Flavor Symmetries: Models and Implications

Neutrino mass squared splittings and angles Lisa L. Everett Nakatani, 1936 Talks by Mohapatra, Valle U. Wisconsin, Madison the ﬁrst who made snow crystal in a laboratory

!"#$%& '()*#+*,

Absolute neutrino mass scale? The symmetry group of

!"#"$#%""& '()*)+,-./0+'1'23"&0+456(5.0+1'7 8 is D6 , one of the ﬁnite groups. Introduction/Motivation

Neutrino Oscillations:

2 i∆mij L 2E να νβ (L) = iα i∗β j∗α jβe− P → U U U U ij ! 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!) fits: Schwetz, Tortola, Valle ’08 The Data: Neutrino Masses

Homestake, Kam, SuperK,KamLAND,SNO, SuperK, MINOS,miniBOONE,...

∆m2 m2 m2 Assume: 3 neutrino mixing ij ≡ i − j

2 2 +0.23 5 2 Solar: ∆m = ∆m12 = 7.65 0.20 10− eV ! | | − × (best ﬁt 1 σ ) 2 +0.12 3 2 ± Atmospheric: ∆m31 = 2.4 0.11 10− eV ± − ×

Normal Hierarchy Inverted Hierarchy 3 2 1 2 1 3

Cosmology (WMAP): mi < 0.7 eV i ! ﬁt: Schwetz, Tortola, Valle ’08 The Data: Lepton Mixing Homestake, Kam, SuperK,KamLAND,SNO, SuperK, Palo Verde, CHOOZ, MINOS...

MNSP = 1(θ ) 2(θ13, δMNSP) 3(θ ) Maki, Nakagawa, Sakata U R ⊕ R R " P Pontecorvo

cos θ sin θ " ! ! MNSP cos θ sin θ cos θ cos θ sin θ |U | !  − ⊕ ! ⊕ ! ⊕  sin θ sin θ sin θ cos θ cos θ  ⊕ ! − ⊕ ! ⊕   

Solar: θ = θ12 = 33.4◦ 1.4◦ ! ± +4.0 (best ﬁt 1 σ ) Atmospheric: θ = θ23 = 45.0◦ 3.4 ± ⊕ − +3.5 Reactor: ! = sin θ13, θ13 = 5.7◦ 5.7 − 2 large angles, 1 small angle (no constraints on CP violation) Compare: Quark Mixing

Cabibbo; Kobayashi, Maskawa = (θCKM) (θCKM, δ ) (θCKM) UCKM R1 23 R2 13 CKM R3 12

CKM Mixing Angles: θ = 13.0◦ 0.1◦ Cabibbo angle θ 12 ± c CKM θ = 2.4◦ 0.1◦ 23 ± CKM θ = 0.2◦ 0.1◦ 13 ± Jarlskog J Im( ∗ ∗ ) CP violation: ≡ UαiUβjUβiUαj Dunietz, Greenberg, Wu J (CKM) sin 2θCKM sin 2θCKM sin 2θCKM sin δ CP ! 12 23 13 CKM 5 J 10− δCKM = 60◦ 14◦ ∼ ± 3 small angles, 1 large phase 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 2 large, 1small

Step 2 for theory: understanding lepton mixing pattern Step 1: Origin of Neutrino Mass Scale

Charged Fermions: Dirac mass terms

Y H ψ¯ ψ ij · Li Rj parametrized by Yukawa couplings

Neutrinos: beyond physics of Yukawas!

Assuming SM Higgs sector: λ 1 = Y L¯ Hν + ij (L H)(L H) + (M ν¯ (ν )c + h.c.) −Lν νij Li Rj Λ Li Lj 2 ij Ri Rj Majorana masses: Minkowski;Yanagida; Prototype: Type I seesaw Gell-Mann, Ramond, Slansky;...

m2 m (100 GeV) m1 0 m ∼ O ∼ M = ν m M M m m2 M m1 M ! ∼ " m ! " ν ν + ν 1,2 ∼ L,R M R,L but many other possibilities... Type II seesaw (Higgs triplets), Type III seesaw (triplet fermions), double seesaw, higher-dimensional operators, supersymmetry +R-parity violation,...

12 Dirac masses: issue: Yukawa suppression Y 10− ν ∼ options: extra dimensions, ﬂavor symmetries, supersymmetry breaking effects,...

