Interplay of flavour and CP symmetries

Transcription

1 p. 1/79 Interplay of flavour and CP symmetries C. Hagedorn EC Universe, TUM, Munich, Germany ULBPhysTh Seminar, Université Libre de Bruxelles, , Brussels, Belgium

2 p. 2/79 Outline lepton mixing: parametrization and data combination of flavour and CP symmetries general idea examples: G f = S 4 and G f = (48) predictions for leptogenesis conclusions & outlook

3 p. 3/79 Parametrization of lepton mixing charged lepton and (Majorana) neutrino mass terms e c am e,ab l b and ν a m ν,ab ν b cannot be diagonalized simultaneously going to the mass basis U em em e U e = diag(m 2 e,m 2 µ,m 2 τ) and U T ν m ν U ν = diag(m 1,m 2,m 3 ) leads to non-diagonal charged current interactions lw / U PMNS ν with U PMNS = U eu ν

4 p. 4/79 Parametrization of lepton mixing Parametrization (PDG) U PMNS = Ũ diag(1,e iα/2,e i(β/2+δ) ) with Ũ = c 12 c 13 s 12 c 13 s 13 e iδ s 12 c 23 c 12 s 23 s 13 e iδ c 12 c 23 s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 c 12 c 23 s 13 e iδ c 12 s 23 s 12 c 23 s 13 e iδ c 23 c 13 and s ij = sinθ ij, c ij = cosθ ij Jarlskog invariant J CP J CP = Im [ U PMNS,11 U PMNS,13U PMNS,31U PMNS,33 ] = 1 8 sin2θ 12sin2θ 23 sin2θ 13 cosθ 13 sinδ

5 p. 5/79 Parametrization of lepton mixing Parametrization (PDG) U PMNS = Ũ diag(1,e iα/2,e i(β/2+δ) ) with Ũ = c 12 c 13 s 12 c 13 s 13 e iδ s 12 c 23 c 12 s 23 s 13 e iδ c 12 c 23 s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 c 12 c 23 s 13 e iδ c 12 s 23 s 12 c 23 s 13 e iδ c 23 c 13 and s ij = sinθ ij, c ij = cosθ ij Majorana invariant I 1 I 1 = Im [ ( ) ] UPMNS,12 2 U 2 PMNS,11 = sin 2 θ 12 cos 2 θ 12 cos 4 θ 13 sinα

6 p. 6/79 Parametrization of lepton mixing Parametrization (PDG) U PMNS = Ũ diag(1,e iα/2,e i(β/2+δ) ) with Ũ = c 12 c 13 s 12 c 13 s 13 e iδ s 12 c 23 c 12 s 23 s 13 e iδ c 12 c 23 s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 c 12 c 23 s 13 e iδ c 12 s 23 s 12 c 23 s 13 e iδ c 23 c 13 and s ij = sinθ ij, c ij = cosθ ij Majorana invariant I 2 I 2 = Im [ ( ) ] UPMNS,13 2 U 2 PMNS,11 = sin 2 θ 13 cos 2 θ 12 cos 2 θ 13 sinβ

7 p. 7/79 Data on lepton mixing Latest global fits (Capozzi et al. ( 13))

8 p. 8/79 Data on lepton mixing Latest global fits (Capozzi et al. ( 13))

9 p. 9/79 Data on lepton mixing Latest global fits NH [IH] (Capozzi et al. ( 13)) best fit and 1σ error 3σ range sin 2 θ 13 = [9] [1] [21] [8] sin 2 θ [300] sin 2 θ 23 = sin 2 θ 12 = sin 2 θ [37] [59] 0.027[9] [0.531 sin 2 θ ] 0.357[63] sin 2 θ [59] δ = 1.39[5]π +0.33[24]π 0.27[39]π α, β 0 δ 2π unconstrained

10 p. 10/79 Data on lepton mixing Latest global fits NH [IH] (Capozzi et al. ( 13)) U PMNS [39] [5] 0.40[2] [4] and no information on Majorana phases Mismatch in lepton flavour space is large!

