Models of Neutrino Masses in View of the Large 13 Discovery Mu-Chun - - PowerPoint PPT Presentation

Models of Neutrino Masses in View of the Large θ13 Discovery

Mu-Chun Chen, University of California at Irvine

Flavor Physics and CP Violation (FPCP2013), Búzios, Brazil, May 19-24, 2013

Where Do We Stand?

• Exciting Time in ν Physics: recent hints of large θ13 from T2K, MINOS, Double Chooz, Daya

Bay and RENO

• Latest 3 neutrino global analysis (including recent results from reactor experiments):

2

Fogli, Lisi, Marrone, Montanino, Palazzo, Rotunno (2012)

P(νa νb) = ⇤ ⇤ νb|ν, t ⇥⇤ ⇤2 ⇥ sin2 2θ sin2 ⌅∆m2 4E L ⇧

Parameter Best fit 1σ range 2σ range 3σ range δm2/10−5 eV2 (NH or IH) 7.54 7.32 – 7.80 7.15 – 8.00 6.99 – 8.18 sin2 θ12/10−1 (NH or IH) 3.07 2.91 – 3.25 2.75 – 3.42 2.59 – 3.59 ∆m2/10−3 eV2 (NH) 2.43 2.33 – 2.49 2.27 – 2.55 2.19 – 2.62 ∆m2/10−3 eV2 (IH) 2.42 2.31 – 2.49 2.26 – 2.53 2.17 – 2.61 sin2 θ13/10−2 (NH) 2.41 2.16 – 2.66 1.93 – 2.90 1.69 – 3.13 sin2 θ13/10−2 (IH) 2.44 2.19 – 2.67 1.94 – 2.91 1.71 – 3.15 sin2 θ23/10−1 (NH) 3.86 3.65 – 4.10 3.48 – 4.48 3.31 – 6.37 sin2 θ23/10−1 (IH) 3.92 3.70 – 4.31 3.53 – 4.84 ⊕ 5.43 – 6.41 3.35 – 6.63 δ/π (NH) 1.08 0.77 – 1.36 — — δ/π (IH) 1.09 0.83 – 1.47 — —

Theoretical Challenges

(i) Absolute mass scale: Why mν << mu,d,e?

• seesaw mechanism: most appealing scenario ⇒ Majorana
• GUT scale (type-I, II) vs TeV scale (type-III, double seesaw)
• TeV scale new physics (extra dimension, U(1)´) ⇒ Dirac or Majorana

(ii) Flavor Structure: Why neutrino mixing large while quark mixing small?

• neutrino anarchy: no parametrically small number
• near degenerate spectrum, large mixing
• predictions strongly depend on choice of statistical measure
• still alive and kicking
• family symmetry: there’s a structure, expansion parameter (symmetry effect)
• mixing result from dynamics of underlying symmetry
• for leptons only (normal or inverted)
• for quarks and leptons: quark-lepton connection ↔ GUT (normal)
• Alternative?
• In this talk: assume 3 generations, no LSND/MiniBoone/Reactor Anomaly

3

Hall, Murayama, Weiner (2000); de Gouvea, Murayama (2003) de Gouvea, Murayama (2012)

Planck 2013 Data Release: Neff = 3.26 ± 0.35 ⇒ sterile neutrino marginally consistent

Origin of Mass Hierarchy and Mixing

• In the SM: 22 physical quantities which seem unrelated
• Question arises whether these quantities can be related
• No fundamental reason can be found in the framework of SM
• less ambitious aim ⇒ reduce the # of parameters by imposing symmetries
• SUSY Grand Unified Gauge Symmetry
• GUT relates quarks and leptons: quarks & leptons in same GUT multiplets
• one set of Yukawa coupling for a given GUT multiplet ⇒ intra-family relations
• seesaw mechanism naturally implemented
• Family Symmetry
• relate Yukawa couplings of different families
• inter-family relations ⇒ further reduce the number of parameters

4

eV keV MeV GeV TeV meV

t c u b s d µ ! e

"1 "2 "3 "2 "1 "1 "3 "2 "3 normal hierarchy inverted hierarchy nearly degenerate

Mass spectrum of elementary particles

LMA-MSW solution

⇒ Experimentally testable correlations among physical observables

Quarks vs Leptons, CKM vs PMNS

• Quark mixings are small
• Lepton mixings are large
• How to realize this when quarks and leptons are unified??

5

0.12 - 0.17

Origin of Flavor Mixing and Mass Hierarchy

• Several models have been constructed based on
• GUT Symmetry [SU(5), SO(10)] ⊕ Family Symmetry GF
• Family Symmetries GF based on continuous groups:
• U(1)
• SU(2)
• SU(3)
• Recently, models based on discrete family symmetry groups have been constructed
• A4 (tetrahedron)
• T´ (double tetrahedron)
• S3 (equilateral triangle)
• S4 (octahedron, cube)
• A5 (icosahedron, dodecahedron)
• ∆27
• Q4

6 u u u d d d e e e s s s t t t b b b ! ! !µ

µ µ

" " " µ µ µ ! ! !"

" "

c c c ! ! !e

e e

SU(2)F SU(10)

GUT Symmetry SU(5), SO(10), ... family symmetry (T′, SU(2), ...)

