A New Family Symmetry: Discrete Quaternion Group

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A New Family Symmetry Discrete Quaternion Group
XLth Rencontres de Moriond Electroweak
Interactions and Unified theories La Thuile,
Italy, March 9th, 2005
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
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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?

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Data on neutrino oscillations
Maltoni, Schwetz, Tortola, Valle, hep-ph/ /0405172
If such symmetry is the discrete quaternion group
Q8
Assuming normal ordering of the mass spectrum
Assuming inverted ordering of the mass spectrum
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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 2 - 3 sector ?q23 ltlt
  ?l23 .
• In first approximation ?q23 ? 0 and ?q13 ? 0 .

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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).

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Discrete Family Symmetry

• Standard Model 13 free parameters in the Yukawa
  sector.
• Non-zero Majorana (Dirac) neutrino masses
  additional 9 (7) parameters in the neutrino mass
  matrix.
• Family (or flavor or horizontal) symmetry
  relations among masses and mixing of the 3
  fermion generations.
• Discrete groups have more low dimensional
  representations than continuous Lie groups.
• Features of fermion mixing related to the
  structure of the EW Higgs sector.

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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/9409330
• P.H.Frampton and O.C.W.Kong, hep-ph/9502395
• P.H.Frampton and A.Rasin, hep-ph/9910522,
• Fermion Mass Matrices
• K.S.Babu and J.Kubo, hep-ph/0411226,
• SUSY Flavor Model

• Real numbers a ? (R, )
• Z2 1,-1 ? U(1) ?? ? , ??? ???
• Complex numbers aib ? (C, )
• Z4 ?1, ?i ? U(1) ??i ? ? i ??i
• Quaternion numbers ai1bi2ci3d ? (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

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Quaternions on hep-ph

• D.Chang, W.-Y. Keung and G.Senjanovic, Neutrino
  Magnetic Moment, PRD 42 (1990) 1599
• R.Anderson and G.C.Joshi, History of Quaternions
  in Physics, 9208222
• R.Dahm, Hadron Classification, 9601207
• P.H.Frampton and T.W.Kephart, Fermion Masses,
  9409330
• P.H.Frampton and O.C.W.Kong, Quark Mass Matrices,
  9502395
• C.S.Lim, Extra Families and CP, 9704294
• P.H.Frampton and A.Rasin, Fermion Masses
  including Neutrinos, 9910522
• K.S.Babu and J.Kubo, SUSY Flavor Model, 0411226

• 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/9409330
• P.H.Frampton and O.C.W.Kong, hep-ph/9502395
• P.H.Frampton and A.Rasin, hep-ph/9910522,
• Fermion Mass Matrices
• K.S.Babu and J.Kubo, hep-ph/0411226,
• SUSY Flavor Model

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Fermion assignments under Q8

• Irreducible representations 1 , 1 ? , 1?
  , 1? ? , 2 Two parities distinguish the
  1-dim irreps.
• The 3 generation of fermions transform
  as 3SU(2)
  1? ? 1? 1 ? , 1SU(2) 1 ,
  2SU(2) 2

Basic tensor product rule 2 ? 2 1
( 1? ? 1? 1 ? )
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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
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The neutrino sector

• Majorana mass term na Mab nb

• Mab depends on which are the superheavy fields.
• Higgs triplet VEVs lt xi0 gt (Type II seesaw)
  Yab La Lb xi h.c. ui lt xi0 gt v2 /
  Mx
• To obtain Mab phenomenologically viable, e.g.
  x1 1 x2 1- (x3 x4)
  2 (in general, x1 and x2 in 2
  different 1-dim irreps)

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Maximal mixing in 2-3 sector
Scalar Fields Ml M?
?1 ? 1 ?, ?2 ? 1? ? ?1 ? 1? , ?2 ? 1? ?
?1 ? 1 , ?2 ? 1? ? ?1 ? 1 ?, ?2 ? 1? ?
?1 ? 1 , ?2 ? 1? ? ?1 ? 1? , ?2 ? 1? ?
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Q8 neutrino mass matrices

• Neutrino sector depends on the choice of
  superheavy fields. Contributions to Mn from
  scalar triplet VEVs ltxigt Yab La Lb xi h.c.
• x1 and x2 in two different Q8 1-dim irreps
  (remember that doublets are F1 1 , F2 1
  )
• The solar mixing requires (x3 , x4) 2

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Q8 predictions for neutrinos (I)
Scenarios (1) or (2)

• Two texture zeros or one zero and one equality
• Inverted hierarchy (with m3 gt 0.015 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 gt
  0.02 eV

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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 0.035 eV lt m3 lt 0.065 eV
• sinq13 lt 0.2 ? sin22q23 gt 0.98
• No neutrinoless 2b decay mee 0

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Phenomenology of Q8 Higgs sector

• 2 Higgs doublets distinguished by a parity
• F1 1 , F2 1 , lt Fi gt vi
• FCNCs in quark 1-2 sector DmK, DmD at tree
  level

• For mh100GeV, DmK/mK 10-15 (exp. 7 10-15),
  DmD/mD 10-15 (exp. lt 2.5 10-14).
• 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
  ).

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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 -.