11 p. 11/79 General idea interpret this mismatch in lepton flavour space as mismatch of residual symmetries G ν and G e if we want to predict lepton mixing, we have to derive this mismatch let us assume that there is a symmetry, broken to G ν and G e this symmetry is in the following a combination of a finite, discrete, non-abelian symmetry G f and CP (Feruglio et al. ( 12, 13), Holthausen et al. ( 12), Grimus/Rebelo ( 95)) [Masses do not play a role in this approach.]

12 p. 12/79 General idea interpret this mismatch in lepton flavour space as mismatch of residual symmetries G ν and G e if we want to predict lepton mixing, we have to derive this mismatch let us assume that there is a symmetry, broken to G ν and G e this symmetry is in the following a combination of a finite, discrete, non-abelian symmetry G f and CP (Feruglio et al. ( 12, 13), Holthausen et al. ( 12), Grimus/Rebelo ( 95)) [Masses do not play a role in this approach.]

13 p. 13/79 General idea interpret this mismatch in lepton flavour space as mismatch of residual symmetries G ν and G e if we want to predict lepton mixing, we have to derive this mismatch let us assume that there is a symmetry, broken to G ν and G e this symmetry is in the following a combination of a finite, discrete, non-abelian symmetry G f and CP (Feruglio et al. ( 12, 13), Holthausen et al. ( 12), Grimus/Rebelo ( 95)) [Masses do not play a role in this approach.]

14 p. 14/79 General idea interpret this mismatch in lepton flavour space as mismatch of residual symmetries G ν and G e if we want to predict lepton mixing, we have to derive this mismatch let us assume that there is a symmetry, broken to G ν and G e this symmetry is in the following a combination of a finite, discrete, non-abelian symmetry G f and CP (Feruglio et al. ( 12, 13), Holthausen et al. ( 12), Grimus/Rebelo ( 95)) [Masses do not play a role in this approach.]

15 p. 15/79 General idea Idea: Relate lepton mixing to how G f and CP are broken Interpretation as mismatch of embedding of different subgroups G ν and G e into G f and CP ւ neutrinos G ν G f & CP ց charged leptons G e

16 p. 16/79 General idea Idea: Relate lepton mixing to how G f and CP are broken Interpretation as mismatch of embedding of different subgroups G ν and G e into G f and CP ւ neutrinos assume 3 generations of Majorana neutrinos G f & CP ց charged leptons distinguish 3 generations

17 p. 17/79 General idea Idea: Relate lepton mixing to how G f and CP are broken Interpretation as mismatch of embedding of different subgroups G ν and G e into G f and CP neutrinos G ν = Z 2 CP ւ G f & CP ց charged leptons G e = Z N with N 3 An example: µτ reflection symmetry (Harrison/Scott ( 02, 04), Grimus/Lavoura ( 03))

18 p. 18/79 General idea neutrinos G ν = Z 2 CP ւ G f & CP ց charged leptons G e = Z N with N 3 Further requirements two/three non-trivial mixing angles irred 3-dim rep of G f "maximize" predictability of approach

19 p. 19/79 General idea Definition of generalized CP transformation (see e.g. Branco et al. ( 11)) with X is unitary and symmetric φ i CP X ij φ j

20 p. 20/79 General idea Definition of generalized CP transformation (see e.g. Branco et al. ( 11)) with X is unitary and symmetric; apply CP twice φ i CP X ij φ j φ CP Xφ CP XX φ = φ

21 p. 21/79 General idea Definition of generalized CP transformation (see e.g. Branco et al. ( 11)) φ i CP X ij φ j with X is unitary and symmetric. Realize direct product of Z 2 G f and CP

22 p. 22/79 General idea Definition of generalized CP transformation (see e.g. Branco et al. ( 11)) φ i CP X ij φ j with X is unitary and symmetric. Realize direct product of Z 2 G f and CP ; Z generates Z 2 φ CP Xφ Z 2 XZ φ and φ Z 2 Zφ CP ZXφ

23 p. 23/79 General idea Definition of generalized CP transformation (see e.g. Branco et al. ( 11)) φ i CP X ij φ j with X is unitary and symmetric. Realize direct product of Z 2 G f and CP ; Z generates Z 2 XZ ZX = 0

