CP Violation Predictions from Flavour Symmetries

Transcription

1 CP Violation Predictions from Flavour Symmetries Arsenii V. Titov in collaboration with Ivan Girardi and Serguey T. Petcov SISSA and INFN, Trieste, Italy Neutrino Oscillation Workshop 016 September 6, 016, Otranto, Lecce, Italy

2 Outline 3-Neutrino Mixing Discrete Flavour Symmetry Approach General Set-up Dirac Phase Sum Rules Predictions Statistical Analysis Majorana Phases Sum Rules Predictions Generalised CP Symmetry Neutrinoless Double Beta Decay Conclusions September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy

3 3-Neutrino Mixing U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix Parameter Best fit 3σ range sin θ sin θ 3 (NO) sin θ 3 (IO) sin θ 13 (NO) sin θ 13 (IO) δ π (NO) δ π (IO) m ev m ev (NO) m ev (IO) Capozzi et. al., NPB 908 (016) 18 Symmetry behind this? NO IO King and Luhn, RPP 76 (013) September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 3

4 3-Neutrino Mixing Talk by Neutrino 016, London, July 9, 016 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 4

5 Discrete Flavour Symmetry Approach G f Flavour symmetry group (non-abelian discrete) G e G ν Residual symmetries (Abelian) of the charged lepton and neutrino mass matrices M e and M ν ρ is a unitary representation of G f under which LH fields are transformed If G e = Z k, k > or Z m Z n, m, n and G ν = Z Z, the matrices U e and U ν are fixed (up to permutations of columns and right multiplication by diagonal phase matrices) U = U e U ν is fixed September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 5

6 Discrete Flavour Symmetry Approach G f = A 4 /T, S 4, A 5 possess a 3-dimensional ρ (unification of 3 flavours at high energies, where G f is unbroken) Examples: Bimaximal mixing (S 4 ) Tri-bimaximal mixing (A 4 /T, S 4 ) These mixing forms per se are excluded by the data (θ 13 = 0) However, perturbative corrections are sufficient to reconstitute compatibility of, e.g., tri-bimaximal mixing with the data If G e = 1 (G f is fully broken in the charged lepton sector), then U e is not fixed, and it provides the requisite corrections (charged lepton corrections) For different breaking patterns see Girardi, Petcov, Stuart, Titov, NPB 90 (016) 1 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 6

7 Discrete Flavour Symmetry Approach G ν = Z Z U ν is fixed (up to permutations of columns and right multiplication by a diagonal phase matrix): Symmetry Forms of U ν R ij is a rotation matrix in the i-j plane ν ν ν Symmetry form Group θ 1 θ 3 θ 13 Tri-bimaximal (TBM) A 4 /T sin 1 1/ 3 35 Bi-maximal (BM) S 4 π/4 = 45 Golden ratio A (GRA) A 5 sin 1 1/ + r 31 π/4 = 45 0 Golden ratio B (GRB) D 10 sin 1 3 r/ = 36 Hexagonal (HG) D 1 π/6 = 30 r is the golden ratio: r = September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 7

8 General Set-up In general, U e and U ν are CKM-like matrices Frampton, Petcov, Rodejohann, NPB 687 (004) 31 Considered Cases Case U e U ν e A1 R 1 θ 1 e A R 13 θ 13 e e B1 R 1 θ 1 R 3 θ 3 e e B R 13 θ 13 R 3 θ 3 e C1 R 1 θ 1 e C R 13 θ 13 ν ν R 3 θ 3 R 1 θ 1 ν ν ν R 3 θ 3 R 13 θ 13 R 1 θ 1 U e e = R 3 θ 3 leads to θ 13 = 0 for U ν containing rotations θ 13 = θ ν 13 for U ν containing 3 rotations In the case of U e e e = R 1 θ 1 R 13 θ 13 and U ν containing rotations, a free phase parameter ω enters resulting sum rules for the CP-violating phases September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 8

9 Dirac Phase: Sum Rules Petcov, NPB 89 (015) 400; Girardi, Petcov, Titov, EPJC 75 (015) 345 In cases A1 and A for θ ν 3 = π 4, s s 13, i.e., θ 3 π 4 In cases B1 and B the best fit values of all the three mixing angles can be reproduced September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 9