Motivation: Tri-bimaximal (TBM) neutrino mixing

Discrete gauge anomaly: Araki, Kobayashi, Kubo, Ramos-Sanchez, Ratz, Vaudrevange (2008)

Anomaly-free discrete R-symmetries: simultaneous solutions to mu problem and proton decay problem, naturally small Dirac neutrino mass, M.-C.C, M. Ratz, C. Staudt, P . Vaudrevange, (2012)

Tri-bimaximal Neutrino Mixing

• Neutrino Oscillation Parameters
• Latest Global Fit (3σ)
• Tri-bimaximal Mixing Pattern
• Leading Order: TBM (from symmetry) + holomorphic Corrections/contributions
• Is TBM still a good starting point?

Harrison, Perkins, Scott (1999)

ts sin2 θatm, TBM = 1/2 an

ts sin2 θ⇥,TBM = 1/3

⇤ ⌥ d sin θ13,TBM = 0.

⇧ UMNS =

• ⇤

1 c23 s23 −s23 c23 ⇥ ⌅

• ⇤

c13 s13e−iδ 1 −s13eiδ c13 ⇥ ⌅

• ⇤

c12 s12 −s12 c12 1 ⇥ ⌅

P(νa νb) = ⇤ ⇤ νb|ν, t ⇥⇤ ⇤2 ⇥ sin2 2θ sin2 ⌅∆m2 4E L ⇧

7

Fogli, Lisi, Marrone, Montanino, Palazzo, Rotunno (2012)

sin2 θatm = 0.386 (0.331 − 0.637)

sin2 θ = 0.307 (0.259 − 0.359)

sin2 θ13 = 0.0241 (0.0169 − 0.0313)

Tri-bimaximal Neutrino Mixing

8

θ12 θ13 θ23 TBM prediction: arctan √ 0.5

• ≈ 35.3◦

45◦ Best fit values (±1σ):

• 33.6+1.1

−1.0

• 8.93+0.46

−0.48

• 38.4+1.4

−1.2

• Fogli, Lisi, Marrone, Montanino, Palazzo, Rotunno, 2012

Non-Abelian Finite Family Symmetry A4

• TBM mixing matrix: can be realized with finite group family

symmetry based on A4

• A4: even permutations of 4 objects

S: (1234) → (4321) T: (1234) → (2314)

• a group of order 12
• Invariant group of tetrahedron

9

Ma, Rajasekaran (2001); Babu, Ma, Valle (2003); ...

Invariant Group of Tetrahedron T: (1234) → (2314) S: (1234) →(4321)

10

[Animation Credit: Michael Ratz]

Non-Abelian Finite Family Symmetry A4

• TBM mixing matrix: can be realized with finite group family

symmetry based on A4

• A4: even permutations of 4 objects

S: (1234) → (4321) T: (1234) → (2314)

• a group of order 12
• Invariant group of tetrahedron
• Problem: A4 does not give rise to quark mixing

11

Ma, Rajasekaran (2001); Babu, Ma, Valle (2003); ...

GUT Compatibility ⇒ SU(5) x T´

• Double Tetrahedral Group T´
• Symmetries ⇒ 10 parameters in Yukawa sector ⇒ 22 physical observables
• neutrino mixing angles from group theory (CG coefficients)
• TBM: misalignment of symmetry breaking patterns
• neutrino sector: T’ → GTST2 , charged lepton sector: T’ → GT
• GUT symmetry ⇒ deviation from TBM related to quark mixing θc
• complex CG’s of T´ ⇒ Novel Origin of CP Violation
• CP violation in both quark and lepton sectors entirely from group theory
• connection between leptogenesis and CPV in neutrino oscillation

12

M.-C.C, K.T. Mahanthappa

• Phys. Lett. B652, 34 (2007);
• Phys. Lett. B681, 444 (2009)

M.-C.C, K.T. Mahanthappa,

• Phys. Lett. B681, 444 (2009)

Predictions: a SUSY SU(5) x T´ Model

• Charged Fermion Sector: 7 parameters ⇒ 9 masses, 3 angles, 1 phase

Georgi-Jarlskog relations at GUT scale ⇒ Vd,L ≠ I

⇧e

12 ⇧

↵ me mµ ⇧ 1 3 ↵md ms ⇤ 1 3⇧c . the tri-bimaximal mixing pattern

⇧ ⌦ y θc ⇧ ⇤ ⇤⌦ md/ms eiα⌦ mu/mc ⇤ ⇤ ⇤ ⌦ md/ms, wh ling constants. Even though is of the size of the

13

M.-C.C, K.T. Mahanthappa

• Phys. Lett. B652, 34 (2007);
• Phys. Lett. B681, 444 (2009)

spinorial representations in charged fermion sector ⇒ complex CGs ⇒ CPV in quark and lepton sectors

ly, md ⌃ 3me, y, mµ ⌃ 3ms is

SU(5) ⇒ Md = (Me)T ⇒ corrections to TBM related to θc

quark CP phase: γ = 45.6 degrees

Predictions: a SUSY SU(5) x T´ Model

• Neutrino Sector:

(2+1 parameters)

• Prediction for MNS matrix: (for )
• sum rule among absolute masses:

tan2 θ⇤ ⌃ tan2 θ⇤,T BM + 1 2θc cos δ

neutrino mixing angle

1/2

quark mixing angle complex CGs: leptonic Dirac CPV

⌅13 ⌅ ⌅c/3 ⇧ 2

CGs of SU(5) & T´

⇒ connection between leptogenesis & leptonic CPV at low energy

14

normal hierarchy predicted

M.-C.C, K.T. Mahanthappa

• Phys. Lett. B652, 34 (2007);
• Phys. Lett. B681, 444 (2009)

