Transcript S_3, Q_8, SO(10)

Discrete and Unified
Ideas on Fermion Mixing
symmetries for the understanding
of neutrino data
Michele Frigerio
University of California, Riverside
AHEP Seminar @ IFIC
Valencia, March 14, 2005
The Flavor Problem
• Standard Model contains 13 free parameters in the
Yukawa sector.
• Majorana (Dirac) neutrino masses require
additional 9 (7) parameters in the neutrino mass
matrix.
• Two pathways to obtain relations among masses
and mixing:
– Different fermion generations related by a symmetry
(“flavor” or “family” or “horizontal”)
– Fermions in the same generation related by an
enlarged gauge symmetry (GUTs).
Comparing two approaches to
fermion mixing
•
Discrete Family symmetries:
1. Discrete groups have more low dimensional
representations than continuous Lie groups.
2. Non-Abelian groups can relate different generations,
because of irreducible representations with dimension
larger than one.
3. Features of fermion mixing can be related to the
structure of the EW Higgs sector.
•
Grand Unification symmetries:
1. Different SM fermions fit into the same large
representation of a larger gauge group.
2. Quark-lepton relations induced by gauge structure.
3. Connection between the GUT scale and the seesaw
scale, where neutrino masses are generated.
Data on neutrino oscillations
Maltoni, Schwetz,
Tortola, Valle,
hep-ph/0405172
New J. Phys. 6 :
122, 2004
Assuming
inverted
normal
ordering
of the
mass
spectrum
A look at fermion mixing values
CKM
PMNS
• No hierarchies between quark and lepton 12 and 13 .
• Large disparity in the 2 - 3 sector: q23 << l23 .
• In first approximation q23  0 and q13  0 .
• S3 : LARGE NEUTRINO MIXING AND NORMAL MASS
HIERARCHY: A DISCRETE UNDERSTANDING.
Shao-Long Chen, M.F. and Ernest Ma,
hep-ph/0404084, Phys. Rev. D 70, 073008 (2004)
• Q8 :
QUATERNION FAMILY SYMMETRY OF QUARKS AND
LEPTONS.
M.F., Satoru Kaneko, Ernest Ma and Morimitsu Tanimoto,
hep-ph/0409187, Phys. Rev. D 71, 011901(R) (2005)
• Z2Z2 :
SMALLNESS OF LEPTONIC 13 AND DISCRETE
SYMMETRY.
Shao-Long Chen, M.F. and Ernest Ma,
hep-ph/0412018, to appear in Phys. Lett. B
• SO(10) :
FERMION MASSES IN SUSY SO(10) WITH TYPE II
SEESAW: A NON-MINIMAL PREDICTIVE SCENARIO.
Stefano Bertolini, M.F. and Michal Malinsky,
hep-ph/0406117, Phys. Rev. D 70, 095002 (2004)
Perfect Geometric Solids in All Dimensions
D=2 :
Triangle
Z_3; D_3 = S_3
D=3 :
Tetrahedron
[4]
A_4
Icosahedron
[20]
A_5
Octahedron
[8]
S_4
Cube
[6]
S_4
Dodecahedron
[12]
A_5
4-Crosspolytope
[16]
Q_8; ...
Hyper-Icosahedron
[600]
Q_120; ...
Hyper-Diamond
[24]
Q_24; ...
4-Cube
[8]
Hyper-Dodecahedron
[120]
4-Simplex
[5]
no; S_5
D=4 :
5-Simplex
[6]
.
...
. .
5-Crosspolytope
[32]
...
.
Square
Z_4; D_4
Pentagon
Z_5; D_5
...
5-Cube
[10]
.
...
.
• Symmetry of the solid: subgroup of SO(3) for D=2 and D=3,
SO(4) for D=4, …
• For D=2 (4,8) vertices can form a group as a subset of the complex
(quaternionic, octonionic) units, that is U(1) (SU(2) , S7 ~ SU(3)).
Why to worry about the origin
of maximal 2-3 mixing?
(no precision measurements in next generation experiments: T2K?)
• For all possible mass spectra and all choices of CP
phases, 23 ~ 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 23 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: q23  2º).
