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Phenomenology of Discrete Flavour Symmetries

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Phenomenology of Discrete Flavour Symmetries Universita` degli studi di Padova Facolta` di Scienze MM.FF.NN. Dipartimento di Fisica \G. Galilei" SCUOLA DI DOTTORATO DI RICERCA IN FISICA CICLO XXII Phenomenology of Discrete Flavour Symmetries Coordinatore: Ch.mo Prof. ATTILIO STELLA arXiv:1004.2211v3 [hep-ph] 15 Dec 2010 Supervisore: Ch.mo Prof. FERRUCCIO FERUGLIO Dottorando: Dott. LUCA MERLO February 1, 2010 Abstract The flavour puzzle is an open problem both in the Standard Model and in its possible supersymmetric or grand unified extensions. In this thesis, we discuss possible explana- tions of the origin of fermion mass hierarchies and mixings by the use of non-Abelian discrete flavour symmetries. We present two realisations in which the flavour symmetry 0 contains either the double-valued group T or the permutation group S4: the spontaneous breaking of the flavour symmetry produces realistic fermion mass hierarchies, the lepton 2 2 mixing matrix close to the so-called tribimaximal pattern (sin θ12 = 1=3, sin θ23 = 1=2 and θ13 = 0) and the quark mixing matrix comparable to the Wolfenstein parametrisation. The exact tribimaximal scheme deviates from the experimental best-fit angles for 0 values at most of the 1σ level. In the T - and S4-based models, the symmetry breaking accounts for such discrepancies, by introducing corrections to the tribimaximal pattern of the order of λ2, being λ the Cabibbo angle. On the experimental side, the present measurements do not exclude θ13 ∼ λ and therefore, if it is found that θ13 is close to its present upper bound, this could be interpreted as an indication that the agreement with the tribimaximal mixing is accidental. Then a scheme where instead the bimaximal mixing 2 2 (sin θ12 = 1=2, sin θ23 = 1=2 and θ13 = 0) is the correct first approximation modified by terms of O(λ) could be relevant. This recalls the well-known empirical quark-lepton complementarity, for which θ12 + λ ∼ π=4. We present a flavour model based on the spontaneous breaking of the S4 discrete group which naturally leads to the bimaximal mixing at the leading order and, after the introduction of the breaking terms, to θ13 ∼ λ and θ12 + O(λ) ∼ π=4, which we call \weak" complementarity relation. Masses and mixings are evaluated at a very high energy scale and for a comparison with experimental measurements we illustrate a general analysis on the stability under the renormalisation group running to evolve these observables to low energies. We consider also the constraints on flavour violating processes arising from introduc- ing a flavour symmetry: in particular we concentrate on the lepton sector, analysing some lepton flavour violating decays and the discrepancy between the theoretical prediction and the experimental measurement of the anomalous magnetic moment of the muon. We develop the study both in the Standard Model scenario and in its minimal super- symmetric extension, using at first an effective operator approach and then a complete loop computation. Interesting hints for the scale of New Physics and for the forthcoming experimental results from LHC are found. Finally we discuss the impact of an underlining flavour symmetry on leptogenesis in order to explain the baryon asymmetry of the universe. Riassunto della Tesi Lo studio del sapore nella fisica particellare `etutt'oggi un problema aperto sia nel Modello Standard sia nelle sue estensioni supersimmetriche o grande unificate. In questa tesi, affrontiamo la questione dell'origine della gerarchia di massa nei fermioni e dei loro angoli di mescolamento, utilizzando simmetrie discrete di sapore non Abeliane. In partico- lare, illustriamo due modelli in cui la simmetria di sapore contiene il gruppo T 0 o il gruppo S4: la rottura spontanea della simmetria di sapore produce come effetto delle gerarchie di massa realistiche per i fermioni, la matrice di mescolamento leptonica con la cosiddetta 2 2 struttura tribimassimale (sin θ12 = 1=3, sin θ23 = 1=2 e θ13 = 0) con piccole correzioni e la matrice di mescolamento per i quark che ben si confronta con la parametrizzazione di Wolfenstein. La struttura tribimassimale presenta delle deviazioni al massimo ad 1σ dai valori cen- 0 trali trovati sperimentalmente. Nei modelli basati sui gruppi T e S4, la rottura della sim- metria compensa a queste piccole deviazioni, introducendo delle correzioni alla struttura tribimassimale dell'ordine di λ2, dove λ rappresenta l'angolo di Cabibbo. Sperimental- mente, le misure attuali non escludono θ13 ∼ λ e quindi se il valore dell'angolo di reattore risulter`avicino al suo attuale limite superiore, questo potrebbe essere interpretato come un'indicazone che l'accordo con la struttura tribimassimale `esolo accidentale. In