Dihedral flavor symmetries [Elektronische Ressource] / put forward by Alexander Simon Blum

Dissertationsubmitted to theCombined Faculties for the Natural Sciences and forMathematicsof the Ruperto-Carola University of Heidelberg, Germanyfor the degree ofDoctor of Natural SciencesPut forward byAlexander Simon Blum, Master of ScienceBorn in: Oakland, California, USAOral Examination: June 10th 2009Dihedral Flavor SymmetriesReferees:Prof. Dr. Manfred LindnerProf. Dr. Berthold StechZusammenfassungDieseDissertationbehandeltdieM¨oglichkeitdenFlavorsektordesStandardmodellsderTeilchen-physik (mit Neutrinomassen), das heißt die Massen und Mischungsmatrizen der Fermionen,mithilfe einer diskreten, nicht-abelschen Flavorsymmetrie zu beschreiben. Insbesondere wer-den massenunabh¨angige Texturen untersucht, bei denen ein oder mehrere Mischungswinkelallein durch die Gruppentheorie bestimmt werden. Zu diesem Zweck wird eine systematischeAnalyse einer großen Klasse von diskreten Symmetrien, der Diedergruppen, durchgefuhrt.¨ DerUrsprung von massenunabh¨angigen Texturen, die aus solchen Symmetrien entstehen, wird un-tersucht und es wird gezeigt, dass solche Strukturen auf naturlic¨ he Weise aus der Minimierungvon skalaren Potentialen folgt, wobei die Skalare Eichsingletts, sogenannte Flavons, sind, dienur unter der Flavorsymmetrie nicht-trivial transformieren. Aus diesen Vorgaben werden zweiModelle konstruiert, eines beschreibt die Leptonen und beruht auf der Gruppe D , das andere4die Quarks und verwendet die Gruppe D .
Publié le : jeudi 1 janvier 2009
Lecture(s) : 30
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Source : ARCHIV.UB.UNI-HEIDELBERG.DE/VOLLTEXTSERVER/VOLLTEXTE/2009/9557/PDF/THESIS_BLUM.PDF
Nombre de pages : 158
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Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for
Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Alexander Simon Blum, Master of Science
Born in: Oakland, California, USA
Oral Examination: June 10th 2009Dihedral Flavor Symmetries
Referees:
Prof. Dr. Manfred Lindner
Prof. Dr. Berthold StechZusammenfassung
DieseDissertationbehandeltdieM¨oglichkeitdenFlavorsektordesStandardmodellsderTeilchen-
physik (mit Neutrinomassen), das heißt die Massen und Mischungsmatrizen der Fermionen,
mithilfe einer diskreten, nicht-abelschen Flavorsymmetrie zu beschreiben. Insbesondere wer-
den massenunabh¨angige Texturen untersucht, bei denen ein oder mehrere Mischungswinkel
allein durch die Gruppentheorie bestimmt werden. Zu diesem Zweck wird eine systematische
Analyse einer großen Klasse von diskreten Symmetrien, der Diedergruppen, durchgefuhrt.¨ Der
Ursprung von massenunabh¨angigen Texturen, die aus solchen Symmetrien entstehen, wird un-
tersucht und es wird gezeigt, dass solche Strukturen auf naturlic¨ he Weise aus der Minimierung
von skalaren Potentialen folgt, wobei die Skalare Eichsingletts, sogenannte Flavons, sind, die
nur unter der Flavorsymmetrie nicht-trivial transformieren. Aus diesen Vorgaben werden zwei
Modelle konstruiert, eines beschreibt die Leptonen und beruht auf der Gruppe D , das andere4
die Quarks und verwendet die Gruppe D . Im zweiten Modell ist es das Element V der14 ud
Quark-Mischungsmatrix - im wesentlichen der Cabibbo-Winkel - das in erster N¨aherung allein
durch die Gruppentheorie vorhergesagt wird. Abschließend wird die M¨oglichkeit diskutiert, die
diskrete Flavorgruppe als Untergruppe einer kontinuierlichen Eichsymmetrie zu beschreiben
und es wird gezeigt, dass dafur¨ die ursrprunglic¨ he Eichsymmetrie von verh¨altnism¨aßig großen
Darstellungen gebrochen werden muss.
Abstract
This thesis deals with the possibility of describing the flavor sector of the Standard Model
of Particle Physics (with neutrino masses), that is the fermion masses and mixing matrices,
with a discrete, non-abelian flavor symmetry. In particular, mass independent textures are
considered, where one or several of the mixing angles are determined by group theory alone
and are independent of the fermion masses. To this end a systematic analysis of a large class of
discrete symmetries, the dihedral groups, is analyzed. Mass independent textures originating
from such symmetries are described and it is shown that such structures arise naturally from
the minimization of scalar potentials, where the scalars are gauge singlet flavons transforming
non-trivially only under the flavor group. Two models are constructed from this input, one
describing leptons, based on the group D , the other describing quarks and employing the4
symmetry D . In the latter model it is the quark mixing matrix element V - basically the14 ud
Cabibbo angle - which is at leading order predicted from group theory. Finally, discrete flavor
groups are discussed as subgroups of a continuous gauge symmetry and it is shown that this
implies that the original gauge symmetry is broken by fairly large representations.Lark on the moon, singing -
sweet song
of non-attachment.
