Non-zero Ue3 and TeV-leptogenesis through A4 symmetry breaking

Transcription

1 Non-zero Ue3 and TeV-leptogenesis through A4 symmetry breaking National Tsing-Hua Unv. Chian-Shu Chen (NCKU/AS) with Y.H. Ahn & S.K. Kang 11/1/009

2 ouline Introduction A 4 symmetry The model Neutrino mass generation How to realize Tri-bimaximal mixings Leptogenesis with flavour effects in our model Conclusion

3 Introduction The tiny neutrino masses have been observed in experiments which is contradicted with the prediction of SM in particle physics. The elements of neutrino mixing matrix are measured except the third mixing angle θ 13, however, the data have showed its structure are quite different from CKM in quark sectors. If neutrinos are Majorana fermions, lepton number is broken which provides a opportunity to explain the baryon asymmetry in our universe through the mechanism so-called Leptogenesis.

4 Current neutrino data

5 Neutrino oscillation : Mass eigenstates and flavour eigenstates ν α = 3 U αi ν i i=1 U = c 3 s 3 0 s 3 c 3 c 13 0 s 13 e iα s 13 e iα 0 c 13 c 1 s 1 0 s 1 c e iφ e iφ ( ) U : PMNS mixing matrix Pontecorvo, Sov. Phys. JETP6,49(1958), 33, 549(1967)

6 Current neutrino oscillation data m Sol / 5 ev sin θ 1 U e3 sin θ 3 m Atm / 3 ev Best-fit σ σ <

7 Neutrino angles A very good first approximation : tri-bimaximal ansatz (Harrison,Perkins & Scott, 00) Corresponding to tan θatm = 1, tan θsolar = 1/, sinθ13 = 0 A4 symmetry : E. Ma ; G. Altarelli T symmetry : Frampton μ-τ symmetry, S4, Δ(54),...

8 On the other hand, neutrinos are predicted to be massless in SM Dirac or Majorana If Lepton Number is Violation : Many realiztions: (1)Seesaw mechanism:typei,ii,iii ()Radiative models : Zee, Babu,LQ,.. (3)SUSY neutrino masses : R-parity violation.

9 A4 symmetry The alternating group of order 4 : a set of all 1 even permutations of 4 objects. It is isomorphic to the group of three dimensional rotations of a regular tetrahedron It has 3 3 representations for the 1 group Elements 1: r1: r: r3:

10 There are 4 irreducible representations : 1, 1, 1, 3 The tensor products are Let s and denote the basis vectors of 3 s Under c and a :

11 The model A radiative seesaw mechanism implemented in A4 symmetry The model content Inert doulet SM Higgs 4 4 L Y Field l L l R,l R,l R N R ϑ η ψ χ Φ A 4 3 1, 1, Z Z 4 1 i i 1 i 1 1 i SU() L U(1) Y (, 1) (1, ) (1, 0) (1, 0) (, 1) (1, 0) (1, 0) (, 1) Neutrino mass generation CP phases and θ13 (SM A4 effective operator) Charged lepton masses

12 Charged lepton masses If with Remember that that is and Taking the breaking scale of discrete symmetry above the EW scale to be

13 The Dirac Yukawa couplings of neutrino sectors are After discrete symmetry breaking, the coupling matrix can be written And the Majorana masses of right-handed neutrinos

14 For simplicity, we take Thus Diagonalizing MR with

15 And In the basis of both and mass matrices are diagonal with

16 Neutrino mass generation (m ν ) αβ = i Ỹ ναi Ỹ νβi O(λ Φη )v 16π ω( M i M i m η ) with ω(z i )=( z i 1 z i )[1 + z i ln z i 1 z i ] Radiative seesaw mechanism

17 Tri-bimaximal mixing In the limit of μ-τ symmetry and overall scale of neutrino masses Majorana phases

18 Deviation from Tri-bimaximal We obtain m eff = m 0 U TB P ν ω 1 a + eiφ x ω xeiφ ( ω 1 a + ω ) eiφ x ω xeiφ ( ω 1 a eiφ x ω + ω ) ω + x e iφ ( ω 1 a + ω 3 b ) xeiφ ( ω 3 b + ω ) xe iφ ( ω 3 b + ω ) ω 3 b + eiφ x ω P ν T U TB T Can not be diagonalized by UTB Neutrino masses to first order of x :