questo 2 2 caso, la struttura bimassimale (sin θ12 = 1=2, sin θ23 = 1=2 and θ13 = 0) potrebbe essere in prima approssimazione una migliore scelta se poi intervengono delle correzioni dell'ordine di λ. Questo meccanismo ricorda l'osservazione del tutto empirica per cui θ12 + λ ∼ π=4, che va sotto il nome di relazione di complementariet`a.Studiamo questa alternativa in un modello basato sulla rottura spontanea del gruppo S4 che presenta la struttura bimassimale in prima approssimazione e, dopo l'introduzione dei termini di rottura, θ13 ∼ λ e θ12 + O(λ) ∼ π=4, che chiamiamo complementariet`a\debole". In questi modelli, le masse e gli angoli di mescolamento sono tipicamente studiati a energie molto alte e per il confronto con le misure sperimentali sviluppiamo uno studio sulla stabilit`adi questi osservabili durante l'evoluzione a bassa scala dovuta al gruppo di rinormalizzazione. Inotre consideriamo i limiti su processi con violazione di sapore che sorgono dall'uso di una simmetria di sapore: in particolare analizziamo alcuni decadimenti con violazione di sapore leptonico e la discrepanza tra la predizione teorica e la misura sperimentale del momento magnetico anomalo del muone. Sviluppiamo l'analisi sia nel Modello Stan- dard sia nella sua estensione supersimmetrica minimale, usando prima un approccio di Lagrangiana efficace e poi uno studio quantistico a un loop. Troviamo interessanti indi- cazioni sulla scala di energia della nuova fisica, specialmente in previsione dei prossimi risultati a LHC. In fine discutiamo l'impatto dell'introduzione di una simmetria di sapore sulla lepto- genesi, utilizzata per spiegare l'asimmetria barionica nell'universo. Contents Introduction V 1 The Standard Model and Beyond 1 1.1 The Standard Model and the Neutrino Masses . .1 1.1.1 The See-Saw Mechanisms . .3 1.1.2 The Physical Basis and the Mixing Matrices . .7 1.2 Supersymmetry . 14 1.3 Grand Unified Theories . 18 2 The Flavour Puzzle 23 2.1 The Minimal (Lepton) Flavour Violation . 25 2.2 Flavour Structure Approach . 27 2.2.1 The Bimaximal Mixing Pattern . 29 2.2.2 The Tribimaximal Mixing Pattern . 30 2.3 Flavour Symmetry Overview . 31 2.3.1 Discrete Flavour Symmetries . 32 2.3.2 The Flavour Symmetry Breaking . 34 3 Flavour Models with the Tribimaximal Mixing 37 3.1 A4-Based Model . 39 3.1.1 The Neutrino Mass Spectrum . 42 3.1.2 The Next-To-Leading Order Contributions . 44 3.1.3 Type I See-Saw Realisation . 46 3.1.4 Extension to Quarks . 49 3.2 T 0-Based Model . 50 3.2.1 Outline of the Model . 51 3.2.2 Fermion Masses and Mixings at the Leading Order . 54 3.2.3 Higher-Order Corrections . 56 3.3 S4-Based Model . 59 3.3.1 The Lepton Sector . 61 3.3.2 The Quark Sector . 65 I 3.3.3 Higher Order Corrections . 67 3.3.4 See-Saw Extensions . 68 3.4 Conclusions of the Chapter . 69 4 A Flavour Model with the Bimaximal Mixing 71 4.1 The Structure of the Model . 72 4.1.1 The Light Neutrino Spectrum and the Value of r .......... 76 4.2 The Next-To-Leading Order Corrections . 77 4.3 Phenomenological Implications . 79 4.4 Extension to Quarks: a GUT Realisation . 81 4.5 Conclusions of the Chapter . 84 5 Running Effects on Flavour Models 85 5.1 Running Effects on Neutrino Mass Operator mν ............... 86 5.1.1 Analytical Approximation to the Running Evolution of mν ..... 88 5.2 Flavour Symmetries and Running Effects . 90 5.2.1 Running Effects on Neutrino Mixing Patterns . 91 5.2.2 Running Effects in the Charged Lepton Sector . 94 5.2.3 Full Running Effects on the Tribimaximal Mixing Pattern . 95 5.3 Running Effects in the Altarelli-Feruglio Model . 97 5.3.1 Running of the Angles . 97 5.3.2 Running of the Phases . 99 5.4 Conclusions of the Chapter . 100 6 Flavour Violation 101 6.1 Low-Energy Effective Approach . 103 6.1.1 The Dipole Matrix . 104 6.1.2 Dipole Transitions . 105 6.1.3 Comparison with Minimal Flavour Violation . 108 6.2 Explicit Supersymmetric A4 Model . 110 6.3 Slepton Masses in the Physical Basis . 112 6.4 Results in the Mass Insertion Approximation . 114 6.4.1 Analytic results . 115 6.5 Numerical analysis . 117 6.5.1 Framework . 118 6.5.2 Results for Radiative Leptonic Decays . 119 6.5.3 Results for Anomalous Magnetic Moment of the Muon . 123 6.6 Conclusions of the Chapter . 124 II 7 Leptogenesis 127 7.1 The Basic Framework . 128 7.1.1 General Remarks on Leptogenesis . 129 7.2 CP asymmetry and Exact U0 Mixing Without any FS . 129 c 7.3 Implications of Flavour Symmetries on να .................. 130 7.4 Conclusions of the Chapter . 132 8 Summary and Final Remarks 133 Acknowledgments 135 A Group Theory Details 137 A.1 The Group A4 .................................. 137 A.2 The Group T 0 .................................. 139 A.3 The Group S4 { I Version . 141 A.4 The Group S4 { II Version . 143 B Vacuum Alignment 147 B.1 The Group T 0 .................................
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