Flavor Symmetries: Models and Implications

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1 Flavor Symmetries: Models and Implications Lisa L. Everett Nakatani, 1936 U. Wisconsin, Madison squared splittings and angles ra, Valle the first who made snow crystal in a laboratory *, The symmetry group of '()*)+,-./0+'1'3"&0+456(5.0+1'7 8 is D6, one of the finite groups.

2 Neutrino Oscillations: Introduction/Motivation P να ν β (L) = ij U iα U iβu jαu jβ e i m ij L E massive neutrinos observable lepton mixing First particle physics evidence for physics beyond SM! SM flavor puzzle ν SM flavor puzzle Ultimate goal: satisfactory and credible flavor theory (very difficult!)

3 The Data: Neutrino Masses fits: Schwetz, Tortola, Valle 08 Homestake, Kam, SuperK,KamLAND,SNO, SuperK, MINOS,miniBOONE,... m ij m i m j Assume: 3 neutrino mixing Solar: m = m 1 = ev Atmospheric: m 31 = ± ev (best fit ±1σ) Normal Hierarchy Cosmology (WMAP): 3 1 Inverted Hierarchy m i < 0.7 ev i 1 3

4 The Data: Lepton Mixing fit: Schwetz, Tortola, Valle 08 Homestake, Kam, SuperK,KamLAND,SNO, SuperK, Palo Verde, CHOOZ, MINOS... U MNSP = R 1 (θ )R (θ 13, δ MNSP )R 3 (θ )P U MNSP cos θ sin θ ɛ cos θ sin θ cos θ cos θ sin θ Maki, Nakagawa, Sakata Pontecorvo sin θ sin θ sin θ cos θ cos θ Solar: Atmospheric: Reactor: θ = θ 1 = 33.4 ± 1.4 θ = θ 3 = ɛ = sin θ 13, θ 13 = (best fit ±1σ) large angles, 1 small angle (no constraints on CP violation)

5 Compare: Quark Mixing U CKM = R 1 (θ CKM 3 )R (θ CKM 13, δ CKM )R 3 (θ CKM 1 ) Cabibbo; Kobayashi, Maskawa Mixing Angles: θ CKM 1 = 13.0 ± 0.1 Cabibbo angle θ c θ CKM 3 =.4 ± 0.1 θ CKM 13 = 0. ± 0.1 CP violation: J (CKM) CP J 10 5 J Im(U αi U βj U βiu αj) sin θ CKM 1 sin θ CKM 3 sin θ CKM 13 sin δ CKM δ CKM = 60 ± 14 3 small angles, 1 large phase Jarlskog Dunietz, Greenberg, Wu

6 A paradigm shift Strikingly different flavor patterns for quarks and leptons! Mass scales, hierarchies of neutral and charged fermions: Step 1 for theory: suppressing neutrino mass scale Mixing Angles: quarks small, leptons large, 1small Step for theory: understanding lepton mixing pattern

7 Step 1: Origin of Neutrino Mass Scale Charged Fermions: Dirac mass terms Y ij H ψ Li ψ Rj parametrized by Yukawa couplings Neutrinos: beyond physics of Yukawas! Assuming SM Higgs sector: L ν = Y νij LLi Hν Rj + λ ij Λ (L LiH)(L Lj H) + 1 (M ij ν Ri (ν Rj ) c + h.c.)

8 Majorana masses: Prototype: Type I seesaw Minkowski;Yanagida; Gell-Mann, Ramond, Slansky;... M ν = ( 0 m m M but many other possibilities... ) m O(100 GeV) M m m 1 m M m M m 1 ν 1, ν L,R + m M ν R,L Type II seesaw (Higgs triplets), Type III seesaw (triplet fermions), double seesaw, higher-dimensional operators, supersymmetry +R-parity violation,... Dirac masses: issue: Yukawa suppression Y ν 10 1 options: extra dimensions, flavor symmetries, supersymmetry breaking effects,... Note: in both cases, many mechanisms exploit SM singlet nature of ν R

