1. Duality in Left-Right Symmetric Seesaw Mechanism Michele Frigerio Service de Physique Théorique, CEA/Saclay Rencontres de Physique des Particules March 3, 2006 - Institut Henri Poincaré Evgeny Kh. Akhmedov in collaboration with Evgeny Kh. Akhmedov Phys. Rev. Lett. 96, 061802 (2006) [hep-ph/0509299] & in preparation

2. Neutrino mass in Left-Right models Left-Right gauge symmetry: SU(2) L x SU(2) R x U(1) B-L  SU(2) L x U(1) Y extensions: SU 422, SO(10), … Pati, Salam, Mohapatra, Senjanovic, Georgi, Fritzsch, Minkowski Leptons: L = ( L l L ) T L c = ( R c l R c ) T Mass matrix in ( L, R c ) basis: Yukawa couplings: VEVs: - v R = breaks SU 221 into SU 21 - v = breaks SU 21 into U(1) em - v L =  v 2 /v R is induced by EW breaking

3. Seesaw in Left-Right models Effective mass matrix of light neutrinos: f determines heavy masses and mixing f directly contributes to light masses Seesaw mechanisms: - v << v R (Type I seesaw) - v L << v (Type II seesaw) Minkowski, Gell-Mann, Ramond, Slansky, Yanagida, Glashow, Mohapatra, Senjanovic Magg, Wetterich, Lazarides, Shafi, Mohapatra, Senjanovic, Schecter, Valle Type I and II contributions are strictly intertwined.

4. A very well motivated framework Seesaw explains (i) smallness of mass;(ii) baryogenesis via leptogenesis Left-Right models (i) incorporate naturally RH neutrinos; (ii) explain maximal parity violation Several completely realistic models (for unification, fermion masses, p-decay, …) do not contain sources of mass other than type I and II seesaw (fermion triplets, double seesaw, radiative mechanisms, extra dimensions, …) Supersymmetry can be easily incorporated. If only (B - L)-even Higgs bosons acquire VEVs, then R-parity is automatically unbroken. Let us take the LR seesaw formula seriously!

5. LR seesaw: the parameter space v 2 = (174 GeV) 2 (EWSB) 0  v L  GeV(  ≈ - 2 v L 2 / v 2 ) TeV  v R  M Pl (no RH weak currents) 0  (m ) ij  eV : partially known from oscillations data 0  y ij  1 : in general unknown Yukawa couplings, but -Minimal SUSY LR: y = tan  y e -Minimal SO(10): y = y u -Seesaw + mSUGRA:y ij << 1 to suppress, e.g.,    0  f ij  1 : completely unknown Yukawa couplings Bottom-up approach: what is the structure of the matrix f ? To what extent we can reconstruct the seesaw heavy sector ?

6. Seesaw duality Non-linear matrix equation in f  Multiple solutions Different structures of f are viable physical options One generation: f 2 - (m /v L ) f - v 2 y 2 /(v L v R ) = 0  f = f ± Three generations: f = f 1±, …, f 4± 4 pairs of dual f structures reproduce the same m Duality: f solution if and only if f is ^

7. Ambiguity on the seesaw type Suppose type II dominates: Suppose type I dominates: LR-symmetry  y = y T  duality holds  f II + f I = m f II is a solution if and only if f I is! If in a model f = f II, there is always another model where f = f I (with the same values for v L,R, m, y)

8. A realistic numerical example Tribimaximal mixing: tan 2  23 = 1 tan 2  12 = 0.5 tan 2  13 = 0 No CP violation v L v R = v 2 (natural when scalar potential couplings are of order 1) Neglect CKM-like rotations (both charged lepton and neutrino Yukawa couplings diagonal in the same basis) y 1 = 10 -2 y 2 = 10 -1 y 3 = 1 (inter-generation hierarchy slightly weaker than for charged fermions) Eigenvalues: -0.1, 0.2, 0.9 Normal hierarchy with  m 2 sol /  m 2 atm = 0.038 EX EX TH There are 4 dual pairs of f structures such that:

9. Features of the solutions

10. One seesaw type dominance in m 12, m 22, m 23 : type II in the case of f 4, type I in the dual. Mixed seesaw in m 11, m 13, m 33. Consider a given pair of dual solutions: Seesaw Duality: f 4 structure has dominant  -block; large (but non-maximal) 2-3 mixing Dual structure is hierarchical, with dominant 33-entry; small 2-3 mixing

11. Perspectives Identification of flavor symmetries in the structures of f. Radiative stability of the LR seesaw formula: –Running below the LR-symmetry breaking scale Baryogenesis via Leptogenesis: lepton asymmetry from the decays of R or  L –The matrix f determines masses and mixing of R ’s as well as couplings of  L to leptons –Each solution for f leads to different asymmetry More options for model-building –Different forms of f available to accommodate  mass –Extra symmetries of the model as selection criterion

12. Summary Neutrino mass in Left-Right symmetric models Analysis of the LR symmetric seesaw formula –8 structures of f reproduce the same m and y –Duality among solutions: f  (m  / v L - f) –Criteria to identify the dominant seesaw type –General analytic method to solve for f Spin-off both in phenomenology and theory E. Kh. Akhmedov, M.Frigerio & E. Kh. Akhmedov, PRL 96, 061802 (2006) [hep-ph/0509299] and in preparation