1. Resonant Leptogenesis In S4 Model Nguyen Thanh Phong Cantho University In cooperation with Prof. CSKim, SKKang and Dr. YHAhn (work in progress)

2. Outline 1.Introduction 2.S4 model 3.S4 model with a soft breaking term 4.Resonant Flavored Leptogenesis through soft breaking 5.Summary

3. 1. Introduction oscillations 1σ (G.L.Fogli …,hep-ph:0809.2936; 0806.2649) Bi-large Mixing Angles No information about all 3 CP phases: ,  1,  2 Tri-Bimaximal mixing pattern-TBM (Harrison, Perkin, Scott, 2002) Need deviations from the TBM…

4. 1. Introduction mass spectrum Absolute scale of neutrino mass? 1.Effective e from single  decay Current limit from Mainz+Troitsk (Mainz) m  <1.8(2.2) eV Future Katrin may lower this down to 0.25eV Current available lower bound limit 0.2eV Future lower bound sensitivity 0.01eV 3.Cosmological limit  m i < 0.61eV Also the cosmological observations show that the baryon is not symmetry in our Universe and the baryon to photon ratio is obtained 2. Effective Majorana mass from 0 2  decay

5. 2. S4 model How to obtain TBM mixing matrix? @ … based on continuous group: SO(3) - (SFKing, JHEP0508105); SU(3) - (G.G. Ross, NPB733, 31 (2006))…. @ … based on discrete symmetry group: A4 – E. Ma, K.S. Babu, G. Altarelli, F. Feruglio, M. Hirsch, A. See, S.L. Chen, S.K. Kang….. @ … based on discrete symmetry group: S4 – E. Ma, C.S. Lam, Mohapatra, H. Ishimori, Luca Merlo et al., Y. Cai, Y. Koide, Ding… S4 is given by permutations of 4 objects …defined by 2 generators S & T which satisfy S 4 = T 3 = (ST 2 ) 2 =I …composed by 24 elements => 5 irreducible representations 1 1, 1 2, 2, 3 1, 3 2 Multiplication rules for S4

6. 2. S4 model Recent paper on S4 symmetry by Ding ( Nucl.Phys.B827:82-111,2010 ) After S4 breaking and spontaneous breaking one get diagonal charged lepton matrix and Dirac and Majorana neutrino mass matrices are After seesawing, the neutrino mass matrix is exact diagonalized by TBM matrix Majorana CP phases The light neutrino masses can be both normal or inverted hierarchy. Since mixing matrix has TBM pattern, only  1 contributes to effective Majorana mass in 0 2  decay:

7. 2. S4 model Allowed parameter region by 1  low energy experimental data: cos  >0 (red) and cos  <0 (green) correspond to normal and inverted ordering of light neutrino massess The prediction of m ee as a function of  and r The correlation between high energy phase  and the Majorana phase  1

8. 2. S4 model In the basis where M R is real and diagonal, the Yukawa coupling matrix is given. The combination of Yukawa coupling matrix which is relevant to leptogenesis is then obtained to be real unflavored leptogenesis could not work. Besides, for leptogenesis to be viable the exact degenerate of heavy Majorana neutrino masses have to be lifted. Solution? Consider the contributions from the next to leading order terms. Introducing a soft breaking term: this is our method in this work. others…

9. 3. S4 model with a soft breaking term… Introduce a soft-breaking term in a single element ( ,  ) of M R : to simplify our discussion we only consider an element (2,2) there are 9 possibilities After seesawing, the light neutrino masses and lepton mixing matrix are modified to be: a i are functions of r and  It is interesting that U 13 has non-zero value which indicates non zero of the mixing angle  13 and Dirac CP phase  ; and  12(23) are also lifted from their TBM value s. By a suitable choice of ,  13 can be obtained a value that can be measured by fut ure short and long baseline neutrino oscillation experiments. In the follow figures we show the mixing angles and Dirac CP phase with  = 0.1

10. 3. S4 model with a soft breaking term… The red (green) color corresponds to the normal (inverted) ordering of light neutri no masses which is cos  >0 (cos  90 0 (90 0 -> 270 0 ).

11. 4. Resonant Flavored Leptogenesis through soft breaking For almost degenerate heavy Majorana masses, the CP asymmetry by N i given by The decay width of N j ; Mass slitting parameter With soft breaking term, the masses of heavy neutrinos are obtained Then the mass slitting parameters are approximated about  N ~ . The Dirac Yukawa coupling matrix in this case is modified to be where the diagonalizing matrix of M R given as The parameter  is assumed very small, then we find that H matrix is almost the same as before, as a result the contributions from N 3 decay to lepton asymmetry are negligible since

12. 4. Leptogenesis… Before going to detail discussion of leptogenesis, notice that since  << 10 -6, as can see later, the effects of soft breaking on low energy observables are negligible, hence the low energy observables are given as without soft breaking. The CP flavored asymmetries are then calculated as Here r and  are determined in the above,  is arbitrary and a is determined once m 0 and M a re known (a 2 =m 0 M/v u 2 ). Thus in our numerical calculation we can take M and  as indepen dent inputs, however the lepton asymmetries given above depend on quantity M/ . We can see that the lepton asymmetries can be arbitrary enhanced by lowering , however, the perturbation parameter  is constrained from the condition of resonant leptogenesis Taking the seesaw scale M = 10 6 GeV, then it requires  » 10 -10

13. 4. Leptogenesis… Besides CP asymmetries, in order to calculate baryon asymmetry we need to calculate the washout parameters The final formula for the baryon asymmetry with wash-out factor We can see in the above, up to the first order, the CP asymmetries have a relation and the washout factor for  and  are also equal, then the value of baryon asymmetry can be obtained as

14. 4. Leptogenesis… From numerical calculation we obtained the washout factors for normal and inverted hierarchy of light neutrino masses as, respectively is needed for successful leptogenesis leading to   10 -6 is required for M = 10 6 GeV The predictions of  B as a function of |  m ee  | for M = 10 6 GeV, tan  = 2.5 for normal hier-archy (left figure) and inverted hierarchy (right figure) of neutrino masses. The green (upper) and the red (lower) patterns correspond to  = 10 -7 and  = 10 -6, resp ectively.

15. 5. Summary We study the S4 model in the context of a seesaw model which naturally leads to the TBM form of the lepton mixing matrix. In the model, the combination of Yukawa coupling matrix, which is relevant for leptogenesis, is real. Besides, the heavy right-handed masses are exactly degenerate in the model. Those reasons forbid the leptogenesis to occur. By introducing a soft breaking term in heavy Majorara neutrino mass matrix, the mixing angles are lifted from their TBM values and the none-zero Dirac CP phase is obtained. And also the exact degenerate heavy Majorara neutrino masses are lifted leading to flavored leptogenesis become viable. Interestingly that we find a direct link between leptogenesis and the neutrino- less double beta decay parameter |  m ee  | through a high energy phase  We also show that our predictions for |  m ee  | can be constrained by the current observation of baryon asymmetry  B = 6.1  10 -10 The needed scale of heavy right-handed neutrino mass is small enough in our work so that the gravitino problem is safely avoided.