1. A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia NO-VE 2006: ``Ultimate Goals’’

2. Two fundamental issues

3. Hierarchy of masses: |m 2 /m 3 | ~ 0.2 |sin  | 0 0.2 0.4 0.6 0.8 1-3 1-2 2-3 11 1-2,  12 2-3,  23 1-3,  13 Quarks Leptons 13 o 2.3 o 45 o ~ 0.5 o 34 o <10 o Mixing |m  /m  | = 0.06 Neutrinos Charged leptons |m s /m b | ~ 0.02 - 0.03Down quarks |m c /m t | ~ 0.005 Up-quarks

4. up down charged neutrinos quarks quarks leptons 10 -1 10 -2 10 -3 10 -4 10 -5 1 m u m t = m c 2 V us V cb ~ V ub at m Z Regularities? Koide relation

5.    permutation symmetry Neutrino mass matrix in the flavor basis: A B B B C D B D C Discrete symmetries S 3, D 4 Can both features be accidental? Often related to equality of neutrino masses For charged leptons: D = 0 Are quarks and leptons fundamentally different? Can this symmetry be extended to quark sector?

6. A. Joshipura, hep-ph/0512252 Smallness of V cb Maximal (large) 2-3 leptonic mixing 2-3 symmetry X A A A B C A C B Universal mass matrices +  m Quarks, charged leptons: B ~ C, X << A << B Neutrinos: B >> C, X ~ B - Hierarchical mass spectrum - Small quark mixing - Degenerate neutrino mass spectrum; - Large lepton mixing additional symmetries are needed to explain hierarchies/equalities of parameters 2-3 symmetry Does not contradict mass hierarchy

7. A B B B C D B D C    permutation symmetry Matrix for the best fit values of parameters (in meV) 3.2 6.0 0.6 24.8 21.4 30.7 Bari group sin 2  13  = 0.01 sin 2  23 = 0.43 Substantial deviation from symmetric structure Structure of mass matrix is sensitive to small deviations Of 1-3 mixing from zero and 2-3 mixing from maximal

8. Similar gauge structure, correspondence Similar gauge structure, correspondence Very different mass and mixing patterns Very different mass and mixing patterns Particular symmetries in leptonic (neutrino) sector? Q-L complementarity? Additional structure exists which produces the difference. Is this seesaw? Something beyond seesaw? Symmetry correspondence

9. Majorana masses Majorana masses Q  = 0 Q c = 0 Is this enough to explain all salient properties of neutrinos? mix with singlets of the SM mix with singlets of the SM Neutrality Basis of seesaw mechanism Dynamical effects

10. l l L lRlR Standard Model... H S Window to hidden world? M S s R A A Planck scale physics s S

11. Screening of the Dirac structure Induced effects of new neutrino states

12. Correspondence: u r, u b, u j d r, d b, d j e color Symmetry: Leptons as 4 th color Unification: form multiplet of the extended gauge group, in particular, 16-plet of SO(10) Pati-Salam Can it be accidental? More complicated connection between quarks and leptons? Complementarity?

13. Provide with all the ingredients necessary for seesaw mechanism Large mass scale Lepton number violation RH neutrino components Give relations between masses of leptons and quarks generically m b = m  b -  unification In general: ``sum rules’’ But - no explanation of the flavor structure large 2-3 leptonic mixing

15. is realized in terms of the mass matrices (matrices of the Yukawa couplings) and not in terms of observables – mass ratios and mixing angles. Y U = Y D = Y D = Y L = Y 0 Universal structure for mass matrices of all quarks and leptons in the lowest approximation: Y f = Y 0 +  Y f ( Y 0 ) ij >> (  Y f ) ij f = u, d, L, D, M approximate Mass matrices M = Y V Mass matrices M = Y V diagonalization Eigenvalues = masses Eigenstates = mixing Small perturbations: I. Dorsner, A.S. NPB 698 386 (2004)

16. Small perturbations allow to explain large difference in mass hierarchies and mixings of quarks and leptons Unstable with respect to small perturbations Y f ij = Y 0 ij (1 +  f ij ) Universal singular f = u, d, e, Perturbations  ~ 0.2 – 0.3 4 3 2 Y 0 = 3 2 2  ~ 0.2 - 0.3 Form of perturbations is crucial Important example:

17. Nearly singular matrix of RH neutrinos leads to - enhancement of lepton mixing - flip of the sign of mixing angle, so that the angles from the charged leptons and neutrinos sum up Seesaw: m ~ 1/M