24 p. 24/79 General idea neutrino sector: Z 2 CP preserved neutrino mass term ν a m ν,ab ν b is invariant under ν α Z αβ ν β is invariant under generalized CP transformation ν α X αβ ν β charged lepton sector: Z N, N 3, preserved charged lepton mass term e c am e,ab l b is invariant under l α Q e,αβ l β

25 p. 25/79 General idea neutrino sector: Z 2 CP preserved neutrino mass matrix m ν fulfills Z T m ν Z = m ν and Xm ν X = m ν charged lepton sector: Z N, N 3, preserved charged lepton mass matrix m e fulfills Q em em e Q e = m em e

26 p. 26/79 General idea neutrino sector: Z 2 CP preserved and generated by (ν = Ω ν ν ) X = Ω ν Ω T ν and Z = Ω ν Z diag Ω ν Z diag = diag( 1,1, 1) and Ω ν unitary charged lepton sector: Z N, N 3, preserved charged lepton mass matrix m e fulfills Q em em e Q e = m em e

27 p. 27/79 General idea neutrino sector: Z 2 CP preserved neutrino mass matrix m ν fulfills Z diag [Ω T νm ν Ω ν ]Z diag = [Ω T νm ν Ω ν ] and [Ω T νm ν Ω ν ] = [Ω T νm ν Ω ν ] charged lepton sector: Z N, N 3, preserved charged lepton mass matrix m e fulfills Q em em e Q e = m em e

28 p. 28/79 General idea neutrino sector: Z 2 CP preserved neutrino mass matrix m ν is diagonalized by Ω ν (X,Z)R(θ)K ν charged lepton sector: Z N, N 3, preserved charged lepton mass matrix m e fulfills Q em em e Q e = m em e

29 p. 29/79 General idea neutrino sector: Z 2 CP preserved neutrino mass matrix m ν is diagonalized by Ω ν (X,Z)R(θ)K ν charged lepton sector: Z N, N 3, preserved and generated by Q e = Ω e Q diag e Ω e with Ω e unitary Q diag e = diag(ω n e N,ωn µ N,ωn τ N ) and n e n µ n τ and ω N = e 2πi/N

30 p. 30/79 General idea neutrino sector: Z 2 CP preserved neutrino mass matrix m ν is diagonalized by Ω ν (X,Z)R(θ)K ν charged lepton sector: Z N, N 3, preserved charged lepton mass matrix m e fulfills Ω e(q e )m em e Ω e (Q e ) is diagonal

31 p. 31/79 General idea neutrino sector: Z 2 CP preserved neutrino mass matrix m ν is diagonalized by Ω ν (X,Z)R(θ)K ν charged lepton sector: Z N, N 3, preserved charged lepton mass matrix m e fulfills Ω e(q e )m em e Ω e (Q e ) is diagonal conclusion: PMNS mixing matrix reads U PMNS = Ω eω ν R(θ)K ν in lw / U PMNS ν

32 p. 32/79 General idea U PMNS = Ω eω ν R(θ)K ν 3 unphysical phases are removed by Ω e Ω e K e U PMNS contains one parameter θ permutations of rows and columns of U PMNS possible Predictions: Mixing angles and CP phases are predicted in terms of one parameter θ only, up to permutations of rows/columns

33 p. 33/79 General idea: consistency conditions We want to consistently combine G f and the generalized CP transformation φ i CP X ij φ j "closure" relations have to hold

34 p. 34/79 General idea: consistency conditions We want to consistently combine G f and the generalized CP transformation φ i CP X ij φ j "closure" relations have to hold: assume φ transforms as 3-dim rep of G f, then φ CP Xφ G f XA φ CP XA X φ = (X AX) φ

35 p. 35/79 General idea: consistency conditions We want to consistently combine G f and the generalized CP transformation φ i CP X ij φ j "closure" relations have to hold: (X AX) = A with in general A A and A, A G f

36 p. 36/79 General idea: consistency conditions We want to consistently combine G f and the generalized CP transformation φ i CP X ij φ j "closure" relations have to hold: (X AX) = A with in general A A and A, A G f compare to relation for having direct product ofz 2 andcp XZ ZX = 0