9 Step : Origin of Large Lepton Mixings Standard paradigm: spontaneously broken flavor symmetry Recall for quarks: Y ij H ψ Li ψ Rj ( ϕ M ) nij H ψli ψ Rj Froggatt, Nielsen hierarchical masses, small mixings: continuous family symmetries CKM matrix: small angles and/or alignment flavon fields U CKM = U u U d 1 + O(λ) λ ϕ M Wolfenstein parametrization: λ sin θ c = 0. suggests Cabibbo angle may be a useful flavor expansion parameter

10 Flavor Model Building in the ν SM Main issue: what is U MNSP in limit of exact symmetry? for the leptons, large angles suggest U MNSP = U e U ν W + O(λ ) bare mixing angles (θ 0 1, θ 0 13, θ 0 3) flavor expansion parameter useful, and motivated in unified/string scenarios, to take λ = λ sin θ c ideas of Cabibbo haze and quark-lepton complementarity

11 Aside: Lepton Mixing Angles are non-generic Classify scenarios by the form of U MNSP in symmetry limit note: lepton mixing angle pattern has the most challenges (w/3 families) 3 small angles diagonal M ν 1 large, small RankM ν < 3 3 large angles anarchical M ν large, 1 small fine-tuning, non-abelian Issues: size of θ 13, origin of non-maximal θ 1 large angles may suggest discrete non-abelian family symmetries!

12 U MNSP = U e U ν W + O(λ ) ν Classify models by form of W(θ1, 0 θ13, 0 θ3) 0 : In general: θ 0 3 = 45 θ 0 13 = 0 (reasonable) More variety in choice of bare solar angle : θ 0 1 bi-maximal mixing tri-bimaximal mixing golden ratio mixing (quark-lepton complementarity) Harrison, Perkins, Scott (HPS) φ = (1 + 5)/ or other options...

13 Scenario 1. Bi-maximal Mixing bare solar angle θ 0 1 = 45 tan θ 0 1 = 1 U (BM) MNSP = Requires large perturbations: θ 1 = θ O(λ) π 4 θ c quark-lepton complementarity Raidal; Minakata, Smirnov; Frampton, Mohapatra; Xing; Ferrandis, Pakvasa; King; L.E., Ramond; Plentinger, Lindner; Dighe, Rodejohann, many, many others...

14 Bimaximal mixing scenarios: useful framework for exploring Cabibbo effects in quark+lepton sectors m u m t λ 8 m c m t λ 4 m b m d m b λ 4 m s m b λ m b m e m τ λ 5 m µ m τ λ 1 λ 3 (GUT scale) m τ m t θ CKM 1 λ θ CKM 3 λ θ CKM 13 λ 3 m m λ θ 13 O(λ) θ 3 < O(λ) but implementation in full grand unified theories: very challenging recent work in context of discrete non-abelian family symmetries Altarelli, Feruglio, and Merlo, 09,...

15 Scenario II. Tri-bimaximal (HPS) Mixing U (HPS) MNSP = bare solar angle tan θ1 0 = 1 θ1 0 = 35.6 Harrison, Perkins, Scott Does not require large perturbations! θ 1 = θ O(λ ) amusing note: MNSP looks like Clebsch-Gordan coeffs Meshkov; Zee,...

16 Naturally obtained from discrete non-abelian symmetries (subgroups of SO(3), SU(3) ) A Few Examples: A 4 (tetrahedron) Ma and collaborators (earliest in 01), Altarelli, Feruglio, Babu and He, Valle, Hirsch et al., King et al., many, many others... S 4 (cube) T (3n ) A 5 (icosahedron) Most popular scenario! Ma; Hagedorn, Lindner, Mohapatra; Cai, Yu; Zhang,... Aranda, Carone, Lebed; Chen, Mahanthappa,... Luhn, Nasri, Ramond; Ma; King, Ross,... L.E., Stuart many models, elegant results issues: incorporating quarks, vacuum alignment of flavon fields