18. In some (universality) basis in the first approximation all the mass matrices but M l (for the charged leptons) are diagonalized by the same matrix V: V + M f V = D f For the charged leptons, the mass V T M l V * = D l is diagonalized by V * M l = M d T V for u, d, V * for l Diagonalization: SU(5) type relation Quark mixing: V CKM = V + V = I Lepton mixing: V PMNS = V T V In the first approximation A Joshipura, A.S. hep-ph/0512024 Another version is when neutrinos have distinguished rotation: V for u, d, l V * for

19. V CKM = V’ + V V PMNS = V T V’ In general, up and down fermions can be diagonalized by different matrices V’ and V respectively V PMNS = V T V V CKM + = V 0 PMNS V CKM + V PMNS V CKM = V T V Quark and lepton rotations are complementary to VV T V 0 PMNS = V T V - symmetric, characterized by 2 angles; - close to the observed mixing for  /2 ~  ~ 20 – 25 o - 1-3 mixing near the upper bound - gives very good description of data - predicts sin  13 > 0.08 V PMNS (with CKM corr.)

20. M u, ~ m D * A D * Universal mixing and universal matrices M d ~ m D * A D M l ~ m D A D * D = diag(1, i, 1) A is the universal matrix: A ~                           i ~ 0.2 – 0.3 Can be embedded in to SU(5) and SO(10) with additional assumptions

22. 11  l   q 12   sol  C = 46.7 o +/- 2.4 o A.S. M. Raidal H. Minakata H. Minakata, A.S. Phys. Rev. D70: 073009 (2004) [hep-ph/0405088]  l   q 23   atm  V  cb = 45 o +/- 3 o 2-3 leptonic mixing is close to maximal because 2-3 quark mixing is small Difficult to expects exact equalities but qualitatively 1-2 leptonic mixing deviates from maximal substantially because 1-2 quark mixing is relatively large

23. Quark-lepton symmetry Existence of structure which produces bi-maximal mixing Existence of structure which produces bi-maximal mixing sin  C = 0.22 as ``quantum’’ of flavor physics sin  C = 0.22 as ``quantum’’ of flavor physics sin  C = m  /m  sin  C ~ sin  13 Appears in different places of theory Mixing matrix weakly depends on mass eigenvalues In the lowest approximation: V quarks = I, V leptons =V bm m 1 = m 2 = 0 ``Lepton mixing = bi-maximal mixing – quark mixing’’

24. F. Vissani V. Barger et al U PMNS = U bm U bm = U 23 m U 12 m Two maximal rotations ½ ½ -½ ½ ½ ½ -½ ½ - maximal 2-3 mixing - zero 1-3 mixing - maximal 1-2 mixing - no CP-violation Contradicts data at (5-6)  level In the lowest order? Corrections? U PMNS = U’ U bm U’ = U 12 (  ) Generates simultaneously Deviation of 1-2 mixing from maximal Non-zero 1-3 mixing 0 U bm = As dominant structure? Zero order?

25. Charged leptons Charged leptons sin(  C ) + 0.5sin  C ( 2 - 1) Neutrinos q-l symmetry tan 2  12 = 0.495 H. Minakata, A.S. R. Mohapatra, P. Frampton, C. W.Kim et al., S. Pakvasa … Maximal mixing CKM mixing m D ~ m u Maximal mixing m l ~ m d q-l symmetry m D T M -1 m D sin(  C ) CKM mixing QLC-2QLC-1 sin  12 = sin  13 = sin  C / 2 ~ V ub

26. 29 31 33 35 37 39  12 Fogli et al Strumia-Vissani 33 22 11 SNO (2 ) QLC2 tbm QLC1  - analysis does change bft but error bars become smaller  12  +  C ~   /4 90% 99% U tbm = U tm U m 13 U QLC1 = U C U bm Give the almost same 12 mixing coincidence

27. P. F. Harrison D. H. Perkins W. G. Scott U tbm = U 23 (  /4)U 12 - maximal 2-3 mixing - zero 1-3 mixing - no CP-violation U tbm = 2/3 1/3 0 - 1/6 1/3 1/2 1/6 - 1/3 1/2 2  is tri-maximally mixed 3 is bi-maximally mixed sin 2  12 = 1/3 in agreement with 0.315 Mixing parameters - some simple numbers 0, 1/3, 1/2 S 3 group matrix Relation to group matrices? In flavor basis… relation to masses? No analogy in the Quark sector? Implies non-abelian symmetry L. Wolfenstein