37 p. 37/79 General idea: consistency conditions We want to consistently combine G f and the generalized CP transformation φ i CP X ij φ j "closure" relations have to hold: (X AX) = A with in general A A and A, A G f compare to relation for having direct product ofz 2 andcp (X ZX) = Z

38 p. 38/79 General idea: consistency conditions fulfilling these conditions ensures a consistent theory, but can lead to enlarged symmetry group additional requirement in order not to change representation content of G f (Chen et al. ( 14)): all representations transform into complex conjugate under CP

39 p. 39/79 General idea: consistency conditions fulfilling these conditions ensures a consistent theory, but can lead to enlarged symmetry group additional requirement in order not to change representation content of G f (Chen et al. ( 14)): all representations transform into complex conjugate under CP [mathematically: mapping induced via X has to be classinverting automorphism (A A 1 )]

40 p. 40/79 General idea: consistency conditions fulfilling these conditions ensures a consistent theory, but can lead to enlarged symmetry group additional requirement in order not to change representation content of G f (Chen et al. ( 14)): all representations transform into complex conjugate under CP if not fulfilled or not possible to fulfill for G f constraints on representations [S 4 fulfilled; (48) not fulfilled in general, only for certain representations]

41 p. 41/79 Study of S 4 and CP Generators in rep. 3 : ( ω = e 2πi/3 ) S = , T = ω ω, U = which fulfill S 2 = 1, T 3 = 1, U 2 = 1, (ST) 3 = 1, (SU) 2 = 1, (TU) 2 = 1, (STU) 4 = 1

42 p. 42/79 Study of S 4 and CP A transformation X in rep. 3 for Z = S is X 3 = which fulfills XX = XX = 1 (X AX) = A, XZ ZX = 0

43 p. 43/79 Study of S 4 and CP A transformation X in rep. 3 for Z = S is X 3 = which fulfills XX = XX = 1 (X AX) = A, XZ ZX = 0 Residual symmetry G e is generated by T.

44 p. 44/79 Study of S 4 and CP Maximal θ 23 and δ from G e = Z 3, Z = S and X 3 (Harrison/Scott ( 02, 04), Grimus/Lavoura ( 03), Feruglio et al. ( 12, 13)) U PMNS = 1 6 2cosθ cosθ +i 3sinθ cosθ i 3sinθ 2 2sinθ 2 sinθ i 3cosθ 2 sinθ +i 3cosθ K ν sin 2 θ 13 = 2 3 sin2 θ, sin 2 θ 12 = and sinδ = 1, J CP = sin2θ cos2θ, sin2 θ 23 = 1 2, sinα = 0, sinβ = 0

45 p. 45/79 Study of S 4 and CP Maximal θ 23 and δ from G e = Z 3, Z = S and X 3 (Harrison/Scott ( 02, 04), Grimus/Lavoura ( 03), Feruglio et al. ( 12, 13)) U PMNS = 1 6 2cosθ cosθ +i 3sinθ cosθ i 3sinθ 2 2sinθ 2 sinθ i 3cosθ 2 sinθ +i 3cosθ K ν sin 2 θ , sin 2 θ , sin 2 θ 23 = 1 2 and sinδ = 1, J CP , sinα = 0, sinβ = 0 for θ 0.185

46 p. 46/79 Study of S 4 and CP Maximal θ 23 and δ from G e = Z 3, Z = S and X 3 (Feruglio et al. ( 12, 13)) 1.0 sin 2 Θ 12 3Σ Θ 0 Θ bf Θ Π 6 Θ Π 4 Θ Π 3 3Σ 0.2 sinθ

47 p. 47/79 Study of S 4 and CP Maximal θ 23 and δ from G e = Z 3, Z = S and X 3 (Feruglio et al. ( 12, 13)) 0.10 J CP 3Σ Θ 0 Θ Π 2 _ 0.05 Θ bf Θ Π 6 Θ Π 4 sin 2 Θ Θ Π

