Leptogenesis and LFV in type I+II seesaw mechanism Stéphane Lavignac (SPhT Saclay) • introduction • reconstruction of the right-handed neutrino spectrum • implications for leptogenesis • implications for lepton flavour violation • conclusions [based on P. Hosteins, S. L. and C. Savoy, hep-ph/0606078, NPB to appear] 4th “Flavour in the LHC era” meeting CERN, 9-11 October 2006 Introduction Why consider the type I+II seesaw mechanism? • Both the type I (heavy right-handed neutrino exchange) and the type II (heavy SU(2)L triplet exchange) are present in many extensions of the SM, such as left-right symmetric theories and SO(10) GUTs. • Right-handed neutrinos are suggestive of grand unification. However, SO(10) models with type I seesaw mechanism generally fail to accommodate successful leptogenesis (Yν ∝ Yu ⇒ very hierarchical right-handed neutrino masses, with M1 ~ 10⁵ GeV). • More generally, studies of leptogenesis and LFV are usually done in the framework of the type I seesaw mechanism, or assume dominance of one of the two seesaw mechanisms. It is interesting to investigate whether the generic situation where both contributions are comparable in size can lead to qualitatively different results. Type I+II seesaw mechanism: ΔL = SU(2)L triplet with couplings fLij to lepton doublets 2 v T −1 II I Mν = fLvL − Y fR Y ≡ Mν + Mν vR Right-handed neutrino mass matrix: MR = fRvR vR ≡〈ΔR〉 scale of B-L breaking ΔR = SU(2)R triplet with couplings fRij to right-handed neutrinos ∆ 2 2 vL is small since it is an induced vev: vL ≡ " L# ∼ v vR/M∆L In theories with underlying left-right symmetry (such as SO(10) with a 1 2 6 H ), T one has Y = Y and f L = f R ⇒ 2 matrices of couplings Y and f In a fundamental theory, expect Y to be related to other Yukawa couplings ⇒ for phenomenological studies, need to reconstruct the fij as a function of the Yij (for a given set of low-energy neutrino parameters) For n generations, there are 2 n different solutions [Akhmedov, Frigerio] Reconstruction of the heavy neutrino mass spectrum 2 v −1 The left-right symmetric seesaw formula Mν = fvL − Y f Y vR with f, Y complex symmetric matrices (Y invertible), can be rewritten as 2 −1 v Z = αX − βX α ≡ vL , β ≡ vR −1 −1 T −1 −1 T T with Z = NY Mν (NY ) , X = NY f(NY ) , NY such that Y = NY NY Z complex symmetric ⇒ can be diagonalized by a complex orthogonal matrix OZ if its eigenvalues zi are all distinct: T T Z = OZ Diag (z1, z2, z3)OZ , OZ OZ = 1 Then X can be diagonalized by the same orthogonal matrix as Z, and its eigenvalues are the solutions of: −1 zi = αxi − βxi (i = 1, 2, 3) + − 2 solutions x i , x i for each i ⇒ 2³ = 8 solutions for X, hence for f: x1 0 0 T T ± f = NY OZ 0 x2 0 OZ NY , xi = xi 0 0 x3 [see Akhmedov, Frigerio for an alternative reconstruction procedure] Properties of the solutions −1 We denote the 2 solutions of z i = α x i − β x i by: 2 ± zi ± !zi + 4αβ 2 x ≡ |z3| ! 4αβ i 2α + + + + + − (+,+,+) refers to the solution ( x 1 , x 2 , x 3 ) , (+,+,–) to ( x 1 , x 2 , x 3 ) , etc 2 In the large vR limit ( 4 α β ! |z 1 | ): zi + , , Mν xi ! (“type II branch”) f (+ + +) −→ α vL β v2 − (−,−,−) −→ − −1 xi ! − (“type I branch”) f Y Mν Y zi vR The remaining 6 solutions correspond to mixed cases in which Mν receives significant contributions from both seesaw mechanisms 2 In the small vR limit ( | z 3 | ! 4 α β ): ± (±,±,±) −→ ± xi ! ± !β/α f !