Non-zero Ue3, Leptogenesis in A4 Symmetry 2 nd International Workshop on Dark Matter, Dark Energy and Matter-Antimatter Asymmetry Y.H.Ahn (Academia Sinica) based on the paper with C.S.Kim and S. Oh 1
Outline Present Knowledges and Tri-Bimaximal Mixing A Hint for Non-zero Ue3? A4 symmetry+tbm in type-iii Seesaw Deviations from TBM Alternatives to deviations from TBM, realized in type-iii Seesaw Leptogenesis Conclusion 2
Present Knowledges Neutrino oscillation (PRL101,141801 G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A.M. Rotunno) Bi-Large mixing angles theta13>0 Nothing is known about all three CP-vilating phases δ, CP ϕ, 1 ϕ2 Cosmological limit (including WMAP 3-years result) upper bound on neutrino masses (JCAP10,104 : Uros Seljak, Anze Slosar, Patrick McDonald) Starting to disfavor the degenerate spectrum of neutrinos BAU η B #of baryons(n ) = 6.2 10 #of photons(n ) B 10 γ 3
All data can be explained in terms of oscillation between just 3 known species : Two possible orderings of neutrino masses Earth matter effects(lbl) Quasi-Degenerate case 4
Pontecorvo-Maki-NaKagawa-Sakata (PMNS) Matrix U PMNS 1 0 0 c 0 s e c s 0 iδcp 13 13 12 12 = 0 c23 s23 0 1 0 s12 c12 0 iδcp 0 s23 c 23 s13e 0 c 13 0 0 1 Atmospheric and SBL reactor Solar and LBL accelerator LBL reactor P ν Majorana phases P ν Neutrinoless Double beta decay 5
Tri-Bimaximal The current neutrino oscillation data in 3σ are well described by so called Tri-Bimaximal mixing matrix (Harrison, Perkins and Scott; see also Wolfenstein(1970) and He and Zee) It suggests that flavor structure for mixing should be divorced from trying to understand the mass eigenvalues. m U m U A B B A + B+ D A B D TBM * diag? 1 1 ν = TBM ν TBM =. 2 ( ) 2 ( + ) 1.. 2 ( A+ B+ D) where 1 1 A= (2 m1 + me ), B= ( me 3 3 2 m), D= me 2iα 2iα 2iβ 2 2 1 3 Correlations between mass matrix elements It is suggestive of a flavor symmetry. 6
Deviations from Tri-Bimaximal (J.Phys.Conf.Ser.203:012103,2010 ; G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo and A.M. Rotunno) 7
Deviations from Tri-Bimaximal Solar mixing Angle sin 2 U θ = 1 U 2 e2 12 2 e3 Tri-maximal assumption 2 U e2 = 13 sin 2 1 1 θ 12 = 2 31 3 e ( U 3 ) disfavored in 1σ due to 1 2 sin θ 12 = 0.331< 3 This situation becomes worse when U e3 2 is non-zero We don t know yet 2 1 2 1 2 1 sin θ23 = or sin θ23 > or sin θ23 < 2 2 2 8
Seesaw A simple and attractive explanation of the smallness of ν mass : Origin of B L operator SEESAW MECHANISM: (i) SM+RH ν (EW singlet) (ii) SM+SU(2) Triplet Higgs (iii) SM+SU(2) Triplet fermions while Ł of the EW int. keeps invariant SU(2) U(1) * d d 3 3 Seesaw model has 18 parameters: 12 real+6 phases (cf. Casas Ibarra YD = UPMNS mν R MR υ ) Integrating out the heavy fermions leaves us with observable mass matrix (9 observables: 6 real+3 phases) Half of the parameters of the model get lost at low-e The 3 low-e CP-violating depend, in general, on all 6 seesaw phases. The effects of high-e CP-violating phases control the generation of the BAU in the leptogenesis scenario, in <m> and in the leptonic CP-violating rephasing invariant Jcp. In approaches to reconstruct the high-energy physics from low-energy data, one can assume an flavor symmetry, which may reduce the unknown parameters. 9
A4 Unless flavor symmetries are assumed, particle masses and mixings are generally undetermined in gauge theory: To understand the present neutrino oscillation data we consider A4 flavor symmetry. (E.Ma and G.Rajarasekaran; G.Altarelly and F.Feruglio; X.G.He, Y.Y.Keum and R.Volkas) Tri-Bimaximal Non-zero U e3 : Higher order corrections and very small theta13 In seesaw-i or -III + A4 : Leptogenesis scale ~ 10^13-10^15 GeV, due to the equal size of moduli of neutrino Yukawa couplings (strong wash-out) Without the aid of higher order corrections or soft-breaking term, Non-zero theta13 as well as Leptogenesis are not possible TeV-scale Resonant-Leptogenesis cf) Branco et al [PRD79,093007]: theta13=0 + Resonant-leptogenesis cf)ahn et al [To appear in PRD] Leptogenesis and neutrinoless double beta decay in S4 symmetry 10