48 p. 48/79 Study of S 4 and CP Maximal θ 23 and δ from G e = Z 3, Z = S and X 3 (Feruglio et al. ( 12, 13)) 0.10 J CP 3Σ Θ 0 _ Θ Π Θ bf Θ Π 3 Θ Π 6 Θ Π 4 sinθ

49 p. 49/79 Study of (48) and CP Generators in rep. 3: (Miller et al. ( 16), Fairbairn et al. ( 64), Luhn et al. ( 07)) ( ω = e 2πi/3 ) a = ω ω 2, c = , d = a 1 ca which satisfy a 3 = 1, c 4 = 1, d 4 = 1, cd = dc, aca 1 = c 1 d 1

50 p. 50/79 Study of (48) and CP A transformation X in rep. 3 for Z = c 2 is (Ding/Zhou ( 13)) X 3 = d which fulfills XX = XX = 1 (X AX) = A, XZ ZX = 0 Residual symmetry G e is generated by a.

51 p. 51/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13)) U PMNS = ( 2+ 6)cos2θ 1 4+( 2+ 6)cos2θ cos2θ ( 2+ 6)cos2θ ( 2+ 6)cos2θ 2 2 2cos2θ sin 2 θ 13 = 1 ( 4+( 2+ ) 6)cos2θ, sin 2 θ 12 = 12 ( sin 2 θ 23 = 1 ) 6( 3 1)cos2θ ( 2+ 6)cos2θ, J CP = sin2θ ( 2+ 6)cos2θ,

52 p. 52/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13)) U PMNS = ( 2+ 6)cos2θ 1 4+( 2+ 6)cos2θ cos2θ ( 2+ 6)cos2θ ( 2+ 6)cos2θ 2 2 2cos2θ sinα = sinβ = cos2θ +( 1+ 3)sin2θ 4+( 2+ 6)cos2θ, 2sin2θ 4+(2+ 3)cos 2 2θ

53 p. 53/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13)) U PMNS = ( 2+ 6)cos2θ 1 4+( 2+ 6)cos2θ cos2θ ( 2+ 6)cos2θ ( 2+ 6)cos2θ 2 2 2cos2θ sin 2 θ , sin 2 θ , sin 2 θ , J CP , and sinδ 0.735, sinα 0.732, sinβ 1 for θ 1.437

54 p. 54/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13))

55 p. 55/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13))

56 p. 56/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13))

57 p. 57/79 Study of (48) and CP Angles and phases from G e = Z 3, Z = c 2 and X 3 (Ding/Zhou ( 13)) 1.0 sin 2 Β Θ bf Θ 5Π 6 Θ 2Π 3 Θ 3Π 4 Θ Π 3 Θ Π 4 Θ Π 6 sin 2 Α Θ 0 Θ Π

58 p. 58/79 Basics of leptogenesis baryon asymmetry of the Universe is measured well Y B = n B n B s = (8.77±0.24) (WMAP ( 08), Planck ( 13)) 0 this asymmetry can be explained by decay of heavy right-handed neutrinos (Fukugita/Yanagida ( 86)) the three Sakharov conditions are fulfilled (Sakharov ( 67))

59 p. 59/79 Basics of leptogenesis baryon asymmetry of the Universe is measured well Y B = n B n B s = (8.77±0.24) (WMAP ( 08), Planck ( 13)) 0 this asymmetry can be explained by decay of heavy right-handed neutrinos (Fukugita/Yanagida ( 86)) the three Sakharov conditions are fulfilled (Sakharov ( 67)) C and CP violation: Yukawa couplings of right-handed neutrinos provide source of CP violation

60 p. 60/79 Basics of leptogenesis baryon asymmetry of the Universe is measured well Y B = n B n B s = (8.77±0.24) (WMAP ( 08), Planck ( 13)) 0 this asymmetry can be explained by decay of heavy right-handed neutrinos (Fukugita/Yanagida ( 86)) the three Sakharov conditions are fulfilled (Sakharov ( 67)) departure from thermal equilibrium: Yukawa interactions of right-handed neutrinos have slow enough rate Γ < H