28. 0 0.01 0.02 0.03 0.04 0.05sin 2  13 Fogli et al Strumia-Vissani 33 22 11 T2K 90% 99% Double CHOOZ CC  m 21 2 /  m 32 2 Non-zero central value (Fogli, et al): Atmospheric neutrinos, SK spectrum of multi-GeV e-like events Lower theoretical bounds: Planck scale effects RGE- effects V.S. Berezinsky F. Vissani M. Lindner et al In agreement with 0 value QLC1

29. 1). Superheavy M S >> v EW - decouple 2). Heavy: v EW >> m S >> m 3). Light: m S ~ m  play role in dynamics of oscillations

30. 0 m D 0 m = m D T 0 M D T 0 M D M S If M D = A -1 m D m = m D M D -1 M S M D -1 m D m = A 2 M S Structure of the neutrino mass matrix is determined by M S -> physics at highest (Planck?) scale immediately m D similar (equal) to quark mass matrix - cancels m D << M D << M S Double (cascade) seesaw M R = - M D T M S -1 M D Additional fermions A ~ v EW /M GU N S R. Mohapatra PRL 56, 561, (1986) M S – Majorana mass matrix of new fermions S M. Lindner M. Schmidt A.S. JHEP0507, 048 (2005) A.S. PRD 48, 3264 (1993) R. Mohapatra. J. Valle

31. leads to quasi-degenerate spectrum if e.g. M S ~ I, leads to quasi-degenerate spectrum if e.g. M S ~ I, Structure of the neutrino mass matrix is determined by origin of ``neutrino’’ symmetry origin of ``neutrino’’ symmetry origin of maximal (or bi-maximal) mixing origin of maximal (or bi-maximal) mixing MSMS M S ~ M Pl ? Q-l complementarity Reconciling Q-L symmetry and different mixings of quarks and leptons Seesaw provides scale and not the flavor structure of neutrino mass matrix

32. Mixing with sterile states change structure of the mass matrix of active neutrinos Consider one state S which has - Majorana mass M and - mixing masses with active neutrinos, m iS (i = e, ,  ) Active neutrinos acquire (e.g. via seesaw) the Majorana mass matrix m a After decoupling of S the active neutrino mass matrix becomes (m ) ij = (m a ) ij - m iS m jS /M induced mass matrix sin  S = m S /M m ind = sin  S 2 M

33. sin  S 2 M > 0.02 – 0.03 eV Induced matrix can reproduce the following structures of the active neutrino mass sin  S 2 M ~ 0.003 eV Sub-leading structures for normal hierarchy sin  S 2 M < 0.001 eV Effect is negligible Dominant structures for normal and inverted hierarchy

34. In the case of normal mass hierarchy m tbm ~ m 2 /3 + m 3 /2 1 1 1 0 0 0 0 1 -1 0 -1 1 m 2 =  m sol 2 Assume the coupling of S with active neutrinos is flavor blind (universal): m iS = m S = m 2 /3 Then m ind can reproduce the first matrix m tbm = m a +  m ind m a is the second matrix Two sterile neutrinos can reproduce whole tbm-matrix

35. R. Zukanovic-Funcal, A.S. in preparation Two regions are allowed: M S ~ 0.1 – 1 eVM S > (0.1 – 1) GeV and

36. Q & L: - strong difference of mass and mixing pattern; - possible presence of the special leptonic (neutrino) symmetries; - quark-lepton complementarity Still approximate quarks and leptons universality can be realized. Mixing with new neutrino states can play the role of this additional structure: - screening of the Dirac structure - induced matrix with certain symmetries. This may indicate that q & l are fundamentally different or some new structure of theory exists (beyond seesaw)

37. 0.2 0.3 0.4 0.5 0.6 0.7sin 2  23 Fogli et al 33 22 11 T2K SK (3 ) 90% QLC1 SK (3 ) - no shift from maximal mixing 1). in agreement with maximal 2). shift of the bfp from maximal is small 3). still large deviation is allowed: (0.5 - sin 2  23 )/sin  23 ~ 40% 22 maximal mixing Gonzalez-Garcia, Maltoni, A.S. sin 2 2  23 > 0.93, 90% C.L. QLC2

38. R. Zukanovic-Funcal, A.S. in preparation

40. Smallness of mass Large mixing Possibility that their properties are related to very high scale physics Can propagate in extra dimensions Manifestations of non- QFT features? Violate fundamental symmetries, Lorentz inv. CPT, Pauli principle?