 ✓ ◆ MD = @ 2ξ0 + η0 −ξ0 −ξ0 + η00 −ξ0 2ξ0 + η00 −ξ0 + η0 −ξ0 + η00 −ξ0 + η0 2ξ0 1 A ζ0ζ0

0vu

MRR = @ 1 1 1 1 A s0Λ

η00

0 = 0

> 0

Numerical Results: Neutrino Sector

• For :
• Neutrino Masses:
• Leptonic CP violation from CG coefficients:

⇧ |UMNS| =   0.838 0.542 0.0583 0.362 0.610 0.705 0.408 0.577 0.707  

prediction for Dirac CP phase: δ = 197 degrees (in standard parametrization) 3 independent parameters in neutrino sector predicted 3 masses and 3 angles: all agree with exp within 1σ

15

DD 0.824259 0.542816 0.161084 0.264063 0.609846 0.747234 0.500867 0.577441 0.644743 * Abs Abs@Vmns DD sin2 θ12 = 0.30 sin2 θ23 = 0.43 sin2 θ13 = 0.026

m1 = 0.0036 eV m2 = 0.0093 eV m3 = 0.051 eV

Two Majorana CPV measures:

S1 ⌘ Im

• UMNS, e1U ⇤

MNS, e3

= 0.034

S2 ⌘ Im

• UMNS, e2U ⇤

MNS, e3

= 0.029

η00

0 6= 0

ξ0 = 0.051, η0 = 0.23, η00

0 = 0.055

Sum Rules: Quark-Lepton Complementarity

• QLC-I
• QLC-II
• testing these sum rules could be a more robust way to distinguish different models

mixing parameters

best fit 3σ range θq

23

2.36o 2.25o - 2.48o θq

12

12.88o 12.75o - 13.01o θq

13

0.21o 0.17o - 0.25o

mixing parameters

best fit 3σ range θe

23

38.4o 35.1o - 52.6o θe

12

33.6o 30.6o - 36.8o θe

13

8.9o 7.5o - 10.2o

Quark Mixing Lepton Mixing

θc + θsol ≅ 45o tan2θsol ≅ tan2θsol,TBM + (θc / 2) * cos δe θq23 + θe23 ≅ 45o

Raidal, ‘04; Smirnov, Minakata, ‘04 Ferrandis, Pakvasa; King; Dutta, Mimura; M.-C.C., Mahanthappa

θe13 ≅ θc / 3√2

(BM) (TBM)

16

measuring leptonic mixing parameters to the precision of those in quark sector

need improved δθ12 measurement

Sum Rules: Quark-Lepton Complementarity

• QLC-I
• QLC-II
• testing these sum rules could be a more robust way to distinguish different models

mixing parameters

best fit 3σ range θq

23

2.36o 2.25o - 2.48o θq

12

12.88o 12.75o - 13.01o θq

13

0.21o 0.17o - 0.25o

mixing parameters

best fit 3σ range θe

23

38.4o 35.1o - 52.6o θe

12

33.6o 30.6o - 36.8o θe

13

8.9o 7.5o - 10.2o

Quark Mixing Lepton Mixing

θc + θsol ≅ 45o tan2θsol ≅ tan2θsol,TBM + (θc / 2) * cos δe θq23 + θe23 ≅ 45o

Raidal, ‘04; Smirnov, Minakata, ‘04 Ferrandis, Pakvasa; King; Dutta, Mimura; M.-C.C., Mahanthappa

θe13 ≅ θc / 3√2

(BM) (TBM)

17

measuring leptonic mixing parameters to the precision of those in quark sector

need improved δθ12 measurement

☜ Too small

Correlations among other Observables

• SUSY GUTs: Lepton flavor violating charged lepton decays
• five viable SUSY SO(10) models with dark matter constraints:

18

• individual branching

fraction: strong dependence on soft SUSY parameters

• correlations between

branching fractions: strong dependence on flavor structure

C.H. Albright, M.-C.C (2008)

“Large” Deviations from TBM in A4

• Generically: corrections on the order of (θc)2
• from charged lepton sector:
• through GUT relations
• from neutrino sector:
• higher order holomorphic contributions in superpotential
• Modifying the Neutrino sector: Different symmetry breaking patterns
• TBM: misalignment of
• A4 → GTST2 and A4 → GT
• A4: group of order 12 ⇒ many subgroups
• systematic study of breaking into other A4 subgroups: no viable solutions were found
• complete breaking of A4 required to generate realistic masses and sizable θ13

19

M.-C.C, J. Huang, J. O’Bryan, A. Wijangco, F. Yu, (2012)

“Large” Deviations from TBM in A4

• Subgroup preserving VEVs:

20

normal inverted

M.-C.C, J. Huang, J. O’Bryan, A. Wijangco, F. Yu, (2012)

“Large” Deviations from TBM in A4

• Orthogonal VEV patterns:

21

inverted normal

M.-C.C, J. Huang, J. O’Bryan, A. Wijangco, F. Yu, (2012)

“Large” Deviations from TBM in A4

• Predictions for Dirac CP phase

22

normal

M.-C.C, J. Huang, J. O’Bryan, A. Wijangco, F. Yu, (2012)

Flavor Model Structure: A4 Example

• interplay between the symmetry breaking patterns

in two sectors lead to lepton mixing (BM, TBM, ...)