Maximal mixing in 2 2 matrices
Ml
Mn
Flavor
Symmetry
Models
U(1), Zn
m2atm from
symmetry-breaking
non-Abelian
Q8
non-Abelian
(if any)
m from
simmetry-breaking
non-Abelian
S3 (+ = - p/2)
Q8 (+ = p ,  = )
non-Abelian
Q8
QUATERNION GROUPS FOR FLAVOR PHYSICS
• D.Chang, W.-Y. Keung
and G.Senjanovic,
Quaternions:
Group
Theory
Basics
PRD 42 (1990) 1599, Neutrino Magnetic Moment
• P.H.Frampton
hep-ph/9409330;
• Real
numbers: aand
 (T.W.Kephart,
R, ·)
P.H.Frampton and O.C.W.Kong, hep-ph/9502395;
Z2P.H.Frampton
= {+1,-1}  and
U(1)A.Rasin,
: +hep-ph/9910522,
+ ,  
• Complex
a+ib  (C, ·)
Fermion numbers:
Mass Matrices
and J.Kubo, hep-ph/0411226,
Z•4K.S.Babu
= {1, i}
 U(1) : i   i i
SUSY Flavor Model
• Quaternion numbers: a+i1b+i2c+i3d  (Q, ·)
( ij )2 = -1 , ij ik = ejkl il : non Abelian !
Q8 = {1, i1, i2, i3}  SU(2)
(8 vertices of the hyper-octahedron on the 4-sphere)
(1 2)T  i j (1 2)T
Fermion assignments under Q8
• Irreducible representations:
1+ + , 1+  , 1 + , 1  , 2
•
f
Two parities distinguish the 1-dim irreps (Z2  Z2);
1+  , 1 + and 1  are interchangeable;
2 is realized by ± 12 , ± i 1 , ± 2 , ± i 3 .
• The 3 generation of fermions transform as
3SU(2)= 1  + 1 + + 1+  ,
• Basic tensor product rule:
1SU(2)= 1+ + ,
2  2 = 1+ + + ( 1  + 1 + + 1+  )
2SU(2)= 2
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
The neutrino sector
• Majorana mass term: na Mab nb
• Mab depends on which are the superheavy fields.
Higgs triplet VEVs < xi0 > (Type II seesaw):
Yiab La Lb xi + h.c.
< xi0 > ~ v2 / Mx
• To obtain Mab phenomenologically viable:
x1 and x2 in two different 1-dim irreps;
(x3 x4) ~ 2 to generate the 1-2 mixing.
Q8 predictions for neutrinos (I)
Scenarios (1) or (2):
(x1 ~ F1 , x2 ~ F2 or ~ F2)
• Two texture zeros or
one zero and one equality
• Inverted hierarchy (with
m3 > 0.015 eV) or quasidegeneracy (masses up to
present upper bound)
• Atmospheric mixing
related to 1-3 mixing:
23 = p/4  13 = 0
• Observable neutrinoless
2b-decay:
mee = a > 0.02 eV
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):
(x1 ~ F1 , x2 ~ F2)
•
•
•
•
One texture zero and one equality
Normal hierarchy: 0.035 eV < m3 < 0.065 eV
sin13 < 0.2  sin2223 > 0.98
No neutrinoless 2b decay: mee = 0
Phenomenology of Q8 Higgs sector
• 2 Higgs doublets distinguished by a parity:
F1 ~ 1+ + , F2 ~ 1+ –, < Fi > = vi
• FCNCs in quark 1-2 sector: mK, mD at tree level:
For mh=100GeV, mK/mK ~ 10-15 (exp.: 7 10-15),
mD/mD ~ 10-15 (exp.: < 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
 +   with comparable strength (~ mt / mW ).
SO(3)
Z2  Z2
3
1+ + 1+ + 1 
5, 7, …
…
13 = 0 with a Z2  Z12 symmetry1++
• sin213 < 0.047 at 3 C.L.