10 Dirac Phase: Predictions δ [ ], using the best fit values of the neutrino mixing angles for NO θ ν 3 = π 4 The values in square brackets are those of θ ν ν 13, θ 1 a = sin 1 1 3, b = sin r, c = sin 1 1 3, d = sin 1 3 r ν Non-zero values of θ 13: Bazzocchi, arxiv: ; Toorop, Feruglio, Hagedorn, PLB 703 (011) 447; Rodejohann and Zhang, PLB 73 (014) 174 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 10

11 Dirac Phase: Statistical Analysis Likelihood: Present: χ i are the 1-dimensional projections from the global analysis performed in Capozzi et. al., PRD 89 (014) Future: x i are the current best fit values of sin θ 1, sin θ 13 and sin θ 3 σ xi are the prospective 1σ uncertainties: 0.7% for sin θ 1 (JUNO) 3% for sin θ 13 (Daya Bay) 5% for sin θ 3 (NOvA and TK) September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 11

12 Dirac Phase: Statistical Analysis Case B1: U e e e = R 1 θ 1 R 3 θ 3 Present Future Girardi, Petcov, Titov, NPB 894 (015) 733 RG corrections to sum rule predictions are negligible within the SM extended by the Weinberg (dimension 5) operator, see Gehrlein, Petcov, Spinrath, Titov, arxiv: September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 1

13 Dirac Phase: Statistical Analysis Case B: U e e e = R 13 θ 13 R 3 θ 3 Present Future Girardi, Petcov, Titov, EPJC 75 (015) 345 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 13

14 Rephasing Invariant J CP : Statistical Analysis Case B1: U e e e = R 1 θ 1 R 3 θ 3 J CP determines the magnitude of CP-violating effects in neutrino oscillations Krastev and Petcov, PLB 05 (1988) 84 NO case B1 IO case B1 NO global fit IO global fit Relatively large CP-violating effects in neutrino oscillations in the cases of TBM, GRA, GRB, HG: J CP 0.03, J CP 3σ and suppressed effects in the case of BM: J CP 0 Girardi, Petcov, Titov, NPB 894 (015) 733 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 14

15 Majorana Phases: Sum Rules In these expressions U is in the standard parametrisation, and the corresponding sum rules for sin θ 3 and δ (slide 9) should be used The phases κ 1 and κ 31 are 0 or π and known when the angles θ ij ν are fixed for all the cases, but B1 and B, for which κ 31 = 0 π + β, where β is a free phase parameter Girardi, Petcov, Titov, arxiv: September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 15

16 Majorana Phases: Predictions α 1 ξ 1 / [ ], using the best fit values of the neutrino mixing angles for NO First number corresponds to δ = cos 1 cos δ, second is for δ = π cos 1 cos δ θ ν 3 = π 4 The values in square brackets are those of θ ν ν 13, θ 1 a = sin 1 1 3, b = sin r, c = sin 1 1 3, d = sin 1 3 r September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 16

17 Majorana Phases: Predictions α 31 ξ 31 1 [ ] ( α 31 ξ 31 1 β [ ] in cases B1 and B), using the best fit values of the neutrino mixing angles for NO First number corresponds to δ = cos 1 cos δ, second is for δ = π cos 1 cos δ θ ν 3 = π 4 The values in square brackets are those of θ ν ν 13, θ 1 a = sin 1 1 3, b = sin r, c = sin 1 1 3, d = sin 1 3 r September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 17

18 Generalised CP Symmetry X are generalised CP transformations Generalised CP symmetry should be consistent with (residual) flavour symmetry: It can be shown that Thus, the phases ξ i are known once U ν is fixed by G ν, and X consistent with G ν are identified September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 18

19 Generalised CP Symmetry Example: G f = A 4 S = T 3 = ST 3 = 1 G ν = Z S Z acc (Z acc is a μ τ symmetry which arises accidentally) leads to tri-bimaximal mixing in the neutrino sector The generalised CP transformations consistent with the preserved S generator are X = ρ(1) and X = ρ S. Then Thus, the phases ξ i, and hence ξ 1 and ξ 31, can be either 0 or π A similar situation takes place for G f = S 4 and A 5 (BM and GRA mixing forms, respectively) September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 19