• symmetry breaking achieved through flavon VEVs
• each sector preserves different residual symmetry
• full Lagrangian does not have these residual

symmetries

• general approach: include high order terms in

holomorphic superpotential

• possible to construct models where higher order

holomorphic superpotential terms vanish to ALL

• rders
• useful tool: Hilbert basis method
• quantum correction?

⇒ uncertainty in predictions due to Kähler corrections

23

GF Ge Gν

charged lepton sector e.g. Z2 subgroup of A4 neutrino sector e.g. Z3 subgroup of A4 〈Φe〉 〈Φν〉

〈 Φe〉∝ (1,0,0) 〈 Φν〉∝ (1,1,1)

e.g. A4

Leurer, Nir, Seiberg (1993); Dudas, Pokorski, Savoy (1995); Dreiner, Thomeier (2003);

Kappl, Ratz, Staudt (2011)

Kähler Corrections

• Superpotential: holomorphic
• Kähler potential: non-holomorphic
• Canonical Kähler potential
• Correction

24

• can be induced by flavon VEVs
• important for order parameter ~ θc
• can lead to non-trivial mixing

Kcanonical ⊃

• Lf† δfg Lg +
• Rf† δfg Rg

∆K =

• Lf† (∆KL)fg Lg +
• Rf† (∆KR)fg Rg

K = Kcanonical + ∆K ,

Wleading = 1 Λ(Φe)gf Lg Rf Hd + 1 Λ Λν (Φν)gf Lg Hu Lf Hu

Weff = (Ye)gf Lg Rf Hd + 1 4κgf Lg Hu Lf Hu

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

• rder parameter

<flavon vev> / Λ ~ θc

Kähler Corrections

• Consider infinitesimal change, x :
• rotate to canonically normalized L’:

⇒ corrections to neutrino mass matrix

25

K = Kcanonical + ∆K = L† (1 − 2x P) L

e κ · v2

u = 2mν with

L → L′ = (1 − x P) L .

Wν = 1 2(L · Hu)T κν(L · Hu) ' 1 2[(1 + xP)L0 · Hu]T κν[(1 + xP)L0 · Hu] ' 1 2(L0 · Hu)T κνL0 · Hu + x(L0 · Hu)T (P T κν + κνP)L0 · Hu

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Kähler Corrections

• Consider infinitesimal change, x :
• rotate to canonically normalized L’:

⇒ corrections to neutrino mass matrix ⇒ differential equation

• same structure as the RG evolutions for neutrino mass operator
• analytic understanding of evolution of mixing parameters
• size of Kähler corrections can be substantially larger (no loop suppression)

26

• S. Antusch, J. Kersten, M. Lindner, M. Ratz (2003)

K = Kcanonical + ∆K = L† (1 − 2x P) L

L → L′ = (1 − x P) L .

mν(x) ≃ mν + x P T mν + x mν P .

dmν dx = P T mν + mν P

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Back to A4 Example

• Kähler corrections due to flavon field:
• quadratic in flavon
• such terms cannot be forbidden by any (conventional) symmetry
• Kähler corrections once flavon fields attain VEVs
• additional parameters diminish predictivity of the scheme
• possible to forbid all contributions from RH sector as well as

with additional symmetries in the particular A4 model considered

27

(LΦν)†(LΦν)

• r (LΦe)†(LΦe)

and

(LΦν)†(LΦe),

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

• Contributions from Flavon VEVs (1,0,0) and (1,1,1)
• five independent “basis” matrices
• RG correction: essentially along PIII = diag(0,0,1) direction due to yτ dominance
• Kähler corrections can be along different directions than RG

28

PI =   1   ,  

  PII =   1   ,     PIII =   1    

  PIV =   1 1 1 1 1 1   ,  

  PV =   i −i −i i i −i  

Back to A4 Example

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Enhanced θ13

• consider change due to correction along PV direction
• Kähler metric:
• Contributions of flavon VEV:
• Corrections to the leading order TBM prediction ( )
• Complex matrix P ⇒ CP violation induced
• for the example considered:

29

rm KL = 1 − 2xP, w

〈Φ〉= (1, 1, 1) υ

ere me π mµ π mτ has to the complex P m

I Due to th

δ ¥ π/2.

with

∆θ13

• ≃

κV · v2 Λ2 · 3 √ 6 m1 m1 + m3

• PV =

  i −i −i i i −i  

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

An Example: Enhanced θ13

30

0.00 0.02 0.04 0.06 0.08 0.10 2 4 6 8

m1 [eV] ∆θ13 [◦] ∆θ13 an. ∆θ13 num.