• If (ni , li), lic ~ (+,), (,+), (,),
F1 , x1 ~ (+,+),
F2 ~ (+,), x2 ~ (,), then:
SOME RECENT DISCRETE MODELS FOR 13=0
• C.Low, hep-ph/0404017 & 0501251,
Abelian
(the sameSymmetries
structure canClassification
be obtained with type I seesaw
• W.Grimus,
A.S.Joshipura,
if the heaviest
RH neutrinoS.Kaneko,
decouples)L.Lavoura, M.Tanimoto,
hep-ph/0407112, Non-Abelian Model
 - t Lepton Flavor Violation
L=23
• Z2  Z2 Higgs potential is CP conserving: H0 (SM-like)
and h0 mix (both CP-even), A0 (CP-odd) and h are
mass eigenstates.
• Lepton Flavor Violation mediated by h0, A0, h :
–BR(t3) ~ 5 ·10-9 (exp < 2 ·10-6)
–BR(t) ~ 2 ·10-12 (exp < 1 ·10-6)
–(g - 2)/2 ~ 6 · 10-13 (theory-exp discrepancy ~ 3 ·10-9)
(but they scale with tan b).
Minimal SUSY SO(10)
• SO(10) as the unified gauge group after MSSM
RGE evolution of SU321 couplings
• Choice of Higgs multiplets should allow
– to break SO(10) spontaneously to SU321
– to reproduce observed fermion masses
• Minimal number of couplings (as many as
MSSM) if the choice is 10+126+126+210
Aulakh, Bajc, Melfo, Senjanovic, Vissani 2003
• R-parity is automatically guaranteed
• Two related seesaw contributions to neutrino
masses:
SO(10) Constraints on Flavor
Only two Yukawa matrices
contribute to fermion masses:
16f  16f = 10 + 120 + 126
Lepton mass matrices can be
expressed as a function of
quark parameters:
The bidoublet components in
10 and 126 Higgs multiplets
take VEVs.
Dominant type II seesaw is
assumed.
Is large mixing generated in
the neutrino sector?
Is minimal SO(10) viable?
• b-t unification is related with large atm ! (dominant
t-block in Mn)
• 2-3 large mixing is generated, however we need also
m2sol << m2atm: this implies sol ≈ p/4 or,
alternatively, atm < p/4 significantly
• Numerical analysis of the real case: agreement with
data only allowing 2 deviations from central values
of both quarks and neutrino parameters
• Possible wayouts:
– Including CP phases: value of dCKM? Work in progress...
– Including type I seesaw: correlations do not allow
improvements.
The role of 120 Higgs
•
•
•
Missing renormalizable contribution to the Yukawa
sector: antisymmetric coupling to fermions:
dMu,d,l = Y120 (v120)u,d,l
120 has no role in breaking SO(10)  SU321: its
mass parameter can be at the cutoff: M120 ~ MPl
(“extended survival hypothesis”)
F - flatness of the superpotential implies that 120
bidoublet VEVs are suppressed by MGUT / MPl ~
10-3 (decoupling):
W  M120120H2 + l10H120H210H
 M120(1,2,2)1202 + l(1,2,2)10(1,2,2)120(1,1,1)210
<W / (1,2,2)120> = 0 
 <(1,2,2)120> ~ MGUT / MPl <(1,2,2)10>
Numerical fit with and without 120
120 Higgs corrections to fermion
mass matrices are small, but
important for first generation
masses and, being antisymmetric,
also mixing angles are modified
significantly in a predictive way!
sin22sol
Ue3
m2sol /m2atm
Summary
•
•
•
•
Data on lepton masses and mixing are nowadays very
constraining; the largest mass difference (2-3 sector) is
associated with the largest mixing!
Discrete symmetries are suitable to decode the flavor
problem: they can
– explain texture zeros or equalities in the mass matrix
– accomodate maximal 2-3 mixing (if non-Abelian)
– explain zero 1-3 mixing
– constrain the neutrino mass spectrum and mee
– require an extended EW Higgs sector with definite
phenomenology (FCNCs, LFV, …)
After few decades the exploration of Grand Unification
models is still fruitful and the constraints from neutrinos are a
powerful guideline.
Selection rules: we need to know Ue3 , the deviation from
atm = p/4, the type of mass spectrum and 0n2bdecay rate.