19 Neutrino masses-square differences m 1 m m 1 m 0 m 3 m 3 m m 0 { ω ω 1 a + x ( ω + ω 1 ) cos φ} a { ω 3 b ω x ( ω + ω 1 ) } cos φ 4 a m η Light neutrino masses νdiag.( ω1 gν 16π M a, ω, ω 3 b ) P T ν ω(z i )=( z i 1 z i )[1 + z i ln z i 1 z i ] z i = M i m η,, Right-hande neutrino masses s M(a, 1,b)=(M 1,M,M 3 )

20 Neutrino spectrum g ΑM m i = νm m η 1 α M / m [ η 1+ α M / m η ln αm m η f 8π M α 1 α M / m η 1 α M / m η m Η 0.0 = gνm f (α M m ), η Case-IV Case-III,V ] m m m Case-II Quasi-degenerate m Normal hierarchy (Case-I) Α Case-I M 1, M 3 (a>1 b with ξ = 0): Case-II M 1 >M 3 >M (a > b > 1 with ξ = 0): Case-III M 3 >M >M 1 (b>1 a with ξ = π): Case-IV Case-V M 1 >M >M 3 (a>1 >bwith ξ = 0): M 1 >M M 3 (a>1 b with ξ = 0):

21 Mixing angles and CP phase Atmospheric angle : θ 3 + π 4 6(ω ω 1 a ) sin φ 3(ω ω 1 a ) sin φ + 3(ω + ω 3 b ) cos φ Solar angle : tan θ 1 ω ω 1 + x ( ω a 1 a ω ω 1 4x( ω a 1 a + ω ) cos φ + x {( ω 3 b + ω ) +ω ω 3 b cos φ} + ω ) cos φ + x {( ω 3 b + ω ) +ω ω 3 b cos φ} Θ13 mixing : CP phase :

22 3 Nomal hierarchy spectrum (Case-I) # CP # 300 CP CP # !0 0!0!00!00!300!0!00! !! !! !! !300!! ! !! FIG. FIG. 3: (Upper-panel:) 3: Left-figure Left-figure represents represents that that the atmospheric the atmospheric mixing mixing angle angle θ 3 over θ over the pha the 13

23 ! 1 1! ! x x Quasi-degenerate 3 FIG. 4: Left-figure shows the mixing angle of θ 1 as a function of the parameter x. Right-figur 50 shows θ 13 as a function of the parameter x represents 46 that how the mixing angles θ 1 and0θ 0 13 depend on the parameter x, as can b 46 seen in Eqs. (5-6), in which especially the unknown 0 0 mixing angle θ 13 is very sensitive to the parameter x Quasi-degenerate light neutrino mass spectrum !! CP # !0!00!300 CP # !0!00! !!

24 !! !! FIG. 5: FIG. The same 5: The as same Fig.3. as Fig.3. e ofvalue 38m38 0 with of m 0 with = 0eV m = in 0eV Eq. (3), in Eq. then (3), we then impose we the impose current the experimental current experimenta results 1! 40 1! 40 neutrino on neutrino masses masses and mixings and mixings into theinto hermitian 8 the hermitian 8 matrix m matrix m eff m 36 eff and eff m 36 varying eff and varying all the ameter parameter space {κ(or space a, {κ(or b), φ, a, x, b), g ν }: φ, x, g ν }: κ κ, 3.8 0, φ 0 π φ, π, x<0.4 x<0.4, 0.38, g0.38 ν 0.4 g ν, 0.4,(36) 30 re the where parameter the parameter g ν can be g ν replaced can be replaced by m 0 due by 0 mto 0 0 Eq. due(3). to Eq. (3) x x ! x x 0.3 ig. 5 shows how the mixing angles θ, θ, θ and δ in neutrino oscillation depend 1 13!