κV v2/Λ2 = (0.2)2

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Corresponding Change in θ12

31

0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.1 0.2 0.3 0.4

m1 [eV] ∆θ12 [◦]

κV v2/Λ2 = (0.2)2

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Corresponding Change in θ23

32

0.00 0.02 0.04 0.06 0.08 0.0 0.1 0.2 0.3 0.4 0.00 0.02 0.04 0.06 0.08 0.10 2.6 2.4 2.2 2.0 1.8 1.6 1.4

m1 [eV] ∆θ23 [◦]

κV v2/Λ2 = (0.2)2

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Summary

• Fundamental Origin of fermion mass hierarchy and flavor mixing still not known
• neutrino masses: evidence of physics beyond the SM
• SUSY SU(5) x T′ as a family symmetry
• near tri-bimaximal leptonic mixing & realistic CKM matrix
• complex CG coefficients in T′: origin of CPV both in quark and lepton sectors
• 10 model parameters accommodate observed 22 masses, mixing angles and CP phases

for all quarks and leptons

• Testable predictions:
• interesting leading order sum rules between quark and lepton mixing angles
• lepton flavor violating charged lepton decays and correlations among these processes
• Possible connection to space-time geometry?
• theoretical understanding of Kähler corrections crucial for achieving precision

compatible with experimental accuracy

33

Back-up Slides

34

Where Do We Stand?

• Latest 3 neutrino global analysis include atm, solar, reactor experiments (3σ):
• search for absolute mass scale:
• end point kinematic of tritium beta decays
• WMAP + 2dFRGS + Lyα ∑(mνi) < (0.7-1.2) eV
• neutrinoless double beta decay

35

m" e < 2.2 eV (95% CL) Mainz m" µ <170 keV m"# <15.5 MeV

Tritium → He3 + e− + νe KATRIN: increase sensitivity ~ 0.2 eV

current bound: | < m > | < (0.19 − 0.68) eV (CUORICINO, Feb 2008)

(Global Minima) Discovery phase into precision phase for some oscillation parameters Many great discoveries yet to come

P(νa νb) = ⇤ ⇤ νb|ν, t ⇥⇤ ⇤2 ⇥ sin2 2θ sin2 ⌅∆m2 4E L ⇧

current bound: | ⌅m⇧ | ⇥

• X

i=1,2,3

miU 2

ie

• sin2 θatm = 0.386 (0.331 − 0.637)

sin2 θ = 0.307 (0.259 − 0.359) sin2 θ13 = 0.0241 (0.0169 − 0.0313)

Fogli, Lisi, Marrone, Montanino, Palazzo, Rotunno (2012)

(0.14-0.38) eV (EXO, 2012)

Where Do We Stand?

• Search for absolute mass scale:
• end point kinematic of tritium beta decays
• WMAP + 2dFRGS + Lyα ∑(mνi) < (0.36-1.5) eV
• very model dependent
• neutrinoless double beta decay
• uncertainty in nuclear matrix element
• Effective number of neutrinos:
• WMAP7 + BAO: Neff = 4.34 +0.86-0.88
• BBN: Ns < 1.2

36

m" e < 2.2 eV (95% CL) Mainz m" µ <170 keV m"# <15.5 MeV

Tritium → He3 + e− + νe KATRIN: increase sensitivity ~ 0.2 eV

current bound: | < m > | < (0.19 − 0.68) eV (CUORICINO, Feb 2008)

current bound: | ⌅m⇧ | ⇥

• X

i=1,2,3

miU 2

ie

• Mangano, Serpico, arXiv:1103.1261

Gonzalez-Garcia et al, arXiv:1006.3795 Komatsu et al, arXiv:1001.4538

Planck update

Tri-bimaximal Neutrino Mixing from A4

• fermion charge assignments:
• SM Higgs ~ singlet under A4
• operators for neutrino masses:
• two scalar (flavon) fields for neutrino sector:
• product rules:

37

⇤ ⇧ 1 2 3 ⌅ ⌃

L

⌅ 3, eR ⌅ 1, µR ⌅ 1, ⇧R ⌅ 1

HHLL M ⌃⌅⌥ Λ + ⌃⇥⌥ Λ ⇥ T ⇥ → GT ST 2 :

• duct rules:
• ξ

⇥ = ξ0Λ ⇧ ⌥ 1 1 1 ⌃ ⌦ ⌦ ⌦

• ⇧

T ⇥ − invariant:

⌅ ∼ 3, ⇥ ∼ 1

• η
• = uΛ

L =

A4 A4

3 ⊗ 3 = 1 ⊕ 1′ ⊕ 1′′ ⊕ 3s ⊕ 3a ,

Altarelli, Feruglio (2005)

Tri-bimaximal Neutrino Mixing from A4

• Neutrino Masses: triplet flavon contribution
• Neutrino Masses: singlet flavon contribution
• resulting mass matrix:

38

3S = 1 3   2α1β1 − α2β3 − α3β2 2α3β3 − α1β2 − α2β1 2α2β2 − α1β3 − α3β1   1 = α1β1 + α2β3 + α3β2 1 = α1β1 + α2β3 + α3β2

Mν = λv2 Mx      2ξ0 + u −ξ0 −ξ0 −ξ0 2ξ0 u − ξ0 −ξ0 u − ξ0 2ξ0     

V T

ν MνVν = diag(u + 3ξ0, u, −u + 3ξ0) v2 u

Mx , UTBM =

• ⇧

⇧ ⇧ ⇤ ⌥ 2/3 1/ √ 3 − ⌥ 1/6 1/ √ 3 −1/ √ 2 − ⌥ 1/6 1/ √ 3 1/ √ 2 ⇥ ⌃ ⌃ ⌃ ⌅

Form diagonalizable:

• - no adjustable parameters
• - neutrino mixing from CG coefficients!