20 Neutrinoless Double Beta Decay Effective Majorana mass: Using the best fit values of θ 1, θ 13, m 1, m 31(3) and the predicted values of the Dirac phase and Majorana phases for ξ 1, ξ 31 = 0, 0 IO IO NO NO TBM, GRA, GRB, HG β 0, π Girardi, Petcov, Titov, arxiv: September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 0

21 Neutrinoless Double Beta Decay Effective Majorana mass: Using the best fit values of θ 1, θ 13, m 1, m 31(3) and the predicted values of the Dirac phase and Majorana phases for ξ 1, ξ 31 = π, π IO IO NO NO TBM, GRA, GRB, HG β 0, π Girardi, Petcov, Titov, arxiv: September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 1

22 Conclusions Exact (within the schemes considered) sum rules for the cosine of the Dirac phase and the Majorana phases were derived and numerical predictions were obtained Sufficiently precise measurements of the Dirac phase and the mixing angles are the key to the possible discrete symmetry origin of the observed pattern of neutrino mixing Relatively large CP-violating effects in neutrino oscillations in the cases of TBM, GRA, GRB, HG and suppressed effects in the case of BM were found Constrained parameter space in neutrinoless double beta decay is predicted September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy

23 Backup September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 3

24 Dirac Phase: Statistical Analysis Case B1: Dependence on the best fit values Larger b.f.v. of sin θ 1 Larger b.f.v. of sin θ 3 s 1 bf = 0.33 s 3 bf = s 13 pbf = s 1 bf = s 3 bf = s 13 pbf = IO neutrino mass spectrum Gonzalez-Garcia et. al., JHEP 1411 (014) 05 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 4

25 Dirac Phase: Statistical Analysis Case C1: Present NO IO θ ν 13, θ ν 1 : Case I = Case IV = π 0, π 4 Case II = π 0, sin r Case V = π 10, π 4 Case III = sin 1 π 0, π 6 1 3, π 4 Girardi, Petcov, Titov, EPJC 75 (015) 345 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 5

26 Dirac Phase: Statistical Analysis Case C1: Future pbf = ν = 0 s 3 θ 13 pbf = π 0 s 3 θ ν 13 = pbf = π 10 s 3 θ ν 13 = s 3 pbf = ν = sin θ 13 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 6

27 Dirac Phase: Statistical Analysis Case C: Present NO IO θ ν 13, θ ν 1 : Case I = π 0, sin 1 Case IV = sin Case II = 1 3, π 4 Case V = π 0, π 4 Case III = π 0, sin 1 3 r π 10, π 4 Girardi, Petcov, Titov, EPJC 75 (015) 345 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 7

28 Dirac Phase: Statistical Analysis Case C: Future pbf = 0.51 ν = 0 s 3 θ 13 pbf = π 0 s 3 θ ν 13 = pbf = π 10 s 3 θ ν 13 = s 3 pbf = ν = sin θ 13 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 8

29 sin θ 3 : Statistical Analysis Case B1 NO case B1 IO case B1 NO global fit IO global fit Girardi, Petcov, Titov, NPB 894 (015) 733 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 9

30 sin θ 3 : Statistical Analysis Case B NO case B IO case B NO global fit IO global fit Girardi, Petcov, Titov, EPJC 75 (015) 345 September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 30

31 Neutrinoless Double Beta Decay Case A1 ( = A in terms of predictions for m ) ξ 1, ξ 31 = 0, 0 ξ 1, ξ 31 = 0, π TBM GRB GRA HG ξ 1, ξ 31 = π, 0 ξ 1, ξ 31 = π, π September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 31

32 Neutrinoless Double Beta Decay Case C1 ξ 1, ξ 31 = 0, 0 ξ 1, ξ 31 = 0, π Case I Case V Case II Case IV ξ 1, ξ 31 = π, 0 ξ 1, ξ 31 = π, π Case III September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 3

33 Neutrinoless Double Beta Decay Case C ξ 1, ξ 31 = 0, 0 ξ 1, ξ 31 = 0, π Case II Case III Case V Case I ξ 1, ξ 31 = π, 0 ξ 1, ξ 31 = π, π Case IV September 6, 016 Arsenii V. Titov NOW 016 Otranto, Lecce, Italy 33