Basho¯Contents
1 Introduction 9
1.1 Flavor in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Neutrino Masses and Leptonic Mixing . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Non-Abelian Discrete Flavor Symmetries 19
2.1 Theoretical Approaches to the Flavor Sector . . . . . . . . . . . . . . . . . . . . . 19
2.2 Properties of Discrete Non-Abelian Flavor Symmetries . . . . . . . . . . . . . . . 22
2.3 The D Scaling Model - a Worked-Out Example . . . . . . . . . . . . . . . . . . 234
2.3.1 Phenomenological Considerations - Scaling . . . . . . . . . . . . . . . . . 23
2.3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Dihedral Flavor Groups 27
3.1 Introduction to Dihedral Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Single-Valued Dihedral Groups . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Double-Valued groups . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Non-Trivial Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Breaking Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Mass Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.2 Conventions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.3 Three Singlet Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.4 Two Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
03.4.5 Mass Matrices in D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46n
3.4.6 Majorana Mass Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Diagonalization and Mixing Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Subgroup Mismatch in D 57n
4.1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Derivation of the Non-Trivial Mixing Angle . . . . . . . . . . . . . . . . . 57
4.1.2 Comparison with Phenomenology . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 A D Example from the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 624
4.3 D and the Cabibbo Angle - a Worked-Out Example . . . . . . . . . . . . . . . . 637
4.3.1 Quark Sector of the D Model . . . . . . . . . . . . . . . . . . . . . . . . 637
4.3.2 Higgs of the D Model . . . . . . . . . . . . . . . . . . . . . . . . . 647
5 Flavons and Flavon Potentials 67
5.1 Flavon Potentials in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.1 One Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.2 Several Doublets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Flavon Potentials in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
78 CONTENTS
5.2.1 What Changes in Supersymmetry? . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 One Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.3 R-Symmetry and Driving Fields . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.4 Several Doublets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.5 Further Aspects of Flavon Models . . . . . . . . . . . . . . . . . . . . . . 80
6 Putting It All Together:
Two Worked-Out Examples 83
6.1 D in the Leptonic Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834
6.1.1 Lepton Masses and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1.2 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.3 Flavon Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.4 Next-to-Leading Order Corrections . . . . . . . . . . . . . . . . . . . . . . 90
6.2 D in the Quark Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9314
6.2.1 Quark Masses and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.2 Flavon Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Outlook: Where Could Such a Symmetry Come From? 107
8 Summary and Conclusions 119
Acknowledgements 123
A Glossary 125
A.1 Theory of Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B Mathematical Details of Dihedral Groups 129
B.1 Kronecker Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.2 Clebsch Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B.2.1 ...for D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130n
0B.2.2 ...for D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131n
B.3 Real Representation Matrices for D . . . . . . . . . . . . . . . . . . . . . . . . . 132n
C Results and Tables 135
C.1 Decomposition under Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
0C.2 Breaking Chains for D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140n
C.3 Possible Forms of V from Subgroup Mismatch. . . . . . . . . . . . . . . . . . . . 140
C.4 D Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417
C.5 D Flavon Potential at Next-to-Leading Order . . . . . . . . . . . . . . . . . . . 14314
Bibliography 150Chapter 1
Introduction
Who ordered that? Isidor Isaac Rabi’s exclamation upon the discovery of the muon is still
unanswered, more than fifty years later. And to top it off: Whoever ordered the muon found
that he/she liked it and decided to order a lot more.
Physicists may not have been able to answer the (possibly unanswerable) question concerning
the fundamental reason for the existence of the muon. At least the discovery of a full set of
threegenerations,thatisthreecopiesofeachfundamentalfermionoccurringinordinaryatomic
matter (including the neutrino), along with the precise experimental determination of the mass
hierarchies and the mixing among these generations, have allowed us to at least formulate the
question more precisely. The fundamental one remains: Why are there three generations of fer-
mions? The most satisfying answer we can give to date is that three generations is the minimal
number needed to allow for CP violation, a discovery for which the 2008 Nobel Prize in physics
was awarded to Makoto Kobayashi and Toshihide Maskawa.
But even if we humbly accept that three generations of fermions exist, we still need to explain
how the observed masses and mixings come to be. It is the answer to this question to which
this doctoral thesis will attempt to contribute a small part.
