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.