17 Scenario III. Golden Ratio Mixing Idea: solar angle related to golden ratio φ = (1 + 5)/ Two proposed scenarios: tan θ 1 = 1 φ θ 1 = 31.7 L.E., Stuart 08, + work in progress icosahedral flavor symmetry I (A 5 ) cos θ 1 = φ θ 1 = 36 Adulpravitchai, Blum, Rodejohann 09 dihedral flavor symmetry D 10

18 Scenario III: tan θ 1 = 1 φ Ramond et al., 03 (footnote), Kajiyama, Raidal, Strumia 07 Z Z L.E. and Stuart, 08 and continuing... A 5 U (GR1) MNSP = φ 5 1 5φ φ 5 5φ φ 5 1 5φ A 5 isomorphic to icosahedral group, A 5 I I

elements: Rotations which take vertices to vertices, i.e., by 0, π 5, 4π 5, π 3, π Rotation by each angle forms

19 The (Rotational) Icosahedral Group, I ~ A5 Properties of the icosahedron: 0 faces 30 edges 1 vertices (equilateral triangles) (3 sides/triangle, triangles/edge) (3 vertices/triangle, 5 vertices/edge) Group elements: Rotations which take vertices to vertices, i.e., by 0, π 5, 4π 5, π 3, π Rotation by each angle forms a conjugacy class: e, 1C 5, 1C 5, 0C 3, 15C (Schoenflies: C k n = πk n rotation) order=number of elements: = 60

20 The (Rotational) Icosahedral Group, I ~ A5 Theorem: group order = sum of squares of irred. reps = 60 = (two triplets!) Conjugacy classes: characterized by trace (character) Character Table I e C 5 1 φ 1 φ C5 1 1 φ φ C C

21 The (Rotational) Icosahedral Group, I ~ A5 From character table, deduce tensor product decomposition: 3 3= = = = = = = = = = Not enough for flavor model building. Need explicit representations! I not a crystallographic point group, so there was work to be done...

22 Lepton Flavor Model Building with A5 Mass terms: L m = a ij M L ihl j H + Y (e) ij L i ē j H Charge assignments: L 3, ē 3 LL : 3 3 = 1 3 5, Lē : 3 3 = 4 5 Leading order: charged leptons massless, neutrinos degenerate... Fix it at higher order with flavon sector: ξ 5 ψ 5, χ 4 LL Lē L mass = α ijk M L ihl j Hξ k + β ijk M L iē j Hψ k + γ ijl M L iē j Hχ l

23 Lepton Flavor Model Building with A5 (continued) Explicit toy example with assumed flavon vevs: specific golden prediction for solar mixing angle, plus neutrinoless double beta decay: m ββ = m 1φ 5 + m φ 5. neutrino masses (normal hierarchy) hierarchical charged lepton masses In progress: dynamics of flavon sector, quark flavor mixing,... Rich and virtually unexplored model building territory!

24 Scenario III: cos θ 1 = φ Rodejohann 08, Adulpravitchai, Blum, and Rodejohann, 09 D 10 U (GR) MNSP = φ 1 φ 0 5 φ φ 1 5 φ φ 1 complete flavor theory based on dihedral symmetry (solar angle prediction based on exterior angle of decagon)

25 Conclusions/Outlook The ν flavor puzzle is intriguing and very rich: Many options for neutrino mass scale suppression, each with implications for particle/astroparticle physics Many theoretically motivated mixing patterns: Bi-maximal, tri-bimaximal, mixing, golden ratio,... Themes: Dirac v. Majorana? role of family symmetries? quark-lepton unification? Data will of course continue to be crucial! May provide our best window to ultrahigh scale physics!

Lepton Flavor and CPV

Lepton Flavor and CPV Alexander J. Stuart 25 May 2017 Based on: L.L. Everett, T. Garon, and AS, JHEP 1504, 069 (2015) [arxiv:1501.04336]; L.L. Everett and AS, arxiv:1611.03020 [hep-ph]. The Standard Model