Altarelli, Feruglio (2005)

Tri-bimaximal Neutrino Mixing from A4

• charged lepton sector -- without quarks
• operators for charged lepton masses
• scalar sector: flavon triplet for charged lepton masses
• resulting charged lepton mass matrix = diagonal
• leptonic mixing matrix = tri-bimaximal
• seesaw realization with three RH neutrinos (N1, N2, N3) ~ 3

39

(⌃)1eR(1) + (⌃)1µR(1) + (⌃)1⇧R(1)

 1 = α1β1 + α2β3 + α3β2 1 = α3β3 + α1β2 + α2β1 1 = α2β2 + α1β3 + α3β1

T ⇥ → GT :

⇒

• φ
• = φ0Λ

     1     

VMNS = V †

e,LVν = I · UT BM = UT BM

A4

Altarelli, Feruglio (2005)

The Model

• Symmetry: SUSY SU(5) x T′
• Particle Content
• additional symmetry:
• predictive model: only 11 operators allowed up to at least dim-7
• vacuum misalignment: neutrino sector vs charged fermion sector
• mass hierarchy: lighter generation masses allowed only at higher dim
• forbids Higgsino mediated proton decay

40

tional Z12 ×Z

12 sym

a 10(Q, uc, ec)L anda 5(dc, ◆)L parameter ⌃ = eiπ/6.

T3 Ta F N H5 H0

5

∆45 φ φ0 ψ ψ0 ζ ζ0 ξ η η00 S SU(5) 10 10 5 1 5 5 45 1 1 1 1 1 1 1 1 1 1 T 0 1 2 3 3 1 1 10 3 3 20 2 100 10 3 1 100 1 Z12 ω5 ω2 ω5 ω7 ω2 ω2 ω5 ω3 ω2 ω6 ω9 ω9 ω3 ω10 ω10 ω10 ω10 Z0

12

ω ω4 ω8 ω5 ω10 ω10 ω3 ω3 ω6 ω7 ω8 ω2 ω11 1 1 1 ω2

Neutrino Sector

• Operators:
• symmetry breaking
• resulting mass matrices

T ⇥ → GT ST 2 :

= ⇧ ⌥ 1 1 1 ⌃ 0Λ ⇧ ⌃ T ⇥ − invariant:

⇤η⌅ = η0Λ

MRR = @ 1 1 1 1 A s0Λ

• nly vector representations

⇒ all CG are real

41

⇧S⌃ = S0

 Wν = λ1NNS + 1 Λ3  H5FNζζ0 ✓ λ2ξ + λ3η + λ4η00 ◆

 ✓ ◆ MD = @ 2ξ0 + η0 −ξ0 −ξ0 + η00 −ξ0 2ξ0 + η00 −ξ0 + η0 −ξ0 + η00 −ξ0 + η0 2ξ0 1 A ζ0ζ0

0vu

hη00i = η00

0Λ

: diagonalized by TBM; ⇒ deviation from TBM η00

0 = 0

η00

0 6= 0

Mν = MDM 1

RRM T D

Mν

[Note: m2 → (1,1,1) unchanged]

Up Quark Sector

• Operators:
• top mass: allowed by T′
• lighter family acquire masses thru operators with higher dimensionality
• dynamical origin of mass hierarchy
• symmetry breaking:
• Mass matrix:
• WT T

= ytH5T3T3 + 1 Λ2 H5  ytsT3Taψζ + ycTaTbφ2

• + 1

Λ3 yuH5TaTbφ⇥3 

• no contributions to

elements involving 1st family; true to all levels

both vector and spinorial reps involved ⇒ complex CG

T → GT

⌥

• ⌥
• ⌦ =

⇧ ⌥ 1 ⌃ ⌦0Λ , ↵ = 1 ⇥ ↵0Λ ,

• ⇥

dim-6

T ⇥ → GT ST 2 :

• φ⇥

= φ

0Λ

⇧ ⌥ 1 1 1 ⌃ ⌦ ⌦ ⌦

• ⌃

dim-7

Mu = ⌅

• ⌃

iφ⇥3

1i 2 φ⇥3 1i 2 φ⇥3

φ⇥3

0 + (1 i 2)φ2

y⇥ψ0ζ0 y⇥ψ0ζ0 1 ⇧ ⌥ ytvu ,

42

Down Quark & Charged Lepton Sectors

• operators:
• generation of b-quark mass: breaking of T′ : dynamical origin for hierarchy between

mb and mt

• lighter family acquire masses thru operators with higher dimensionality
• symmetry breaking:
• mass matrix:
• consider 2nd, 3rd families only: TBM exact
• Georgi-Jarlskog relations:

T ⇥ → GT :

⌥

• ⌥
• ⌦ =

⇧ ⌥ 1 ⌃ ⌦0Λ , ↵ = 1 ⇥ ↵0Λ ,

• ⇥

T ⇥ → nothing:

⌥

• ψ⇥

= ψ

0Λ

⇧ ⌥ 1 1 ⌃

• ⇥
• ⇥

complex CG

ly, md ⌃ 3me, y, mµ ⌃ 3ms is

corrections to TBM

43



• WT F

= 1 Λ2 ybH⇥

5FT3φζ + 1

Λ3  ys∆45FTaφψζ⇥ + ydH50FTaφ2ψ⇥

• 

✓ ◆

  Md =   (1 + i)⌅0⇧⇥ −(1 − i)⌅0⇧⇥ ⇧0⇥ ⌅0⇧⇥ ⌅0⇧⇥   ydvd⌅0  

  Me =   −(1 − i)⌅0⇧⇥ ⌅0⇧⇥ (1 + i)⌅0⇧⇥ −3⇧0⇥ ⌅0⇧⇥   ydvd⌅0

• T ⇥ ⌅ GS :