61 p. 61/79 Basics of leptogenesis baryon asymmetry of the Universe is measured well Y B = n B n B s = (8.77±0.24) (WMAP ( 08), Planck ( 13)) 0 this asymmetry can be explained by decay of heavy right-handed neutrinos (Fukugita/Yanagida ( 86)) the three Sakharov conditions are fulfilled (Sakharov ( 67)) baryon number violation: Majorana masses violate lepton number so that lepton asymmetry is generated which is partially converted into baryon asymmetry via sphaleron processes

62 p. 62/79 Basics of leptogenesis baryon asymmetry of the Universe is measured well Y B = n B n B s = (8.77±0.24) (WMAP ( 08), Planck ( 13)) 0 this asymmetry can be explained by decay of heavy right-handed neutrinos (Fukugita/Yanagida ( 86)) the three Sakharov conditions are fulfilled (Sakharov ( 67)) simplest scenario: thermal leptogenesis in which asymmetry stems from N 1 decay (with no flavour effects)

63 p. 63/79 Basics of leptogenesis baryon asymmetry of the Universe is measured well Y B = n B n B s = (8.77±0.24) (WMAP ( 08), Planck ( 13)) 0 this asymmetry can be explained by decay of heavy right-handed neutrinos (Fukugita/Yanagida ( 86)) the three Sakharov conditions are fulfilled (Sakharov ( 67)) simplest scenario: Y B 10 3 ǫη with ǫ CP asymmetry, η washout factor

64 p. 64/79 Basics of leptogenesis CP asymmetry ǫ ǫ αα = Γ(N 1 Hl α ) Γ(N 1 H lα ) Γ(N 1 Hl)+Γ(N 1 H l) diagrammatically: the CP asymmetry arises from interference of tree-level diagram N 1 y D H l α

65 p. 65/79 CP asymmetry ǫ Basics of leptogenesis ǫ αα = Γ(N 1 Hl α ) Γ(N 1 H lα ) Γ(N 1 Hl)+Γ(N 1 H l) diagrammatically: the CP asymmetry arises from interference of tree-level diagram and one-loop diagrams H N 1 l β N j yd y D y D H l α + N 1 y D H y D N j y l D β l α H

66 p. 66/79 Basics of leptogenesis CP asymmetry ǫ ǫ αα = Γ(N 1 Hl α ) Γ(N 1 H lα ) Γ(N 1 Hl)+Γ(N 1 H l) computation of ǫ in case of unflavoured leptogenesis ǫ = 1 8π j 1 Im ( ) (Ŷ D Ŷ D )2 j1 (ŶDŶ D ) 11 f(x j ) with Ŷ D = U R Y D and U R M RU R = diag(m 1,M 2,M 3 )

67 p. 67/79 Leptogenesis in flavour models leptogenesis has been studied in several models with A 4 or S 4 flavour symmetry (Jenkins/Manohar ( 08), H et al. ( 09), Bertuzzo et al. ( 09), Aristizabal Sierra et al. ( 09)) G f G e in charged lepton sector and m e is diagonal G f G ν = Z 2 ( Z 2 ) in neutrino sector and M R encodes mixing, while Y D has trivial flavour structure

68 p. 68/79 Leptogenesis in flavour models leptogenesis has been studied in several models with A 4 or S 4 flavour symmetry (Jenkins/Manohar ( 08), H et al. ( 09), Bertuzzo et al. ( 09), Aristizabal Sierra et al. ( 09)) G f G e in charged lepton sector and m e is diagonal G f G ν = Z 2 ( Z 2 ) in neutrino sector and M R encodes mixing, while Y D has trivial flavour structure for generations in 3 and Y D invariant under G f ǫ vanishes

69 p. 69/79 Leptogenesis in flavour models leptogenesis has been studied in several models with A 4 or S 4 flavour symmetry (Jenkins/Manohar ( 08), H et al. ( 09), Bertuzzo et al. ( 09), Aristizabal Sierra et al. ( 09)) G f G e in charged lepton sector and m e is diagonal G f G ν = Z 2 ( Z 2 ) in neutrino sector and M R encodes mixing, while Y D has trivial flavour structure if residual G ν is broken at level ε, ǫ ε 2 for unflavoured leptogenesis [ǫ ε for flavoured leptogenesis]

