1. A New Family Symmetry: Discrete Quaternion Group
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, (R) (2005) [hep-ph/ ]

2. A motivation: the form of Mn
What we know on neutrinos: Dm2sol and q12, Dm2atm and q23, upper bound on mi and q13.
All these data (as well as CP phases) are encoded in the structure of the 3x3 Majorana neutrino mass matrix Mn.
Many viable ideas on the possible structure of Mn : 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 Mn?

3. A look at data on fermion mixing
CKM
PMNS
Quark - Lepton Symmetry ?
No hierarchies between quark and lepton 12 and 13 .
Large disparity in the sector: q23 << l23 .
In first approximation q23  0 and q13  0 .

4. 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, q23 ~ p/4 determines the dominant structure of the mass matrix Mn (exception: Mn  I3).
Mn structure is stable under radiative corrections  RGE running from GUT to EW scale cannot generate large q23 from small (exception: Mn  I3).
(na la)T is SU(2)L isodoublet  flavor alignment expected between na and la : cancellation between mixing in Mn and Ml (that is the case for quarks: qq23  2º).

5. Quaternions: Group Theory Basics
QUATERNION GROUPS FOR FLAVOR PHYSICS
D.Chang, W.-Y. Keung and G.Senjanovic,
PRD 42 (1990) 1599, Neutrino Magnetic Moment
P.H.Frampton and T.W.Kephart, hep-ph/ ;
P.H.Frampton and O.C.W.Kong, hep-ph/ ;
P.H.Frampton and A.Rasin, hep-ph/ ,
Fermion Mass Matrices
K.S.Babu and J.Kubo, hep-ph/ ,
SUSY Flavor Model
Quaternions: Group Theory Basics
Real numbers: a  (R, ·)
Z2 = {+1,-1}  U(1) : + + ,  
Complex numbers: a+ib  (C, ·)
Z4 = {1, i}  U(1) : i   i i
Quaternion numbers: a+i1b+i2c+i3d  (Q, ·)
( ij )2 = -1 , ij ik = ejkl il : non Abelian !
Q8 = {1, i1, i2, i3}  SU(2) :
(1 2)T  i j (1 2)T

6. Fermion assignments under Q8
Irreducible representations: , 1+  , 1 + , 1  , Two parities distinguish the 1-dim irreps.
The 3 generation of fermions transform as SU(2)= 1  + 1  , 1SU(2)= , 2SU(2)= 2
• Basic tensor product rule:
2  2 = ( 1  + 1  )

7. Yukawa coupling structure
Yukawa couplings: Ykij i cj Fk The matrix structure depends on Fk assignments.
Two Higgs doublets: F1 ~ 1+ + , F2 ~ 1+ –
Quark sector
Charged lepton sector
Only 1-2 Cabibbo mixing
Only 2-3 maximal mixing

8. The neutrino sector Majorana mass term: na Mab nb
Mab depends on which are the superheavy fields.
Higgs triplet VEVs < xi0 > (Type II seesaw): Yab La Lb xi + h.c ui = < xi0 > ~ v2 / Mx
To obtain Mab phenomenologically viable, e.g.: x1 ~ x2 ~ (x3 x4) ~ (in general, x1 and x2 in 2 different 1-dim irreps)

9. Q8 predictions for neutrinos (I)
Scenarios (1) or (2):
Two texture zeros or one zero and one equality
Inverted hierarchy (with m3 > eV) or quasi-degeneracy (masses up to present upper bound)
Atmospheric mixing related to 1-3 mixing: q23 = p/4  q13 = 0
Observable neutrinoless 2b-decay: mee = a > 0.02 eV

10. Q8 predictions for neutrinos (II)
One cannot tell scenario (1) from (2): they are distinguished by the Majorana phase between m2 and m3, which presently cannot be measured!
Scenario (3):
One texture zero and one equality
Normal hierarchy: eV < m3 < eV
sinq13 <  sin22q23 > 0.98
No neutrinoless 2b decay: mee = 0

11. Phenomenology of Q8 Higgs sector
2 Higgs doublets distinguished by a parity:
F1 ~ 1+ + , F2 ~ 1+ –, < Fi > = vi
FCNCs in quark 1-2 sector: DmK, DmD at tree level:
For mh=100GeV, DmK/mK ~ (exp.: ), DmD/mD ~ (exp.: < ).
No FCNCs in lepton 2-3 sector: maximal mixing implies diagonal couplings to both Higgs doublets.
The non-standard Higgs h0 decays into t + t - and m + m - with comparable strength (~ mt / mW ).

12. Summary
Data on lepton masses and mixing are nowadays very constraining; the largest mass difference (2-3 sector) is associated with the largest mixing!
Q8 is the smallest non-trivial SU(2) subgroup; Q8 accommodates in different representations the 3 generations of quarks and leptons.
Neutrino Q8 phenomenology:
Accomodates maximal 2-3 mixing (and all other data);
Constrains mass spectrum and mee;
Explains texture zeros or equalities in the mass matrix.
Higgs Q8 phenomenology:
Two doublets model with parity symmetry;
Non-standard decay rates into t + t - and m + m -.