As the title of this thesis already suggests, our approach to this question is based on sym-
metries. The symmetry principle is one of the foundations of the Standard Model of Particle
Physics (SM). The symmetries of space-time and the internal gauge symmetries constrain the
allowedtermsintheStandardModelLagrangianinsuchawaythatonlyasmallnumberoffree
parameters is left to be determined from experiment. The success of this approach is spectac-
ular. And it is interesting that the SM has its weakest points where the symmetry principle is
faced with its limitations. For one, the electroweak gauge symmetry is broken in nature. This
breaking is performed in the SM through the Higgs Mechanism, which leaves many questions
unanswered, both experimentally (the Higgs boson is the only particle in the SM which has not
been observed in expt) and theoretically (most notably the hierarchy problem).
There is also one aspect of the Standard Model which remains virtually untouched by the sym-
metry principle altogether: the flavor sector, that is the masses of and the mixing among the
three fermion generations. This sector contains by far the majority of the free parameters in
the SM Lagrangian and can be considered the part of the model that most resembles a semi-
empirical hodgepodge, rather than a fundamental theory. Physicists have thus been looking for
a symmetry governing flavor for quite a while. In this thesis we join this search by analyzing
a large class of possible symmetries, the dihedral groups, with respect to their suitability as
symmetries of flavor.
To go about this, we will start by reviewing the Standard Model of Particle Physics (SM), as
far as it deals with flavor physics, in section 1.1. We review the origin of masses and mixing,
which are determined by a large number of free parameters in the SM. We go on to discuss
neutrino masses and mixing in section 1.2.
910 CHAPTER 1. INTRODUCTION
Chapter 2 is then devoted to presenting the general approach of this thesis towards the flavor
puzzle. Theexperimentalresultspresentedinchapter1areanalyzedinsection2.1. Someofthe
observed regularities cry out for an explanation, and we highlight some theoretical approaches
towardsexplainingthem, before, ininsection2.2, presentingourmethodofchoice, augmenting
the SM by a discrete non-abelian flavor symmetry G and thereby constraining the free flavorf
parameters of the SM and finding deeper reasons for non-trivial correlations among them. To
further illustrate this general description, we present a worked-out flavor symmetry model in
section 2.3. From these considerations, we are led to a systematic approach to the problem of
findingbotha suitable group G andthetransformation properties of the involvedfields, basedf
mainly on the existence of residual subgroups which remain exact after spontaneous breaking
of the flavor symmetry.
We perform such a systematic analysis for the set of single-valued and double-valued dihedral
groups in chapter 3. This analysis reveals a very interesting case of subgroup mismatch, dis-
cussed in c 4, first from a general perspective (section 4.1), then through the discussion
ofaninstance, whereithasbeenimplicitlyusedintheliterature(section4.2), andfinallyalong
the lines of a worked-out example, in which this subgroup mismatch leads to the prediction of
the Cabibbo angle (section 4.3). In chapter 5, we discuss how to separate the scales of elec-
troweak and flavor symmetry breaking, leading to models with G broken at a high scale byf
gauge singlet flavon fields. The dynamics involved in this breaking are discussed in a general
manner both for the SM (section 5.1) and for the Minimal Supersymmetric Standard Model
(MSSM),easilythemostpopularcandidateforphysicsbeyondtheSM(section5.2). Weobtain
the result that the conservation of the subgroups necessary for the subgroup mismatch, can be
obtained as a prediction from minimizing very general scalar potentials invariant under G . Allf
the elements gathered in the preceding sections are then put together in two exemplary models
in chapter 6. The first model is based on the group D and uses the subgroup mismatch to4
predict maximal atmospheric mixing in the lepton sector (section 6.1), the second one uses D14
for a prediction of the Cabibbo angle (section 6.2).
Wefinishwithanoutlookonthepossibleoriginofaflavorsymmetry G inchapter7, focussingf
on the possibility that the discrete flavor group may be a subgroup of a spontaneously broken
gauge symmetry, and finally offer our conclusions in chapter 8. A glossary of abbreviations
and group theory terms is given in appendix A, while additional mathematical information on
the dihedral groups can be found in appendix B and several bulky results have been moved to
appendix C. Parts of this thesis have already been published in [1–5].
1.1 Flavor in the Standard Model
The SM contains 45 Weyl fermions, all of which transform non-trivially under the gauge group
of the SM, SU(3) ×SU(2) ×U(1) . They can be divided into three generations of 15 Weylc L Y
fermions each. The transformation properties under the gauge group repeat themselves in each
generation and are given, for the first generation, in table 1.1. Only the names change for the
second generation, with ( c, s, ν , μ ) instead of ( u, d, ν , e), and (t, b, ν , τ) for the thirdμ e τ
(c)generation. When talking about all three generations, we will often alternatively write u ,i
(c) cd , ν , e and e with i = 1,2,3 and α = e,μ,τ, since for left-handed leptons the numericalα α ii
indices are reserved for the mass eigenstates, as we will discuss below. We will also use the
notation Q and l for the quark and lepton SU(2) doublets, respectively.i α L

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