⇧⌅⌃ = ⌅0 , ⇧⌅⇥⌃ = ⌅⇥

ybvd⌦0 ybvd⌦0

⇧⌅⇥⌃ = ⌅⇥

0Λ

Model Predictions

• Charged Fermion Sector (7 parameters)
• model parameters:

Vcb Vub

Georgi-Jarlskog relations ⇒ Vd,L ≠ I SU(5) ⇒ Md = (Me)T ⇒ corrections to TBM related to θc ⇧e

12 ⇧

↵ me mµ ⇧ 1 3 ↵md ms ⇤ 1 3⇧c . the tri-bimaximal mixing pattern

⇧ ⌦ y θc ⇧ ⇤ ⇤⌦ md/ms eiα⌦ mu/mc ⇤ ⇤ ⇤ ⌦ md/ms, wh ling constants. Even though is of the size of the

44

ybvd⌦0

b ⇤ φ0ψ⇥

0/ζ0 = 0.00304

c ⇤ ψ0ζ⇥

0/ζ0 = 0.0172

k ⇤ y⇥ψ0ζ0 = 0.0266 h ⇤ φ2

0 = 0.00426

g ⇤ φ⇥3

0 = 1.45 ⇥ 105

7 parameters in charged fermion sector

n yt/ sin β = 1.25 ≃ ybφ0ζ0/ cos β ≃ 0.011, tan β = 10

Numerical Results

• Experimentally:
• CKM Matrix and Quark CPV measures:

c c

mu : mc : mt = θ7.5

c

: θ3.7

c

: 1 . md : ms : mb = θ4.6

c

: θ2.7

c

: 1 ,

|VCKM| = ⇤ ⇧ 0.974 0.227 0.00412 0.227 0.973 0.0412 0.00718 0.0408 0.999 ⌅ ⌃

β ≡ arg −VcdV ⇥

cb

VtdV ⇥

tb

⇥ = 23.6o, sin 2β = 0.734 ,

α ⌅ arg VtdV ∗

tb

VudV ∗

ub

⇥ = 110o , γ ⌅ arg VudV ∗

ub

VcdV ∗

cb

⇥ = δq = 45.6o , J ⌅ Im(VudVcbV ∗

ubV ∗ cs) = 2.69 ⇤ 10−5 ,

= 0 798

⌅ A = 0.798 ρ = 0.299 η = 0.306

45

CPV entirely from CG coefficients

Direct measurements @ 3σ (CKMFitter, ICHEP2012)

predicting: 9 masses, 3 mixing angles, 1 CP Phase; all agree with exp within 3σ

Recent LHCb result on gamma angle:

New results push the combined best- fit value to a lower value of rB.

value for gamma going down!

sin 2β = 0.691+0.060

0.047

γ (degree) = 66+36

30

α (degree) = 89+21

13

Model Predictions

• Neutrino Sector (3 parameters)
• with

Georgi-Jarlskog relations ⇒ Vd,L ≠ I SU(5) ⇒ Md = (Me)T ⇒ corrections to TBM related to θc

UMNS = V †

e,LUTBM =

• ⇤

1 −θc/3 ∗ θc/3 1 ∗ ∗ ∗ 1 ⇥ ⌅

• ⇤

⇧ 2/3 1/ √ 3 − ⇧ 1/6 1/ √ 3 −1/ √ 2 − ⇧ 1/6 1/ √ 3 1/ √ 2 ⇥ ⌅ (1)

tan2 θ⇤ ⌃ tan2 θ⇤,T BM + 1 2θc cos δ

neutrino mixing angle

1/2

quark mixing angle CG: leptonic Dirac CPV

⌅13 ⌅ ⌅c/3 ⇧ 2

CGs of SU(5) & T´

46

⇧ ⌦ y θc ⇧ ⇤ ⇤⌦ md/ms eiα⌦ mu/mc ⇤ ⇤ ⇤ ⌦ md/ms, wh ling constants. Even though is of the size of the

⇧e

12 ⇧

↵ me mµ ⇧ 1 3 ↵md ms ⇤ 1 3⇧c . the tri-bimaximal mixing pattern

η00

0 = 0 ξ0 = 0.0791 , η0 = 0.1707 , s0Λ = 1012 GeV |m1| = 0.00134 eV, |m2| = 0.00882 eV, |m3| = 0.0504 eV

S0

Model Predictions

• Neutrino Sector (3 parameters)
• with
• sum rules that exist in case are modified

UMNS = V †

e,LUTBM =

• ⇤

1 −θc/3 ∗ θc/3 1 ∗ ∗ ∗ 1 ⇥ ⌅

• ⇤

⇧ 2/3 1/ √ 3 − ⇧ 1/6 1/ √ 3 −1/ √ 2 − ⇧ 1/6 1/ √ 3 1/ √ 2 ⇥ ⌅ (1)

