XLth Rencontres de Moriond Electroweak Interactions and Unified theories La Thuile, Italy, March 9th, 2005

A New Family Symmetry: Discrete Quaternion Group

Michele Frigerio University of California, Riverside M.F., Satoru Kaneko, Ernest Ma and Morimitsu Tanimoto,

QUATERNION FAMILY SYMMETRY OF QUARKS AND LEPTONS

Phys. Rev. D 71, 011901(R) (2005)

[hep-ph/0409187]

A motivation: the form of M

n • What we know on neutrinos: D

m 2 sol

q 23 , upper bound on

m i

and q 13 .

and q 12 , D

m 2 atm

and • All these data (as well as

CP phases

) are encoded in the structure of the 3x3 Majorana neutrino mass matrix

M

n .

• Many viable ideas on the

possible structure of M

n quasi-degeneracy, mt : symmetry, bimaximal mixing, dominant mt -block, flavor democracy, texture zeros … • Search for the

underlying family symmetry

, if any.

Why to consider a quaternion symmetry?

What is the corresponding form of M

n

?

A look at data on fermion mixing

CKM PMNS • No hierarchies between quark and lepton q

12

and q

13

• Large disparity in the 2 - 3 sector: q

q 23

<< q

l 23

.

• In first approximation q

q

23  0 and q

q

13  0 .

.

Why to worry about the origin of maximal 2-3 mixing?

(no precision measurements in next generation experiments) • For any possible neutrino mass spectrum is, q

23

~ p

/4 determines the dominant structure of the mass matrix M

n (exception: M n  I 3 ).

• M n structure is stable under radiative corrections 

RGE running from GUT to EW scale cannot generate large

q

23 from small

(exception: M n  I 3 ).

• ( n a l a ) T is SU(2) L isodoublet 

expected between

n a mixing in M n and M l

and l

a

flavor alignment

: cancellation between (that is the case for quarks: q q 23  2 º ).

• •

QUATERNION GROUPS FOR FLAVOR PHYSICS

•

Quaternions: Group Theory Basics

PRD 42 (1990) 1599, Neutrino Magnetic Moment Real

Z

2 numbers: a 

P.H.Frampton and O.C.W.Kong, hep-ph/9502395;

= {+1,-1}  (

R

, ·) U(1) :  +   + ,     

Complex

Z

• 4 = {  1, numbers: a+ib 

K.S.Babu and J.Kubo, hep-ph/0411226,

 i}  U(1) :   i (

C

, ·)   i   i

Quaternion ( i j ) 2 = -1 , i

numbers: a+i 1 b+i 2 c+i 3 d 

j i k =

e

jkl i l

(

: non Abelian !

Q

, ·)

Q

8 (



1

= {



1,



i

1

,



i

2

,



i

3

}

 

2 ) T

 

i



j (



1



2 )

SU(2)

T

:

Fermion assignments under Q

8

•

Irreducible representations

:

1 + +

,

1 +

 ,

1



+

,

1

  ,

2

Two parities distinguish the 1-dim irreps.

• The 3 generation of fermions transform as

3

SU(2) =

1

  +

1



+

+

1 +

 ,

1

SU(2) =

1 + +

,

2

SU(2) =

2

• Basic

tensor product 2



2

=

1 + +

+ (

1

  rule: +

1



+

+

1 +



)

Yukawa coupling structure

• Yukawa couplings:

Y k ij



i



c j

F

k

The matrix structure depends on F

k

assignments.

• Two Higgs doublets: F

1 ~ 1 + + ,

F

2 ~ 1 + – Quark

sector

Charged lepton

sector Only

1-2 Cabibbo

mixing Only

2-3 maximal

mixing

•

The neutrino sector

Majorana mass term

: n a

M

ab n b •

M

ab depends on which are the superheavy fields.

Higgs triplet VEVs < x i 0 > (

Type II seesaw

):

Y

ab

L

a

L

b x i

+ h.c.

u

i

= <

x i 0

>

~

v

2

/ M

x • To obtain

M

ab x 1 ~ 1 ++ (in general, x 1 phenomenologically viable, e.g.: x 2 ~ 1 and x 2 + ( x 3 x 4 ) ~ 2 in 2 different 1-dim irreps)

Q

8

predictions for neutrinos (I)

Scenarios (1) or (2):

• • Two

texture zeros

or one zero and one

equality Inverted hierarchy

m 3 (with > 0.015 eV) or

quasi degeneracy

(masses up to present upper bound) • Atmospheric mixing related to 1-3 mixing: q 2

m 23

b

ee =

p/4  q -decay:

13 = = a > 0.02 eV

0 • Observable neutrinoless

Q

8

predictions for neutrinos (II)

One cannot tell

scenario (1) from (2)

: they are distinguished by the

Majorana phase

between m 2 m 3 , which presently cannot be measured!

and

Scenario (3):

• One texture

zero

and one

equality

•

Normal hierarchy

: 0.035 eV < m 3 • sin q 13 < 0.2 

sin 2 2

q

23 >

0.98

• No neutrinoless 2 b decay:

m ee = 0

< 0.065 eV

Phenomenology of Q

8

Higgs sector

•

2 Higgs doublets

distinguished by a parity: F

1 ~ 1 + + ,

F

2 ~ 1 + – , <

F

i > = v i

• FCNCs in quark 1-2 sector: D

m K ,

D

m D

at tree level: • For m h =100GeV, D m K /m K D m D /m D ~

10 -15

~

10

(exp.: < 2.5 10

-15

-14 ).

(exp.: 7 10 -15 ),

No FCNCs in lepton 2-3 sector

: maximal mixing implies diagonal couplings to both Higgs doublets.

• The non-standard Higgs h 0 m + m 

decays into

with comparable strength (~ m t t / m + W t ).



and

• • • •

Summary

Data on

lepton masses and mixing

are nowadays very constraining; the largest mass difference (

2-3 sector

) is associated with the largest mixing!

Q 8

Q 8 is the smallest non-trivial SU(2) subgroup; accommodates in different representations the

generations of quarks and leptons

.

3 Neutrino Q 8 phenomenology

: – Accomodates

maximal 2-3 mixing

– Constrains

mass spectrum and m ee

; – Explains

texture zeros or equalities

(and all other data); in the mass matrix. –

Higgs Q 8 phenomenology Two doublets

: model with parity symmetry; – Non-standard

decay rates into

t + t 

and

m + m  .