Discrete and Unified Ideas on Fermion Mixing
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Transcript of Discrete and Unified Ideas on Fermion Mixing
Discrete and Unified Ideas on Fermion Mixing
symmetries for the understanding of neutrino data
Michele FrigerioUniversity of California, Riverside
AHEP Seminar @ IFICValencia, 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 oscillationsMaltoni, Schwetz,Tortola, Valle,hep-ph/0405172New J. Phys. 6 :122, 2004
Assumingnormal
orderingof themass
spectrum
Assuminginvertedordering
of themass
spectrum
A look at fermion mixing values
• No hierarchies between quark and lepton 12 and 13 .• Large disparity in the 2 - 3 sector: q
23 << l23 .
• In first approximation q and q
CKM PMNS
• 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)
• Z2Z2 : 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
. ... . . ... .
5-Simplex[6]
5-Crosspolytope[32]
4-Simplex[5]
no; S_5
4-Crosspolytope[16]
Q_8; ...
Hyper-Icosahedron[600]
Q_120; ...
Tetrahedron[4]
A_4
Icosahedron[20]A_5
Hyper-Diamond[24]
Q_24; ...
Octahedron[8]
S_4
TriangleZ_3; D_3 = S_3
. ... .
5-Cube[10]
4-Cube[8]
Cube[6]
S_4
SquareZ_4; D_4
Hyper-Dodecahedron[120]
Dodecahedron[12]A_5
PentagonZ_5; D_5
...D=2 :
D=3 :
D=4 :
• 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
/4 determines the dominant structure of the mass matrix M(exception: M I3).
• M structure is stable under radiative corrections RGE running from GUT to EW scale cannot generate large 23 from small (exception: M I3).
• (l)T is SU(2)L isodoublet flavor alignment expected between and l cancellation between mixing in Mand Ml (that is the case for quarks: q
23 2º).
Maximal mixing in 2 2 matrices
Q8non-Abelian
S3 (+ = - /2)
Q8 (+ = , = )non-Abelian
m from
simmetry-breaking
non-Abelian(if any)
Q8non-Abelian
m2atm from
symmetry-breakingU(1), Zn
ModelsFlavor
SymmetryMMl
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 = jkl 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
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
Fermion assignments under Q8
• Irreducible representations: 1+ + , 1+ , 1 + , 1 , 2 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 .
• f
• 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+ )
Yukawa coupling structure
• Yukawa couplings: Ykij i c
j k The matrix structure depends on k assignments.
• Two Higgs doublets: 1 ~ 1+ + , 2 ~ 1+ –
Quark sector Charged lepton sector
Only 1-2 Cabibbo mixing
Only 2-3 maximal mixing
The neutrino sector
• M depends on which are the superheavy fields.
Higgs triplet VEVs < i> (Type II seesaw):
YiLLi + h.c. < i
0 > v2 / M
• To obtain M phenomenologically viable: and in two different 1-dim irreps;to generate the 1-2 mixing.
• Majorana mass term: M
Q8 predictions for neutrinos (I)
• Two texture zeros or one zero and one equality
• Inverted hierarchy (with m3 > 0.015 eV) or quasi-degeneracy (masses up to present upper bound)
• Atmospheric mixing related to 1-3 mixing: 23 = 13 =
• Observable neutrinoless 2-decay: mee = a > 0.02 eV
Scenarios (1) or (2):or
Q8 predictions for neutrinos (II)
• One texture zero and one equality
• Normal hierarchy: 0.035 eV < m3 < 0.065 eV
• sin sin2223 >
• No neutrinoless 2decay: mee = 0
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):
Phenomenology of Q8 Higgs sector
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 and with comparable strength (~ m / mW ).
• 2 Higgs doublets distinguished by a parity:
1 ~ 1+ + , 2 ~ 1+ –, < i > = vi • FCNCs in quark 1-2 sector: mK, mD at tree level:
= 0 with a Z2 Z2 symmetry
• If (i , li), lic ~ (+,), (,+), (,), 1 , 1 ~
(+,+), 2 ~ (+,), 2 ~ (,), then:
(the same structure can be obtained with type I seesaw if the heaviest RH neutrino decouples)
SO(3) Z2 Z2
1 1++
3 1+ + 1+ + 1
5, 7, … …
SOME RECENT DISCRETE MODELS FOR 13=0• C.Low, hep-ph/0404017 & 0501251, Abelian Symmetries Classification• W.Grimus, A.S.Joshipura, S.Kaneko, L.Lavoura, M.Tanimoto, hep-ph/0407112, Non-Abelian Model
• sin213 < 0.047 at 3 C.L.
- Lepton Flavor Violation
• Z2 Z2 Higgs potential is CP conserving: H0 (SM-like) and h0 mix (both CP-even), A0 (CP-odd) and h are mass eigenstates.
L=23
• Lepton Flavor Violation mediated by h0, A0, h :–BR(3) ~ 5 ·10-9 (exp < 2 ·10-6)
–BR() ~ 2 ·10-12 (exp < 1 ·10-6)
–(g - 2)/2 ~ 6 · 10-13 (theory-exp discrepancy ~ 3 ·10-9)
(but they scale with tan .
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
• R-parity is automatically guaranteed• Two related seesaw contributions to neutrino
masses:
Aulakh, Bajc, Melfo, Senjanovic, Vissani 2003
SO(10) Constraints on FlavorOnly two Yukawa matricescontribute to fermion masses:
16f 16f = 10 + 120 + 126
The bidoublet components in10 and 126 Higgs multiplets take VEVs.Dominant type II seesaw isassumed.
Lepton mass matrices can beexpressed as a function ofquark parameters:
Is large mixing generated in the neutrino sector?
Is minimal SO(10) viable?
• b- unification is related with large atm(dominant -block in M
• 2-3 large mixing is generated, however we need also m2sol
<< m2atm: this implies sol ≈ or, alternatively, atm <
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 CKM? 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:
Mu,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 M120120H
2 + 10H120H210H
M120(1,2,2)1202 + (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 120120 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!
Ue3
sin22sol
m2sol /m2
atm
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 = , the type of mass spectrumand 02decay rate.