A4 A4 is the symmetry group of the tetrahedron and the finite groups of all twelve the even permutation of four objects: its irreducible representations contain one triplet 3 and three singlets 1,1,1 with the multiplication rules 3 3=3+3+1+1 +1 and 1 1 =1 Let s denote two A4 triplets and where The 12 representation matrices for 3 Identity matrix : I Reflection matrices : r1=diag.(1,-1,-1), r2=diag.(-1,1,-1), r3=diag.(-1,-1,1) Cyclic, Anticyclic matrices : 0 0 1 0 1 0 0 1 0 1 0 0 1 1 c = a = 1 0 0, a = c = 0 0 1, rcr i i, rar i i 11
Type-III See-saw (R.Foot, H.Lew, X.-G. He, G.C. Joshi 1989) In the framework of an extension of the SM, consisting of a Y=0 complex isospin triplet of fermion fields To generate Seesaw neutrino mass matrix Tr[ Σ idσ] Gauge interactions LHC! & Resonant-Leptogenesis! The scalar sector, apart from the usual SM Higgs doublet, is extended through the introduction of the two type of scalar fields To incorporate A4 symmetry 12
Under SU(2) U(1) A4 Z2 Construction of Lagrangian Hence its Yukawa interaction in the lepton sector Z2: forbidden 13
In the charged lepton sector: Assumption: the VEVs of A4 triplets can be equally aligned, i.e, Tri-maximal Charged lepton mass matrix comes from and has the form U(w) Diag.(arbitrary eigenvalues) In the neutrino sector: The Yukawa interaction, after EW symmetry breaking : Z2 symmetry breaking : unit matrix No Leptogenesis and No CP-violation( ) 14
Taking the scale of A4 symmetry breaking to be above EW scale, and assuming the vacuum alignment of heavy singlet scalar, Note that, RH neutrino term Heavy charged lepton term of 3a due to indentically vanishes due to Majorana property Heavy triplet lepton mass : where (M: real and positive) It will give rise to Bi-maximal For convenience sake, re-basing the light charge lepton and heavy fermion parts to be diagonal 15
Then, the Yukawa interaction and charged gauge interaction lagrangian in a weak eigenstate basis where These relative size will determine the mass ordering of light neutrinos. (cf. Y unit matrix) Bi-maximal These high-energy CP-violating Majorana phases are correlated to low-energy CP-violating Majorana phases: Leptogenesis 0νββ -decay (PRD ) 16
The couplings of N and E with light leptons and scalar η The neutrino mass matrix with The charged lepton mass matrix 17
which can be diagonalized by the unitary matrices and the transformation matrices between fields in weak interaction basis and in the mass eigenstate basis With the seesaw assumption M>>v, up to order the components of mixing matrices is the diagonalizing matrix of the light neutrino matrix forming Seesaw-I formula which can be diagonailzed by =Tri-Bimaximal 18
with the tri-bimaximal mixing matrix are the diagonalizing hermitian matrices, respectively 19
In the mass eigenstate the charged gauge interaction term From the light charged lepton current Since can be diagonalized by, and and 20
Deviations TBM from higher dimensional operators In the Lagrangian level, assume that above the cutoff scaleλ there is no CP violation term in the Dirac Yukawa neutrino and charged lepton Yukawa interactions, which for scale below Λ is expressed in terms of 5-dimentional operators. In the presence of 5-D operators driven byχ -VEV alignment, the Yukawa interactions in the lepton sector, which is invariant under SU(2) U(1) A4 Z2 21
The corrected neutrino Yukawa matrix : where High-energy and Low-energy CP-violation sources (Leptogenesis Ue3) The corrected light charged lepton mass matrix : where 22
For the most natural case of hierarchical charged lepton Yukawa couplings the corrected off-diagonal terms are not larger than the diagonal ones in size Only the mixing matrix takes part in PMNS mixing matrix where Low-energy CP-violation 23
For convenience sake, re-basing the light charge lepton and heavy fermion parts to be Diagonal Then, the Yukawa interaction and charged gauge interaction lagrangian in a weak eigenstate basis here same corrected by 5-D operators In a mass eigenstate basis of the charged gauge interaction term which can not be diagonalized by TBM 24
An additional mixing matrix to diagonalize θ=π/4 +δ for δ«1 with the mixing angle 25