Note: in both cases, many mechanisms exploit SM singlet nature of νR Step 2: Origin of Large Lepton Mixings

Standard paradigm: spontaneously broken flavor symmetry ϕ nij Froggatt, Nielsen Y H ψ¯ ψ H ψ¯LiψRj ij · Li Rj M · ! " “flavon” fields Recall for quarks: hierarchical masses, small mixings: continuous family symmetries • CKM matrix: small angles and/or alignment

CKM = u † 1 + (λ) ϕ d λ U U U ∼ O ∼ M Wolfenstein parametrization: λ sin θ = 0.22 ≡ c suggests Cabibbo angle may be a useful ﬂavor expansion parameter Flavor Model Building in the ν SM

Main issue: what is MNSP in limit of exact symmetry? • U for the leptons, large angles suggest

= † + (λ!) UMNSP UeUν ∼ W O “bare” mixing angles ﬂavor expansion 0 0 0 parameter (θ12, θ13, θ23)

useful, and motivated in uniﬁed/string scenarios, to take

λ! = λ sin θ ≡ c ideas of “Cabibbo haze” and quark-lepton complementarity Aside: Lepton Mixing Angles are “non-generic”

Classify scenarios by the form of MNSP in symmetry limit U note: lepton mixing angle pattern has the most challenges (w/3 families)

3 small angles diagonal ν ∼ M 1 large, 2 small Rank ν < 3 • M large angles “anarchical”

large, small ﬁne-tuning, non-Abelian

Issues: size of θ 13 , origin of non-maximal θ 12

large angles may suggest discrete non-Abelian family symmetries! ν = † + (λ!) UMNSP UeUν ∼ W O

Classify models by form of ( θ 0 , θ 0 , θ 0 ) : W 12 13 23

0 0 In general: θ = 45◦ θ = 0 (reasonable) • 23 13 ◦

0 More variety in choice of bare solar angle θ 12 :

“bi-maximal” mixing (quark-lepton complementarity) “tri-bimaximal” mixing Harrison, Perkins, Scott (HPS) “golden ratio” mixing φ = (1 + √5)/2

or other options... Scenario 1. Bi-maximal Mixing

0 0 “bare” solar angle θ12 = 45◦ tan θ12 = 1

1 1 0 √2 − √2 (BM) 1 1 1 =   UMNSP 2 2 − √2 1 1 1  2 2   √2    Requires large perturbations: “quark-lepton 0 π θ12 = θ + (λ) θc complementarity” 12 O ∼ 4 −

Raidal; Minakata, Smirnov; Frampton, Mohapatra; Xing; Ferrandis, Pakvasa; King; L.E., Ramond; Plentinger, Lindner; Dighe, Rodejohann, many, many others... Bimaximal mixing scenarios: useful framework for exploring Cabibbo effects in quark+lepton sectors

m md me 5 2 u λ8 λ4 λ ∆m m ∼ mb ∼ mτ ∼ 2! λ t !∆m ∼ m m m ⊕ c λ4 s λ2 µ λ2 mt ∼ mb ∼ mτ ∼ θ (λ) 13 ≤ O mb mb 3 1 λ (GUT scale) ∆θ23 < (λ) mτ ∼ mt ∼ O θCKM λ θCKM λ2 θCKM λ3 12 ∼ 23 ∼ 13 ∼ but implementation in full grand uniﬁed theories: very challenging

recent work in context of discrete non-Abelian family symmetries Altarelli, Feruglio, and Merlo, ’09,... Scenario II. Tri-bimaximal (HPS) Mixing

0 1 0 “bare” solar angle tan θ = θ12 = 35.26◦ 12 √2 Harrison, Perkins, Scott ’02

2 1 0 3 √3 (HPS) − =  $1 1 1  UMNSP √6 √3 − √2  1 1 1   √6 √3 √2   

Does not require large perturbations! θ = θ0 + (λ2) 12 12 O amusing note: MNSP looks like Clebsch-Gordan coeffs Meshkov; Zee,... Naturally obtained from discrete non-Abelian symmetries (subgroups of S O (3) , S U (3) )

Ma and collaborators (earliest in ’01), Altarelli, Feruglio, A Few Examples: 4 Babu and He, Valle, Hirsch et al., King et al., (tetrahedrA on) many, many others...