25 Leptogenesis Why low-scale leptogenesis is difficult? Decay width : Hubble expansion parameter : We define wash-out factor: K i = Γ i H(M i ) 3 16 gν( GeV M i )δ Nη

26 On the other hand CP asymmetry with function The amount of asymmetry leads Usually κ = 1 K

27 Three possibile enhancement mechanisms 1. Mass degeneracy : CP asymmetry induced by self-energy diagram display an interesting resonant behavior when the masses of the decaying particles are nearly degenerate.. Hierarchy of couplings : Assuming two particles (A,B) decaying into the same decay products. The lighter one A with the suppressed coupling ga to reach the out-of-equilibrium condition while the heavier one B with unsuppressed coupling gb will produce large CP asymmetry through one-loop. 3. Phase space suppression. Instead of tuning down the Yukawa couplings, we use phase space suppression to tune the wash-out factor.

28 In our case, the low-energy data prefer hierarchical spectrum in right-handed neutrinos --- resonance enhancement Since A 4 symmetry, we have the Im{H ij (Ỹν) αi (Ỹν) αj }, with H = g ν H Ỹ ν Ỹν = V R Y ν Y νv R, we can t use hierarchical couplings method! 1+ x xe i ϕ 1 cos φ x ϕ ei ϕ 1 xe i ϕ 1 cos φ 1+x xe i ϕ x ϕ ei 1 ϕ xe i ϕ cos φ 1+ x cos φ ( We are forced to use phase space suppression. However, we have the mixing matrix so we can calculate leptogenesis by considering the flavour effects due to the diffferent Yukawa couplings among charged leptons.

29 Flavor-depending Wash-out factor K α i = Γ(N i ηl α ) H(M i ) = K i (Ỹ ν ) αi(ỹν) αi (Ỹ ν Ỹν) ii K i = Γ i H(M i ) 3 16 gν( GeV )δnη δ M Ni η 1 1, i z i Flavor-depending CP asymmetry ε α i = = Γ(N i l α η) Γ(N i l α η ) α [Γ(N i l α η)+γ(n i l α η )] 1 { ( M 8π(Ỹ ν Ỹν) Im (Ỹ ν Ỹν) ij (Ỹν) αi (Ỹν) j αj }g ii j i M i ) Im { (Ỹ ν Ỹν) ij (Ỹν) αi (Ỹν) αj } = g ν 1+ x xe i ϕ 1 cos φ x ϕ ei ϕ 1 xe i ϕ 1 cos φ 1+x xe i ϕ x ϕ ei 1 ϕ xe i ϕ cos φ 1+ x cos φ (

30 Baryon asymmetry η B ( 151 [ε e i κ N i with washout factor [ ( ) ( 179 Ke i ) + ε µ i κ ( Kµ i In different spectrums approximately which are given constrained 3, for x by 1, by oscillating data In the case of M 3 m η, ) ( ε τ i κ 537 Kτ i equilibrium, ( 8.5 simultaneously ( K α ) 1.16 ) 1 κ + i protecting the N 3 lepton thoughklarge i α e, 0. µ-and τ-yukawa couplings to N 1,,3 ex (Case-I,IV,V) K3 e = x gν m δn 3 η 6bM, Kµ 3 g ν m δn 3 η x (1 + sin φ), bm 3 ε e 3 ε µ 3 ε τ 3 = bx g ν 3aπ (x sin φ + a sin φ), θ 3 in neutrino oscillations. with A 4 symmetry, Eq. (44) explicitly shows how th determined by Eq. (3) allows for a heavy Majorana bxg ν 64aπ {4a 3 cos φ x[ 3 + a cos φ( 3 cos φ + sin φ)]}, bxg ν 64aπ { 4a 3 cos φ + x[ 3 + a cos φ( 3 cos φ sin φ)]} ε e 3 ε µ 3 ε τ 3 )] α = bx m = ( gν ) 1 45(x 3aπ 8 π 5 g sin φ + a sin φ), M Pl GeV. bxg ν 64aπ {4a 3 cos φ x[ 3 + a cos K3 τ g ν m δn 3 η x (1 sin φ), bm 3 bxg ν 64aπ { 4a 3 cos φ + x[ 3 + a co As can be seen in Eqs. (44,45), since the lepton asymm opposite in sign to the first order, i.e. ε µ 3 ετ 3, satisf parameters in µ and τ are almost equal K µ 3 Kτ 3 related with N 3 can play a crucial role in a successful