The neutrino mass eigenvalues are given as From the above the solar and atmospheric mass squared differences are written in a good approximation as 26
PMNS Mixing matrix that is, for the charged lepton correction is negligible In the limit of exact TBM is recovered. 3σ(1σ) Exp. Bounds this is disfavored in 1σ Exp. Results due to the upper bounds Therefore, sizable contributions from the charged lepton sector are required to reach accordance with the 1σ Exp. results 27
PMNS Mixing (charged lepton correction) Using where For and we have Interestingly enough, for which means in this limit only we can have λ 0, and in turn can be corrected sizably by the light charged lepton part. 28
PMNS Mixing (charged lepton correction) For a negative (positive) value of λ, we have For λ 0 and >0 <0 29
Alternatives to deviation from TMB In the case of, that is, taking the scale of A4 symmetry breaking to be much lower than the cutoff scale Λ the contributions from 5-dimensional operators driven by χ-vev alignment can be negligible. Alternatively, we introduce a scalar filed Ψhaving SU(2) doublet, and instead of the bare mass M, a singlet scalar θ filed. We take the scale of A4 Z2 Z4 to be lower than the cutoff scale Λ. 30
Alternatives to deviation from TMB For convenience sake, re-basing the light charge lepton and heavy fermion parts to be Diagonal Then, the Yukawa interaction and charged gauge interaction lagrangian in a weak eigenstate basis The charged lepton part Low-energy CP-violation where 31
Alternatives to deviation from TMB With the seesaw assumption M>>v, up to order the components of mixing matrices are the diagonalizing hermitian matrices, respectively 32
Alternatives to deviation from TMB In the mass eigenstate the charged gauge interaction term From the light charged lepton current : can be sizable whose diagonalization matrix, in general, Since and The PMNS mixing matrix is forming as 33
Another very attractive feature of Seesaw? In addition to the explanation of neutrino masses, seesaw has another appearing feature so-called Leptogenesis η B nb s n n Σi L nb s = κ s nγ s n Σ n i L n γ Lepton asymmetry: Br * ( Σi + η) Br( Σi + η ) MATTER antimatter n n α εi = εi 0 n Σ i α ε α i 2 Im{( YY) ij ( Y) i ( Y) j } M ν ν ν α ν α j = f 2 j i 8 π ( YY ν ν) ii M i It is independent of mixing angles and CP phases of neutrino oscillation. The loop function includes the 1-loop vertex and self energy corrections to the heavy neutrino decay amplitude. x x f( x) = x 1 + (1 + x) ln + 1 + x 1 x 34
Another very attractive feature of Seesaw? wash-out effects: The generated asymmetry survives if decays takes place out-of-equilbrium at otherwise, inverse decay and scattering processes cancel the asymmetry T M i We are in the energy scale where A4 symmetry is broken but the SM gauge group remains unbroken. Choose at or around TeV scale 2 3 2 T Γ α 5 10 yt α < H for T> T M Pl (PRD49,6394 James M. Cline, Kimmo Kainulainen, Keith A. Olive) Flavor effects: EQ wash-out factor with 35
A very attractive feature of Seesaw? The major quantitative difference between the type-i Seesaw leptogenesis and type-iii leptogenesis lies in the fact that fermion triplets couple to SM gauge bosons (NPB 695(2004)169 T.Hambye et.al., JHEP10(2010)036 D. A. Sierra et al) Eq Gauge reactions decouple at T when Γ H = γ n H 1. Eq At this stage, if inverse decay process η Σ are decouple as well ( Γ D H = γ D nσ H 1 ) the CP-violating out-of equilibrium decays of the triplets will produce a A A Σ B L asymmetry. 4 Since γ ~, as M A γd g Mm Σ Σ decreases only large value of m are available to maintain inverse decays active after gauge reaction decoupling takes place. The correct amount of BAU can only be generated for In seesaw-iii + A4 : leptogenesis scale ~ Resonant-Leptogenesis TeV scale M Σ 1.6TeV 36
Conclusions We showed type-iii seesaw+a4 naturally gives rise to TBM. The deviations from TBM can be fitted to 1σ Experimental data through the phase effects of higher dimensional operators. In modified Type-III seesaw, we can have the deviations from TBM. In A4 symmetry implemented in type-iii seesaw, the high-enegry CP-violation can be linked directly to low-energy CP-violation. 37