70 p. 70/79 Leptogenesis in flavour models leptogenesis has been studied in several models with A 4 or S 4 flavour symmetry (Jenkins/Manohar ( 08), H et al. ( 09), Bertuzzo et al. ( 09), Aristizabal Sierra et al. ( 09)) G f G e in charged lepton sector and m e is diagonal G f G ν = Z 2 ( Z 2 ) in neutrino sector and M R encodes mixing, while Y D has trivial flavour structure if residual G ν is broken at level ε, ǫ ε 2 for unflavoured leptogenesis if CP is also a symmetry of the theory, constraints on phases and e.g. on sign of ǫ are expected

71 p. 71/79 Leptogenesis in models with flavour and CP Consider the following scenario G f & CP G e in charged lepton sector and m e is diagonal G f & CP G ν = Z 2 CP in neutrino sector and M R encodes mixing, while Y D has trivial flavour structure assume small breaking in Y D at level ε which is real; e.g. for our example G f = S 4 Y 0 D +δy D = y 0 +aε y 0 +bε 0 y 0 +cε 0 fit of reactor mixing angle requires 0.16 θ 0.21

72 p. 72/79 Leptogenesis in models with flavour and CP Result for ǫ from N 1 decays vs lightest neutrino mass m 0 ε = λ ; normal ordering and best fit values of m 2 ij (Capozzi et al. ( 13)) assumed Ε m

73 p. 73/79 Leptogenesis in models with flavour and CP Consider the following scenario G f & CP G e in charged lepton sector and m e is diagonal G f & CP G ν = Z 2 CP in neutrino sector and M R encodes mixing, while Y D has trivial flavour structure assume small breaking in Y D at level ε which is real; e.g. for our example G f = (48) Y 0 D+δY D = y 0 +(s+2t)ε y 0 +(s t 3u)ε y 0 +(s t+ 3u)ε fit of reactor mixing angle constrains θ: 1.40 θ 1.48

74 p. 74/79 Leptogenesis in models with flavour and CP Result for ǫ from N 1 decays vs lightest neutrino mass m 0 ε = λ ; normal ordering and best fit values of m 2 ij (Capozzi et al. ( 13)) assumed Ε m Notice: phases in K ν can change sign of ǫ

75 p. 75/79 Leptogenesis in models with flavour and CP We can understand this behaviour: Look at Im ( ) (Ŷ D Ŷ D )2 j1 ; for j = 2 ( 2( 1) k 1 y0 2 ε 2 t 2 2tu+u 2 ) 2(t 2 +u 2 )cos2θ +(t 2 2tu u 2 )sin2θ +O(ε 3 ) and for j = 3 4( 1) k 2 y 2 0 ε2 ( t 2 +u 2) sin2θ +O(ε 3 )

76 p. 76/79 Leptogenesis in models with flavour and CP We can understand this behaviour: Now expand for θ = π/2+κ up to κ; for j = 2 ( 2( 1) k 1 y0 2 ε 2 t 2 ( κ)+2tu( 1+2κ)+u 2 (1+ ) 2+2κ) +O(κ 2 ) and for j = 3 8( 1) k 2 y 2 0 ε 2 (t u)(t+u)κ

77 p. 77/79 Leptogenesis in models with flavour and CP We can understand this behaviour: Now expand for θ = π/2+κ up to κ; for j = 2 2( 1) k 1 y 2 0 ε 2 (t κ u κ) +O(κ 2 ) and for j = 3 suppressed by κ The loop function f(x j ) acts as weighting factor of the different contributions.

78 p. 78/79 Conclusions & outlook approach with flavour and CP symmetry strongly constrains lepton mixing results for G f = S 4 or G f = (48) are encouraging leptogenesis can be studied in this approach

79 p. 79/79 Conclusions & outlook continue study of different groups G f ( (3n 2 ) and (6n 2 )) and CP: new mixing patterns, consistent definition of CP,... explore more phenomena which involve CP phases: 0νββ, electric dipole moments, phases of soft supersymmetry breaking terms, CKM phase,... Thank you for your attention.