) + λ ([ H † H † ] [ H ′ H ] + h.c. ) λ ([ H † H ′ ] [ H ′′ † H ′′ ] + h.c. ) + λ ([ H † H ′′ ] [ H ′ † H ′′ ] + h.c. )+ λ ([ H † H ′ † ] [ H ′′ H ′′ ] + h.c. ) + λ ([ H † H ′′ † ] [ H ′ H ′′ ] + h.c. )+ λ ([ H † H ′ ] [ H ′′ † H ′′ ] + h.c. ) + λ ([ H † H ′′ ] [ H ′ † H ′′ ] + h.c. )+ λ ([ H † H ′ † ] [ H ′′ H ′′ ] + h.c. ) + λ ([ H † H ′′ † ] [ H ′ H ′′ ] + h.c. )+ λ [ H † H ] [ H † H ] + λ [ H † H † ] [ H H ] + λ ′ [ H † H † ] [ H H ] + λ [ H † H ] + λ [ H † H † ] [ H H ] + λ ([ H † H ] [ H † H ] + h.c. )+ λ [ H † H ] [ H † H ] + λ [ H † H † ] [ H H ] + λ [ H ′ † H ′ ] [ H ′ † H ′ ] + λ [ H ′ † H ′ † ] [ H ′ H ′ ] + λ ′ [ H ′ † H ′ † ] [ H ′ H ′ ] + λ [ H ′ † H ′ ] + λ [ H ′ † H ′ † ] [ H ′ H ′ ] + λ ([ H ′ † H ′ ] [ H ′ † H ′ ] + h.c. )+ λ [ H ′ † H ′ ] [ H ′ † H ′ ] + λ [ H ′ † H ′ † ] [ H ′ H ′ ] + λ [ H ′′ † H ′′ ] [ H ′′ † H ′′ ] + λ [ H ′′ † H ′′ † ] [ H ′′ H ′′ ] + λ ′ [ H ′′ † H ′′ † ] [ H ′′ H ′′ ] + λ [ H ′′ † H ′′ ] + λ [ H ′′ † H ′′ † ] [ H ′′ H ′′ ] + λ ([ H ′′ † H ′′ ] [ H ′′ † H ′′ ] + h.c. )+ λ [ H ′′ † H ′′ ] [ H ′′ † H ′′ ] + λ [ H ′′ † H ′′ † ] [ H ′′ H ′′ ] + λ [ H † H ] [ H ′ † H ′ ] + λ ([ H † H ′ ] [ H † H ′ ] + h.c. )+ λ [ H † H ′ † ] [ H H ′ ] + λ ([ H † H † ] [ H ′ H ′ ] + h.c. )+ λ [ H † H ] [ H ′ † H ′ ] + λ ([ H † H ′ ] [ H † H ′ ] + h.c. )+ λ [ H † H ′ † ] [ H H ′ ] + λ ([ H † H † ] [ H ′ H ′ ] + h.c. )+ λ ′ [ H † H ′ † ] [ H H ′ ] + λ ′ ([ H † H † ] [ H ′ H ′ ] + h.c. )+ λ ([ H † H ] [ H ′ † H ′ ] + h.c. ) + λ ([ H † H ′ ] [ H † H ′ ] + h.c. )+ λ [ H † H ] [ H ′ † H ′ ] + λ ([ H † H ′ ] [ H † H ′ ] + h.c. )+ λ [ H † H ′ † ] [ H H ′ ] + λ ([ H † H † ] [ H ′ H ′ ] + h.c. )+ λ [ H † H ] [ H ′′ † H ′′ ] + λ ([ H † H ′′ ] [ H † H ′′ ] + h.c. )+ λ [ H † H ′′ † ] [ H H ′′ ] + λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. )+ λ [ H † H ] [ H ′′ † H ′′ ] + λ ([ H † H ′′ ] [ H † H ′′ ] + h.c. ) λ [ H † H ′′ † ] [ H H ′′ ] + λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. )+ λ ′ [ H † H ′′ † ] [ H H ′′ ] + λ ′ ([ H † H † ] [ H ′′ H ′′ ] + h.c. )+ λ ([ H † H ] [ H ′′ † H ′′ ] + h.c. ) + λ ([ H † H ′′ ] [ H † H ′′ ] + h.c. )+ λ [ H † H ] [ H ′′ † H ′′ ] + λ ([ H † H ′′ ] [ H † H ′′ ] + h.c. )+ λ [ H † H ′′ † ] [ H H ′′ ] + λ ([ H † H † ] [ H ′′ H ′′ ] + h.c. )+ λ [ H ′ † H ′ ] [ H ′′ † H ′′ ] + λ ([ H ′ † H ′′ ] [ H ′ † H ′′ ] + h.c. )+ λ [ H ′ † H ′′ † ] [ H ′ H ′′ ] + λ ([ H ′ † H ′ † ] [ H ′′ H ′′ ] + h.c. )+ λ [ H ′ † H ′ ] [ H ′′ † H ′′ ] + λ ([ H ′ † H ′′ ] [ H ′ † H ′′ ] + h.c. )+ λ [ H ′ † H ′′ † ] [ H ′ H ′′ ] + λ ([ H ′ † H ′ † ] [ H ′′ H ′′ ] + h.c. )+ λ ′ [ H ′ † H ′′ † ] [ H ′ H ′′ ] + λ ′ ([ H ′ † H ′ † ] [ H ′′ H ′′ ] + h.c. )+ λ ([ H ′ † H ′ ] [ H ′′ † H ′′ ] + h.c. ) + λ ([ H ′ † H ′′ ] [ H ′ † H ′′ ] + h.c. )+ λ [ H ′ † H ′ ] [ H ′′ † H ′′ ] + λ ([ H ′ † H ′′ ] [ H ′ † H ′′ ] + h.c. )+ λ [ H ′ † H ′′ † ] [ H ′ H ′′ ] + λ ([ H ′ † H ′ † ] [ H ′′ H ′′ ] + h.c. ) ppendix BCounting Relativistic Degrees ofFreedom A notable hurdle in a dark matter relic density calculation is determining the num-ber of relativistic degrees of freedom. This factor, g ⋆ , can be calculated using the for-mula from Ref. 128: g ⋆ = X i =bosons g i (cid:18) T i T γ (cid:19) + 78 X j =fermions g j (cid:18) T j T γ (cid:19) , (B.1)where T γ is the photon temperature. In the modern universe, with a temperature ofapproximately . ◦ K this number is quite small because so few objects fit the require-ments, basically the photon and the neutrinos: g ⋆ = (2 H × b ) γ + (3 e,µ,τ × M × L ×
78 f ×
411 E ) ν ≈ . , (B.2)where we have assumed Majorana neutrinos (Dirac neutrinos typically yield a valueof g ⋆ = 3 . . In this demonstration, we have signified with subscripts the causesfor several contributing factors. The subscripts b and f indicate boson or fermion,respectively. The subscript L indicates the left-handed (or right-handed for subscript R ) helicity state. A bar indicates a factor from antiparticles, while a subscript M is forMajorana fermions (who are their own antiparticle). Here, the subscript E indicates acontribution from entropy, which only becomes a factor on very recent cosmological86cales (long after dark matter freezes out). Two subscripts, unused here, but stillimportant, are C for single colored variants and Sp from the massive bosons. Below,in Fig. B, we have included a tabulation for all of the potential relativistic degrees offreedom for the model detailed in Ch. 5 at some arbitrarily high temperature, yieldinga value of g ⋆ = 119 . (The MSM produces a value of g SM = 104 . for Majorananeutrinos and g SM = 106 . for Dirac neutrinos). Depending on when dark matterfreeze-out occurs this leads to a range of possible values above the MSM, up to . ,but to find the extremes of our theory we assume the maximum allowed value. Particles Multipliers DoF u c td s b × ¯2 × L , R × C ×
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