4 Ma; Hagedorn, Lindner, Mohapatra; Cai, Yu; Zhang,... (cube)S

! Aranda, Carone, Lebed; Chen, Mahanthappa,... T 2 ∆(3n ) Luhn, Nasri, Ramond; Ma; King, Ross,... 5 L.E., Stuart (icosahedrA on) Most popular scenario! many models, elegant results issues: incorporating quarks, “vacuum alignment” of ﬂavon ﬁelds Scenario III. Golden Ratio Mixing

Idea: solar angle related to “golden ratio” φ = (1 + √5)/2

Two proposed scenarios: • 1 tan θ = θ = 31.72 L.E., Stuart ’08, + 12 φ 12 ◦ work in progress icosahedral ﬂavor symmetry ( ) I A5

φ Adulpravitchai, Blum, cos θ12 = θ = 36 2 12 ◦ Rodejohann ’09 dihedral ﬂavor symmetry D10 1 tan θ = Scenario III: 12 φ

Ramond et al., ’03 (footnote), Kajiyama, Raidal, Strumia ’07 Z2 × Z2 L.E. and Stuart, ’08 and continuing... A5

φ 1 0 √5 − √5φ (GR1) $ $ φ =  1 1 1 1  UMNSP √2 √5φ √2 √5 − √2    1 $ 1 1 $ φ 1     √2 √5φ √2 √5 √2   $ $ 

isomorphic to icosahedral group, 5 I A ! I The (Rotational) Icosahedral Group, I ~ A5 Properties of the icosahedron: 20 faces (equilateral triangles) 30 edges (3 sides/triangle, 2 triangles/edge) 12 vertices (3 vertices/triangle, 5 vertices/edge)

Group elements: 2π 4π 2π Rotations which take vertices to vertices, i.e., by 0, , , , π 5 5 3 Rotation by each angle forms a conjugacy class: 2πk e, 12C , 12C2, 20C , 15C (Schoenﬂies: Ck = rotation) 5 5 3 2 n n

order=number of elements: 1 + 12 + 12 + 15 + 20 = 60 The (Rotational) Icosahedral Group, I ~ A5

Theorem: group order = sum of squares of irred. reps 1 + 12 + 12 + 15 + 20 = 60 = 12 + 32 + 32 + 42 + 52. (two triplets!)

Conjugacy classes: characterized by trace (character)

Character Table

1 3 3! 4 5 I e 1 3 3 4 5 12C 1 φ 1 φ -1 0 5 − 12C2 1 1 φ φ -1 0 5 − 20C3 1 0 0 1 -1 15C2 1 -1 -1 0 1 The (Rotational) Icosahedral Group, I ~ A5 From character table, deduce tensor product decomposition: 3 3=1 3 5 ⊗ ⊕ ⊕ 3! 3! =1 3! 5 ⊗ ⊕ ⊕ 3 3!=4 5 ⊗ ⊕ 3 4=3! 4 5 ⊗ ⊕ ⊕ 3! 4=3 4 5 ⊗ ⊕ ⊕ 3 5=3 3! 4 5 ⊗ ⊕ ⊕ ⊕ 3! 5=3 3! 4 5 ⊗ ⊕ ⊕ ⊕ 4 4=1 3 3! 4 5 ⊗ ⊕ ⊕ ⊕ ⊕ 4 5=3 3! 4 5 5 ⊗ ⊕ ⊕ ⊕ ⊕ 5 5=1 3 3! 4 4 5 5 ⊗ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Not enough for ﬂavor model building. Need explicit representations! I not a crystallographic point group, so there was work to be done... Lepton Flavor Model Building with A5

a Mass terms: = ij L HL H + Y (e)L e¯ H −Lm M i j ij i j

Charge assignments: L 3, e¯ 3! → → LL : 3 3 = 1 3 5, Le¯ : 3 3! = 4 5 ⊗ ⊕ ⊕ ⊗ ⊕ Leading order: charged leptons massless, neutrinos degenerate... Fix it at higher order with ﬂavon sector: χ 4 ξ 5 ψ 5, → LL→ →Le¯ α β γ = ijk L HL Hξ + ijk L e¯ Hψ + ijl L e¯ Hχ −Lmass M 2 i j k M i j k M i j l 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:

m1φ m2 mββ = + . √5 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! φ cos θ = Scenario III: 12 2

Rodejohann ’08, Adulpravitchai, Blum, and Rodejohann, ’09 D10

φ 1 √5 0 2 − 2 φ (GR2) $φ =  1 5 1  UMNSP 2 2φ 2√2 − √2    1 $ 5 φ 1   2 2φ √ √   2 2 2   $ 

complete ﬂavor theory based on dihedral symmetry

(solar angle prediction based on exterior angle of decagon) 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!