β/α Y If Y is hierarchical, f i − → ! β / α y i holds for all 8 solutions Application to SO(10) models with two 10’s and a 1 2 6 in the Higgs sector ! (1) (2) W Yij 16i16j101 + Yij 16i16j102 + fij 16i16j126 If the doublets in the 1 2 6 have no vev, then, in the basis of charged lepton mass eigenstates: mu 0 0 Y v = U T 0 m 0 U U = P V P q c q q u CKM d 0 0 mt m1 0 0 " Mν = Ul 0 m2 0 Ul Ul = PeUP MNSPν 0 0 m3 For a given choice of the yet unmeasured neutrino parameters (including the Majorana phases contained in Pν) and of the high energy phases contained in Pu, Pd and Pe, Y and Mν are known and f can be reconstructed as a function of the B-L breaking scale vR and of β/α β/α = v²/vLvR ~ (MΔL/vR)² depends on details of the model. Assume β/α = 1. Perturbativity constraint: require that the fij remain perturbative up to the Landau pole of the SO(10) gauge coupling (~ 2 x 10¹⁷ GeV) ⇒ constrains β/α ≤ O(1) and restricts the range of vR Mi Case !!! Mi Case "!! 1014 1014 13 10 13 10 12 10 1012 11 10 1011 10 10 1010 9 9 10 VR !GeV" 10 VR !GeV" 1012 1013 1014 1012 1013 1014 Mi Case !"! Mi Case ""! 15 10 1015 14 10 1014 13 10 1013 12 10 1012 11 10 1011 1010 1010 9 9 10 VR !GeV" 10 VR !GeV" 1012 1013 1014 1012 1013 1014 1015 Mi Case !!" Mi Case "!" 15 14 10 10 14 13 10 10 13 12 10 10 12 11 10 10 1011 10 10 1010 109 V !GeV" 9 12 13 14 R 10 VR !GeV" 8 10 10 10 12 13 14 15 10 108 10 10 10 10 7 10 107 106 106 105 105 Mi Case !"" Mi Case """ 15 1015 10 14 1014 10 13 1013 10 12 1012 10 1011 1011 1010 1010 9 9 10 V !GeV" 10 VR !GeV" 12 13 14 15 R 12 13 14 15 16 17 108 10 10 10 10 10810 10 10 10 10 10 107 107 106 106 105 105 Figure 1: Right-handed neutrino masses as a function of vR for each of the 8 solutions (+, +, +) −3 to ( , , ) in the reference case of a hierarchical light neutrino mass spectrum with m1 = 10 − − − u d ν e eV, β = α and no CP violation beyond the CKM phase (δ = Φi = Φi = Φi = Φi = 0). The range of variation of vR is restricted by the requirement that f3 1. Dotted lines indicate a fine-tuning greater than 10% in the (3, 3) entry of the light neutri≤no mass matrix. 9 Mi Case !!! Mi Case "!! 1014 1014 13 10 13 10 12 10 1012 11 10 1011 10 10 1010 9 9 10 VR !GeV" 10 VR !GeV" 1012 1013 1014 1012 1013 1014 Mi Case !"! Mi Case ""! 15 10 1015 14 10 1014 13 10 1013 12 10 1012 11 10 1011 1010 1010 9 9 10 VR !GeV" 10 VR !GeV" 1012 1013 1014 1012 1013 1014 1015 Mi Case !!" Mi Case "!" 15 14 10 10 14 13 10 10 13 12 10 10 12 11 10 10 1011 10 10 1010 109 V !GeV" 9 12 13 14 R 10 VR !GeV" 8 10 10 10 12 13 14 15 10 108 10 10 10 10 7 10 107 106 106 105 105 Mi Case !"" Mi Case """ 15 1015 10 14 1014 10 13 1013 10 12 1012 10 1011 1011 1010 1010 9 9 10 V !GeV" 10 VR !GeV" 12 13 14 15 R 12 13 14 15 16 17 108 10 10 10 10 10810 10 10 10 10 10 107 107 106 106 105 105 Figure 1: Right-handed neutrino masses as a function of vR for each of the 8 solutions (+, +, +) −3 to ( , , ) in the reference case of a hierarchical light neutrino mass spectrum with m1 = 10 − − − u d ν e eV, β = α and no CP violation beyond the CKM phase (δ = Φi = Φi = Φi = Φi = 0). The range of variation of vR is restricted by the requirement that f3 1. Dotted lines indicate a fine-tuning greater than 10% in the (3, 3) entry of the light neutri≤no mass matrix. 9 Mi Case !!! Mi Case "!! 1014 1014 13 10 13 10 12 10 1012 11 10 1011 10 10 1010 9 9 10 VR !GeV" 10 VR !GeV" 1012 1013 1014 1012 1013 1014 Mi Case !"! Mi Case ""! 15 10 1015 14 10 1014 13 10 1013 12 10 1012 11 10 1011 1010 1010 9 9 10 VR !GeV" 10 VR !GeV" 1012 1013 1014 1012 1013 1014 1015 Mi Case !!" Mi Case "!" 15 14 10 10 14 13 10 10 13 12 10 10 12 11 10 10 1011 10 10 1010 109 V !GeV" 9 12 13 14 R 10 VR !GeV" 8 10 10 10 12 13 14 15 10 108 10 10 10 10 7 10 107 106 106 105 105 Mi Case !"" Mi Case """ 15 1015 10 14 1014 10 13 1013 10 12 1012 10 1011 1011 1010 1010 9 9 10 V !GeV" 10 VR !GeV" 12 13 14 15 R 12 13 14 15 16 17 108 10 10 10 10 10810 10 10 10 10 10 107 107 106 106 105 105 Figure 1: Right-handed neutrino masses as a function of vR for each of the 8 solutions (+, +, +) −3 to ( , , ) in the reference case of a hierarchical light neutrino mass spectrum with m1 = 10 − − − u d ν e eV, β = α and no CP violation beyond the CKM phase (δ = Φi = Φi = Φi = Φi = 0).