31 B!9!!11!1!!11!1!13 B!9!!11!1!13!14!13!14! ! 13! ! ! 50 3! show the predictions of η B for MStrong 3 1TeV washout and δ N3 η = 5. Left-figure shows η B 13. Right-figure!9 shows η B as a function of θ 3. The horizontal!9 dotted lines in both d to the phenomenologically acceptable current measurement, and the horizontal nts the best-fit value η B!1 =6. of current measurement from WMAP [1]. dotted lines represent the!13 experimental bounds in 1σ of the mixing angles θ 13 and scillations. B!!11!1!13 8 1! 13!14 B!9!! try, Eq. (44) explicitly shows how! the Yukawa coupling matrix Eq.! (13) 13 Eq. (3) allows for a heavy Majorana neutrino to decay relatively out of B! ! he same sa Fig. 8 except for Mweak 3 1TeV washout and δ N3 η = 6.!!!11!1 3!14! 3 3

32 In the case of M m η, (Case-II) K e g ν m 3M (1 + x cos φ)δ N η, K µ g νm 3M (1 x cos φ x 3 sin φ)δ N η K τ g νm 3M (1 x cos φ + x 3 sin φ)δ N η all the K factors are almost equal due to light neutrinos degenerate spectrum this case refer to. ε e = (b a)x g ν 3abπ sin φ, ε µ xg ν 3abπ { a 3+x(a b)( 3 cos φ + sin φ)} cos φ, ε τ xg ν 3abπ {a 3 x(a b)( 3 cos φ sin φ)} cos φ,

33 ndn, given strong the condition the wash-out initialfor thermal regime K, the Kabundance resulting 1(α = of baryon-to-photon e, Nµ, τ), given theratio initial η B thermal approximately abundance give nd to-photon the In strong condition ratio wash-out ηfor K α B approximately regime, the resulting K α given 1baryon-to-photon as ratio η B approximately give η B (ε τ ) x sin φ εµ ) x sin φ (K η B (ε τ εµ ) x sin µ ), 1.16 φ (K µ, (5) (K µ ), ) 1.16 here x sin φ is from the common factor in K µ,τ. In weak 1.16 wash-out regime K α < 1,,τ here µ,. τ), Inxthe weak sin φresulting wash-out is frombaryon-to-photon the regime common K α < factor 1(α ratio in = K η µ,τ. In weak wash-out regime K α B can be simply given as < 1, can µ, τ), bethe simply resulting givenbaryon-to-photon as ratio η B can be simply given as η B 3. 3 (ε τ εµ )K K µ x sin φ, ε µ )K K µ x sin φ, η B 3. 3 (53) (ε τ εµ )K K µ here x sin φ comes from the common factor in K µ,τ x sin φ,, K µ K τ and K = K e +!9!!9 Kµ + K µ,τ B here If we, xk take, sin µ φ comes for K τ example, and from K = the δ Kcommon e + Kµ + factor K τ. in K µ,τ, K µ K τ and K = K e + N3 η =1.5 6!!11, the! magnitude of washout factors are Kµ giv + e If magnitude we take,!11 for of washout example, factors δ N3 η =1.5 are given 6 as, the magnitude of washout factors are giv K3 e!1!11 =3 3.7, Kµ 3 =3 4.9, Kτ 3 =. 3.7.!13!1.9, K3 τ!13!14 =.K 3 e 3.7 =3. 3.7, Kµ 3 =3 (54) 4.9, Kτ!14! =. 3.7.! ! x!15! ! x X is small in the degenerate spectrum B FIG. : The same as Fig.6 except for M = 1TeV and δ N η =1.5 6.!9 B FIG. : The same as Fig.6 except for M = 1TeV and δ N η =1.5 6.!1!14

34 Conclusion A model with discrete symmetry A 4 implemented in radiative seesaw mechanism is invented. Tri-bimaximal mixings can be realized in this model as the neutrino masses are generated in one-loop level. Non-zero U e3 is generated by introducing a effective operator which obey the symmetry under SU()L U(1)Y A4. By realizing the mixing matrix under the discrete symmetry and by constraining from oscillation data, leptogenesis with flavour effects are considered.