47

η00

0 6= 0

Uν

new contribution does not change the eigenvector corresponds to m2 η00

0 = 0

S0 = 1012 GeV

ξ0 = 0.051, η0 = 0.23, η00

0 = 0.055

< D

MatrixForm=

0.808875

• 0.57735

0.111303

• 0.308046 -0.57735 -0.756158
• 0.500829 -0.57735

0.644854 0, << D DD.vecnu 0.539098 DD Abs @Vmns DD

sin θ

MNS

13

' θc 3 p 2 + θν

13 + κθc

3

: contributions from η00

0 6= 0

+ θν

13 ,

+ κ + κ0

,

+ κ

: related to deviation of θ23 from π/4

tan2 θ ' 1 2 + ✓1 2 + κ0 ◆ θc cos δ

Numerical Results: Neutrino Sector

• Diagonalization matrix for charged leptons:
• MNS Matrix
• Neutrino Masses:
• Leptonic CP violation from CG coefficients:

⇧ |UMNS| =   0.838 0.542 0.0583 0.362 0.610 0.705 0.408 0.577 0.707  

prediction for Dirac CP phase: δ = 197 degrees (in standard parametrization) 3 independent parameters in neutrino sector predicted 3 masses and 3 angles: all agree with exp within 1σ

48

  0.997ei177o 0.0823ei131o 1.31 ⇤ 10−5e−i45o 0.0823ei41.8o 0.997ei176o 0.000149e−i3.58o 1.14 ⇤ 10−6 0.000149 1   DD 0.824259 0.542816 0.161084 0.264063 0.609846 0.747234 0.500867 0.577441 0.644743 * Abs Abs@Vmns DD

sin2 θ12 = 0.30 sin2 θ23 = 0.43 sin2 θ13 = 0.026

m1 = 0.0036 eV m2 = 0.0093 eV m3 = 0.051 eV

Two Majorana CPV measures:

S1 ⌘ Im

• UMNS, e1U ⇤

MNS, e3

= 0.034

S2 ⌘ Im

• UMNS, e2U ⇤

MNS, e3

= 0.029

“Large” Deviations from TBM in A4

• Consider VEV patterns for triplet flavon in neutrino sector
• to satisfy all constraints on two squared masses and mixing angles

⇒additional non-trivial singlets (1’, 1’’) contributions in RH neutrino sector

49

M.-C.C, J. Huang, J. O’Bryan, A. Wijangco, F. Yu, (2012)

Back to A4 Example

• Kähler corrections due to flavon field:
• linear in flavon:
• possible to forbid these terms with additional symmetries

50

∆Klinear =

• i ∈{a,s}
• κ(i)

Φν

Λ ∆K(i)

L† (L⊗Φν)3i + κ(i) Φe

Λ ∆K(i)

L† (L⊗Φe)3i

• + κξ

Λ ∆KξL†L+h.c.

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)

Back to A4 Example

• Kähler corrections due to flavon field 𝜓 :
• six possible contractions:

51

∆K ⊃

6

X

i=1

κ(i) ∆K(i)

(Lχ)†

X(Lχ)X + h.c.

∆K(1)

(Lχ)†

1(Lχ)1

= (L†

1 χ† 1 + L† 2 χ† 3 + L† 3 χ† 2)(L1 χ1 + L2 χ3 + L3 χ2) ,

(3.4a) ∆K(2)

(Lχ)†

10(Lχ)10

= (L†

3 χ† 3 + L† 1 χ† 2 + L† 2 χ† 1)(L3 χ3 + L1 χ2 + L2 χ1) ,

(3.4b) ∆K(3)

(Lχ)†

100(Lχ)100

= (L†

2 χ† 2 + L† 1 χ† 3 + L† 3 χ† 1)(L2 χ2 + L1 χ3 + L3 χ1) ,

(3.4c ∆K(4)

(Lχ)†

31(Lχ)31

= (L†

1 χ† 1 + ω2 L† 2 χ† 3 + ω L† 3 χ† 2)(L1 χ1 + ω L2 χ3 + ω2 L3 χ2)

+ (L†

3 χ† 3 + ω2 L† 1 χ† 2 + ω L† 2 χ† 1)(L3 χ3 + ω L1 χ2 + ω2 L2 χ1)

+ (L†

2 χ† 2 + ω2 L† 1 χ† 3 + ω L† 3 χ† 1)(L2 χ2 + ω L1 χ3 + ω2 L3 χ1)(3.4d)

∆K(5)

(Lχ)†

32(Lχ)32

= (L†

1 χ† 1 + ω L† 2 χ† 3 + ω2 L† 3 χ† 2)(L1 χ1 + ω2 L2 χ3 + ω L3 χ2)

+ (L†

3 χ† 3 + ω L† 1 χ† 2 + ω2 L† 2 χ† 1)(L3 χ3 + ω2 L1 χ2 + ω L2 χ1)

+ (L†

2 χ† 2 + ω L† 1 χ† 3 + ω2 L† 3 χ† 1)(L2 χ2 + ω2 L1 χ3 + ω L3 χ1)(3.4e

∆K(6)

(Lχ)†

31(Lχ)32

= (L†

1 χ† 1 + ω2 L† 2 χ† 3 + ω L† 3 χ† 2)(L1 χ1 + ω2 L2 χ3 + ω L3 χ2)

+ (L†

3 χ† 3 + ω2 L† 1 χ† 2 + ω L† 2 χ† 1)(L3 χ3 + ω2 L1 χ2 + ω L2 χ1)

+ (L†

2 χ† 2 + ω2 L† 1 χ† 3 + ω L† 3 χ† 1)(L2 χ2 + ω2 L1 χ3 + ω L3 χ1)

M.-C.C., M. Fallbacher, M. Ratz, C. Staudt (2012)