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Discrete and Unified Ideas on Fermion Mixing

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 .

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Discrete and Unified Ideas on Fermion Mixing

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  1. 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

  2. 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).

  3. Comparing two approaches to fermion mixing • Discrete Family symmetries: • Discrete groups have more low dimensional representations than continuous Lie groups. • Non-Abelian groups can relate different generations, because of irreducible representations with dimension larger than one. • Features of fermion mixing can be related to the structure of the EW Higgs sector. • Grand Unification symmetries: • Different SM fermions fit into the same large representation of a larger gauge group. • Quark-lepton relations induced by gauge structure. • Connection between the GUT scale and the seesaw scale, where neutrino masses are generated.

  4. Data on neutrino oscillations Maltoni, Schwetz, Tortola, Valle, hep-ph/0405172 New J. Phys. 6 : 122, 2004 Assuming normal ordering of the mass spectrum Assuming inverted ordering of the mass spectrum

  5. 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 . Quark - Lepton Symmetry ?

  6. 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 q13AND 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)

  7. Perfect Geometric Solids in All Dimensions 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)).

  8. 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, q23 ~ p/4 determines thedominant 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). • (nala)T is SU(2)L isodoublet flavor alignment expected between naand la cancellation between mixing in Mn and Ml(that is the case for quarks: qq23  2º).

  9. Maximal mixing in 2 2 matrices Ml M Flavor Symmetry Models U(1), Zn m2atm from symmetry-breaking non-Abelian Q8 non-Abelian (if any) m from simmetry-breaking non-Abelian S3 (+ = - /2) Q8 (+ =  ,  = ) Q8 non-Abelian

  10. 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 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) (8 vertices of the hyper-octahedron on the 4-sphere) (1 2)T i j (1 2)T

  11. 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 s1 , ± s2 , ± i s3 . • 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+  )

  12. Yukawa coupling structure • Yukawa couplings: Ykij i cj FkThe matrix structure depends on Fk assignments. • Two Higgs doublets: F1 ~ 1+ + , F2 ~ 1+ –

  13. The neutrino sector • Majorana mass term: naMabnb • 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: x1and x2 in two different 1-dim irreps; (x3 x4)~ 2 to generate the 1-2 mixing.

  14. Q8 predictions for neutrinos (I) Scenarios (1) or (2): (x1 ~ F1 , x2 ~ F2or ~ F2) • 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: q23 = p/4  q13 = 0 • Observable neutrinoless 2b-decay: mee = a > 0.02 eV

  15. 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! • One texture zero and one equality • Normal hierarchy: 0.035 eV < m3 < 0.065 eV • sinq13 < 0.2 sin22q23 > 0.98 • No neutrinoless 2b decay: mee = 0 Scenario (3): (x1 ~ F1 , x2 ~ F2)

  16. 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~ 10-15 (exp.: 7 10-15), DmD/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 h0decays into t + t - and m + m -with comparable strength (~ mt / mW ).

  17. q13 = 0 with a Z2 Z2 symmetry • sin2q13 < 0.047 at 3s C.L. • If (ni , li), lic ~ (+,), (,+), (,), F1 , x1 ~ (+,+), F2 ~ (+,), x2 ~ (,), then: (the same structure can be obtained with type I seesaw if the heaviest RH neutrino decouples) • SOME RECENT DISCRETE MODELS FOR q13=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

  18. m - t Lepton Flavor Violation qL=q23 • Z2 Z2 Higgs potential is CP conserving: H0 (SM-like) and h0mix (both CP-even), A0 (CP-odd) and h are mass eigenstates. • Lepton Flavor Violation mediated by h0, A0, h : • BR(3m) ~ 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 b).

  19. 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

  20. Only two Yukawa matrices contribute to fermion masses: 16f  16f = 10 + 120 + 126 The bidoublet components in 10 and 126 Higgs multiplets take VEVs. Dominant type II seesaw is assumed. Lepton mass matrices can be expressed as a function of quark parameters: Is large mixing generated in the neutrino sector? SO(10) Constraints on Flavor

  21. Is minimal SO(10) viable? • b-t unification is related with large qatm !(dominant mt-block in Mn) • 2-3 large mixing is generated, however we need also Dm2sol << Dm2atm: this implies qsol ≈p/4or, alternatively, qatm <p/4significantly • Numerical analysis of the real case: agreement with data only allowing 2s 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.

  22. 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>

  23. Numerical fit with and without 120 Ue3 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! Dm2sol /Dm2atm sin22qsol

  24. 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 qatm = p/4, the type of mass spectrumand 02b-decay rate.

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