Obtaining a nonzero 13 in lepton models based on SO(3) A 4 Yuval - - PowerPoint PPT Presentation

Obtaining a nonzero 13 in lepton models based on SO(3)  A4

Yuval Grossman and Wee Hao Ng, Cornell University Phenomenology 2014 arXiv:1404.1413 [hep-ph]

1

Outline

• Tri-bimaximal mixing found and lost; how do

A4 models cope?

• Basics of SO(3)  A4 model
• Nonzero 13 in SO(3)  A4 model
• Summary

2

Tri-bimaximal mixing and A4 models

• For long time, PMNS matrix thought to be

consistent with TBM

• Highly specific pattern… discrete lepton flavor

symmetries?

TBM

2 1 3 3 1 1 1 U 6 3 2 1 1 1 6 3 2                     

 

13

iδ 13

sin θ e

3

Tri-bimaximal mixing and A4 models

• Simplest A4 lepton models

– Components

• SM Higgs and leptons
• Right handed neutrinos
• Scalar flavons , ’

– Features

• Additional Z2 symmetry so flavons sectorized
• Flavons gain VEV  = (v, v, v), ’ = (v’, 0, 0)

Representations of A4

 TBM!

4

Tri-bimaximal mixing… lost

• Recently, 13 found to be much larger!
• Daya Bay (2012):
• RENO (2012):
• TBM ruled out!

 

2 13

sin 2θ 0.092 0.016(stat) 0.005(syst)   

 

2 13

sin 2θ 0.113 0.013(stat) 0.019(syst)   

5

How do A4 models cope?

• Actually many A4 models already allow this!
• Approaches:

– Higher dimension operators, e.g. [Altarelli and Meloni (2009)] – More flavons, e.g. [Chen et al (2013)] – Perturb flavon alignments, e.g. [King (2011)] – Radiative corrections, e.g. [Antusch et al (2003)]

6

Continuous symmetry  A4

• Class of models based on continuous

symmetry  A4.

• Originally motivated to explain origin of A4.
• Specific example: SO(3)  A4 [Berger and

Grossman (2009)]

• Turns out 13 already nonzero at tree level!
• Even better: 13 related to

A4 scale SO(3) scale

7

Basics of SO(3)  A4 model

Left-handed leptons Right-handed leptons Scalars

8

Basics of SO(3)  A4 model

• Most general Lagrangian for leptons
• Flavons gain VEV

a a a b ab T a abc bc 5 abc a bd cd e l e m l m m l m m l 5 m a a a b ab T a abc bc 5 abc a bd cd e f e m f m m f m m f 5 m l

H H H H L y ψ ψ y ψ ψ y ψ T ψ y ψ ψ Λ Λ Λ Λ y ψ ψ y ψ ψ y ψ T ψ y ψ ψ

 

                 ò ò

ca a ca b c abc a a ν ν n n n n ν l n

x Mψ ψ ψ ψ T y ψ Hψ L Λ



    

5 5 (a b c) T 5 5 5 5 5

v v v v T v y z v , v v , v v v x ,                                           

 6  6 mass matrices

9

3D coordinate basis vectors

Obtaining UPMNS

• UPMNS characterize charged-current

interactions between light eigenstates.

• 6  6 mass matrices:

– Block-diagonalise 6  6 mass matrices. – Then diagonalise 3  3 mass matrices.

• Neutrinos:

10

2 2 ν H 2 2 2 2 2 3 3 ν H ν ν H T ν 2 2 2 2 2 2 2 2 2 2 ν T ν T 2 2 2 2 2 ν ν H T ν H 2 2 2 2 2 2 2 2 2 2 ν T ν T

y v M y Mv Λ y x v v v Λ M M Λ x v v M Λ x v v y x v v v Λ y Mv Λ M Λ x v v M Λ x a , of form 0 b v v c c b



                                           

Obtaining UPMNS

• Charged leptons:

H H 6 6 l

v v A B M Λ Λ C D



          

5 2 5 2 e m m 5 m m 5 5 2 5 2 2 2 e m m 5 m m 5 2 5 2 2 5 2 e m m 5 m m 5

y v y v y v (ω ω) y v y v (ω ω ) A y v y v y v (ω ω) ω y v y v (ω ω ) ω , ω ω ω ω y v y v y v (ω ω) ω y v y v (ω ω ) a b c

• fform a

b c a b c ω                                                           

2π i 3

ω e 

T 5 5 m m T m m 5 m 5 T m m T m 5 T 5 T m m 5 m T m m 5 m T

y v 2y v y v y v y v B y v 2y v y v y v y v y v y v y v y v                 

5 2 5 2 e m m 5 m m 5 5 2 5 2 2 2 e m m 5 m m 5 2 5 2 2 5 2 e m m 5 m m 5

y v y v y v (ω ω) y v y v (ω ω ) C y v y v y v (ω ω) ω y v y v (ω ω ) ω , ω ω ω ω y v y v y v (ω ω) ω y v y v (ω ω ) a b c

• fform a

b c a b c ω

     

                                                                   

T 5 5 m m T m m 5 m 5 T m m T m 5 T 5 T m m 5 m T m m 5 m T

y v 2y v y v y v y v D y v 2y v y v y v y v y v y v y v y v

       

                      

11

Nonzero 13

• Block diagonalize
• Useful to define
• Then
• Expanding in small parameter

– LO: – NLO:

6 6 6 6 † l l

M (M )

 

2 3 3 3 3 † † † † † † † 1 † † H l l 2

v M (M ) AA BB (AC BD )(CC DD ) (CA DB ) Λ

  

         

5 T

A,C,E~O(v,v ), B,D~O(v )

T T m m

(y / E y )D B



 

T T

O(v / v )~O(v / v )   ò

T 3 3 H m l T m

v y M A C Λ y

 

       

13 still 0!

T 3 3 1 H m l T m

v y M A C ED C Λ y

  

        

Suggests 13 ~ !

A4 scale SO(3) scale

12

Nonzero 13

• Actual contribution to 13 from deviation turns out to

be ~ (m/m)  ~ 10

• Random simulation results

In agreement with predictions!

13

Summary and further work

• TBM ruled out, but many A4 models allow for

this scenario

• In particular, SO(3)  A4 model allow nonzero

tree-level 13 of size

• % level separation of A4 and SO(3) breaking

scales

τ μ

m A4 ~ m scale SO(3) scale

14

Thank you!

15